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Skills available for New Hampshire high school math standards

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New Hampshire College and Career Ready Standards: Algebra 1 New Hampshire College and Career Ready Standards: Algebra 1
New Hampshire College and Career Ready Standards: Algebra 2 New Hampshire College and Career Ready Standards: Algebra 2
New Hampshire College and Career Ready Standards: All High School Pathways New Hampshire College and Career Ready Standards: All High School Pathways
New Hampshire College and Career Ready Standards: Fourth Courses New Hampshire College and Career Ready Standards: Fourth Courses
New Hampshire College and Career Ready Standards: Geometry New Hampshire College and Career Ready Standards: Geometry
New Hampshire College and Career Ready Standards: Mathematical Practices New Hampshire College and Career Ready Standards: Mathematical Practices
New Hampshire College and Career Ready Standards: Mathematics I New Hampshire College and Career Ready Standards: Mathematics I
New Hampshire College and Career Ready Standards: Mathematics II New Hampshire College and Career Ready Standards: Mathematics II
New Hampshire College and Career Ready Standards: Mathematics III New Hampshire College and Career Ready Standards: Mathematics III
New Hampshire College and Career Ready Standards: Precalculus New Hampshire College and Career Ready Standards: Precalculus

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HSN-RN The Real Number System

HSN-RN.A Extend the properties of exponents to rational exponents.
- HSN-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
- HSN-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
HSN-RN.B Use properties of rational and irrational numbers.
- HSN-RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

HSN-Q Quantities

HSN-Q.A Reason quantitatively and use units to solve problems.
- HSN-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
- HSN-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.
- HSN-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

HSN-CN The Complex Number System

HSN-CN.A Perform arithmetic operations with complex numbers.
- HSN-CN.A.1 Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.
  - Introduction to complex numbers (A2-K.1)
- HSN-CN.A.2 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
- HSN-CN.A.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
HSN-CN.B Represent complex numbers and their operations on the complex plane.
- 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.
- 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.
- 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-W.7)
  - Distance in the complex plane (PC-W.8)
HSN-CN.C Use complex numbers in polynomial identities and equations.
- HSN-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions.
- HSN-CN.C.8 Extend polynomial identities to the complex numbers.
- HSN-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
  - Fundamental Theorem of Algebra (A2-Q.8)
  - Fundamental Theorem of Algebra (PC-F.7)

HSN-VM Vector and Matrix Quantities

HSN-VM.A Represent and model with vector quantities.
- 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).
- 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.
- HSN-VM.A.3 Solve problems involving velocity and other quantities that can be represented by vectors.
HSN-VM.B Perform operations on vectors.
- HSN-VM.B.4 Add and subtract vectors.
  - 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.
  - 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-Y.9)
  - 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.
- HSN-VM.B.5 Multiply a vector by a scalar.
  - 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 subscript x, v subscript y) = (cv subscript x, cv subscript y).
    - Multiply a vector by a scalar (PC-Y.10)
    - Scalar multiples of three-dimensional vectors (PC-Z.4)
  - 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-Y.11)
    - Determine the direction of a vector scalar multiple (PC-Y.12)
HSN-VM.C Perform operations on matrices and use matrices in applications.
- HSN-VM.C.6 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
- 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.
- HSN-VM.C.8 Add, subtract, and multiply matrices of appropriate dimensions.
- 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.
- 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.
- 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.
- 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.

HSG-CO Congruence

HSG-CO.A Experiment with transformations in the plane
- HSG-CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
  - Angle vocabulary (G-C.1)
- HSG-CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
- HSG-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
  - Transformations that carry a polygon onto itself (G-H.15)
- HSG-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
  - Rotate polygons about a point (G-H.7)
- HSG-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
HSG-CO.B Understand congruence in terms of rigid motions
- HSG-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
- HSG-CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
- HSG-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
HSG-CO.C Prove geometric theorems
- HSG-CO.C.9 Prove theorems about lines and angles.
- HSG-CO.C.10 Prove theorems about triangles.
- HSG-CO.C.11 Prove theorems about parallelograms.
HSG-CO.D Make geometric constructions
- HSG-CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
- HSG-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

HSG-SRT Similarity, Right Triangles, and Trigonometry

HSG-SRT.A Understand similarity in terms of similarity transformations
- HSG-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor:
  - HSG-SRT.A.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
    - Dilations: graph the image (G-I.1)
    - Dilations and parallel lines (G-I.6)
  - HSG-SRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
- HSG-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
- HSG-SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
  - Similarity rules for triangles (G-M.8)
  - Similar triangles and indirect measurement (A1)
HSG-SRT.B Prove theorems involving similarity
- HSG-SRT.B.4 Prove theorems about triangles.
- HSG-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
HSG-SRT.C Define trigonometric ratios and solve problems involving right triangles
- HSG-SRT.C.6 Understand 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.
- HSG-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.
- HSG-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
HSG-SRT.D Apply trigonometry to general triangles
- HSG-SRT.D.9 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
- HSG-SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.
- HSG-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).

HSG-C Circles

HSG-C.A Understand and apply theorems about circles
- HSG-C.A.1 Prove that all circles are similar.
  - Similarity of circles (G-M.11)
- HSG-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords.
- HSG-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
- HSG-C.A.4 Construct a tangent line from a point outside a given circle to the circle.
  - Construct a tangent line to a circle (G-X.1)
HSG-C.B Find arc lengths and areas of sectors of circles
- HSG-C.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

HSG-GPE Expressing Geometric Properties with Equations

HSG-GPE.A Translate between the geometric description and the equation for a conic section
- HSG-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.
- HSG-GPE.A.2 Derive the equation of a parabola given a focus and directrix.
- 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.
HSG-GPE.B Use coordinates to prove simple geometric theorems algebraically
- HSG-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically.
  - Classify shapes on the coordinate plane: justify your answer (G-O.13)
- HSG-GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
- HSG-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
- HSG-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

HSG-GMD Geometric Measurement and Dimension

HSG-GMD.A Explain volume formulas and use them to solve problems
- HSG-GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
- 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.
- HSG-GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
HSG-GMD.B Visualize relationships between two-dimensional and three-dimensional objects
- HSG-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.
  - Cross sections of three-dimensional figures (G-T.4)
  - Solids of revolution (G-T.5)

HSG-MG Modeling with Geometry

HSG-MG.A Apply geometric concepts in modeling situations
- HSG-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
  - Area and perimeter: word problems (A1-D.1)
  - Pythagorean theorem (G-P.1)
- HSG-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
  - Calculate density, mass, and volume (G-V.9)
- HSG-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

HSS-ID Interpreting Categorical and Quantitative Data

HSS-ID.A Summarize, represent, and interpret data on a single count or measurement variable
- HSS-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
- HSS-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- HSS-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
- HSS-ID.A.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
HSS-ID.B Summarize, represent, and interpret data on two categorical and quantitative variables
- HSS-ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
  - Find probabilities using two-way frequency tables (A1-KK.3)
  - Find conditional probabilities using two-way frequency tables (A1-KK.4)
- HSS-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
  - HSS-ID.B.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
  - HSS-ID.B.6b Informally assess the fit of a function by plotting and analyzing residuals.
    - Interpret a scatter plot (A1-JJ.1)
  - HSS-ID.B.6c Fit a linear function for a scatter plot that suggests a linear association.
    - Write equations for lines of best fit (A1-JJ.5)
HSS-ID.C Interpret linear models
- HSS-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
- HSS-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
- HSS-ID.C.9 Distinguish between correlation and causation.

HSS-IC Making Inferences and Justifying Conclusions

HSS-IC.A Understand and evaluate random processes underlying statistical experiments
- HSS-IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
- HSS-IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
HSS-IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies
- HSS-IC.B.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
  - Experiment design (A2-RR.8)
  - Experiment design (PC-FF.8)
- HSS-IC.B.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
- HSS-IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
  - Analyze the results of an experiment using simulations (A2-RR.9)
  - Analyze the results of an experiment using simulations (PC-FF.9)
- HSS-IC.B.6 Evaluate reports based on data.

HSS-CP Conditional Probability and the Rules of Probability

HSS-CP.A Understand independence and conditional probability and use them to interpret data
- HSS-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
- HSS-CP.A.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
- HSS-CP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
- HSS-CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
- HSS-CP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
HSS-CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model
- HSS-CP.B.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
- HSS-CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
- HSS-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.
- HSS-CP.B.9 Use permutations and combinations to compute probabilities of compound events and solve problems.

HSS-MD Using Probability to Make Decisions

HSS-MD.A Calculate expected values and use them to solve problems
- 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.
- 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 (A2-PP.5)
  - Expected values of random variables (PC-DD.5)
- 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.
- 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.
HSS-MD.B Use probability to evaluate outcomes of decisions
- HSS-MD.B.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
  - HSS-MD.B.5a Find the expected payoff for a game of chance.
    - Expected values for a game of chance (A2-PP.9)
    - Expected values for a game of chance (PC-DD.9)
  - HSS-MD.B.5b Evaluate and compare strategies on the basis of expected values.
    - Choose the better bet (A2-PP.10)
    - Choose the better bet (PC-DD.10)
- HSS-MD.B.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
- HSS-MD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
  - Choose the better bet (A2-PP.10)
  - Choose the better bet (PC-DD.10)

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HSN-RN The Real Number System

HSN-RN.A Extend the properties of exponents to rational exponents.

HSN-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

HSN-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

HSN-RN.B Use properties of rational and irrational numbers.

HSN-RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

HSN-Q Quantities

HSN-Q.A Reason quantitatively and use units to solve problems.

HSN-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

HSN-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

HSN-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

HSN-CN The Complex Number System

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

HSN-CN.A.1 Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.

HSN-CN.A.2 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

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

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

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.

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.

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.

HSN-CN.C Use complex numbers in polynomial identities and equations.

HSN-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions.

HSN-CN.C.8 Extend polynomial identities to the complex numbers.

HSN-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

HSN-VM Vector and Matrix Quantities

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

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).

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.

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

HSN-VM.B Perform operations on vectors.

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

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.

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

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

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 subscript x, v subscript y) = (cv subscript x, cv subscript y).

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).

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

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

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.

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

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.

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.

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.

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.

HSA-SSE Seeing Structure in Expressions

HSA-SSE.A Interpret the structure of expressions

HSA-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.

HSA-SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients.

HSA-SSE.A.1b Interpret complicated expressions by viewing one or more of their parts as a single entity.

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

HSA-SSE.B Write expressions in equivalent forms to solve problems

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

HSA-SSE.B.3a Factor a quadratic expression to reveal the zeros of the function it defines.

HSA-SSE.B.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

HSA-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions.

HSA-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.

HSA-APR Arithmetic with Polynomials and Rational Expressions

HSA-APR.A Perform arithmetic operations on polynomials

HSA-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

HSA-APR.B Understand the relationship between zeros and factors of polynomials

HSA-APR.B.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

HSA-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

HSA-APR.C Use polynomial identities to solve problems

HSA-APR.C.4 Prove polynomial identities and use them to describe numerical relationships.

HSA-APR.C.5 Know and apply the Binomial Theorem for the expansion of (x + y) to the n power 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.

HSA-APR.D Rewrite rational expressions

HSA-APR.D.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

HSA-CED Creating Equations

HSA-CED.A Create equations that describe numbers or relationships

HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems.

HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

HSA-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

HSA-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

HSA-REI Reasoning with Equations and Inequalities

HSA-REI.A Understand solving equations as a process of reasoning and explain the reasoning

HSA-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

HSA-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.