AR High School Pre-Calculus Curriculum

AR High School Pre-Calculus Curriculum

• CC.HSN.CN.B.4Represent 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.
  • Polar form of complex numbers
• CC.HSN.CN.B.5Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
  • Operations on complex numbers in polar form
• CC.HSN.CN.B.6Calculate 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.
  • Midpoint formula: $M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)$
• CC.HSA.CED.A.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
  • Point-slope form: y - y_1 = m(x - x_1)
  • Graphing quadratic inequalities in two variables
  • Graphing systems of quadratic inequalities
• CC.HSA.REI.D.11Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
  • Graphing linear functions using table of values
  • Graphing exponential functions
  • Graphing logarithmic functions
  • The inverse of 3 x 3 matrices with matrix row operations
  • The inverse of 3 x 3 matrix with determinants and adjugate
• CC.HSF.TF.A.1Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
  • Converting between degrees and radians
  • Trigonometric ratios of angles in radians
  • Radian measure and arc length
• CC.HSF.TF.A.2Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
  • Unit circle
• CC.HSF.TF.A.3Use 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 the exact value of trigonometric ratios
  • Solving expressions using 45-45-90 special right triangles
  • Solving expressions using 30-60-90 special right triangles
• CC.HSF.TF.B.5Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
  • Graphing transformations of trigonometric functions
  • Determining trigonometric functions given their graphs
• CC.HSF.TF.B.6Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
  • Finding inverse trigonometric function from its graph
  • Finding inverse reciprocal trigonometric function from its graph
  • Inverse reciprocal trigonometric function: finding the exact value
• CC.HSF.TF.B.7Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
  • Evaluating inverse trigonometric functions
• CC.HSF.TF.C.8Prove the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
  • Pythagorean identities
• CC.HSF.TF.C.9Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
  • Sum and difference identities
  • Double-angle identities
  • Cofunction identities
• CC.HSG.SRT.D.9Derive 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.
  • Area of triangles: 1/2 a*b sin(C)
• CC.HSG.SRT.D.10Prove the Laws of Sines and Cosines and use them to solve problems.
  • Law of sines
  • Law of cosines
• CC.HSG.SRT.D.11Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.
  • Applications of the sine law and cosine law
• CC.HSA.APR.C.4Prove polynomial identities and use them to describe numerical relationships.
  • Solving polynomials with unknown coefficients
  • Multiplicities of polynomials
  • Imaginary zeros of polynomials
• CC.HSA.APR.C.5Know 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.
  • Pascal's triangle
  • Binomial theorem
  • Determining the equation of a polynomial function
• CC.HSA.APR.D.7Understand 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.
  • Negative exponent rule
  • Multiplying rational expressions
  • Dividing rational expressions
  • Descartes' rule of signs
• CC.HSN.VM.A.1Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.
  • Use sine ratio to calculate angles and sides (Sin = o / h)
  • Introduction to vectors
  • Magnitude of a vector
  • Direction angle of a vector
• CC.HSN.VM.A.2Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
  • Distance formula: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
  • Scalar multiplication of vectors
  • Equivalent vectors
• CC.HSN.VM.A.3Solve problems involving velocity and other quantities that can be represented by vectors.
  • Word problems relating guy wire in trigonometry
  • Word problems on vectors
• CC.HSN.VM.B.4Add and subtract vectors.
  • Slope equation: $m = \frac{y_2-y_1}{x_2- x_1}$
  • Adding and subtracting vectors in component form
  • Operations on vectors in magnitude and direction form
• CC.HSN.VM.B.5Multiply a vector by a scalar.
  • Slope intercept form: y = mx + b
  • Unit vector
• CC.HSN.VM.C.6Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
  • Notation of matrices
  • Adding and subtracting matrices
• CC.HSN.VM.C.7Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
  • Scalar multiplication of matrices
• CC.HSN.VM.C.8Add, subtract, and multiply matrices of appropriate dimensions.
  • Matrix multiplication
  • The three types of matrix row operations
• CC.HSN.VM.C.9Understand 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 matrix multiplication
• CC.HSN.VM.C.10Understand 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.
  • Zero matrix
  • Identity matrix
  • The determinant of a 2 x 2 matrix
  • The determinant of a 3 x 3 matrix (General & Shortcut Method)
  • The Inverse of a 2 x 2 matrix
• CC.HSN.VM.C.11Multiply 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.
  • Transforming vectors with matrices
  • Transforming shapes with matrices
• CC.HSN.VM.C.12Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
  • Finding the transformation matrix
• CC.HSA.REI.C.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
  • System of linear-quadratic equations
  • Nature of roots of quadratic equations: The discriminant
  • Applications of quadratic equations
  • Solving quadratic inequalities
• CC.HSA.REI.C.8Represent a system of linear equations as a single matrix equation in a vector variable.
  • Representing a linear system as a matrix
  • Notation of matrices
  • Adding and subtracting matrices
  • Scalar multiplication of matrices
• CC.HSA.REI.C.9Find 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).
  • The Inverse of a 2 x 2 matrix
  • Solving linear systems using 2 x 2 inverse matrices
  • Matrix multiplication
  • The three types of matrix row operations
• CC.HSG.C.A.4Construct a tangent line from a point outside a given circle to the circle.
  • Tangent properties
• CC.HSG.GPE.A.2Derive the equation of a parabola given a focus and directrix.
  • Conics - Parabola
• CC.HSG.GPE.A.3Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
  • Conics - Ellipse
  • Conics - Hyperbola
• CC.HSN.RN.B.3Explain 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.
  • Rational vs. Irrational numbers
  • Solving radical equations
  • Converting repeating decimals to fractions
• CC.HSN.Q.A.1Use 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.
  • Metric systems
  • Imperial systems
  • Scale diagrams
  • Conversions involving squares and cubic
• CC.HSN.Q.A.2Define appropriate quantities for the purpose of descriptive modeling.
  • Conversions between metric and imperial systems
  • Squares and square roots
  • Pythagorean theorem
  • Estimating square roots
  • Using the pythagorean relationship
  • Applications of pythagorean theorem
• CC.HSN.Q.A.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
  • Upper and lower bound
  • Cubic and cube roots
  • Ratios
  • Rates
  • Proportions
  • Percents, fractions, and decimals
• CC.HSS.IC.A.1Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
  • Sampling methods
• CC.HSS.IC.A.2Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
  • Sampling distributions
  • Rare event rule
• CC.HSS.IC.B.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
  • Census and bias
• CC.HSS.IC.B.4Use 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.
  • Margin of error
  • Confidence intervals to estimate population mean
  • Making a confidence interval
• CC.HSS.IC.B.5Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
  • Traditional hypothesis testing
  • P-value hypothesis testing
  • Analysis of variance (ANOVA)
• CC.HSS.IC.B.6Evaluate reports based on data.
  • Influencing factors in data collection
• CC.HSS.CP.A.5Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
  • Comparing experimental and theoretical probability
• CC.HSS.CP.B.7Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.
  • Addition rule for
• CC.HSS.MD.A.1Define 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.
  • Probability distribution - histogram, mean, variance & standard deviation
• CC.HSS.MD.A.2Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
  • Properties of expectation
• CC.HSS.MD.A.3Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.
  • Binomial distribution
  • Mean and standard deviation of binomial distribution
• CC.HSS.MD.B.7Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
  • Type 1 and type 2 errors
• CC.HSG.GMD.A.2Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
  • Surface area and volume of spheres