PA High School Algebra II Curriculum

PA High School Algebra II Curriculum

• CC.HSN.RN.A.1Explain 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.
  • Convert between radicals and rational exponents
  • Exponents: Product rule (a^x)(a^y) = a^(x+y)
  • Exponents: Division rule: a^x / a^y = a^(x-y)
  • Exponents: Power rule: (a^x)^y = a^(xy)
  • Exponents: Negative exponents
  • Exponents: Zero exponent: a^0 = 1
  • Exponents: Rational exponents
• CC.HSN.RN.A.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.
  • Solving for exponents
  • Operations with radicals
  • Conversion between entire radicals and mixed radicals
  • Converting radicals to mixed radicals
  • Converting radicals to entire radicals
  • Adding and subtracting radicals
  • Multiplying and dividing radicals
  • Rationalize the denominator
  • Evaluating and simplifying radicals
• CC.HSF.IF.A.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
  • Domain and range of a function
  • Function notation (advanced)
  • Identifying functions
• CC.HSF.IF.A.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  • Function notation
  • Difference quotient: applications of functions
• CC.HSF.IF.A.3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
  • Greatest common factors (GCF)
  • Introduction to sequences
  • Sigma notation
• CC.HSF.IF.B.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • Word problems of graphing linear functions
  • Characteristics of quadratic functions
  • Relationship between two variables
  • Understand relations between x- and y-intercepts
  • Combining transformations of functions
  • Reflection across the y-axis: y = f(-x)
  • Reflection across the x-axis: y = -f(x)
  • Transformations of functions: Horizontal stretches
  • Transformations of functions: Vertical stretches
• CC.HSF.IF.B.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  • Point of discontinuity
  • Even and odd functions
• CC.HSF.IF.B.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
  • Rate of change
  • Direct variation
• CC.HSF.IF.C.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
  • Graphing linear functions using a single point and slope
  • Graphing quadratic functions: General form VS. Vertex form
  • Graphing exponential functions
  • Graphing logarithmic functions
  • Graphing from slope-intercept form y=mx+b
  • Graphing transformations of exponential functions
  • Graphing transformations of logarithmic functions
  • Sine graph: y = sin x
  • Cosine graph: y = cos x
  • Tangent graph: y = tan x
  • Cotangent graph: y = cot x
  • Secant graph: y = sec x
  • Cosecant graph: y = csc x
• CC.HSF.IF.C.8Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  • Slope intercept form: y = mx + b
  • General form: Ax + By + C = 0
  • Point-slope form: y - y_1 = m(x - x_1)
  • Converting from general to vertex form by completing the square
  • Adding functions
  • Subtracting functions
  • Multiplying functions
  • Dividing functions
  • Operations with functions
• CC.HSF.IF.C.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
  • Parallel and perpendicular lines in linear functions
  • Graphs of rational functions
  • Inequalities of combined functions
• CC.HSF.BF.A.1Write a function that describes a relationship between two quantities.
  • Applications of linear relations
  • Finding an exponential function given its graph
  • Finding a logarithmic function given its graph
• CC.HSF.BF.A.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
  • Arithmetic sequences
  • Geometric sequences
  • Arithmetic series
  • Geometric series
  • Infinite geometric series
• CC.HSF.BF.B.3Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
  • Transformations of quadratic functions
  • Transformations of functions: Horizontal translations
  • Transformations of functions: Vertical translations
• CC.HSF.LE.A.1Distinguish between situations that can be modeled with linear functions and with exponential functions.
  • Introduction to nonlinear equations
  • Solving exponential equations using exponent rules
  • Exponential decay: Half-life
  • Exponential growth and decay by percentage
• CC.HSF.LE.A.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • Graphing linear functions using table of values
• CC.HSA.APR.A.1Understand 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.
  • Multiplying binomial by binomial
  • Multiplying polynomial by polynomial
  • Polynomial functions
  • Factoring trinomials
  • Factoring difference of cubes
  • Factoring sum of cubes
• CC.HSA.APR.B.2Know 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).
  • Greatest common factors (GCF)
  • Remainder theorem
  • Polynomial long division
  • Polynomial synthetic division
• CC.HSA.APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
  • Factoring polynomials: x^2 + bx + c
  • Characteristics of polynomial graphs
  • Factor theorem
  • Rational zero theorem
• 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.6Rewrite simple rational expressions in different forms.
  • Applications of polynomial functions
  • Solving polynomial inequalities
  • Fundamental theorem of algebra
• 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.HSA.REI.A.2Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
  • Evaluating and simplifying radicals
  • Solving radical equations
  • Square and square roots
  • Cubic and cube roots
• CC.HSF.LE.A.3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
  • Parallel line equation
  • Quadratic function in general form: y = ax^2 + bx + c
• CC.HSF.LE.A.4For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
  • Quotient rule of logarithms
  • Combining product rule and quotient rule in logarithms
  • Power rule of logarithms
  • Evaluating logarithms using logarithm rules
• CC.HSF.BF.B.4Find inverse functions.
  • Finding the quadratic functions for given parabolas
  • Inverse functions
• CC.HSF.BF.B.5Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
  • Product rule of logarithms
  • Solving exponential equations with logarithms
  • What is a logarithm?
  • Converting from logarithmic form to exponential form
  • Evaluating logarithms without a calculator
  • Common logarithms
  • Natural log: ln
  • Evaluating logarithms using change-of-base formula
  • Converting from exponential form to logarithmic form
• 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.SSE.B.4Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
  • Geometric series
  • Factor by taking out the greatest common factor
  • Factor by grouping
  • Factoring difference of squares: $x^2 - y^2$
• CC.HSN.CN.A.1Know there is a complex number i such that i^2 = -1, and every complex number has the form a + bi with a and b real.
  • Understanding the number systems
  • Introduction to imaginary numbers
• CC.HSN.CN.A.2Use the relation i^2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
  • Combining the exponent rules
  • Complex numbers and complex planes
  • Adding and subtracting complex numbers
  • Multiplying and dividing complex numbers
• CC.HSN.CN.A.3Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
  • Complex conjugates
  • Distance and midpoint of complex numbers
  • Angle and absolute value of complex numbers
• 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.HSN.CN.C.7Solve quadratic equations with real coefficients that have complex solutions.
  • Nature of roots of quadratic equations: The discriminant
  • Using quadratic formula to solve quadratic equations
  • Applications of quadratic equations
• CC.HSN.CN.C.8Extend polynomial identities to the complex numbers.
  • Solving polynomials with unknown coefficients
  • Factoring polynomials: x^2 + bx + c
  • Applications of polynomials: x^2 + bx + c
  • Solving polynomials with the unknown
  • Factoring polynomials: ax^2 + bx + c
  • Factoring perfect square trinomials: (a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 - 2ab + b^2
  • Find the difference of squares: (a - b)(a + b) = (a^2 - b^2)
• CC.HSN.CN.C.9Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
  • Word problems of polynomials
  • Fundamental theorem of algebra
• 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.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.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.HSS.ID.A.4Use 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.
  • Introduction to normal distribution
  • Normal distribution and continuous random variable
  • Z-scores and random continuous variables
• CC.HSS.ID.B.5Summarize 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.
  • Reading and drawing Venn diagrams
  • Probability with Venn diagrams
• CC.HSS.CP.A.1Describe 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").
  • Introduction to probability
  • Organizing outcomes
  • Introduction to probability
  • Set notation
  • Set builder notation
  • Intersection and union of 2 sets
  • Intersection and union of 3 sets
• CC.HSS.CP.A.2Understand 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.
  • Probability of independent events
  • Multiplication rule for
  • Probability of independent events
• CC.HSS.CP.A.3Understand 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.
  • Conditional probability
• CC.HSS.CP.B.9Use permutations and combinations to compute probabilities of compound events and solve problems.
  • Permutations
  • Combinations
  • Probability involving permutations and combinations