CA High School Algebra II Curriculum

CA High School Algebra II Curriculum

• CA.A2.N.CN.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.
  • Introduction to imaginary numbers
• CA.A2.N.CN.2Use the relation i^2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
  • Adding and subtracting complex numbers
  • Multiplying and dividing complex numbers
  • Complex conjugates
• CA.A2.N.CN.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
  • Distance and midpoint of complex numbers
• CA.A2.N.CN.8(+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x – 2i).
  • Complex numbers and complex planes
  • Angle and absolute value of complex numbers
  • Polar form of complex numbers
  • Operations on complex numbers in polar form
• CA.A2.N.CN.9(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
  • Fundamental theorem of algebra
• CA.A2.A.SSE.1Interpret expressions that represent a quantity in terms of its context.
  • Characteristics of quadratic functions
  • What is a polynomial?
  • Polynomial components
  • Evaluating polynomials
• CA.A2.A.SSE.2Use the structure of an expression to identify ways to rewrite it.
  • Factor by taking out the greatest common factor
  • Common factors of polynomials
  • Factoring polynomials by grouping
  • Solving polynomials with unknown coefficients
  • Solving polynomials with unknown constant terms
• CA.A2.A.SSE.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
  • Infinite geometric series
• CA.A2.A.APR.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.
  • Adding and subtracting rational expressions
  • Multiplying rational expressions
  • Multiplying monomial by monomial
  • Multiplying monomial by binomial
  • Multiplying binomial by binomial
  • Multiplying polynomial by polynomial
  • Applications of polynomials
  • Adding and subtracting polynomials
• CA.A2.A.APR.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).
  • Remainder theorem
• CA.A2.A.APR.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.
  • Rational zero theorem
  • Characteristics of polynomial graphs
  • Multiplicities of polynomials
  • Imaginary zeros of polynomials
• CA.A2.A.APR.4Prove polynomial identities and use them to describe numerical relationships.
  • Factor theorem
  • 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
• CA.A2.A.APR.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.
  • Binomial theorem
  • Pascal's triangle
• CA.A2.A.APR.6Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
  • Polynomial long division
  • Polynomial synthetic division
• CA.A2.A.APR.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.
  • Dividing rational expressions
  • Applications of rational equations
  • Simplifying complex fractions
  • Partial fraction decomposition
• CA.A2.A.CED.1Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • Solving quadratic inequalities
  • Absolute value functions
  • Solving absolute value equations
• CA.A2.A.CED.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • Quadratic function in general form: y = ax^2 + bx + c
  • Graphing reciprocals of linear functions
  • Graphing reciprocals of quadratic functions
• CA.A2.A.CED.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
  • Graphing systems of linear inequalities
  • Graphing linear inequalities in two variables
  • Graphing quadratic inequalities in two variables
  • Graphing systems of quadratic inequalities
  • Applications of inequalities
• CA.A2.A.CED.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
  • Solving rational equations
  • Solving literal equations
• CA.A2.A.REI.2Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
  • Solving radical equations
• CA.A2.A.REI.3.1Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in context.
  • Solving absolute value inequalities
• CA.A2.A.REI.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 quadratic functions: General form VS. Vertex form
  • Determining number of solutions to linear equations
  • Solving systems of linear equations by graphing
  • Solving systems of linear equations by elimination
  • Solving systems of linear equations by substitution
  • System of linear-quadratic equations
  • System of quadratic-quadratic equations
• CA.A2.F.IF.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.
  • Relationship between two variables
  • Understand relations between x- and y-intercepts
  • Domain and range of a function
  • Identifying functions
  • Function notation
• CA.A2.F.IF.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  • Simplifying rational expressions and restrictions
• CA.A2.F.IF.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.
  • Graphing exponential functions
  • Rate of change
• CA.A2.F.IF.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
  • Graphing logarithmic functions
  • Polynomial functions
  • Radical functions and transformations
  • Basic radical functions
  • Graphing transformations of logarithmic functions
• CA.A2.F.IF.8Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  • Converting from general to vertex form by completing the square
  • Adding functions
  • Subtracting functions
  • Multiplying functions
  • Dividing functions
  • Operations with functions
• CA.A2.F.IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
  • Finding the quadratic functions for given parabolas
  • Function notation (advanced)
  • Difference quotient: applications of functions
• CA.A2.F.BF.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.
  • Graphing transformations of exponential functions
  • Combining transformations of functions
  • Transformations of functions: Horizontal translations
  • Transformations of functions: Vertical translations
  • 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
  • Even and odd functions
• CA.A2.F.BF.4Find inverse functions.
  • Natural log: ln
  • Inverse functions
• CA.A2.F.LE.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.
  • Evaluating logarithms using change-of-base formula
  • What is a logarithm?
  • Converting from logarithmic form to exponential form
  • Evaluating logarithms without a calculator
  • Common logarithms
  • Converting from exponential form to logarithmic form
  • Solving exponential equations with logarithms
• CA.A2.F.LE.4.1Prove simple laws of logarithms.
  • Product rule of logarithms
  • Quotient rule of logarithms
  • Combining product rule and quotient rule in logarithms
  • Power rule of logarithms
• CA.A2.F.LE.4.3Understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.
  • Evaluating logarithms using logarithm rules
  • Solving logarithmic equations
• CA.A2.F.LE.5Interpret the parameters in a linear or exponential function in terms of a context.
  • Exponential growth and decay by a factor
• CA.A2.F.TF.1Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
  • Reference angle
  • Converting between degrees and radians
  • Trigonometric ratios of angles in radians
  • Radian measure and arc length
• CA.A2.F.TF.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
• CA.A2.F.TF.2.1Graph all 6 basic trigonometric 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
• CA.A2.F.TF.5Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
  • Graphing transformations of trigonometric functions
  • Determining trigonometric functions given their graphs
• CA.A2.F.TF.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
• CA.A2.G.GPE.3.1Given a quadratic equation of the form ax^2 + by^2 + cx + dy + e = 0, use the method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola and graph the equation. [In Algebra II, this standard addresses only circles and parabolas.]
  • Conics - Parabola
  • Conics - Circle
• CA.A2.S.ID.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.
  • Mean
  • Range and outliers
  • Application of averages
• CA.A2.S.IC.1Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
  • Data collection
  • Organizing outcomes
• CA.A2.S.IC.2Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
  • Comparing experimental and theoretical probability
  • Probability of independent events
• CA.A2.S.IC.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
  • Classification of data
  • Sampling methods
• CA.A2.S.IC.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.
  • Frequency distribution and histograms
  • Center of a data set: mean, median, mode
  • Spread of a data set - standard deviation & variance
  • Measures of relative standing - z-score, quartiles, percentiles
• CA.A2.S.IC.5Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
  • Bivariate, scatter plots and correlation
  • Regression analysis
  • Equation of the best fit line
• CA.A2.S.IC.6Evaluate reports based on data.
  • Critiquing data presentation
  • Reading and drawing bar graphs
  • Reading and drawing histograms
  • Reading and drawing line graphs
  • Box-and-whisker plots and scatter plots
  • Reading and drawing Venn diagrams
  • Stem and leaf plots
  • Frequency tables and dot plots
• CA.A2.S.MD.6(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
  • Introduction to probability
  • Introduction to probability
  • Addition rule for
  • Multiplication rule for
  • Conditional probability
  • Probability involving permutations and combinations
• CA.A2.S.MD.7(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
  • Probability distribution - histogram, mean, variance & standard deviation
  • Binomial distribution
  • Mean and standard deviation of binomial distribution
  • Introduction to normal distribution
  • Normal distribution and continuous random variable
  • Z-scores and random continuous variables