Algebra II (Massachusetts Curriculum Frameworks)

The Complex Number System (AII.N-CN)

• AII.N-CN.A.1 Know i² = −1 and that every complex number can be written a + bi with a and b real.
• AII.N-CN.A.2 Add, subtract and multiply complex numbers using i² = −1 and the properties of operations.
• AII.N-CN.C.7 Solve quadratics with real coefficients whose solutions are complex.
• AII.N-CN.C.8 (+) Extend polynomial identities to the complex numbers.
• AII.N-CN.C.9 (+) Know the Fundamental Theorem of Algebra and show it is true for quadratic polynomials.

Vector and Matrix Quantities (AII.N-VM)

• AII.N-VM.A.1 (+) Know vectors have magnitude and direction and use standard notation for vectors and their magnitudes.
• AII.N-VM.A.3 (+) Solve problems with velocity and other quantities shown as vectors.
• AII.N-VM.C.6 (+) Use matrices to represent and work with data, such as payoffs or connections in a network.
• AII.N-VM.C.8 (+) Add, subtract and multiply matrices of suitable dimensions.
• AII.N-VM.C.12 (+) Use 2 × 2 matrices to transform points in the plane and relate the absolute value of the determinant to area.

Seeing Structure in Expressions (AII.A-SSE)

• AII.A-SSE.A.1 Interpret expressions that stand for a quantity in context.
  • a. Explain what terms, factors and coefficients in an expression mean.
  • b. Read a complicated expression by seeing parts of it as single quantities, e.g. P(1 + r)ⁿ as P times a factor.
• AII.A-SSE.A.2 Use the structure of an expression to find ways to rewrite it, e.g. x⁴ − y⁴ as a difference of squares.
• AII.A-SSE.B.4 Derive and use the sum formula for a finite geometric series (ratio not 1), e.g. to calculate mortgage payments.

Arithmetic with Polynomials and Rational Expressions (AII.A-APR)

• AII.A-APR.A.1 Know polynomials are closed under addition, subtraction and multiplication, and add, subtract and multiply them.
  • a. Add, subtract, multiply and divide polynomials and compare the polynomial system with the integers.
• AII.A-APR.B.2 Use the Remainder Theorem: p(a) is the remainder on dividing p(x) by x − a, so p(a) = 0 exactly when x − a is a factor.
• AII.A-APR.B.3 Find zeros of factored polynomials and use them to sketch a rough graph.
• AII.A-APR.C.4 Prove polynomial identities and use them to show number relationships, e.g. generating Pythagorean triples.
• AII.A-APR.C.5 (+) Know and use the Binomial Theorem to expand (x + y)ⁿ, with coefficients from Pascal's Triangle.
• AII.A-APR.D.6 Rewrite simple rational expressions in different forms, e.g. as quotient plus remainder over divisor.
• AII.A-APR.D.7 (+) Know rational expressions are closed under the four operations (non-zero divisor) and add, subtract, multiply and divide them.

Creating Equations (AII.A-CED)

• AII.A-CED.A.1 Write and solve one-variable equations and inequalities, including ones from simple root, rational and exponential functions.
• AII.A-CED.A.2 Write equations in two or more variables for relationships and graph them on labeled, scaled axes.
• AII.A-CED.A.3 Show constraints with equations, inequalities or systems and judge whether solutions make sense in context.

Reasoning with Equations and Inequalities (AII.A-REI)

• AII.A-REI.A.2 Solve one-variable rational and radical equations and show how extra (extraneous) solutions can arise.
• AII.A-REI.D.11 Explain why x-values where y = f(x) and y = g(x) cross solve f(x) = g(x); find solutions approximately.

Interpreting Functions (AII.F-IF)

• AII.F-IF.B.4 Interpret key features of graphs and tables of a relationship in context and sketch graphs from verbal descriptions.
• AII.F-IF.B.5 Relate a function's domain to its graph and to the situation it describes.
• AII.F-IF.B.6 Find, interpret and estimate a function's average rate of change over an interval.
• AII.F-IF.C.7 Graph functions from their equations and show key features, by hand in simple cases and with technology.
  • b. Graph square root and cube root functions.
  • c. Graph polynomial functions showing zeros (when factors are known) and end behavior.
  • e. Graph exponential and log functions (intercepts, end behavior) and trig functions (period, midline, amplitude).
• AII.F-IF.C.8 Rewrite a function in equivalent forms to show different properties.
  • a. Factor polynomial functions to show zeros, extreme values and symmetry, and interpret them in context.
• AII.F-IF.C.9 Compare properties of two functions each shown in a different way (equation, graph, table, words).
• AII.F-IF.C.10 Tell whether a function is polynomial, rational, logarithmic, exponential or trigonometric from its equation, table or graph.

Building Functions (AII.F-BF)

• AII.F-BF.A.1 Write simple rational, radical, log or trig functions to describe how two quantities are related.
  • b. Combine standard function types with arithmetic operations.
• AII.F-BF.B.3 Find the effect of f(x) + k, k·f(x), f(kx) and f(x + k) on graphs of rational, radical, log and trig functions; recognize even and odd functions.
• AII.F-BF.B.4 Find inverse functions.
  • a. Solve f(x) = c for a simple invertible function and write an expression for the inverse.

Linear, Quadratic, and Exponential Models (AII.F-LE)

• AII.F-LE.A.4 Write solutions of ab^(ct) = d as logarithms with base 2, 10 or e and evaluate them with technology.

Trigonometric Functions (AII.F-TF)

• AII.F-TF.A.1 See an angle's radian measure as the length of unit-circle arc it cuts off.
• AII.F-TF.A.2 Explain how the unit circle extends trigonometric functions to all real numbers.
• AII.F-TF.B.5 Choose trigonometric functions to model periodic situations with a given amplitude, frequency and midline.
• AII.F-TF.C.8 Prove sin²θ + cos²θ = 1 and use it, with the quadrant, to find other trig values.

Interpreting Categorical and Quantitative Data (AII.S-ID)

• AII.S-ID.A.4 Fit a normal distribution using mean and standard deviation, estimate percentages and know when this is not appropriate.

Making Inferences and Justifying Conclusions (AII.S-IC)

• AII.S-IC.A.1 Understand statistics as a way to infer things about a population from a random sample.
• AII.S-IC.A.2 Decide whether a model fits results from a data-generating process, e.g. by simulation.
• AII.S-IC.B.3 Know the purposes of and differences between surveys, experiments and observational studies and how randomization fits each.
• AII.S-IC.B.4 Estimate a population mean or proportion from a sample survey and build a margin of error with simulation.
• AII.S-IC.B.5 Compare two treatments in a randomized experiment, using simulation to judge if differences are significant.
• AII.S-IC.B.6 Evaluate reports based on data, spotting weak design, analysis, displays or conclusions.

Using Probability to Make Decisions (AII.S-MD)

• AII.S-MD.B.6 (+) Make fair decisions using probability, e.g. a random number generator.
• AII.S-MD.B.7 (+) Use probability to analyze decisions and strategies, e.g. product testing or medical testing.