Massachusetts flagMassachusetts: Model Algebra II [AII] Math Standards

52 standards · 13 domains

ALGEBRA - ARITHMETIC WITH POLYNOMIALS AND RATIONAL EXPRESSIONS

  • AII.A-APR.A.1.a Perform operations on polynomial expressions (addition, subtraction, multiplication, and division), and compare the system of polynomials to the system of integers when performing operations.
  • AII.A-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).
  • AII.A-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.
  • AII.A-APR.C.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.
  • AII.A-APR.C.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.
  • AII.A-APR.D.6 Rewrite 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.
  • AII.A-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.

ALGEBRA - CREATING EQUATIONS

  • AII.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from simple root and rational functions and exponential functions.
  • AII.A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • AII.A-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. For example, represent equations describing satellites orbiting Earth and constraints on Earth’s size and atmosphere.

ALGEBRA - REASONING WITH EQUATIONS AND INEQUALITIES

  • AII.A-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
  • AII.A-REI.D.11 Explain 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 polynomial, rational, and logarithmic functions.

ALGEBRA - SEEING STRUCTURE IN EXPRESSIONS

  • AII.A-SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
  • AII.A-SSE.A.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)^n as the product of P and a factor not depending on P.
  • AII.A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 – y^2)(x^2 + y^2)and further factored (x-y)(x+y)(x-yi)(x+yi).
  • AII.A-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. For example, calculate mortgage payments.

FUNCTIONS - BUILDING FUNCTIONS

  • AII.F-BF.A.1.b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
  • AII.F-BF.B.3 Identify 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. Include simple rational, radical, logarithmic, and trigonometric functions. Utilize technology to experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
  • AII.F-BF.B.4.a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x)=2x^3 or f(x)=(x + 1)∕(x - 1) for x ≠ 1.

FUNCTIONS - INTERPRETING FUNCTIONS

  • AII.F-IF.B.4 For 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
  • AII.F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
  • AII.F-IF.B.6 Calculate 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.
  • AII.F-IF.C.7.b Graph square root and cube root functions.
  • AII.F-IF.C.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
  • AII.F-IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior; and trigonometric functions, showing period, midline, and amplitude.
  • AII.F-IF.C.8.a Use the process of factoring in a polynomial to function to show zeros, extreme values, and symmetry of the graph and interpret these in terms of a context.
  • AII.F-IF.C.9 Translate among different representations of functions (algebraically, graphically, numerically in tables, or by verbal descriptions). Compare properties of two functions each represented in a different way. For example, given a graph of one polynomial function and an algebraic expression for another, say which has the larger relative maximum and/or smaller relative minimum.
  • AII.F-IF.C.10 Given algebraic, numeric and/or graphical representations of functions, recognize the function as polynomial, rational, logarithmic, exponential, or trigonometric.

FUNCTIONS - LINEAR, QUADRATIC, AND EXPONENTIAL MODELS

  • AII.F-LE.A.4 For 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.

FUNCTIONS - TRIGONOMETRIC FUNCTIONS

  • AII.F-TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
  • AII.F-TF.A.2 Explain 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.
  • AII.F-TF.B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
  • AII.F-TF.C.8 Prove 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.

NUMBER AND QUANTITY - THE COMPLEX NUMBER SYSTEM

  • AII.N-CN.A.1 Know there is a complex number i such that i^2 = −1, and every complex number has the form a + bi with x-a and b real.
  • AII.N-CN.A.2 Use the relation i^2 = –1 and the Commutative, Associative, and Distributive properties to add, subtract, and multiply complex numbers.
  • AII.N-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions.
  • AII.N-CN.C.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x – 2i).
  • AII.N-CN.C.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

NUMBER AND QUANTITY - VECTOR AND MATRIX QUANTITIES

  • AII.N-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).
  • AII.N-VM.A.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.
  • AII.N-VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
  • AII.N-VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.
  • AII.N-VM.C.12 (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

STATISTICS AND PROBABILITY - MAKING INFERENCES AND JUSTIFYING CONCLUSIONS

  • AII.S-IC.A.1 Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population.
  • AII.S-IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of five tails in a row cause you to question the model?
  • AII.S-IC.B.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
  • AII.S-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.
  • AII.S-IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
  • AII.S-IC.B.6 Evaluate reports based on data.

STATISTICS AND PROBABILITY - INTERPRETING CATEGORICAL AND QUANTITATIVE DATA

  • AII.S-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.

STATISTICS AND PROBABILITY - USING PROBABILITY TO MAKE DECISIONS

  • AII.S-MD.B.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
  • AII.S-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 and replacing the goalie with an extra skater).