# Algebra II

Curriculum > High > Beijing > Mathematics
sequence
Standard Benchmark DOK CCSS Sequence
5.1 Extend the properties of exponents to rational exponents 5.1.1.B  Compare and order numbers in the Real Number System by size and/or position on a number line (to include ability to identify equivalent terms). 2   1.1.1
5.2 Use properties of rational and irrational numbers. 5.2.3.A  Justify mathematical procedures and determine how they apply to invented operations using field properties (closure, associative, commutative, distributive, identity and inverse). 1   1.1.2
7.1 Interpret and model a given context using expressions 7.1.2.A  Interpret parts of an expression, such as terms, factors, and coefficients.  1 A.SSE.1.a 1.1.3
7.1.3.A  Interpret complicated expressions by viewing one or more of their parts as a single entity. 1 A.SSE.1.b 1.1.4
7.2 Decompose and recompose algebraic expressions using number properties in the context of solving problems 7.2.1.C  Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.*  2 A.SSE.3 1.1.5
7.5 Demonstrate that the relationship of two or more variables can be represented as an equation or inequality and can be represented graphically 7.5.1.A  Create equations and inequalities in one variable and use them to solve problems. 2 A.CED.1 1.1.6
7.5.2.A  Create equations in two or more variables to represent relationships between quantities: graph equations on coordinate axes with labels and scales. 2 A.CED.2 1.1.7
7.5.3.A  Represent constraints by equations or inequalities, and by systems and/or inequalities. 2 A.CED.3 1.1.8
Stem Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. 1 A.CED.4 1.1.9
7.6 Use algebraic properties and inverse operations to justify the steps in solving equations 7.6.1.A  Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.  3 A.REI.1 1.1.10
7.6.3.A  Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 1 A.REI.3 1.1.11
7.10 Understand 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). 7.10.1.A  Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.  1 F.IF.2 1.2.1
7.10.1.A  Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.  1 F.IF.2 1.2.1
7.10.2.A  Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n‐1) for n ≥ 1. 2 F.IF.3 1.2.2
7.11 Describe characteristics of graphs, tables, and equations that model families of functions. 7.11.1.A  Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 2 F.IF.5 1.2.3
7.11.2.A  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.* 2 F.IF.6 1.2.4
7.13 Know the difference between a recursive rule and an explicit expression for a function. 7.13.1.A  Write a function that describes a relationship between two quantities.* 2 F.BF.1 1.2.5
7.13.1.B  Determine an explicit expression, a recursive process, or steps for calculation from a context.  2 F.BF.1.a 1.2.6
7.9 Knows that a graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
7.9.2A  Graph the solutions to a linear inequality in two variables as a half‐plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding halfplanes. 1 A.REI.12 1.2.7
7.10 Understand 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). 7.10.3.A  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.*  2 F.IF.4 1.2.8
7.9 Knows that a graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
7.9.1A  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. 2 A.REI.11 1.2.9
7.8 Solve systems of equations 7.8.1.A  Solve systems of linear equations exactly and approximately, focusing on approximately, focusing on pairs of linear equations in pairs of linear equations in two variables. 1 A.REI.6 2.1.1
7.8.2.A  Solve a simple system consisting of a linear equation and quadratic equation in two variables algebraically and graphically.  1 A.REI.7 2.1.2
7.8.3.A  Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 3 A.REI.5 2.1.3
STEM(+) Represent a system of linear equations as a single matrix equation in a vector variable. 1 A.REI.8 2.1.4
7.7 Solve quadratic equations in one variable. 7.7.1.A  Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x‐p)2 = q that has the same solutions Derive the quadratic formula from this form. 2 A.REI.4.a 2.1.5
7.7.2.A  Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as  appropriate to the initial form of the equation.  1 A.REI.4.b 2.1.6
7.7.3.A  Recognize when the quadratic formula gives complex solutions and write them as a+‐ bi for real numbers a and b. 1 A.REI.4.b 2.1.7
7.11 Describe characteristics of graphs, tables, and equations that model families of functions. 7.11.2.B  Graph linear and quadratic functions and show intercepts, maxima, and minima. 1 F.IF.7.a 2.1.8
7.11.2.C  Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 2 F.IF.8.a 2.1.9
7.12 Use strategies for interpreting key features of representations. 7.12.3.A  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum 2 F.IF.9 2.1.10
7.2 Decompose and recompose algebraic expressions using number properties in the context of solving problems 7.2.2.C  Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 1 A.SSE.3.b 2.1.11
9.5 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities 9.5.1.A  Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, outcomes, or as unions, complements of other events ("or," "and," "not"). 2 S.CP.1 2.2.1
9.5.2.A  Understand 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 determine if they are 2 S.CP.2 2.2.2
9.5.3.A  Understand 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 given B is the same as the conditional probability of B conditional probability of B probability of B. 2 S.CP.3 2.2.3
STEM(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. 2 S.CP.8 2.2.4
STEM(+) Use permutations and combinations to compute probabilities of compound events and solve problems 2 S.CP.9 2.2.5
9.8 Use probability to evaluate outcomes of decisions 　 STEM(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. a. Find the expected payoff for a game of chance. 2 S.MD.5 2.2.6
STEM(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 2 S.MD.6 2.2.7
9.6 Recognize the concepts of conditional probability and independence in everyday language and everyday situations. 9.6.2.A  Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. 2 S.CP.5 2.2.8
9.6.3.A  Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. 2 S.CP.6 2.2.9
Apply the Addition Rule, P(A or B) = P(A) + P(B) ‐ P(A and B), and interpret the answer in terms of the model. 2 S.CP.7 2.2.10
5.3 Perform arithmetic operations with complex numbers. 5.3.1.A  Know 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.  1 N.CN.1 2.3.1
5.3.2.A  Use the relation i 2 = ‐1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers 1 N.CN.2 2.3.2
5.3.3.A  Solve quadratic equations with real coefficients that have complex solutions. 1 N.CN.7 2.3.3
STEM(+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i). 1 N.CN.8 2.3.4
7.3 Demonstrate that polynomials form a system analogous to the integers, namely, they are closed under the operation of addition, subtraction, and multiplication. 7.3.1.A  Add, subtract, and multiply polynomials. 1 A.APR.1 3.1.1
7.3.2.A  Prove polynomial identities and use them to describe numerical relationships. 2 A.APR.4 3.1.2
7.3.3.A  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. 1 A.APR.6 3.1.3
5.4 Represent complex numbers and their operations on the complex plane. 　 STEM(+) Know the Fundamental Theorem of Algebra show that it is true for quadratic polynomials. 2 N.CN.9 3.1.4
7.4 Demonstrate that polynomials can be decomposed and recomposed. 7.4.1.A  Identify zeros of polynomials when suitable factorizations are defined by the polynomial. 1 A.APR.3 3.1.5
7.4.2.A  Apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by xa is p(a), so p(a) = 0 if and only if (x‐a) is a factor of  p(x). 1 A.APR.2 3.1.6
STEM(+) 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.  2 A.APR.5 3.1.7
7.11 Describe characteristics of graphs, tables, and equations that model families of functions. 7.11.1.B  Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 1 F.IF.7.c 3.1.8
7.15 Knows that manipulating the parameters of the symbolic rule will result in a predictable transformation of
the graph.
7.15.1.A  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. 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. 2 F.BF.3 3.1.9
5.1 Extend the properties of exponents to rational exponents 5.1.1.A  Explain 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. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. 1 N.RN.1 3.2.1
5.1.2.A  Rewrite expressions involving radicals and rational exponents using the properties of exponents. 1 N.RN.2 3.2.2
7.6 Use algebraic properties and inverse operations to justify the steps in solving equations 7.6.2.A  Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 1 A.REI.2 3.2.3
7.2 Decompose and recompose algebraic expressions using number properties in the context of solving problems 7.2.3.C  Use the properties of exponents to transform expressions for exponential functions. 1 A.SSE.3.c 3.2.4
7.11 Describe characteristics of graphs, tables, and equations that model families of functions. 7.11.3.B  Graph square root, cube root, and piecewise‐defined functions, including step functions and absolute value functions. 1 F.IF.7.b 3.2.5
7.11.3.C  Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude 1 F.IF.7.e 3.2.6
7.12 Use strategies for interpreting key features of representations. 7.12.1.A  Use the properties of exponents to interpret expressions for exponential functions. 2 F.IF.8.b 3.2.7
7.12.2.A  Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function 2 F.IF.8 3.2.8
7.13 Know the difference between a recursive rule and an explicit expression for a function. 7.13.2.A  Find inverse functions. 1 F.BF.4 3.2.9
7.13.3.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. 1 F.BF.4.a 3.2.10
7.16 Describe characteristic graph, table, and equation formats for linear, exponential, and quadratic functions. 7.16.1.A  Distinguish between situations that can be modeled with linear functions and with exponential functions. 2 F.LE.1 3.3.1
7.16.1.B  Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table). 2 F.LE.2 3.3.2
7.16.2.A  Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. 3 F.LE.1.a 3.3.3
7.16.2.B  Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 2 F.LE.3 3.3.4
7.16.3.A  Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 2 F.LE.1.b 3.3.5
7.16.3.B  For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. 1 F.LE.4 3.3.6
7.8 Solve systems of equations 　 STEM(+) Represent a system of linear equations as a single matrix equation in a vector variable. 1 A.REI.8 3.3.7
STEM(+) Find 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)  1 A.REI.9 3.3.8
1.7 Perform operations on matrices and use matrices in applications. 　 STEM(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.  2 N.VM.6 4.1.1
STEM(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.    1 N.VM.7 4.1.2
STEM(+) Add, subtract, and multiply matrices of appropriate dimensions. 1 N.VM.8 4.1.3
STEM(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. 2 N.VM.9 4.1.4
STEM(+) Understand 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. 1 N.VM.10 4.1.5
7.3 Demonstrate that polynomials form a system analogous to the integers, namely, they are closed under the operation of addition, subtraction, and multiplication. 　 STEM(+) 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. 1 A.APR.7 4.1.6
7.8 Solve systems of equations 　 STEM(+) Find 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)  1 A.REI.9 4.1.7
7.4 Demonstrate that polynomials can be decomposed and recomposed. 　 STEM(+) 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.  2 A.APR.5 4.1.8