# BPS District Mathematics Standards Book

Completion requirements

### HS Math Courses

#### Algebra II

### Algebra II MAT Standards Domains

This course is the minimum recommendation for students who wish to be prepared for success in credit-bearing college mathematics courses. Algebra II provides a review and expansion of basic terminology, notation, concepts, skills and application
of algebra by a thorough examination of the real number system, including an introduction to statistics. An average of a “C” or better in Algebra I and Geometry or an average of an “A” in Informal Geometry is recommended.
**Note**: Standards Identifiers auto-link to popup a detailed description and links to Proficiency Scales with

*Prioritized Standards Identifiers being highlighted a sky-blue.*

## MAT-12.NO** (NO) **Number and Operations

*Learners will develop a foundational understanding of the number system, operations, and computational
fluency to create connections and solve problems within and across concepts.*

**MAT-12.NO.01**Rewrite complex expressions involving radicals and rational exponents using the properties of exponents.**MAT-12.NO.02**Perform operations on complex radical expressions and simplify radicals to write equivalent expressions.**MAT-12.NO.03**Demonstrate that the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational.**MAT-12.NO.04**Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.**MAT-12.NO.05**Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.**MAT-12.NO.06**Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real. Understand the hierarchal relationships among subsets of the complex number system.**MAT-12.NO.07**Use the definition i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.**MAT-12.NO.08**Use conjugates to find quotients of complex numbers.**MAT-12.NO.09**Apply the Fundamental Theorem of Algebra to determine the number of zeros for polynomial functions. Find all solutions to a polynomial equation.**MAT-12.NO.10**Represent complex numbers on the complex plane in rectangular, trigonometric, and polar forms. Find the modulus (absolute value) of a complex number. Explain why the rectangular, trigonometric, and polar forms of a given complex number represent the same number.**MAT-12.NO.11**Represent addition, subtraction, multiplication, conjugation, powers, and roots of complex numbers geometrically on the complex and/or polar plane; use properties of this representation for computation.**MAT-12.NO.12**Extend polynomial identities to the complex numbers.**MAT-12.NO.13**Apply the Fundamental Theorem of Algebra to find all roots of a polynomial equation and determine the nature (i.e., integer, rational, irrational, real, complex) of the roots.**MAT-12.NO.14**Recognize vector quantities as having both magnitude and direction, writing them in polar form.**MAT-12.NO.15**Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.**MAT-12.NO.16**Solve problems involving magnitude and direction that can be represented by vectors.**MAT-12.NO.17**Add and subtract vectors.**MAT-12.NO.18**Multiply a vector by a scalar.**MAT-12.NO.19**Represent data in a matrix. Perform operations (i.e., addition, subtraction, multiplication) on matrices of appropriate dimensions to solve problems and in context. Know that matrix multiplication is not commutative.

## MAT-12.AR** (AR) **Algebraic Reasoning

*Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to
represent mathematical models while adopting approaches and solutions in novel situations.*

**MAT-12.AR.01**Rearrange multi-variable formulas to highlight a quantity of interest.**MAT-12.AR.02**Use the structure of an expression (to extend to polynomial and rational expressions) to identify ways to rewrite it.**MAT-12.AR.03**Interpret expressions that represent a quantity in context.**MAT-12.AR.04**Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.**MAT-12.AR.05**Add, subtract, multiply, and divide rational expressions. Understand that rational expressions form a system analogous to rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.**MAT-12.AR.06**Rewrite simple rational expressions indifferent forms. Write a(x)/b(x) in the form of 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 technology for the more complicated examples.**MAT-12.AR.07**Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic equations and simple rational and exponential equations.**MAT-12.AR.08**Create equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with proper labels and scales.**MAT-12.AR.09**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.**MAT-12.AR.10**Derive the quadratic formula from the form 0 = ax² + bx + c.**MAT-12.AR.11**Solve quadratic equations with real coefficients that have solutions of the form a + bi and a - bi.**MAT-12.AR.12**Solve simple rational and radical equations in one variable and identify extraneous solutions.**MAT-12.AR.13**Add, subtract, and multiply polynomials beyond quadratics. Understand that polynomials form a system comparable to integers, namely, they are closed under the operations of addition, subtraction, and multiplication.**MAT-12.AR.14**Identify zeros of polynomial equations when suitable factorizations are available. Use the zeros to construct a rough graph of the function defined by the polynomial.**MAT-12.AR.15**Apply the Factor and Remainder Theorems to determine efficiently whether a linear expression is a factor of a polynomial equation. Apply the Remainder Theorem in context.**MAT-12.AR.16**Using graphs, technology, tables, or successive approximations, show that the solution(s) to the equation f(x) = g(x) are the x- value(s) that result in the y-values of f(x) and g(x)being the same.**MAT-12.AR.17**Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.**MAT-12.AR.18**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).**MAT-12.AR.19**Solve a system of equations in three or more variables with matrices (using technology).**MAT-12.AR.20**Apply the Binomial Theorem for the expansion of (ax + by)ⁿ in powers of x and y for a positive integer n and integers a and b.

## MAT-12.AR.F** (AR.F) **Functions

*Learners will develop a foundational knowledge of functions and use them to model relationships between quantities.*

**MAT-12.AR.F.01**Write a function that describes a relationship between two quantities.**MAT-12.AR.F.02**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.**MAT-12.AR.F.03**Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.**MAT-12.AR.F.04**Identify the effect of transformations on the graph of a function by replacing f(x) with af(x), f(bx), f(x - h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the value of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.**MAT-12.AR.F.05**Find inverse functions.**MAT-12.AR.F.06**Apply the inverse relationship between exponents and logarithms to solve problems.**MAT-12.AR.F.07**Compare key features of two functions each represented in a different way(algebraically, graphically, numerically, in tables, or by verbal descriptions).**MAT-12.AR.F.08**Use tables, graphs, verbal descriptions, and equations to interpret and sketch the key features of a function modeling the relationship between two quantities.**MAT-12.AR.F.09**Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.**MAT-12.AR.F.10**Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.**MAT-12.AR.F.11**Analyze and graph functions expressed symbolically (by hand in simple cases and using technology for more complicated cases), identifying key features of the graph.**MAT-12.AR.F.12**Compare the end behavior of linear, quadratic, and exponential functions using graphs and/or tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing as a linear or quadratic function.**MAT-12.AR.F.13**Determine whether a linear,quadratic, polynomial, exponential, logarithmic, or trigonometric model fits a situation. Determine an appropriate mathematical model in context (with or without technology).**MAT-12.AR.F.14**Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.**MAT-12.AR.F.15**Use properties of logarithms to express the solution to abᶜᵗ = d where a, c, and dare real numbers and b is a positive real number.Evaluate the logarithm using technology when appropriate.**MAT-12.AR.F.16**Extend right triangle trigonometry and apply knowledge of the unit circle to determine values of sine, cosine, and tangent for multiples of π/3, π/4, and π/6.**MAT-12.AR.F.17**Use the Pythagorean Identity sin² (θ) + cos² (θ) = 1 to find sin (θ), cos (θ), or tan(θ) given sin (θ), cos (θ), or tan (θ) and the quadrant of the angle.**MAT-12.AR.F.18**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.**MAT-12.AR.F.19**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.**MAT-12.AR.F.20**Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.**MAT-12.AR.F.21**Create a trigonometric function to model periodic phenomena.**MAT-12.AR.F.22**Restrict the domain of a trigonometric function to construct its inverse.**MAT-12.AR.F.23**Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in context.**MAT-12.AR.F.24**Know and apply the addition and subtraction formulas for sine, cosine, and tangent to solve problems.

## MAT-12.GM** (GM) **Geometry

*Learners will use visualization, spatial reasoning, geometric modeling, and measurement to investigate
the characteristics of figures, perform transformations, and construct logical arguments.*

**MAT-12.GM.01**Write the equation of a conic section given its special features. Convert between the standard form and general form equations of conic sections.**MAT-12.GM.02**Identify key features of a conic section given its equation. Apply properties of conic sections in context.**MAT-12.GM.03**Determine and apply appropriate formulas to solve right and non-right triangle problems in context.**MAT-12.GM.04**Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

## MAT-12.DPS **(DPS) **Data Probability and Statistics

*Learners will ask and answer questions by collecting, organizing, and displaying relevant data, drawing inferences and conclusions, making predictions, and understanding and applying
basic concepts of probability*

**MAT-12.DPS.01**Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).**MAT-12.DPS.02**Use the mean and standard deviation of a data set to fit it to a normal distribution and estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate.**MAT-12.DPS.03**Evaluate reports based on data.**MAT-12.DPS.04**Represent data on a scatter plot for two quantitative variables and describe how the variables are related.**MAT-12.DPS.05**Informally assess the fit of a function by plotting and analyzing residuals.**MAT-12.DPS.06**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.**MAT-12.DPS.07**Understand the process of making inferences about population parameters based on a random sample from that population.**MAT-12.DPS.08**Decide if a specified model is consistent with results from a given data-generating process (e.g., using simulation).**MAT-12.DPS.09**Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.**MAT-12.DPS.10**Determine when the order in counting matters and use permutations and combinations to compute probabilities of events accordingly. Determine probability situations as conditional, “or” (union), or “and” (intersection), and determine the probability of an event.**MAT-12.DPS.11**Use permutations and combinations to compute probabilities of compound events and solve problems.**MAT-12.DPS.12**Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space. Graph the corresponding probability distribution using the same graphical displays as for data distributions.**MAT-12.DPS.13**Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.**MAT-12.DPS.14**Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.**MAT-12.DPS.15**Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities are calculated; find the expected value.**MAT-12.DPS.16**Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.**MAT-12.DPS.17**Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).**MAT-12.DPS.18**Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).