Grades 11–12 Math (North Dakota Content Standards)

Number and Operations (NO)

• 11-12.NO.1 Rewrite complex expressions with radicals and rational exponents using exponent properties.
• 11-12.NO.2 Operate on complex radical expressions and simplify radicals into equivalent forms.
• 11-12.NO.3 Show why sums and products of rationals are rational and why mixing a rational with an irrational gives an irrational.
• 11-12.NO.4 Use units and unit analysis to guide multi-step problems, and choose units and scales consistently in formulas and displays. (modeling standard)
• 11-12.NO.5 Report quantities with accuracy or precision that fits the limits of measurement. (modeling standard)
• 11-12.NO.6 Know the unit i with i squared = −1, write complex numbers as a + bi, and see how number subsets nest.
• 11-12.NO.7 Add, subtract and multiply complex numbers using i squared = −1 and the usual properties.
• 11-12.NO.8 Divide complex numbers using conjugates.
• 11-12.NO.9 Use the Fundamental Theorem of Algebra to count a polynomial's zeros and find all solutions of a polynomial equation.
• 11-12.NO.10 Show complex numbers in rectangular, trigonometric and polar form, find the modulus and explain why the forms agree. (plus standard)
• 11-12.NO.11 Show operations, powers and roots of complex numbers geometrically and use this to compute. (plus standard)
• 11-12.NO.12 Extend polynomial identities to complex numbers. (plus standard)
• 11-12.NO.13 Use the Fundamental Theorem of Algebra to locate every root of a polynomial and sort roots as integer, rational, real or complex. (plus standard)
• 11-12.NO.14 Recognize vectors as having size and direction and write them in polar form. (plus standard)
• 11-12.NO.15 Find vector components by subtracting initial point coordinates from terminal point coordinates. (plus standard)
• 11-12.NO.16 Solve size-and-direction problems that vectors can represent. (plus standard)
• 11-12.NO.17 Add and subtract vectors, including showing subtraction graphically and by components. (plus standard)
• 11-12.NO.18 Multiply a vector by a scalar. (plus standard)
• 11-12.NO.19 Put data in a matrix, add, subtract and multiply matrices to solve problems, and know matrix multiplication is not commutative. (plus standard)

Algebraic Reasoning (AR)

• 11-12.AR.1 Solve multi-variable formulas for a quantity of interest. (modeling standard)
• 11-12.AR.2 Use the structure of polynomial and rational expressions to find ways to rewrite them.
• 11-12.AR.3 Interpret expressions that stand for a quantity in context. (modeling standard)
• 11-12.AR.4 Pick and write an equivalent form of an expression that reveals properties of the quantity. (modeling standard)
• 11-12.AR.5 Add, subtract, multiply and divide rational expressions, seeing they are closed like rational numbers.
• 11-12.AR.6 Rewrite a(x)/b(x) as q(x) + r(x)/b(x) by inspection, long division or technology.
• 11-12.AR.7 Write and solve equations and inequalities, including linear, quadratic and simple rational and exponential ones. (modeling standard)
• 11-12.AR.8 Write equations in two or more variables for relationships and graph them with proper labels and scales. (modeling standard)
• 11-12.AR.9 Express constraints with equations, inequalities or systems and judge which solutions are viable in a model. (modeling standard)
• 11-12.AR.10 Derive the quadratic formula from ax² + bx + c = 0. (plus standard)
• 11-12.AR.11 Solve quadratics with real coefficients that have complex solutions a ± bi.
• 11-12.AR.12 Solve simple rational and radical equations and spot extraneous solutions.
• 11-12.AR.13 Add, subtract and multiply polynomials beyond quadratics, seeing they are closed like the integers.
• 11-12.AR.14 Find polynomial zeros from suitable factorizations and use them to sketch a rough graph.
• 11-12.AR.15 Use the Factor and Remainder Theorems to test linear factors and apply the Remainder Theorem in context.
• 11-12.AR.16 Show with graphs, tables, technology or successive approximation that solutions of f(x) = g(x) are where outputs match.
• 11-12.AR.17 Solve a linear-quadratic system in two variables by algebra and by graphing.
• 11-12.AR.18 Find a matrix inverse when it exists and use it to solve linear systems, with technology for 3×3 or larger. (plus standard)
• 11-12.AR.19 Solve systems in three or more variables with matrices and technology. (plus standard)
• 11-12.AR.20 Use the Binomial Theorem to expand (ax + by) to a positive integer power. (plus standard)

Algebraic Reasoning: Functions (AR.F)

• 11-12.AR.F.1 Write a function describing a relationship between two quantities. (modeling standard)
• 11-12.AR.F.2 Find and interpret average rate of change of a function over an interval and estimate it from a graph. (modeling standard)
• 11-12.AR.F.3 Rewrite a function in equivalent forms to reveal and explain its properties. (modeling standard)
• 11-12.AR.F.4 Describe how a, b, h and k transform a graph, find them from a graph, and recognize even and odd functions. (modeling standard)
• 11-12.AR.F.5 Find inverse functions. (modeling standard)
• 11-12.AR.F.6 Use the inverse link between exponents and logarithms to solve problems. (modeling standard)
• 11-12.AR.F.7 Compare key features of two functions shown in different ways. (modeling standard)
• 11-12.AR.F.8 Interpret and sketch key features of a function modeling two quantities from tables, graphs, words or equations. (modeling standard)
• 11-12.AR.F.9 Connect a function's domain to its graph and to the situation it models. (modeling standard)
• 11-12.AR.F.10 Graph symbolic functions showing key features, by hand for simple cases and with technology otherwise. (modeling standard)
• 11-12.AR.F.11 Analyze and graph symbolic functions, by hand or with technology, naming key features. (plus, modeling standard)
• 11-12.AR.F.12 Use graphs or tables to show exponential growth eventually overtakes linear or quadratic growth. (modeling standard)
• 11-12.AR.F.13 Decide whether a linear, quadratic, polynomial, exponential, log or trig model fits a situation and pick a suitable model. (modeling standard)
• 11-12.AR.F.14 Write arithmetic and geometric sequences recursively and explicitly, convert between forms and use them as models. (modeling standard)
• 11-12.AR.F.15 Use log properties to solve a·b^(ct) = d and evaluate the log with technology when suitable. (modeling standard)
• 11-12.AR.F.16 Extend right-triangle trig with the unit circle to find sine, cosine and tangent at multiples of π/3, π/4 and π/6.
• 11-12.AR.F.17 Use sin² + cos² = 1 to find one trig value from another and the angle's quadrant.
• 11-12.AR.F.18 Explain how the unit circle extends trig functions to all real numbers as radian measures. (plus standard)
• 11-12.AR.F.19 Use the unit circle to relate trig values at π − x, π + x and 2π − x to the values at x. (plus standard)
• 11-12.AR.F.20 Use the unit circle to explain odd/even symmetry and periodicity of trig functions. (plus standard)
• 11-12.AR.F.21 Build a trig function to model periodic behavior. (plus standard)
• 11-12.AR.F.22 Restrict a trig function's domain to form its inverse. (plus standard)
• 11-12.AR.F.23 Solve trig equations from models using inverse functions and interpret the solutions in context. (plus, modeling standard)
• 11-12.AR.F.24 Know and use the sum and difference formulas for sine, cosine and tangent. (plus standard)

Geometry and Measurement (GM)

• 11-12.GM.1 Write a conic's equation from its features and convert between standard and general form.
• 11-12.GM.2 Identify a conic's key features from its equation and use conic properties in context. (modeling standard)
• 11-12.GM.3 Pick and apply suitable formulas for right and oblique triangle problems in context.
• 11-12.GM.4 Derive the triangle area formula A = ½ab sin C using an altitude. (plus standard)

Data, Probability and Statistics (DPS)

• 11-12.DPS.1 Explain in context how data sets differ in shape, center and spread, allowing for outliers. (modeling standard)
• 11-12.DPS.2 Fit data to a normal curve with mean and standard deviation to estimate percentages, knowing when not to. (modeling standard)
• 11-12.DPS.3 Critically evaluate reports and claims based on data. (modeling standard)
• 11-12.DPS.4 Make a scatter plot of two numeric variables and describe how they relate. (modeling standard)
• 11-12.DPS.5 Informally judge a function's fit by plotting and studying residuals. (plus, modeling standard)
• 11-12.DPS.6 Estimate a population mean or proportion from a survey and build a margin of error by simulation. (plus, modeling standard)
• 11-12.DPS.7 Understand inference about population parameters from a random sample. (plus, modeling standard)
• 11-12.DPS.8 Judge whether a model fits results from a data-generating process, e.g. by simulation. (plus, modeling standard)
• 11-12.DPS.9 Tell apart surveys, experiments and observational studies and explain the role of randomization in each. (plus, modeling standard)
• 11-12.DPS.10 Decide when order matters, use permutations and combinations for probabilities, and classify problems as conditional, or, or and. (modeling standard)
• 11-12.DPS.11 Compute probabilities of compound events with permutations and combinations. (plus, modeling standard)
• 11-12.DPS.12 Define a random variable over a sample space and graph its probability distribution. (plus, modeling standard)
• 11-12.DPS.13 Find the expected value of a random variable and read it as the distribution's mean. (plus, modeling standard)
• 11-12.DPS.14 Weigh decisions by attaching probabilities to payoffs, finding expected values and comparing strategies. (plus, modeling standard)
• 11-12.DPS.15 Build a probability distribution from theoretical probabilities and find its expected value. (plus, modeling standard)
• 11-12.DPS.16 Build a probability distribution from empirical probabilities and find its expected value. (plus, modeling standard)
• 11-12.DPS.17 Use probability to make fair decisions, e.g. drawing lots or random number generators. (plus, modeling standard)
• 11-12.DPS.18 Analyze decisions and strategies with probability, e.g. product or medical testing. (plus, modeling standard)