High School Math North Dakota Standards

Data, Probability, and Statistics

Geometry and Measurement

Functions

Algebraic Reasoning

Number and Operations

Reasoning and Proof

Connections

Problem-Solving

Math Attributes

Eleventh and Twelfth Grades

Rearrange multi-variable formulas to highlight a quantity of interest.

Derive the quadratic formula from the form 0 = ax² + bx + c.

Solve quadratic equations with real coefficients that have solutions of the form a + bi and a - bi.

Solve simple rational and radical equations in one variable and identify extraneous solutions.

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.

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.

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.

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.

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

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).

Solve a system of equations in three or more variables with matrices (using technology).

Use the structure of an expression (to extend to polynomial and rational expressions) to identify ways to rewrite it.

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.

Interpret expressions that represent a quantity in context.

Interpret parts of an expression, such as terms, factors, and coefficients.

Interpret complicated expressions by viewing one or more of their parts as a single entity.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Factor a quadratic expression to reveal the zeros of the function it defines.

Use the properties of exponents to transform exponential expressions.

Complete the square in a quadratic expression to produce an equivalent expression.

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.

Rewrite simple rational expressions in different 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.

Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic equations and simple rational and exponential equations.

Create equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with proper labels and scales.

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.

Write a function that describes a relationship between two quantities.

Combine standard function types using arithmetic operations.

Compose functions.

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Graph square root, cube root, piecewise-defined, step, and absolute value functions.

Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior.

Graph exponential and logarithmic functions, showing intercepts and end behavior.

Graph f(x) = sin x and f(x) = cos x as representations of periodic phenomena.

Analyze and graph functions expressed symbolically (by hand in simple cases and using technology for more complicated cases), identifying key features of the graph.

Graph rational functions, identifying domain, range, asymptote(s), removable and non-removable discontinuities, intercepts, behavior at the asymptote(s), and end behavior.

Graph trigonometric functions, showing period, midline, phase shift, and amplitude.

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.

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).

Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.

Use properties of logarithms to express the solution to ab^ct = d where a, c, and d are real numbers and b is a positive real number. Evaluate the logarithm using technology when appropriate.

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.

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.

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.

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.

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.

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Create a trigonometric function to model periodic phenomena.

Restrict the domain of a trigonometric function to construct its inverse.

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in context.

Know and apply the addition and subtraction formulas for sine, cosine, and tangent to solve problems.

Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.

Use the process of factoring and completing the square in a quadratic function to show zeros, minimum/maximum, and symmetry of the graph, and interpret these in terms of context.

Use the properties of exponents to interpret expressions for exponential functions.

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.

Find inverse functions.

Verify by composition that one function is the inverse of another.

Recognize that the graph of a function and its inverse are reflection images over the line y = x.

Produce an invertible function from a non-invertible function by restricting the domain.

Apply the inverse relationship between exponents and logarithms to solve problems.

Compare key features of two functions each represented in a different way (algebraically, graphically, numerically, in tables, or by verbal descriptions).

Use tables, graphs, verbal descriptions, and equations to interpret and sketch the key features of a function modeling the relationship between two quantities.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

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.

Use permutations and combinations to compute probabilities of compound events and solve problems.

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.

Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. Evaluate and compare strategies on the basis of expected values.

Find the expected payoff for a game of chance.

Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities are calculated; find the expected value.

Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

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.

Evaluate reports based on data.

Identify and explain misleading use of data, recognize when claims based on data confuse correlation and causation.

Recognize and describe how graphs and data can be distorted to support different points of view.

Represent data on a scatter plot for two quantitative variables and describe how the variables are related.

Fit a function to the data (with or without technology) and interpret the special features (e.g., meaning of a and b in the exponential function y = ab^x) of the function in context.

Use functions fitted to data to solve problems in the context of the data.

Informally assess the fit of a function by plotting and analyzing residuals.

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.

Understand the process of making inferences about population parameters based on a random sample from that population.

Decide if a specified model is consistent with results from a given data-generating process (e.g., using simulation).

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

Write the equation of a conic section given its special features. Convert between the standard form and general form equations of conic sections.

Identify key features of a conic section given its equation. Apply properties of conic sections in context.

Determine and apply appropriate formulas to solve right and non-right triangle problems in context.

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.

Learners can create connections within and across concepts, using supporting evidence to interpret how they originate, extend, and relate to other learning, ideas, and life experiences.

Learners can analyze, execute, critique, and adapt approaches and solutions when problem-solving in novel situations.

Learners can reason logically, citing evidence to critique and explain what they see, think, and conclude through exploration, generalization, and validation.

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.

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.

Extend polynomial identities to the complex numbers.

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.

Recognize vector quantities as having both magnitude and direction, writing them in polar form.

Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Solve problems involving magnitude and direction that can be represented by vectors.

Add and subtract vectors. Represent vector subtraction graphically by connecting the tips in the appropriate order and using the components to perform vector subtraction.

Add vectors end-to-end, component-wise, and by the parallelogram rule. Know that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

Understand that vector subtraction v-w is defined as v+(-w), where -w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction.

Multiply a vector by a scalar.

Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction. Use the components to perform scalar multiplication (e.g., as c(v_x, v_y) = (cv_x , cv_y)).

Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v.

Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

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.

Perform operations on complex radical expressions and simplify radicals to write equivalent expressions.

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.

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.

Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.

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.

Use the definition i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Use conjugates to find quotients of complex numbers.

Apply the Fundamental Theorem of Algebra to determine the number of zeros for polynomial functions.

Data, Probability, and Statistics

Geometry and Measurement

Functions

Algebraic Reasoning

Number and Operations

Reasoning and Proof

Connections

Problem-Solving

Math Attributes

Ninth and Tenth Grades

Use the structure of an expression (i.e., quadratic and exponential) to identify ways to rewrite it.

Solve quadratic equations in one variable by inspection (e.g., for x² = 49) taking square roots, the quadratic formula, and factoring, as appropriate to the initial form of the equation.

Add, subtract, and multiply polynomials.

Rearrange formulas to isolate a quantity or variable(s) of interest using the same reasoning as in solving equations.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.

Create linear and exponential equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.

Justify each step in solving a linear equation that may or may not have a solution.

Solve linear equations and inequalities (to include compound inequalities) in one variable.

Solve a system of linear equations graphically and algebraically. Create and solve a system of linear equations in context.

Graph the solution set to a two-variable system of linear inequalities. bCreate and graph the solution set to a two-variable system of linear inequalities in context.

Solve absolute value equations and inequalities in one or two variables.

Determine whether a relationship is a function given a table, graph, or words, identifying x as an element of the domain and f(x) as an element in the range. Determine the domain and range of a function in context.

Find the inverse of a linear function and describe the relationship between the domain, range, and graph of the function and its inverse in context.

Interpret the parameters in a linear, quadratic, or exponential function in context.

Identify, using graphs or tables, the solution(s) to linear and exponential functions f(x) = g(x) as x-value(s) that result in equivalent y-values.

Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in context.

Sketch the key features (to include intercepts, maximums, minimums, and lines of symmetry, where applicable) of linear, exponential, and quadratic functions modeling the relationship between two quantities using tables, graphs, written descriptions, and equations.

Relate the domain of a linear, quadratic, or exponential function to its graph and, where applicable, to the quantitative relationship it describes.

Calculate and interpret the rate of change of linear, quadratic, or exponential functions (presented algebraically or as a table) over specified intervals. Estimate the rate of change from a graph.

Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.

Use appropriate forms of linear, quadratic, and exponential functions to show zeros, extreme values, and symmetry (where applicable) and interpret them in context.

Use the properties of an exponential function to classify it as growth or decay.

Compare key features of two linear, exponential, or quadratic functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Identify situations that can be modeled with linear, quadratic, and exponential functions. Justify the most appropriate model for a situation based on the rate of change over equal intervals. Include situations in which a quantity grows or decays.

Identify the effect of transformations on the graph of a linear, absolute value, or quadratic function by replacing f(x) with af(x), f(x - h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the value of a, h, and k given the graph of the function.

Represent data with plots on the real number line (dot plots, histograms, and box plots).

Construct and interpret two-way frequency tables of data for two categorical variables. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

Compare the center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets using statistics appropriate to the shape of the data distribution.

Represent data on two quantitative variables on a scatter plot and describe how the variables are related.

Fit a linear function to the data (with or without technology) if appropriate.

Compute (using technology) and interpret the correlation coefficient of a linear fit.

Interpret the meaning of the slope and y-intercept of the linear model in context.

Interpolate and extrapolate the linear model to predict values.

Distinguish between correlation and causation.

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes or as unions, intersections, or complements of other events ("or," "and," "not").

Recognize that event A is independent of event B if the probability of event A does not change in response to the occurrence of event B. Apply the formula P(A and B) = P(A)·P(B) given that events A and B are independent.

Recognize that the conditional probability of an event A given B is the probability that event A will occur given the knowledge that event B has already occurred. Calculate the conditional probability of A given B and interpret the answer in context.

Apply the formula P(A or B) = P(A) + P(B) – P(A and B) and interpret the answer in context.

Determine the number of outcomes using permutations and combinations in context.

Know precise definitions and notations of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, and plane.

Prove and apply theorems about triangles.

Prove and apply theorems about parallelograms.

Make basic geometric constructions (e.g., segment, angle, bisectors, parallel and perpendicular lines) with a variety of tools and methods.

Apply basic constructions to create polygons such as equilateral triangles, squares, and regular hexagons inscribed in circles.

Verify experimentally and justify the properties of dilations given by a center and a scale factor.

Use transformations to decide if two given figures are similar. Apply the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Prove similarity theorems about triangles.

Apply knowledge of congruence and similarity criteria for triangles to solve problems and to prove relationships in various geometric figures.

Recognize how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle.

Explain and use the relationship between the sine and cosine of complementary angles.

Represent transformations in the plane. Describe transformations as functions taking points in the plane as inputs and giving other points as outputs. Compare transformations that preserve distance and angle to those that do not (i.e., rigid versus non-rigid motion).

Solve applied problems involving right triangles using trigonometric ratios, the Pythagorean Theorem, and special right triangles (30°-60°-90° and 45°-45°-90°).

Solve unknown sides and angles of non-right triangles using the Laws of Sines and Cosines.

Apply theorems about relationships between line segments and circles or angles and circles formed by radii, diameter, secants, tangents, and chords to find unknown lengths or angles.

Construct the incenter and circumcenter of a triangle. Relate the incenter and circumcenter to the inscribed and circumscribed circles.

Construct a tangent line from a point outside a given circle to the circle.

Explain and use the formulas for arc length and area of sectors of circles.

Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.

Develop and verify the slope criteria for parallel and perpendicular lines. Apply the slope criteria for parallel and perpendicular lines to solve problems.

Verify simple geometric theorems algebraically using coordinates. Verify algebraically, using coordinates, that a given set of points produces a particular type of triangle or quadrilateral.

Determine the midpoint or endpoint of a line segment using coordinates. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Describe the rotations and reflections of a triangle, rectangle, parallelogram, trapezoid, or regular polygon that map each figure onto itself or another figure.

Compute perimeters of polygons and areas of triangles, parallelograms, trapezoids, and kites using coordinates.

Explain derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone.

Calculate the surface area for prisms, cylinders, pyramids, cones, and spheres to solve problems.

Know and apply volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects and identify three-dimensional objects generated by rotations of two-dimensional objects.

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; scaling a model).

Develop or verify the characteristics of rotations, reflections, and translations in angles, circles, perpendicular lines, parallel lines, and line segments.

Draw the image of a figure that has undergone a series of transformations [rotation(s), reflection(s), or translation(s)] of a geometric figure using a variety of methods (e.g., graph paper, tracing paper, or geometry software).

Predict the effect of a specified rigid motion on a given figure using geometric descriptions of rigid motions. Determine whether two figures are congruent using the definition of congruence in terms of rigid motions.

Use the definition of congruence, based on rigid motions, to show two triangles are congruent if and only if their corresponding sides and corresponding angles are congruent.

Prove two triangles are congruent using the congruence theorems.

Prove and apply theorems about lines and angles.

Learners can create connections within and across concepts, using supporting evidence to interpret how they originate, extend, and relate to other learning, ideas, and life experiences.

Learners can analyze, execute, critique, and adapt approaches and solutions when problem-solving in novel situations.

Learners can reason logically, citing evidence to critique and explain what they see, think, and conclude through exploration, generalization, and validation.

Explain how the definition of rational exponents follows from extending the properties of integer exponents; rewrite simple expressions involving radicals and rational exponents using the properties of exponents.

Perform basic operations on simple radical expressions to write a simplified equivalent expression.

Choose and interpret the scale and the units in graphs and data displays.

Define appropriate quantities and units for the purpose of descriptive modeling.

Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.