Grades 9, 10, 11, 12

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Explain how the definition of rational exponents follows from extending the properties of integer exponents, allowing for a notation for radicals in terms of rational exponents.N.RN.1

Rewrite expressions involving radicals and rational exponents using the properties of exponents.N.RN.2

Explain why the sum or product of two rational numbers is rational; the sum of a rational and an irrational number is irrational; and the product of a nonzero rational and an irrational number is irrational.N.RN.3

Use unit analysis to understand and guide the process of solving multi-step problems; choose and interpret units consistently in formulas; and choose and interpret the scale and origin in graphs and data displays.N.Q.1

Define appropriate quantities for the purpose of descriptive modeling.N.Q.2

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.N.Q.3

Interpret expressions that represent a quantity in terms of its context.A.SSE.1.i

Recognize and use the structure of an expression to identify ways to rewrite it.A.SSE.2.i

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.A.SSE.3

Understand that polynomials form a system closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.A.APR.1

Create equations and inequalities in one variable arising from situations in which linear, quadratic, and exponential functions are appropriate and use them to solve problems.A.CED.1.i

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.A.CED.2.i

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.A.CED.3.i

Rewrite formulas to highlight a quantity of interest, using the same reasoning as in solving equations.A.CED.4.i

Explain each step in solving an 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.A.REI.1

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.A.REI.3

Solve quadratic equations in one variable.A.REI.4.i

Understand the principles of the elimination method.A.REI.5

Solve systems of linear equations exactly and approximately by graphing, focusing on pairs of linear equations in two variables.A.REI.6

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.A.REI.7

Understand that the 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).A.REI.10.i

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, including but not limited to using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, quadratic and exponential.A.REI.11.i

Graph a linear inequality (strict or inclusive) in two variables; graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.A.REI.12

Understand that a function maps each element of the domain to 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 off is the graph of the equation y = f(x).F.IF.1

Use function notation, evaluate functions, and interpret statements that use function notation in terms of a context.F.IF.2

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.F.IF.3

For functions, including linear, quadratic, and exponential, that model 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.F.IF.4.i

Relate the domain of a function to its graph and find an appropriate domain in the context of the problem.F.IF.5.i

Calculate and interpret the average rate of change of a function, both symbolically and from a table over a specified interval. Estimate the rate of change from a graph.F.IF.6

Graph parent functions and their transformations expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.F.IF.7.i

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.F.IF.8

Compare properties of two functions (linear, quadratic and exponential) each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).F.IF.9.i

Write a function (linear, quadratic, and exponential) that describes a relationship between two quantities.F.BF.1.i

Write arithmetic and geometric sequences both recursively and with an explicit formula and use them to model situations.F.BF.2

Identify the effect on the graph of f(x) (linear, exponential, quadratic) replaced with 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 contrasting cases and illustrate an explanation of the effects on the graph using technology.F.BF.3.i

Distinguish between situations that can be modeled with linear functions and with exponential functions.F.LE.1

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).F.LE.2

Recognize, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.F.LE.3

Interpret the parameters in a linear or exponential function in terms of a context.F.LE.5

Represent data with plots on the real number line (dot plots, histograms, and box plots).S.ID.1

Use statistics appropriate to the shape and context of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.S.ID.2

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).S.ID.3

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.S.ID.5

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.S.ID.6

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.S.ID.6.7

Compute (using technology) and interpret the correlation coefficient of a linear fit.S.ID.6.8

Distinguish between correlation and causation.S.ID.6.9

State and apply precise definitions of angle, circle, perpendicular, parallel, ray, line segment, and distance based on the undefined notions of point, line, and plane.G-CO.1

Represent transformations in the plane. (e.g., using transparencies and/or geometry software);G-CO.2

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and/or reflections that map the figure onto itself.G-CO.3

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.G-CO.4

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure, (e.g., using graph paper, tracing paper, or geometry software). Specify a sequence of transformations that will map a given figure onto another.G-CO.5

Use geometric descriptions of rigid motions to transform figures.G-CO.6

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.G-CO.7

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.G-CO.8

Prove theorems about lines and angles. Theorems must include but not limited to: vertical angles are congruent; when a transversal intersects parallel lines, alternate interior angles are congruent and same side interior angles are supplementary (using corresponding angles postulate); points on a perpendicular bisector of a line segment are equidistant from the segment's endpoints.G-CO.9

Prove congruence theorems about triangles. Theorems must include but not limited to: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the mid segment of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.G-CO.10

Prove theorems about parallelograms. Theorems must include but not limited to: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.G-CO.11

Perform geometric constructions with a compass and straightedge. including copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines/segments, constructing a line parallel to a given line through a point not on the line.G-CO.12

Construct an equilateral triangle, a square, and a regular hexagon.G-CO.13

Verify experimentally and apply the properties of dilations as determined by a center and a scale factor.G-SRT.1

Determine whether figures are similar, using the definition of similarity and using similarity transformations.G-SRT.2

Use the properties of similarity transformations to establish similarity theorems. Theorems must include AA, SAS, and SSS.G-SRT.3

Prove theorems about triangles involving similarity. Theorems must include but not limited to: a line parallel to one side of a triangle divides the other two proportionally, and its converse; the Pythagorean Theorem proved using triangle similarity.G-SRT.4

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.G-SRT.5

Define, using similarity, that side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios (sine, cosine, and tangent) for acute angles.G-SRT.6

Explain and use the relationship between the sine and cosine of complementary angles.G-SRT.7

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.G-SRT.8

Prove that all circles are similar.G-C.1

Identify and describe relationships among central angles, inscribed angles, circumscribed angles, radii, and chords.G-C.2

Construct, using a compass and straight edge, the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.G-C.3

Construct a tangent line from a point outside a given circle to the circle.G-C.4

Derive using similarity the length of the arc intercepted by an angle is proportional to the radius.G-C.5

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.G-GPE.1

Use coordinates to prove geometric relationships algebraically. For example, determine whether a figure defined by four given points in the coordinate plane is a rectangle; determine whether the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).G-GPE.2

Define and use the slope criteria for parallel and perpendicular lines. (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).G-GPE.3

Find the point on a directed line segment between two given points that partitions the segment in a given ratio. e.g. Determine the point(s) that divide the segment with endpoints of (-4, 7) and (6, 3) into the ratio 2:3G-GPE.4

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.G-GPE.5

Give an informal argument for the formulas for the volume of a cylinder, pyramid, sphere, and cone. Use dissection arguments, and informal limit arguments.G-GMD.1

Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figuresG-GMD.2

Know and apply volume and surface area formulas for cylinders, pyramids, cones, and spheres for composite figures to solve problems.G-GMD.3

Identify two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.G-GMD.4

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).G-MG.1

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).G-MG.2

Apply geometric concepts to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).G-MG.3

Describe events as subsets of a sample space or as unions, intersections, or complements of other events.S-CP.1

Determine whether two events A and B are independent.S-CP.2

Determine conditional probabilities and interpret independence by analyzing conditional probability.S-CP.3

Construct and interpret two-way frequency tables of data. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.S-CP.4

Recognize and explain the concepts of conditional probability and independence in everyday language and situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.S-CP.5

Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the result.S-CP.6

Apply the Addition Rule, P(A or B), and interpret the result.S-CP.7

Apply the general Multiplication Rule, P(A and B), and interpret the result.S-CP.8

Interpret expressions that represent a quantity in terms of its context.A.SSE.1.ii

Know and apply the Remainder Theorem.A.APR.2

Identify zeros of polynomials by factoring.A.APR.3

Rewrite rational expressions.

Rewrite simple rational expressions in different forms; using inspection, synthetic division, long division, box method or, for the more complicated examples, a computer algebra system.A.APR.6

Create equations and inequalities in one variable and use them to solve problems.A.CED.1.ii

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.A.CED.2.ii

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.A.CED.3.ii

Rewrite formulas to highlight a quantity of interest, using the same reasoning as in solving equations.A.CED.4.ii

Solve rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Rational functions are limited to those whose numerators are of degree at most 1 and denominators of degree at most 2. Radical functions are limited to square roots or cube roots of at most quadratic polynomials.A.REI.2

Select, justify and apply appropriate methods to solve quadratic equations in one variable. Recognize complex solutions and write them as a +/- bi for real numbers a and b.A.REI.4.ii

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, including but not limited to using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.A.REI.11.ii

For functions that model 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 (including even, odd, or neither); end behavior; and periodicity.F.IF.4.ii

Relate the domain of a function to its graph and find an appropriate domain in the context of the problem.F.IF.5.ii

Graph parent functions and their transformations expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.F.IF.7.ii

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).F.IF.9.ii

Write a function that describes a relationship between two quantities.F.BF.1.ii

Identify the effect on the graph of f(x) replaced with 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 contrasting cases and illustrate an explanation of the effects on the graph using technology.F.BF.3.ii

Find inverse functions.F.BF.4.ii

(+)Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.F.BF.5.ii

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.F.LE.4

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.F.TF.1

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions (sine and cosine) to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.F.TF.2

Choose trigonometric functions (sine and cosine) to model periodic phenomena with specified amplitude, frequency, and midline.F.TF.5

Prove the Pythagorean identity sin²(A) + cos²(A) = 1 and use it to calculate trigonometric ratios.F.TF.8

Know there is a complex number i such that i² = –1, and every complex number has the form a + bi where a and b are real numbers.N.CN.1

Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.N.CN.2

Solve quadratic equations with real coefficients that have complex solutions.N.CN.7

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.S.ID.4

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.S.IC.1

Determine whether a specified model is consistent with results from a given data-generating process.S.IC.2

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.S.IC.3

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.S.IC.4

Use data from a randomized experiment to compare two treatment groups; use simulations to decide if differences between parameters are significant.S.IC.5

Evaluate reports based on data.S.IC.6

(+)Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.N.CN.3

(+)Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.N.CN.4

(+)Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.N.CN.5

(+)Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.N.CN.6

(+)Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x - 2i).N.CN.8

(+)Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.N.CN.9

(+)Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudesN.VM.1

(+)Write a vector in component form.N.VM.2

(+)Solve problems involving velocity and other quantities that can be represented by vectors.N.VM.3

(+)Add and subtract vectors.N.VM.4

(+)Represent scalar multiplication graphically by scaling vectors and/or reversing their direction; perform scalar multiplication component-wise.N.VM.5.a

(+)Compute the magnitude of a scalar multiple cv. 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).N.VM.5.b

(+)Use matrices to represent and manipulate data.N.VM.6

(+)Multiply matrices by scalars to produce new matrices.N.VM.7

(+)Add, subtract, and multiply matrices of appropriate dimensions.N.VM.8

(+)Understand that, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.N.VM.9

(+)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. Discover that the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.N.VM.10

(+)Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.N.VM.11

(+)Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.N.VM.12

(+)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.A.APR.5

(+)Discover 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.A.APR.7

(+)Represent a system of linear equations as a single matrix equation in a vector variable.A.REI.8

(+)Use matrices to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).A.REI.9

(+)Solve linear, quadratic, polynomial, and rational inequalities in two variables algebraically and graphically.A.REI.10

(+)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.A.SSE.4

(+)Use summation notation to describe the sums in a series.A.SSE.5

(+)Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.F.IF.7

(+)Write a function that describes a relationship between two quantities.F.BF.1

(+)Find inverse functions.F.BF.4

(+)Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.F.BF.5

(+)Use reciprocal properties to develop definitions for cotangent, cosecant, and secant.F.BF.6

(+)Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and 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.F.TF.3

(+)Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.F.TF.4

(+)Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.F.TF.6

(+)Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.F.TF.7

(+)Prove the addition and subtraction, half-angle, and double-angle formulas for sine, cosine, and tangent and use them to solve problems.F.TF.9

(+)Use fundamental trigonometric identities.F.TF.10

(+)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 and use the formula to solve problems.G.SRT.9

(+)Prove the Laws of Sines and Cosines and use them to solve problems involving right and non-right triangles.G.SRT.10

(+)Analyze conic sections using equations and graphs.G.GPE.3

(+)Use permutations and combinations to compute probabilities of compound events and solve problems.G.GPE.9

(+)Assign a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.S.MD.1

(+)Calculate the expected value of a random variable; understand that it is the mean of the probability distribution.S.MD.2

(+)Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; calculate the expected value.S.MD.3

(+)Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; calculate the expected value.S.MD.4

(+)Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding calculating the expected values.S.MD.5

(+)Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).S.MD.6

(+)Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).S.MD.7

(+)Define polar coordinates and the relationship between polar coordinates and Cartesian coordinates with and without the use of technology.PC.PC.1

(+)Use polar equations to model and solve problems using graphs and algebraic properties.PC.PC.2

(+)Given equations for a parametric function, plot the graph and make conclusions about the geometric figure that result.PC.PE.1

(+)Convert between a pair of parametric equations and an equation in x and y. Model and solve problems using parametric equations.PC.PE.2

(+)Determine if a function is continuous at a point. Find the types of discontinuities of a function and relate them to finding limits of a function. Use the concept of limits to describe discontinuity and end-behavior of the function.PC.L.1

(+)Demonstrate knowledge of both the definition and graphical interpretation of limits of values of functions and sequences. Verify and estimate limits using graphs, tables, and technology.PC.L.2

(+)Evaluate limits of functions and apply properties of limits, including one-sided limits and limits at infinity using algebra.PC.L.3

(+)Define arithmetic and geometric sequences and series. Model and solve word problems involving applications of sequences and series, interpret the solutions and determine whether the solutions are reasonable.PC.S.1