Grades 9, 10, 11, 12

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(+) Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.9-12.VM.B.4.a

(+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.9-12.VM.B.4.b

(+) Demonstrate understanding of vector subtraction v – w as v + (–w), where --w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.9-12.VM.B.4.c

(+) Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).9-12.VM.B.5.a

(+) 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).9-12.VM.B.5.b

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

Use the structure of an expression to identify ways to rewrite it.A.SSE.A.2

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

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.B.4

Demonstrate understanding that polynomials form a system analogous to the integers; namely, they are closed under certain operations.A.APR.A.1

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).A.APR.B.2

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.A.APR.B.3

Prove polynomial identities and use them to describe numerical relationships.A.APR.C.4

(+) 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.C.5

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

(+) Demonstrate understanding 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.D.7

Create one-variable equations and inequalities to solve problems, including linear, quadratic, rational, and exponential functions.A.CED.A.1

Interpret the relationship between two or more quantities.A.CED.A.2

Represent constraints using equations or inequalities and interpret solutions as viable or non-viable options in a modeling context.A.CED.A.3

Represent constraints using systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.A.CED.A.4

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.A.CED.A.5

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 or refute a solution method.A.REI.A.1

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.A.REI.A.2

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

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

Verify 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.A.REI.C.5

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.A.REI.C.6

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

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

(+) 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).A.REI.C.9

Demonstrate understanding that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. Show that any point on the graph of an equation in two variables is a solution to the equation.A.REI.D.10

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. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.A.REI.D.11

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 half-planes.A.REI.D.12

Demonstrate understanding that a function is a correspondence from one set (called the domain) to another set (called the range) that 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).F.IF.A.1

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

Demonstrate that a sequence is a functions, sometimes defined recursively, whose domain is a subset of the integers.F.IF.A.3

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 maxima and minima; symmetries; end behavior; and periodicity.F.IF.B.4

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.F.IF.B.5

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.F.IF.B.6

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

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

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

Given algebraic, numeric and/or graphical representations of functions, recognize the function as polynomial, rational, logarithmic, exponential, or trigonometric.F.IF.C.10

Write a function that describes a relationship between two quantities. Functions could include linear, exponential, quadratic, simple rational, radical, logarithmic, and trigonometric.F.BF.A.1

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.F.BF.A.2

Identify the effect on the graph of replacing f(x) by f(x) + 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. Include, linear, quadratic, exponential, absolute value, simple rational and radical, logarithmic, and trigonometric functions. Utilize technology to experiment with cases and illustrate an explanation of the effects on the graph. Include recognizing even and odd functions from their graphs and algebraic expressions for them.F.BF.B.3

Find inverse functions algebraically and graphically.F.BF.B.4

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

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

Use graphs and tables to demonstrate that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.F.LE.A.3

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.A.4

Interpret the parameters in a linear or exponential function (of the form f(x) = b^x + k) in terms of a context.F.LE.B.5

Demonstrate radian measure as the ratio of the arc length subtended by a central angle to the length of the radius of the unit circle.F.TF.A.1

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.F.TF.A.2

(+) 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.A.3

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

Model periodic phenomena using trigonometric functions with specified amplitude, frequency, and midline.F.TF.B.5

(+) 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.B.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.B.7

Relate the Pythagorean Theorem to the unit circle to discover the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use the Pythagorean identity to find the value of a trigonometric function (sin(θ), cos(θ), or tan(θ)) given one trigonometric function (sin(θ), cos(θ), or tan(θ)) and the quadrant of the angle.F.TF.C.8

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

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.G.CO.A.1

Represent transformations in the plane and describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not.G.CO.A.2

Describe the rotations and reflections that carry a given figure (rectangle, parallelogram, trapezoid, or regular polygon) onto itself.G.CO.A.3

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

Draw the transformation (rotation, reflection, or translation) for a given geometric figure.G.CO.A.5

Specify a sequence of transformations that will carry a given figure onto another.G.CO.A.6

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.G.CO.B.7

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.B.8

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.G.CO.B.9

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent, and conversely prove lines are parallel; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.G.CO.C.10

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent, and conversely prove a triangle is isosceles; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.G.CO.C.11

Prove theorems about parallelograms. Theorems include: 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.C.12

Prove theorems about polygons. Theorems include: the measures of interior and exterior angles; apply properties of polygons to the solutions of mathematical and contextual problems.G.CO.C.12.a

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.) Constructions include: copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.G.CO.D.13

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.G.CO.D.14

Verify experimentally the properties of dilations given by a center and a scale factor.G.SRT.A.1

Use the definition of similarity to decide if two given figures are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.G.SRT.A.2

Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.G.SRT.A.3

Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.G.SRT.B.4

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

Demonstrate understanding that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.G.SRT.C.6

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

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

(+) Derive the formula A = ½ absin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.G.SRT.D.9

(+) Prove the Laws of Sines and Cosines and use them to solve problems.G.SRT.D.10

(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems; resultant forces).G.SRT.D.11

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

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.G.C.A.2

Prove properties of angles for a quadrilateral and other polygons inscribed in a circle, by constructing the inscribed and circumscribed circles of a triangle.G.C.A.3

(+) Construct a tangent line to a circle from a point outside the given circle.G.C.A.4

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.G.C.B.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.A.1

Derive the equation of a parabola given a focus and directrix.G.GPE.A.2

(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.G.GPE.A.3

Use coordinates to prove simple geometric theorems algebraically, including the distance formula and its relationship to the Pythagorean Theorem.G.GPE.B.4

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.G.GPE.B.5

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.G.GPE.B.6

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula).G.GPE.B.7

Give an informal argument for the formulas for the circumference of a circle; area of a circle; volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.G.GMD.A.1

(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.G.GMD.A.2

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.G.GMD.A.3

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

Use geometric shapes, their measures, and their properties to describe objects.G.MG.A.1

Apply concepts of density based on area and volume in modeling situations.G.MG.A.2

Apply geometric methods to solve design problems.G.MG.A.3

Use dimensional analysis for unit conversions to confirm that expressions and equations make sense.G.MG.A.4

Differentiate between count data and measurement variable.S.ID.A.1

Represent measurement data with plots on the real number line (dot plots, histograms, and box plots).S.ID.A.2

Compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different variables, using statistics appropriate to the shape of the distribution for each measurement variable.S.ID.A.3

Interpret differences in shape, center, and spread in the context of the variables accounting for possible effects of extreme data points (outliers) for measurement variables.S.ID.A.4

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.A.5

Represent data on two categorical variables on a clustered bar chart and describe how the variables are related. 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.B.6

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.S.ID.B.7

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.S.ID.C.8

Compute (using technology) and interpret the linear correlation coefficient.S.ID.C.9

Distinguish between (linear) correlation and causation.S.ID.C.10

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

Decide if a specified model is consistent with results from a given data-generating process (e.g., using simulation or validation with given data).S.IC.A.2

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

Use data from a sample survey to estimate a population mean or proportion and a margin of error.S.IC.B.4

Use data from a randomized and controlled experiment to compare two treatments; use margins of error to decide if differences between treatments are significant.S.IC.B.5

Evaluate reports of statistical information based on data.S.IC.B.6

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").S.CP.A.1

Demonstrate understanding 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 independent.S.CP.A.2

Understand the conditional probability of A given B as P(A∩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 probability of A, and the conditional probability of B given A is the same as the probability of B.S.CP.A.3

Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.S.CP.A.4

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.S.CP.A.5

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.S.CP.B.6

Apply the Addition Rule, P(A ∪ B) = P(A) + P(B) - P(A ∪ B), and interpret the answer in terms of the model.S.CP.B.7

(+) Apply the general Multiplication Rule in a uniform probability model P(A ∩ B) = P(A)P(B/A) = P(B)P(A/B), and interpret the answer in terms of the model.S.CP.B.8

(+) Use permutations and combinations to compute probabilities of compound events and solve problems.S.CP.B.9

(+) 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.S.MD.A.1

(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution of the variable.S.MD.A.2

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

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

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

(+) Use probabilities to make objective decisions.S.MD.B.6

(+) Analyze decisions and strategies using probability concepts.S.MD.B.7