9-12. Extend the properties of exponents to rational exponents.
9-12.HS.N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
9-12. Use properties of rational and irrational numbers.
9-12.HS.N-RN.3 Explain why 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.
9-12. Reason quantitatively and use units to solve problems.
9-12.HS.N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
9-12. Represent complex numbers and their operations on the complex plane.
9-12.HS.N-CN.4 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.
9-12.HS.N-VM.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
9-12.HS.N-VM.4.c Understand 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.HS.N-VM.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|·||v||. Compute the direction of cv knowing that when |c|v is not equal to 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
9-12.HS.N-VM.9 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
9-12.HS.N-VM.10 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. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
9-12.HS.A-SSE.4.1 This standard recognizes arithmetic and geometric sequences and series and appropriate application of their formulas as important.
9-12.HS.A-SSE.4.2 In order to derive this formula, students must have mastery of the following prerequisite skills: factoring, solving systems of equations, recognizing and writing the formula for a geometric sequence and series. Note that "derive" can mean "to convince" by showing using a picture/diagram/explanation.
9-12.HS.A-APR Arithmetic with Polynomials and Rational Expressions
9-12. Perform arithmetic operations on polynomials
9-12.HS.A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
9-12.HS.A-APR.4.1 x² + y², x² - y², and 2xy represent the terms of a Pythagorean triple. See the glossary for a table of polynomial identities.
9-12.HS.A-APR.5 Know and apply the Binomial Theorem for the expansion of (x + y) to the n power 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.
9-12.HS.A-APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form 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, for the more complicated examples, a computer algebra system.
9-12.HS.A-APR.7 Understand 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.
9-12.HS.A-REI Reasoning with Equations and Inequalities
9-12. Understand solving equations as a process of reasoning and explain the reasoning
9-12.HS.A-REI.1 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 a solution method.
9-12.HS.A-REI.4 Solve quadratic equations in one variable.
9-12.HS.A-REI.4.a1 Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from this form.
9-12.HS.A-REI.4.a1.1 It is important for all students to be able to apply the quadratic formula to solve a quadratic equation. Asking students to derive the quadratic formula from completing the square helps them develop an appreciation and understanding of the origin of the quadratic formula. In order to derive this formula, students must have mastery of the following pre-requisite knowledge and skills: simplifying radicals, properties of powers and completing the square, manipulating fractions, and solving equations for one variable in terms of other variables.
9-12.HS.A-REI.4.b2 Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
9-12.HS.A-REI.4.b2.2 It is appropriate for all students to be able to recognize when the quadratic formula gives complex solutions. The pre-requisite for this standard is the understanding of complex numbers and their properties
9-12. Solve systems of equations
9-12.HS.A-REI.5 Prove 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.
9-12.HS.A-REI.11 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, e.g., 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.
9-12.HS.A-REI.11.1 Use a graphing calculator to find the approximate solution(s) to the system below. Solutions: x=0 and x=-1 f(0)=4, g(0)=4
9-12.HS.A-REI.12 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.
9-12. Understand the concept of a function and use function notation
9-12.HS.F-IF.1 Understand that a function from one set (called the domain) to another set (called the range) 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).
9-12. Interpret functions that arise in applications in terms of the context
9-12.HS.F-IF.4 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.
9-12.HS.F-IF.7.e.1 (This standard creates an advance expectation for students.) In order to graph exponential, logarithmic, and trigonometric functions, students must have mastery of the following pre-requisite skills: laws of exponents, properties of inverse functions, asymptotic behavior and transformation of functions.
9-12.HS.F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
9-12.HS.F-IF.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
9-12.HS.F-BF.3 Identify the effect on the graph of replacing f(x) by 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 cases and illustrate an explanation of the effects on the graph using technology.
9-12.HS.F-LE.2 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).
9-12.HS.F-LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
9-12.HS.F-LE.4 For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
9-12.HS.F-LE.4.1 (This standards creates an advanced expectation for students.) In an effort to evaluate a logarithm, students are required to write an exponential equation in logarithmic form. The following pre-requisite skills must be mastered: definition and properties of logarithms, inverse functions, and solving equations for a single variable.
9-12. Interpret expressions for functions in terms of the situation they model
9-12.HS.F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
9-12.HS.F-LE.5.1 parameter: A constant or a variable in a mathematical expression, which distinguishes various specific cases. For example, in the equation y = mx + b, m and b are parameters which specify the particular straight line represented by the equation. (From "Mathematics Dictionary, edited by Glenn James and Robert James, 1960, Princeton, New Jersey). (ND)
9-12.HS.F-TF Trigonometric Functions
9-12. Extend the domain of trigonometric functions using the unit circle
9-12.HS.F-TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
9-12.HS.F-TF.1.1 (This standard creates an advanced expectation for students.) This standard assumes that all students know and understand that there are approximately 6.28 radius lengths around the circumference of any circle. Radians are the preferred unit of measure. There should be a natural connection from students' previous knowledge of circumference to radian measure.
9-12.HS.F-TF.2 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.
9-12.HS.F-TF.2.1 (This standard creates an advanced expectation for students.) The following pre-requisite skills must be mastered: right triangle trigonometry and knowledge of radian measure.
9-12.HS.F-TF.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi-x, pi+x, and 2pi-x in terms of their values for x, where x is any real number.
9-12.HS.F-TF.5.1 (This standard creates an advanced expectation for students.) In an effort to model periodic phenomena students must have mastery of the following pre-requisite skills: knowledge of the characteristics of trigonometric graphs and effects of their transformations on an equation
9-12.HS.F-TF.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
9-12.HS.F-TF.8 Prove the Pythagorean identity sin²(theta) + cos²(theta) = 1 and use it to find sin(theta), cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle.
9-12.HS.F-TF.8.1 Students might "prove" by providing a formal proof, demonstrating, or justifying. See the glossary for a definition of mathematical proof.
9-12.HS.F-TF.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
9-12. Experiment with transformations in the plane
9-12.HS.G-CO.1 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.
9-12.HS.G-CO.1.1 For further discussion of precision in mathematics, please see the discussion on Mathematical Practices in the Introduction.
9-12.HS.G-CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; 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 (e.g., translation versus horizontal stretch).
9-12.HS.G-CO.4.1 (This standard creates an advanced expectation for students.) Successive reflections over each of two intersecting lines results in a rotation. Successive reflections over each of two parallel lines results in a translation.
9-12.HS.G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
9-12.HS.G-CO.5.1 Note: Students must be able to perform a transformation as well as describe a series of transformations that have occurred.
9-12. Understand congruence in terms of rigid motions
9-12.HS.G-CO.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.
9-12.HS.G-CO.6.1 Note: Students must be able to predict and recognize rigid motions and use them to justify congruence.
9-12.HS.G-CO.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.
9-12.HS.G-CO.7.1 Congruent: Two plane or solid figures are congruent if one can be obtained from the other by rigid motion (a sequence of rotations, reflections, and translations).
9-12.HS.G-CO.7.2 Rigid motion: A transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions are here assumed to preserve distances and angle measures.
9-12.HS.G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
9-12.HS.G-CO.8.1 Congruent: Two plane or solid figures are congruent if one can be obtained from the other by rigid motion (a sequence of rotations, reflections, and translations).
9-12.HS.G-CO.8.2 Rigid motion: A transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions are here assumed to preserve distances and angle measures.
9-12. Prove geometric theorems
9-12.HS.G-CO.9 Prove theorems about lines and angles.
9-12.HS.G-SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they 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.
9-12.HS.G-C.1.1 Students might "prove" by providing a formal proof, demonstrating, or justifying.
9-12.HS.G-C.1.2 mathematical proof: A carefully reasoned argument for verifying a conjecture that would meet the standards of the broader mathematics community (From Principles and Standards for School Mathematics, NCTM, 2000).
9-12.HS.G-C.2 Identify and describe relationships among inscribed angles, radii, and chords.
9-12.HS.G-C.3.1 Students might "prove" by providing a formal proof, demonstrating, or justifying.
9-12.HS.G-C.3.2 mathematical proof: A carefully reasoned argument for verifying a conjecture that would meet the standards of the broader mathematics community (From Principles and Standards for School Mathematics, NCTM, 2000). (ND)
9-12.HS.G-C.3.3 Students may use technology to perform the constructions.
9-12.HS.G-C.4 Construct a tangent line from a point outside a given circle to the circle.
9-12. Find arc lengths and areas of sectors of circles
9-12.HS.G-C.5 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.
9-12.HS.G-C.5.1 (This standard creates an advanced expectation for students.) "Derive using similarity" denotes arcs and sectors of circles with differing radii. Students must have mastery of the following pre-requisite skill: radian measure.
9-12.HS.G-GPE Expressing Geometric Properties with Equations
9-12. Translate between the geometric description and the equation for a conic section
9-12.HS.G-GPE.1 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.
9-12.HS.G-GPE.1.1 (This standard creates an advanced expectation for students.) It is appropriate for all students to derive the equation of a circle of given center and radius using the Pythagorean Theorem. The following prerequisite skill must be mastered: completing the square in two variables.
9-12.HS.G-GPE.2 Derive the equation of a parabola given a focus and directrix.
9-12. Use coordinates to prove simple geometric theorems algebraically
9-12.HS.G-GPE.4 Use coordinates to prove simple geometric theorems algebraically.
9-12.HS.G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
9-12. Explain volume formulas and use them to solve problems
9-12.HS.G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
9-12.HS.G-GMD.1.1 Cavalier's Principle: 2D: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas.
9-12.HS.G-GMD.1.2 3D: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in crosssections of equal area, then the two regions have equal volumes.
9-12.HS.G-GMD.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
9-12.HS.G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
9-12.HS.G-MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
9-12.HS.G-MG.3 Apply geometric methods 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).
S Statistics and Probability
9-12.HS.S-ID Interpreting Categorical and Quantitative Data
9-12. Summarize, represent, and interpret data on a single count or measurement variable
9-12.HS.S-ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
9-12.HS.S-ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
9-12.HS.S-ID.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.
9-12. Summarize, represent, and interpret data on two categorical and quantitative variables
9-12.HS.S-ID.5 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.
9-12.HS.S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
9-12.HS.S-ID.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
9-12.HS.S-ID.6.b Informally assess the fit of a function by plotting and analyzing residuals.
9-12.HS.S-IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
9-12.HS.S-IC.6 Evaluate reports based on data.
9-12.HS.S-IC.6.1 See introduction to High School - Statistics and Probability for a discussion on specific criteria for evaluating reports.
9-12.HS.S-CP Conditional Probability and the Rules of Probability
9-12. Understand independence and conditional probability and use them to interpret data
9-12.HS.S-CP.1 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").
9-12.HS.S-CP.2 Understand 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.
9-12.HS.S-CP.3 Understand the conditional probability of A given B as P(A and 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.
9-12.HS.S-CP.4 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.
9-12. Calculate expected values and use them to solve problems
9-12.HS.S-MD.1 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.