9-12.1.1 The complex number system includes real numbers and imaginary numbers
9-12.1.1.a Extend the properties of exponents to rational exponents.
9-12.1.1.a.i 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.
9-12.1.1.a.ii Rewrite expressions involving radicals and rational exponents using the properties of exponents.
9-12.1.2.a.iv Describe factors affecting take-home pay and calculate the impact
9-12.1.2.a.v Design and use a budget, including income (net take-home pay) and expenses (mortgage, car loans, and living expenses) to demonstrate how living within your means is essential for a secure financial future
9-12.2 Patterns, Functions, and Algebraic Structures
9-12.2.1 Functions model situations where one quantity determines another and can be represented algebraically, graphically, and using tables
9-12.2.1.a Formulate the concept of a function and use function notation.
9-12.2.1.a.i Explain 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.
9-12.2.1.b Interpret functions that arise in applications in terms of the context
9-12.2.1.b.i 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.
9-12.2.1.e Build new functions from existing functions
9-12.2.1.e.i 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, and find the value of k given the graphs.
9-12.2.1.e.ii Experiment with cases and illustrate an explanation of the effects on the graph using technology.
9-12.2.1.e.iii Find inverse functions.
9-12.2.1.f Extend the domain of trigonometric functions using the unit circle
9-12.2.1.f.i Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
9-12.2.1.f.ii 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.
9-12.2.2 Quantitative relationships in the real world can be modeled and solved using functions
9-12.2.2.a Construct and compare linear, quadratic, and exponential models and solve problems
9-12.2.2.a.i Distinguish between situations that can be modeled with linear functions and with exponential functions.
9-12.2.2.a.i.1 Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
9-12.2.2.a.i.2 Identify situations in which one quantity changes at a constant rate per unit interval relative to another.
9-12.2.2.a.i.3 Identify situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
9-12.2.2.a.iii Use graphs and tables to describe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
9-12.2.2.a.iv 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.
9-12.2.2.b Interpret expressions for functions in terms of the situation they model
9-12.2.2.b.i Interpret the parameters in a linear or exponential function in terms of a context.
9-12.2.3.c Perform arithmetic operations on polynomials
9-12.2.3.c.i Explain 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.
9-12.2.4.b Understand solving equations as a process of reasoning and explain the reasoning
9-12.2.4.b.i 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.
9-12.2.4.c.ii Solve quadratic equations in one variable.
9-12.2.4.c.ii.1 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.
9-12.2.4.e.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.
9-12.2.4.e.iii 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.
9-12.3.1.a.iv Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages and identify data sets for which such a procedure is not appropriate.
9-12.3.1.a.v Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
9-12.3.1.b Summarize, represent, and interpret data on two categorical and quantitative variables
9-12.3.1.b.i 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.
9-12.3.1.b.ii Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
9-12.3.1.b.ii.1 Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
9-12.3.1.b.ii.2 Informally assess the fit of a function by plotting and analyzing residuals.
9-12.3.3.a.ii Explain 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.
9-12.3.3.a.iii Using the conditional probability of A given B as P(A and B)/P(B), interpret the 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.
9-12.3.3.a.iv 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.
9-12.3.3.a.v Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
9-12.3.3.b Use the rules of probability to compute probabilities of compound events in a uniform probability model
9-12.3.3.b.i 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.
9-12.3.3.b.ii Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.
9-12.3.3.c Analyze the cost of insurance as a method to offset the risk of a situation
9-12.4 Shape, Dimension, and Geometric Relationships
9-12.4.1 Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically
9-12.4.1.a Experiment with transformations in the plane
9-12.4.1.a.i State 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.
9-12.4.1.a.ii Represent transformations in the plane using appropriate tools.
9-12.4.1.a.viii Specify a sequence of transformations that will carry a given figure onto another.
9-12.4.1.b Understand congruence in terms of rigid motions
9-12.4.1.b.i Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure.
9-12.4.1.b.ii Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
9-12.4.1.b.iii 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.
9-12.4.1.b.iv Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
9-12.4.1.c Prove geometric theorems
9-12.4.1.c.i Prove theorems about lines and angles.
9-12.4.1.c.ii Prove theorems about triangles.
9-12.4.1.c.iii Prove theorems about parallelograms.
9-12.4.1.d Make geometric constructions
9-12.4.1.d.i Make formal geometric constructions with a variety of tools and methods.
9-12.4.1.d.ii Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
9-12.4.2 Concepts of similarity are foundational to geometry and its applications
9-12.4.2.a Understand similarity in terms of similarity transformations
9-12.4.2.a.i Verify experimentally the properties of dilations given by a center and a scale factor:
9-12.4.2.a.i.1 Show that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
9-12.4.2.a.i.2 Show that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
9-12.4.2.a.iii 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.
9-12.4.2.d Prove and apply trigonometric identities
9-12.4.2.d.i Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1.
9-12.4.2.d.ii Use the Pythagorean identity to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
9-12.4.2.e Understand and apply theorems about circles.
9-12.4.2.e.i Identify and describe relationships among inscribed angles, radii, and chords.
9-12.4.2.e.ii Construct the inscribed and circumscribed circles of a triangle.
9-12.4.2.e.iii Prove properties of angles for a quadrilateral inscribed in a circle.
9-12.4.2.f Find arc lengths and areas of sectors of circles.
9-12.4.2.f.i 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.
9-12.4.2.f.ii Derive the formula for the area of a sector.
9-12.4.3 Objects in the plane can be described and analyzed algebraically
9-12.4.3.a Express Geometric Properties with Equations.
9-12.4.3.a.i Translate between the geometric description and the equation for a conic section
9-12.4.3.a.i.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem.
9-12.4.3.a.i.2 Complete the square to find the center and radius of a circle given by an equation.
9-12.4.3.a.i.3 Derive the equation of a parabola given a focus and directrix.
9-12.4.3.a.ii Use coordinates to prove simple geometric theorems algebraically
9-12.4.3.a.ii.1 Use coordinates to prove simple geometric theorems algebraically.
9-12.4.3.a.ii.2 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.