### 1: Algebra

#### 1.1: Arithmetic with Polynomials and Rational Expressions

1.1.1: Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations.

1.1.2: Know and apply the Division Theorem and the Remainder Theorem for polynomials.

1.1.3: Graph polynomials identifying zeros when suitable factorizations are available and indicating end behavior. Write a polynomial function of least degree corresponding to a given graph.

1.1.4: Prove polynomial identities and use them to describe numerical relationships.

#### 1.2: Creating Equations

1.2.1: Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable.

1.2.2: Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales.

1.2.3: Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation.

1.2.4: Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines.

#### 1.3: Reasoning with Equations and Inequalities

1.3.1: Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original.

1.3.2: Solve simple rational and radical equations in one variable and understand how extraneous solutions may arise.

1.3.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

1.3.4: Solve mathematical and real-world problems involving quadratic equations in one variable.

1.3.4.1: Use the method of completing the square to transform any quadratic equation in 𝑥𝑥 into an equation of the form (x − h)² = k that has the same solutions. Derive the quadratic formula from this form.

1.3.4.2: Solve quadratic equations by inspection, 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.

1.3.5: Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation.

1.3.6: Solve systems of linear equations algebraically and graphically focusing on pairs of linear equations in two variables.

1.3.6.1: Solve systems of linear equations using the substitution method.

1.3.6.2: Solve systems of linear equations using linear combination.

1.3.7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions.

1.3.8: Represent a system of linear equations as a single matrix equation in a vector variable.

1.3.9: Using technology for matrices of dimension 3 × 3 or greater, find the inverse of a matrix if it exists and use it to solve systems of linear equations.

1.3.10: Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.

1.3.11: Solve an equation of the form f(x) = g(x) graphically by identifying the x - coordinate(s) of the point(s) of intersection of the graphs of y = f(x) and y =g(x).

1.3.12: Graph the solutions to a linear inequality in two variables.

#### 1.4: Structure and Expressions

1.4.1: Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions.

1.4.2: Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions.

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

1.4.3.1: Find the zeros of a quadratic function by rewriting it in equivalent factored form and explain the connection between the zeros of the function, its linear factors, the x-intercepts of its graph, and the solutions to the corresponding quadratic equation.

### 2: Functions

#### 2.1: Building Functions

2.1.1: Write a function that describes a relationship between two quantities.

2.1.1.1: Write a function that models a relationship between two quantities using both explicit expressions and a recursive process and by combining standard forms using addition, subtraction, multiplication and division to build new functions.

2.1.1.2: Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations.

2.1.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

2.1.3: Describe the effect of the transformations kf (x), f(x) + k, f(x + k), and combinations of such transformations on the graph of y = f(x) for any real number k. Find the value of k given the graphs and write the equation of a transformed parent function given its graph.

2.1.4: Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, as f and g are inverse functions if and only if f(x) = y and g(y) = x, for all values of x in the domain of f and all values of y in the domain of g, and find inverse functions for one-to-one function or by restricting the domain.

2.1.4.2: If a function has an inverse, find values of the inverse function from a graph or table.

2.1.5: Understand and verify through function composition that exponential and logarithmic functions are inverses of each other and use this relationship to solve problems involving logarithms and exponents.

#### 2.2: Interpreting Functions

2.2.1: Extend previous knowledge of a function to apply to general behavior and features of a function.

2.2.1.1: 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.

2.2.1.2: Represent a function using function notation and explain that f(x) denotes the output of function f that corresponds to the input x.

2.2.1.3: Understand that the graph of a function labeled as f is the set of all ordered pairs (x,y) that satisfy the equation y = f(x).

2.2.3: Define functions recursively and recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

2.2.4: Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity.

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

2.2.6: Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context.

2.2.7: Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases.

2.2.7.4: Graph trigonometric functions, showing period, midline, and amplitude.

2.2.8: Translate between different but equivalent forms of a function equation to reveal and explain different properties of the function.

2.2.8.1: 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.

2.2.8.2: Interpret expressions for exponential functions by using the properties of exponents.

2.2.9: Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal.

#### 2.3: Linear, Quadratic, and Exponential

2.3.1: Distinguish between situations that can be modeled with linear functions or exponential functions by recognizing situations in which one quantity changes at a constant rate per unit interval as opposed to those in which a quantity changes by a constant percent rate per unit interval.

2.3.1.1: Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

2.3.1.2: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

2.3.2: Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables.

2.3.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function.

2.3.5: Interpret the parameters in a linear or exponential function in terms of the context.

#### 2.4: Trigonometry

2.4.2: Define sine and cosine as functions of the radian measure of an angle in terms of the x - and y - coordinates of the point on the unit circle corresponding to that angle and explain how these definitions are extensions of the right triangle definitions.

2.4.2.1: Define the tangent, cotangent, secant, and cosecant functions as ratios involving sine and cosine.

2.4.2.2: Write cotangent, secant, and cosecant functions as the reciprocals of tangent, cosine, and sine, respectively.

2.4.3: 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.

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

2.4.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

2.4.8: Justify the Pythagorean, even/odd, and cofunction identities for sine and cosine using their unit circle definitions and symmetries of the unit circle and use the Pythagorean identity to find sinA, cosA, or tanA, given sinA, cosA, or tanA, and the quadrant of the angle.

2.4.9: Justify the sum and difference formulas for sine, cosine, and tangent and use them to solve problems.

### 3: Geometry

#### 3.1: Circles

3.1.2: Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems.

3.1.3: Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle.

#### 3.2: Congruence

3.2.1: Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects.

3.2.2: Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions.

3.2.3: Describe rotations and reflections that carry a regular polygon onto itself and identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations.

3.2.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

3.2.6: Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other.

3.2.7: Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle- Angle-Side, and Hypotenuse-Leg congruence conditions.

3.2.8: Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following:

3.2.8.1: vertical angles are congruent;

3.2.8.2: when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary;

3.2.9: Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following:

3.2.9.1: measures of interior angles of a triangle sum to 180°;

3.2.9.4: the medians of a triangle meet at a point.

3.2.10: Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following:

3.2.10.1: opposite sides of a parallelogram are congruent;

3.2.10.2: opposite angles of a parallelogram are congruent;

3.2.10.3: diagonals of a parallelogram bisect each other;

3.2.10.4: rectangles are parallelograms with congruent diagonals;

3.2.11: Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships.

#### 3.3: Geometric Measurement and Dimension

3.3.1: Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems.

3.3.2: Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle.

3.3.3: Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications.

#### 3.4: Expressing Geometric Properties with Equations

3.4.1: Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula.

3.4.2: Use the geometric definition of a parabola to derive its equation given the focus and directrix.

3.4.3: Use the geometric definition of an ellipse and of a hyperbola to derive the equation of each given the foci and points whose sum or difference of distance from the foci are constant.

3.4.5: Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope.

3.4.7: Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates.

#### 3.6: Similarity, Right Triangles, and Trigonometry

3.6.1: Understand 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. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

3.6.2: Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other.

3.6.3: Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results.

3.6.4: Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following:

3.6.4.1: A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion.

3.6.4.2: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

3.6.4.3: The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides.

3.6.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

3.6.6: Understand 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.

3.6.8: Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem.

### 4: Number and Quantity

#### 4.3: Complex Number System

4.3.1: 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.

4.3.3: Find the conjugate of a complex number in rectangular and polar forms and use conjugates to find moduli and quotients of complex numbers.

4.3.4: Graph complex numbers on the complex plane in rectangular and polar form and explain why the rectangular and polar forms of a given complex number represent the same number.

4.3.5: Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

#### 4.4: Vector and Matrix Quantities

4.4.1: 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.

4.4.2: Represent and model with vector quantities. Use the coordinates of an initial point and of a terminal point to find the components of a vector.

4.4.4: Perform operations on vectors.

4.4.4.1: Add and subtract vectors using components of the vectors and graphically.

4.4.4.2: Given the magnitude and direction of two vectors, determine the magnitude of their sum and of their difference.

4.4.7: Perform operations with matrices of appropriate dimensions including addition, subtraction, and scalar multiplication.

4.4.11: Apply 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

### 5: Statistics and Probability

#### 5.1: Conditional Probability and Rules of Probability

5.1.2: Use the multiplication rule to calculate probabilities for independent and dependent events. 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.

5.1.3: 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.

5.1.4: 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.

5.1.5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

5.1.6: Calculate the conditional probability of an event A given event B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

5.1.7: Apply the Addition Rule and the Multiplication Rule to determine probabilities, including conditional probabilities, and interpret the results in terms of the probability model.

5.1.8: Use permutations and combinations to solve mathematical and real-world problems, including determining probabilities of compound events. Justify the results.

#### 5.2: Making Inferences and Justifying Conclusions

5.2.1: Understand statistics and sampling distributions as a process for making inferences about population parameters based on a random sample from that population.

5.2.2: Distinguish between experimental and theoretical probabilities. Collect data on a chance event and use the relative frequency to estimate the theoretical probability of that event. Determine whether a given probability model is consistent with experimental results.

5.2.3: Plan and conduct a survey to answer a statistical question. Recognize how the plan addresses sampling technique, randomization, measurement of experimental error and methods to reduce bias.

5.2.4: 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.

5.2.6: Evaluate claims and conclusions in published reports or articles based on data by analyzing study design and the collection, analysis, and display of the data.

#### 5.3: Interpreting Data

5.3.1: Select and create an appropriate display, including dot plots, histograms, and box plots, for data that includes only real numbers.

5.3.2: Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets that include all real numbers.

5.3.3: Summarize and represent data from a single data set. Interpret differences in shape, center, and spread in the context of the data set, accounting for possible effects of extreme data points (outliers).

5.3.4: 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.

5.3.5: Analyze bivariate categorical data using two-way tables and identify possible associations between the two categories using marginal, joint, and conditional frequencies.

5.3.6: Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and use the function to solve problems in the context of the data.

5.3.7: Create a linear function to graphically model data from a real-world problem and interpret the meaning of the slope and intercept(s) in the context of the given problem.

5.3.8: Using technology, compute and interpret the correlation coefficient of a linear fit.

5.3.9: Differentiate between correlation and causation when describing the relationship between two variables. Identify potential lurking variables which may explain an association between two variables.

5.3.10: Create residual plots and analyze those plots to compare the fit of linear, quadratic, and exponential models to a given data set. Select the appropriate model and use it for interpolation.

#### 5.4: Using Probability to Make Decisions

5.4.1: Develop the probability distribution for a random variable defined for a sample space in which a theoretical probability can be calculated and graph the distribution.

5.4.2: Calculate the expected value of a random variable as the mean of its probability distribution. Find expected values by assigning probabilities to payoff values. Use expected values to evaluate and compare strategies in real-world scenarios.

5.4.3: Construct and compare theoretical and experimental probability distributions and use those distributions to find expected values.

5.4.4: Use probability to evaluate outcomes of decisions by finding expected values and determine if decisions are fair.

5.4.5: Use probability to evaluate outcomes of decisions. Use probabilities to make fair decisions.

5.4.6: Analyze decisions and strategies using probability concepts.

### 6: Calculus

#### 6.1: Limits and Continuity

6.1.2: Understand the definition and graphical interpretation of continuity of a function.

6.1.2.2: Classify discontinuities as removable, jump, or infinite. Justify that classification using the definition of continuity.

### 7: Derivatives

#### 7.1: Understand the concept of the derivative of a function geometrically, numerically, analytically, and verbally.

7.1.1: Interpret the value of the derivative of a function as the slope of the corresponding tangent line.

7.1.3: Approximate the derivative graphically by finding the slope of the tangent line drawn to a curve at a given point and numerically by using the difference quotient.

7.1.6: Understand the definition of the derivative and use this definition to determine the derivatives of various functions.

#### 7.2: Apply the rules of differentiation to functions.

7.2.1: Know and apply the derivatives of constant, power, trigonometric, inverse trigonometric, exponential, and logarithmic functions.

#### 7.3: Apply theorems and rules of differentiation to solve mathematical and real-world problems.

7.3.3: Explain the relationship between the increasing/decreasing behavior of f and the signs of f′. Use the relationship to generate a graph of f given the graph of f′, and vice versa, and to identify relative and absolute extrema of f.

Correlation last revised: 1/5/2017

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