### MGSE9-12.N.CN: The Complex Number System

#### 2.1: Perform arithmetic operations with complex numbers.

MGSE9-12.N.CN.1: Understand 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.

MGSE9-12.N.CN.2: Use the relation 𝘪² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

MGSE9-12.N.CN.3: Find the conjugate of a complex number; use the conjugate to find the quotient of complex numbers.

#### 2.2: Use complex numbers in polynomial identities and equations.

MGSE9-12.N.CN.7: Solve quadratic equations with real coefficients that have complex solutions by (but not limited to) square roots, completing the square, and the quadratic formula.

MGSE9-12.N.CN.9: Use the Fundamental Theorem of Algebra to find all roots of a polynomial equation.

### MGSE9-12.A.SSE: Seeing Structure in Expressions

#### 3.1: Interpret the structure of expressions

MGSE9-12.A.SSE.1: Interpret expressions that represent a quantity in terms of its context.

MGSE9-12.A.SSE.1a: Interpret parts of an expression, such as terms, factors, and coefficients, in context.

MGSE9-12.A.SSE.1b: Given situations which utilize formulas or expressions with multiple terms and/or factors, interpret the meaning (in context) of individual terms or factors.

MGSE9-12.A.SSE.2: Use the structure of an expression to rewrite it in different equivalent forms.

#### 3.2: Write expressions in equivalent forms to solve problems

MGSE9-12.A.SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

MGSE9-12.A.SSE.3c.3c: Use the properties of exponents to transform expressions for exponential functions.

### MGSE9-12.A.APR: Arithmetic with Polynomials and Rational Expressions

#### 4.1: Perform arithmetic operations on polynomials

MGSE9-12.A.APR.1: Add, subtract, and multiply polynomials; understand that polynomials form a system analogous to the integers in that they are closed under these operations.

#### 4.2: Understand the relationship between zeros and factors of polynomials

MGSE9-12.A.APR.2: 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).

MGSE9-12.A.APR.3: 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.

#### 4.3: Use polynomial identities to solve problems

MGSE9-12.A.APR.5: Know and apply that the Binomial Theorem gives 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 using Pascal’s Triangle.

### MGSE9-12.A.CED: Creating Equations

#### 5.1: Create equations that describe numbers or relationships

MGSE9-12.A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, simple rational, and exponential functions.

MGSE9-12.A.CED.2: Create linear, quadratic, and exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (The phrase “in two or more variables” refers to formulas like the compound interest formula, in which A = P(1 + r/n)nt has multiple variables.)

MGSE9-12.A.CED.3: Represent constraints by equations or inequalities, and by systems of equation and/or inequalities, and interpret data points as possible (i.e. a solution) or not possible (i.e. a nonsolution) under the established constraints.

MGSE9-12.A.CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

### MGSE9-12.A.REI: Reasoning with Equations and Inequalities

#### 6.1: Understand solving equations as a process of reasoning and explain the reasoning

MGSE9-12.A.REI.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

#### 6.2: Solve equations and inequalities in one variable

MGSE9-12.A.REI.4: Solve quadratic equations in one variable.

MGSE9-12.A.REI.4b: Solve quadratic equations by inspection (e.g., for 𝘹² = 49), taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation.

#### 6.3: Represent and solve equations and inequalities graphically

MGSE9-12.A.REI.11: Using graphs, tables, or successive approximations, show that the solution to the equation 𝘧(𝘹) = 𝑔(𝘹) is the 𝘹-value where the 𝑦-values of 𝘧(𝘹) and 𝑔(𝘹) are the same.

### MGSE9-12.F.IF: Interpreting Functions

#### 7.1: Interpret functions that arise in applications in terms of the context

MGSE9-12.F.IF.4: Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries and end behavior.

MGSE9-12.F.IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

MGSE9-12.F.IF.6: 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.

#### 7.2: Analyze functions using different representations

MGSE9-12.F.IF.7: Graph functions expressed algebraically and show key features of the graph both by hand and by using technology.

MGSE9-12.F.IF.7b: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

MGSE9-12.F.IF.7c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

MGSE9-12.F.IF.7d: Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

MGSE9-12.F.IF.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior.

MGSE9-12.F.IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

MGSE9-12.F.IF.8b: Use the properties of exponents to interpret expressions for exponential functions.

### MGSE9-12.F.BF: Building Functions

#### 8.1: Build a function that models a relationship between two quantities

MGSE9-12.F.BF.1: Write a function that describes a relationship between two quantities.

MGSE9-12.F.BF.1b: Combine standard function types using arithmetic operations in contextual situations. (Adding, subtracting, and multiplying functions of different types).

#### 8.2: Build new functions from existing functions

MGSE9-12.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.

MGSE9-12.F.BF.4: Find inverse functions.

MGSE9-12.F.BF.4b: Verify by composition that one function is the inverse of another.

MGSE9-12.F.BF.4c: Read values of an inverse function from a graph or a table, given that the function has an inverse.

MGSE9-12.F.BF.5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

### MGSE9-12.F.LE: Linear, Quadratic, and Exponential Models

#### 9.1: Construct and compare linear, quadratic, and exponential models and solve problems

MGSE9-12.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.

### MGSE9-12.G.C: Circles

#### 10.1: Understand and apply theorems about circles

MGSE9-12.G.C.2: Identify and describe relationships among inscribed angles, radii, chords, tangents, and secants. 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.

#### 10.2: Find arc lengths and areas of sectors of circles

MGSE9-12.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.

### MGSE9-12.G.GPE: Expressing Geometric Properties with Equations

#### 11.1: Translate between the geometric description and the equation for a conic section

MGSE9-12.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.

#### 11.2: Use coordinates to prove simple geometric theorems algebraically

MGSE9-12.G.GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

### MGSE9-12.G.GMD: Geometric Measurement and Dimension

#### 12.1: Explain volume formulas and use them to solve problems

MGSE9-12.G.GMD.1: Give informal arguments for geometric formulas.

MGSE9-12.G.GMD.1a: Give informal arguments for the formulas of the circumference of a circle and area of a circle using dissection arguments and informal limit arguments.

MGSE9-12.G.GMD.1b: Give informal arguments for the formula of the volume of a cylinder, pyramid, and cone using Cavalieri’s principle.

MGSE9-12.G.GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

### MGSE9-12.S.CP: Conditional Probability and the Rules of Probability

#### 14.1: Understand independence and conditional probability and use them to interpret data

MGSE9-12.S.CP.1: Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (or, and, not).

MGSE9-12.S.CP.2: Understand that if two events A and B are independent, the probability of A and B occurring together is the product of their probabilities, and that if the probability of two events A and B occurring together is the product of their probabilities, the two events are independent.

MGSE9-12.S.CP.3: Understand the conditional probability of A given B as P (A and B)/P(B). Interpret independence of A and B in terms of conditional probability; that is, 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.

#### 14.2: Use the rules of probability to compute probabilities of compound events in a uniform probability model

MGSE9-12.S.CP.6: 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 context.

Correlation last revised: 1/22/2020

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