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.

Points in the Complex Plane
Roots of a Quadratic

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

Points in the Complex Plane

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

Points in the Complex Plane
Roots of a Quadratic

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.

Points in the Complex Plane
Roots of a Quadratic

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

Polynomials and Linear Factors

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.

Compound Interest
Operations with Radical Expressions
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II

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.

Compound Interest
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II

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

Dividing Exponential Expressions
Equivalent Algebraic Expressions I
Equivalent Algebraic Expressions II
Exponents and Power Rules
Multiplying Exponential Expressions
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II
Using Algebraic Expressions

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.

Exponents and Power Rules

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.

Addition and Subtraction of Functions
Addition of Polynomials
Modeling the Factorization of x2+bx+c

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).

Dividing Polynomials Using Synthetic Division
Polynomials and Linear Factors

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.

Polynomials and Linear Factors
Quadratics in Factored Form

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.

Binomial Probabilities

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.

Absolute Value Equations and Inequalities
Arithmetic Sequences
Exploring Linear Inequalities in One Variable
Geometric Sequences
Linear Inequalities in Two Variables
Modeling One-Step Equations
Modeling and Solving Two-Step Equations
Solving Equations on the Number Line
Solving Linear Inequalities in One Variable
Solving Two-Step Equations
Using Algebraic Equations

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.)

Point-Slope Form of a Line
Points, Lines, and Equations
Solving Equations by Graphing Each Side
Standard Form of a Line

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.

Linear Inequalities in Two Variables
Linear Programming
Solving Linear Systems (Standard Form)
Systems of Linear Inequalities (Slope-intercept form)

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

Solving Formulas for any Variable

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.

Radical Functions

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.

Modeling the Factorization of x2+bx+c
Roots of a Quadratic

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.

Absolute Value Equations and Inequalities
Absolute Value with Linear Functions
Circles
Exponential Functions
Parabolas
Point-Slope Form of a Line
Points, Lines, and Equations
Quadratics in Polynomial Form
Quadratics in Vertex Form
Radical Functions
Solving Equations on the Number Line
Standard Form of a Line

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.

Absolute Value with Linear Functions
Cat and Mouse (Modeling with Linear Systems)
Exponential Functions
General Form of a Rational Function
Graphs of Polynomial Functions
Introduction to Exponential Functions
Linear Functions
Logarithmic Functions
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Radical Functions
Slope-Intercept Form of a Line

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

General Form of a Rational Function
Introduction to Functions
Radical Functions
Rational Functions

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.

Distance-Time Graphs
Distance-Time and Velocity-Time Graphs

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.

Absolute Value with Linear Functions
Radical Functions

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

Graphs of Polynomial Functions
Polynomials and Linear Factors
Quadratics in Factored Form
Quadratics in Vertex Form
Roots of a Quadratic
Zap It! Game

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

General Form of a Rational Function
Rational Functions

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.

Compound Interest
Exponential Growth and Decay

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).

Addition and Subtraction of Functions

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.

Exponential Functions
Logarithmic Functions
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions
Zap It! Game

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

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

Logarithmic Functions

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

Logarithmic Functions

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

Logarithmic Functions

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.

Compound Interest

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.

Chords and Arcs
Circumference and Area of Circles
Inscribed Angles

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.

Chords and Arcs

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.

Circles
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard

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.

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.

Circumference and Area of Circles

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

Prisms and Cylinders
Pyramids and Cones

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

Prisms and Cylinders
Pyramids and Cones

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).

Independent and Dependent Events

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.

Independent and Dependent Events

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.

Independent and Dependent Events

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.

Independent and Dependent Events

Correlation last revised: 9/24/2019

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