1: Seeing Structure in Expressions

1.1: Interpret the structure of expressions.

KY.HS.A.1: Interpret expressions that represent a quantity in terms of its context.

KY.HS.A.1.a: Interpret parts of an expression, such as terms, factors and coefficients.

Compound Interest
Operations with Radical Expressions
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II
Solving Formulas for any Variable

KY.HS.A.1.b: Interpret complicated expressions, given a context, by viewing one or more of their parts as a single entity.

Arithmetic Sequences
Arithmetic and Geometric Sequences
Compound Interest
Exponential Growth and Decay
Geometric Sequences
Translating and Scaling Functions
Using Algebraic Expressions

KY.HS.A.2: Use the structure of an expression to identify ways to rewrite it and consistently look for opportunities to rewrite expressions in equivalent forms.

Dividing Exponential Expressions
Equivalent Algebraic Expressions I
Equivalent Algebraic Expressions II
Exponents and Power Rules
Factoring Special Products
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c
Multiplying Exponential Expressions
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II
Simplifying Trigonometric Expressions
Solving Algebraic Equations II
Using Algebraic Expressions

1.2: Write expressions in equivalent forms to solve problems.

KY.HS.A.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

KY.HS.A.3.a: Write the standard form of a given polynomial and identify the terms, coefficients, degree, leading coefficient and constant term.

Addition of Polynomials
Quadratics in Polynomial Form

KY.HS.A.3.b: Factor a quadratic expression to reveal the zeros of the function it defines.

Factoring Special Products
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c
Quadratics in Factored Form

KY.HS.A.3.c: Use the properties of exponents to rewrite exponential expressions.

Dividing Exponential Expressions
Exponents and Power Rules
Multiplying Exponential Expressions

KY.HS.A.3.d: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Quadratics in Vertex Form

2: Arithmetic with Polynomials and Rational Expressions

2.1: Perform arithmetic operations on polynomials.

KY.HS.A.5: Add, subtract and multiply polynomials.

Addition and Subtraction of Functions
Addition of Polynomials
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c

2.2: Understand the relationship between zeros and factors of polynomials.

KY.HS.A.6: Know and apply the Remainder Theorem.

Dividing Polynomials Using Synthetic Division
Polynomials and Linear Factors

2.2.1.2: For a polynomial ??(??) and a number ??, the remainder on division by ?? – ?? is ??(a), so ??(??) = 0 if and only if (?? – ??) is a factor of ??(??).

Dividing Polynomials Using Synthetic Division
Polynomials and Linear Factors

KY.HS.A.7: Identify roots of polynomials when suitable factorizations are available. Know these roots become the zeros (x-intercepts) for the corresponding polynomial function.

Graphs of Polynomial Functions
Modeling the Factorization of x²+bx+c
Polynomials and Linear Factors
Quadratics in Factored Form
Quadratics in Vertex Form

2.3: Use polynomial identities to solve problems.

KY.HS.A.8: Prove polynomial identities and use them to describe numerical relationships.

Factoring Special Products

KY.HS.A.9: Know and apply the Binomial Theorem for 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 for example by Pascal’s Triangle.

Binomial Probabilities

2.4: Rewrite rational expressions.

KY.HS.A.10: Rewrite simple rational expressions in different forms.

Dividing Polynomials Using Synthetic Division

3: Creating Equations

3.1: Create equations that describe numbers or relationships.

KY.HS.A.12: Create equations and inequalities in one variable and use them to solve problems.

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

KY.HS.A.13: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Absolute Value Equations and Inequalities
Cat and Mouse (Modeling with Linear Systems)
Circles
Compound Interest
Linear Functions
Point-Slope Form of a Line
Points, Lines, and Equations
Quadratics in Polynomial Form
Quadratics in Vertex Form
Slope-Intercept Form of a Line
Solving Equations by Graphing Each Side
Solving Equations on the Number Line
Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Slope-Intercept Form)
Solving Linear Systems (Standard Form)
Standard Form of a Line
Using Algebraic Equations

KY.HS.A.14: Create a system of equations or inequalities to represent constraints within a modeling context. Interpret the solution(s) to the corresponding system as viable or nonviable options within the context.

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

KY.HS.A.15: Rearrange formulas to solve a literal equation, highlighting a quantity of interest, using the same reasoning as in solving equations.

Area of Triangles
Solving Formulas for any Variable

4: Reasoning with Equations and Inequalities

4.1: Understand solving equations as a process of reasoning and explain the reasoning.

KY.HS.A.16: Understand 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.

Equivalent Algebraic Expressions I
Equivalent Algebraic Expressions II
Modeling One-Step Equations
Modeling and Solving Two-Step Equations
Solving Algebraic Equations I
Solving Algebraic Equations II
Solving Equations by Graphing Each Side
Solving Equations on the Number Line
Solving Formulas for any Variable
Solving Two-Step Equations

KY.HS.A.17: Solve and justify equations in one variable. Justify the solutions and give examples showing how extraneous solutions may arise.

Radical Functions

KY.HS.A.17.a: Solve rational equations written as proportions in one variable.

Percents and Proportions
Proportions and Common Multipliers
Radical Functions

KY.HS.A.17.b: Solve radical equations in one variable.

Radical Functions

4.2: Solve equations and inequalities in one variable.

KY.HS.A.18: Solve linear equations and inequalities in one variable, including literal equations with coefficients represented by letters.

Area of Triangles
Compound Inequalities
Exploring Linear Inequalities in One Variable
Linear Inequalities in Two Variables
Modeling One-Step Equations
Modeling and Solving Two-Step Equations
Solving Algebraic Equations I
Solving Algebraic Equations II
Solving Equations by Graphing Each Side
Solving Equations on the Number Line
Solving Formulas for any Variable
Solving Linear Inequalities in One Variable
Solving Two-Step Equations

KY.HS.A.19: Solve quadratic equations in one variable.

KY.HS.A.19.a: Solve quadratic equations by taking square roots, 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.

Factoring Special Products
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c
Points in the Complex Plane
Quadratics in Factored Form
Roots of a Quadratic

KY.HS.A.19.b: 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.

Roots of a Quadratic

KY.HS.A.19.c: Solve quadratic equations by completing the square.

Factoring Special Products
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c
Points in the Complex Plane
Roots of a Quadratic

4.3: Solve systems of equations.

KY.HS.A.20: Solve systems of linear equations in two variables.

KY.HS.A.20.a: Understand a system of two equations in two variables has the same solution as a new system formed by replacing one of the original equations with an equivalent equation.

Solving Equations by Graphing Each Side
Solving Linear Systems (Standard Form)

KY.HS.A.20.b: Solve systems of linear equations with graphs, substitution and elimination, focusing on pairs of linear equations in two variables.

Cat and Mouse (Modeling with Linear Systems)
Solving Equations by Graphing Each Side
Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Slope-Intercept Form)
Solving Linear Systems (Standard Form)

KY.HS.A.22: Use matrices to solve a system of equations.

KY.HS.A.22.a: Represent a system of linear equations as a single matrix equation in a vector variable.

Solving Linear Systems (Matrices and Special Solutions)

KY.HS.A.22.b: Find the inverse of a matrix if it exists.

Solving Linear Systems (Matrices and Special Solutions)

KY.HS.A.22.c: Use matrices to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Solving Linear Systems (Matrices and Special Solutions)

4.4: Represent and solve equations and inequalities graphically.

KY.HS.A.23: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.

Absolute Value Equations and Inequalities
Cat and Mouse (Modeling with Linear Systems)
Circles
Ellipses
Hyperbolas
Parabolas
Point-Slope Form of a Line
Points, Lines, and Equations
Solving Equations by Graphing Each Side
Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Slope-Intercept Form)
Solving Linear Systems (Standard Form)
Standard Form of a Line

KY.HS.A.24: Justify that the solutions of the equations f(x) = g(x) are the x-coordinates of the points where the graphs of y = f(x) and y = g(x) intersect. Find the approximate solutions graphically, using technology or tables.

Cat and Mouse (Modeling with Linear Systems)
Point-Slope Form of a Line
Solving Equations by Graphing Each Side
Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Slope-Intercept Form)
Standard Form of a Line

4.4.2.1: Students justify solutions for equations which Include cases where ??(??) and/or ??(??) are linear, polynomial, rational, absolute value, exponential and logarithmic functions.

Cat and Mouse (Modeling with Linear Systems)
Point-Slope Form of a Line
Solving Equations by Graphing Each Side
Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Slope-Intercept Form)
Standard Form of a Line

KY.HS.A.25: Graph linear inequalities in two variables.

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

KY.HS.A.25.a: Graph the solutions to a linear inequality as a half-plane (excluding the boundary in the case of a strict inequality).

Linear Inequalities in Two Variables
Systems of Linear Inequalities (Slope-intercept form)

KY.HS.A.25.b: Graph the solution set to a system of linear inequalities as the intersection of the corresponding half-planes.

Linear Programming
Systems of Linear Inequalities (Slope-intercept form)