Advanced Algebra and Functions

          Module 1: Foundations of Algebraic Expressions

Section 1: Basic Operations and Expressions
- Lesson 1: Understanding Variables and Constants  
  Module 1, Section 1, Lesson 1: Introducing variables, constants, and their roles in algebraic expressions.
- Lesson 2: Combining Like Terms  
  Module 1, Section 1, Lesson 2: Learning how to simplify expressions by grouping similar terms.

Section 2: Properties of Exponents
- Lesson 1: Rules of Exponents  
  Module 1, Section 2, Lesson 1: Exploring the product, quotient, and power rules for exponents.
- Lesson 2: Applications of Exponent Rules  
  Module 1, Section 2, Lesson 2: Applying exponent rules to simplify expressions and solve basic equations.

Section 3: Factoring Fundamentals
- Lesson 1: Common Factoring Techniques  
  Module 1, Section 3, Lesson 1: Introducing greatest common factors and simple factoring techniques.
- Lesson 2: Factoring by Grouping  
  Module 1, Section 3, Lesson 2: Learning to factor expressions by grouping terms effectively.

Section 4: Equation Basics
- Lesson 1: Solving Simple Linear Equations  
  Module 1, Section 4, Lesson 1: Understanding the process of isolating variables in linear equations.
- Lesson 2: Balance Principle and Verification  
  Module 1, Section 4, Lesson 2: Emphasizing the balance principle in equations and methods for checking solutions.

Section 5: Introduction to Linear Functions
- Lesson 1: Concept of a Function  
  Module 1, Section 5, Lesson 1: Defining a function and exploring linear relationships.
- Lesson 2: Graphing Linear Equations  
  Module 1, Section 5, Lesson 2: Learning to graph linear equations on the coordinate plane.

Module 2: Polynomials and Their Operations

Section 1: Polynomial Basics
- Lesson 1: Defining Polynomials  
  Module 2, Section 1, Lesson 1: Introducing polynomials, terms, degrees, and coefficients.
- Lesson 2: Identifying Polynomial Components  
  Module 2, Section 1, Lesson 2: Recognizing leading coefficients, constant terms, and variable parts.

Section 2: Adding and Subtracting Polynomials
- Lesson 1: Combining Like Terms in Polynomials  
  Module 2, Section 2, Lesson 1: Techniques for adding and subtracting polynomial expressions.
- Lesson 2: Practical Applications of Operations  
  Module 2, Section 2, Lesson 2: Applying addition and subtraction in problem‐solving scenarios.

Section 3: Multiplication and Division of Polynomials
- Lesson 1: Multiplying Polynomials  
  Module 2, Section 3, Lesson 1: Learning methods for multiplying binomials and polynomials.
- Lesson 2: Introduction to Polynomial Division  
  Module 2, Section 3, Lesson 2: Basic concepts of dividing polynomials, including long division.

Section 4: Zeros and Graphing Polynomials
- Lesson 1: Finding Zeros of Polynomials  
  Module 2, Section 4, Lesson 1: Techniques for determining polynomial roots.
- Lesson 2: Graphing Based on Zeros  
  Module 2, Section 4, Lesson 2: Understanding how zeros determine the shape of the polynomial graph.

Section 5: Advanced Factoring of Polynomials
- Lesson 1: Factoring Quadratics  
  Module 2, Section 5, Lesson 1: Techniques for factoring quadratic expressions.
- Lesson 2: Special Factoring Patterns  
  Module 2, Section 5, Lesson 2: Learning patterns such as the difference of squares and perfect square trinomials.

Module 3: Advanced Polynomial Functions

Section 1: Quadratic Functions
- Lesson 1: Properties of Quadratic Functions  
  Module 3, Section 1, Lesson 1: Understanding the standard form, vertex, and axis of symmetry.
- Lesson 2: Graphing Quadratic Functions  
  Module 3, Section 1, Lesson 2: Step-by-step techniques for sketching quadratic curves based on key features.

Section 2: Higher-Degree Polynomial Functions
- Lesson 1: Characteristics of Higher-Degree Functions  
  Module 3, Section 2, Lesson 1: Exploring end behavior and turning points in polynomials of degree three and above.
- Lesson 2: Graphing Techniques for Higher-Degree Polynomials  
  Module 3, Section 2, Lesson 2: Practical steps to sketch the graphs using zeros and multiplicities.

Section 3: The Remainder Theorem
- Lesson 1: Understanding the Remainder Theorem  
  Module 3, Section 3, Lesson 1: Learning the theorem and its importance in evaluating polynomials.
- Lesson 2: Applying the Remainder Theorem  
  Module 3, Section 3, Lesson 2: Step-by-step examples of using the remainder theorem to solve problems.

Section 4: Using the Factor Theorem
- Lesson 1: Introduction to the Factor Theorem  
  Module 3, Section 4, Lesson 1: Explaining the connection between factors and zeros of polynomials.
- Lesson 2: Factor Theorem in Practice  
  Module 3, Section 4, Lesson 2: Applying the theorem to factor and solve polynomial equations.

Section 5: Solving Polynomial Equations
- Lesson 1: Strategies for Solving Equations  
  Module 3, Section 5, Lesson 1: Techniques for finding multiple roots and solving complex polynomials.
- Lesson 2: Consolidating Polynomial Equation Methods  
  Module 3, Section 5, Lesson 2: Review and practice with various types of polynomial equations.

Module 4: Rational and Exponential Functions

Section 1: Rational Expressions
- Lesson 1: Defining Rational Expressions  
  Module 4, Section 1, Lesson 1: Introduction to rational expressions and identifying domains.
- Lesson 2: Simplifying Rational Expressions  
  Module 4, Section 1, Lesson 2: Techniques to reduce and simplify expressions through factoring.

Section 2: Graphing Rational Functions
- Lesson 1: Characteristics of Rational Graphs  
  Module 4, Section 2, Lesson 1: Learning about asymptotes, holes, and intercepts in rational functions.
- Lesson 2: Step-by-Step Graph Sketching  
  Module 4, Section 2, Lesson 2: Practical techniques for graphing rational functions with guided examples.

Section 3: Introduction to Exponential Functions
- Lesson 1: Defining Exponential Functions  
  Module 4, Section 3, Lesson 1: Understanding the basics of exponential growth and decay.
- Lesson 2: Comparing Exponential to Linear Functions  
  Module 4, Section 3, Lesson 2: Recognizing the differences in rate of change and behavior of exponential functions.

Section 4: Properties of Exponential Expressions
- Lesson 1: Manipulating Exponential Expressions  
  Module 4, Section 4, Lesson 1: Applying exponent laws to exponential expressions in various contexts.
- Lesson 2: Solving Simple Exponential Equations  
  Module 4, Section 4, Lesson 2: Techniques for isolating the exponential term and solving for variables.

Section 5: Solving Exponential Equations
- Lesson 1: Introduction to Logarithms (Conceptual Overview)  
  Module 4, Section 5, Lesson 1: An accessible explanation of logarithms as the inverse of exponential functions.
- Lesson 2: Techniques for Exponential Equations  
  Module 4, Section 5, Lesson 2: Step-by-step methods to solve exponential equations using introductory logarithmic ideas.

Module 5: Functions and Their Applications

Section 1: Function Notation and Interpretation
- Lesson 1: Understanding Function Notation  
  Module 5, Section 1, Lesson 1: Defining and interpreting function notation and its importance.
- Lesson 2: Evaluating Functions  
  Module 5, Section 1, Lesson 2: Techniques to plug in inputs and compute outputs accurately.

Section 2: Operations on Functions
- Lesson 1: Adding and Subtracting Functions  
  Module 5, Section 2, Lesson 1: Combining functions through addition and subtraction.
- Lesson 2: Multiplying and Dividing Functions  
  Module 5, Section 2, Lesson 2: Exploring operations with functions and understanding restrictions on domains.

Section 3: Inverse Functions
- Lesson 1: Finding the Inverse of a Function  
  Module 5, Section 3, Lesson 1: Steps to determine if a function has an inverse and how to compute it.
- Lesson 2: Graphical Interpretation of Inverses  
  Module 5, Section 3, Lesson 2: Understanding the symmetry of functions and their inverses on the coordinate plane.

Section 4: Function Transformations
- Lesson 1: Shifts and Reflections  
  Module 5, Section 4, Lesson 1: Learning horizontal and vertical shifts as well as reflections of function graphs.
- Lesson 2: Stretching and Compressing  
  Module 5, Section 4, Lesson 2: Exploring how scaling affects the shape and function behavior.

Section 5: Real-World Applications of Functions
- Lesson 1: Modeling Real Situations with Functions  
  Module 5, Section 5, Lesson 1: Applying function concepts to model practical problems from everyday life.
- Lesson 2: Analyzing and Interpreting Function-Based Models  
  Module 5, Section 5, Lesson 2: Concluding methods of interpreting data and results derived from function models.