Advanced Algebraic Thinking

          Module 1: Foundations of Algebraic Thinking

Section 1: Understanding Variables and Constants
- Lesson 1: Introduction to Variables
  Module 1, Section 1, Lesson 1: Explore what variables are and how they represent unknowns in expressions.
- Lesson 2: Recognizing Constants
  Module 1, Section 1, Lesson 2: Understand the role of fixed numbers in algebraic expressions.

Section 2: Basic Operations and Properties
- Lesson 1: Fundamental Operations Overview
  Module 1, Section 2, Lesson 1: Review addition, subtraction, multiplication, and division as tools of algebra.
- Lesson 2: Exploring Operational Properties
  Module 1, Section 2, Lesson 2: Learn about the commutative, associative, and distributive properties in simple contexts.

Section 3: Order of Operations and Expression Evaluation
- Lesson 1: Mastering the Order of Operations
  Module 1, Section 3, Lesson 1: Introduce the hierarchy of operations and its importance in solving expressions.
- Lesson 2: Evaluating Algebraic Expressions
  Module 1, Section 3, Lesson 2: Practice evaluating expressions step by step following the correct order.

Section 4: The Language of Algebra
- Lesson 1: Translating Words to Algebra
  Module 1, Section 4, Lesson 1: Convert verbal descriptions into algebraic expressions and equations.
- Lesson 2: Crafting Simple Equations
  Module 1, Section 4, Lesson 2: Write straightforward equations from given problem situations.

Section 5: Introduction to Equations and Inequalities
- Lesson 1: Understanding the Structure of Equations
  Module 1, Section 5, Lesson 1: Discuss the parts of an equation and the balance between both sides.
- Lesson 2: Basics of Inequalities
  Module 1, Section 5, Lesson 2: Introduce concepts of inequalities and their representation on a number line.

Module 2: Patterns, Relations, and Functions

Section 1: Recognizing Number Patterns
- Lesson 1: Identifying Sequences and Patterns
  Module 2, Section 1, Lesson 1: Observe and describe numerical patterns and sequences.
- Lesson 2: Expressing Patterns Algebraically
  Module 2, Section 1, Lesson 2: Learn to express observed patterns using algebraic formulas.

Section 2: Understanding Functions as Mappings
- Lesson 1: Introducing Functions
  Module 2, Section 2, Lesson 1: Define functions and understand the idea of mapping inputs to outputs.
- Lesson 2: Function Notation and Domain
  Module 2, Section 2, Lesson 2: Learn how to write and interpret function notation and discuss domain basics.

Section 3: Linear Relationships and Graphs
- Lesson 1: Plotting Linear Equations
  Module 2, Section 3, Lesson 1: Learn how to represent linear equations through graphing.
- Lesson 2: Interpreting Slope and Intercept
  Module 2, Section 3, Lesson 2: Understand how slope and intercept affect the shape and position of a graph.

Section 4: Introduction to Nonlinear Patterns
- Lesson 1: Exploring Quadratic Patterns
  Module 2, Section 4, Lesson 1: Begin to introduce quadratic expressions as examples of nonlinear patterns.
- Lesson 2: Recognizing Exponential Trends
  Module 2, Section 4, Lesson 2: Identify and describe exponential growth and decay in simple terms.

Section 5: Applying Patterns in Context
- Lesson 1: Real-World Pattern Identification
  Module 2, Section 5, Lesson 1: Apply pattern recognition to solve real-life problems.
- Lesson 2: Predicting Outcomes Using Patterns
  Module 2, Section 5, Lesson 2: Use identified patterns to make predictions and validate outcomes.

Module 3: Equations and Inequalities in Depth

Section 1: Solving Simple Equations
- Lesson 1: Isolating the Variable
  Module 3, Section 1, Lesson 1: Use inverse operations to isolate and solve for the variable in basic equations.
- Lesson 2: Balancing Equations
  Module 3, Section 1, Lesson 2: Practice maintaining equality by performing operations on both sides.

Section 2: Multi-step Equation Strategies
- Lesson 1: Simplifying by Combining Like Terms
  Module 3, Section 2, Lesson 1: Learn techniques to combine like terms and simplify equations.
- Lesson 2: Solving Equations with Variables on Both Sides
  Module 3, Section 2, Lesson 2: Develop strategies for moving terms and solving equations where variables appear on both sides.

Section 3: Working with Inequalities
- Lesson 1: Solving Basic Inequalities
  Module 3, Section 3, Lesson 1: Apply similar techniques from equations to solve inequalities.
- Lesson 2: Graphing Inequalities on a Number Line
  Module 3, Section 3, Lesson 2: Learn how to visually represent solutions to inequalities.

Section 4: Understanding Absolute Value in Equations
- Lesson 1: Introducing Absolute Value
  Module 3, Section 4, Lesson 1: Understand the concept of absolute value and its significance in algebra.
- Lesson 2: Solving Absolute Value Equations
  Module 3, Section 4, Lesson 2: Practice solving equations that include absolute value expressions.

Section 5: Systems of Equations and Inequalities
- Lesson 1: Solving Systems by Substitution
  Module 3, Section 5, Lesson 1: Learn the substitution method to solve simple systems.
- Lesson 2: Solving Systems by Elimination
  Module 3, Section 5, Lesson 2: Practice the elimination method to find solutions for systems.

Module 4: Algebraic Expressions and Structures

Section 1: Simplifying Algebraic Expressions
- Lesson 1: Combining Like Terms for Simplification
  Module 4, Section 1, Lesson 1: Identify and combine like terms to simplify expressions.
- Lesson 2: Applying the Distributive Property
  Module 4, Section 1, Lesson 2: Use the distributive property to expand and simplify expressions.

Section 2: Factoring Fundamentals
- Lesson 1: Factoring out the Greatest Common Factor
  Module 4, Section 2, Lesson 1: Learn how to factor out common factors from algebraic expressions.
- Lesson 2: Simple Factoring Techniques for Binomials
  Module 4, Section 2, Lesson 2: Introduce basic methods for factoring simple binomials.

Section 3: Exploring Polynomial Expressions
- Lesson 1: Understanding the Structure of Polynomials
  Module 4, Section 3, Lesson 1: Discuss terms, degrees, and the overall structure of polynomial expressions.
- Lesson 2: Operations on Polynomials
  Module 4, Section 3, Lesson 2: Practice adding and subtracting polynomials to reinforce structure concepts.

Section 4: Introduction to Rational Expressions
- Lesson 1: Defining Rational Expressions
  Module 4, Section 4, Lesson 1: Explain what rational expressions are and their role in algebra.
- Lesson 2: Simplifying Rational Expressions
  Module 4, Section 4, Lesson 2: Learn techniques for reducing rational expressions to simplest form.

Section 5: Function Composition Basics
- Lesson 1: Understanding Function Composition
  Module 4, Section 5, Lesson 1: Introduce the idea of composing two functions into a single operation.
- Lesson 2: Evaluating Composite Functions
  Module 4, Section 5, Lesson 2: Practice evaluating and simplifying composite functions step by step.

Module 5: Fundamental Problem-Solving Strategies

Section 1: Logical Reasoning in Algebra
- Lesson 1: Developing Logical Approaches to Problems
  Module 5, Section 1, Lesson 1: Focus on building logical reasoning skills for approaching algebraic problems.
- Lesson 2: Applying Critical Thinking to Algebra
  Module 5, Section 1, Lesson 2: Use critical thinking to assess and solve algebraic challenges.

Section 2: Strategy Development and Planning
- Lesson 1: Breaking Down Complex Problems
  Module 5, Section 2, Lesson 1: Learn to decompose multi-step problems into smaller, manageable parts.
- Lesson 2: Planning Step-by-Step Solutions
  Module 5, Section 2, Lesson 2: Develop systematic strategies for approaching and resolving problems.

Section 3: Problem Modeling with Algebra
- Lesson 1: Translating Real-World Situations to Equations
  Module 5, Section 3, Lesson 1: Practice converting real-life scenarios into algebraic models.
- Lesson 2: Constructing Predictive Models
  Module 5, Section 3, Lesson 2: Create simple algebraic models that can predict outcomes.

Section 4: Verification and Evaluation Techniques
- Lesson 1: Checking and Verifying Solutions
  Module 5, Section 4, Lesson 1: Learn methods to check the validity of solutions.
- Lesson 2: Reflecting on Problem-Solving Strategies
  Module 5, Section 4, Lesson 2: Evaluate various approaches and learn from mistakes in the problem-solving process.

Section 5: Synthesis of Algebraic Concepts
- Lesson 1: Integrating Multiple Concepts in Problem Solving
  Module 5, Section 5, Lesson 1: Combine different algebraic techniques to solve comprehensive problems.
- Lesson 2: Capstone Problem-Solving Exercise
  Module 5, Section 5, Lesson 2: Apply all the fundamental concepts in a final integrated problem that reinforces the skills learned throughout the course.