Advanced Geometry: Transformation and Symmetry

          Module 1: Foundations of Transformational Geometry

Section 1: Understanding Transformation Geometry
- Lesson 1: What is Transformation Geometry?
  - Module 1, Section 1, Lesson 1: What is Transformation Geometry?
- Lesson 2: Historical Context and Importance
  - Module 1, Section 1, Lesson 2: Historical Context and Importance

Section 2: Basic Terminology and Concepts
- Lesson 1: Key Definitions in Transformation Geometry
  - Module 1, Section 2, Lesson 1: Key Definitions in Transformation Geometry
- Lesson 2: Fundamental Properties of Transformations
  - Module 1, Section 2, Lesson 2: Fundamental Properties of Transformations

Section 3: Coordinate Systems and Approaches
- Lesson 1: The Cartesian Plane and Its Role in Geometry
  - Module 1, Section 3, Lesson 1: The Cartesian Plane and Its Role in Geometry
- Lesson 2: Representing Transformations on the Coordinate Plane
  - Module 1, Section 3, Lesson 2: Representing Transformations on the Coordinate Plane

Section 4: Visualizing Transformations
- Lesson 1: Sketching Simple Transformations
  - Module 1, Section 4, Lesson 1: Sketching Simple Transformations
- Lesson 2: Observing Changes in Geometric Figures
  - Module 1, Section 4, Lesson 2: Observing Changes in Geometric Figures

Section 5: Real-World Applications of Transformations
- Lesson 1: Transformations in Art and Architecture
  - Module 1, Section 5, Lesson 1: Transformations in Art and Architecture
- Lesson 2: Everyday Examples of Geometric Transformations
  - Module 1, Section 5, Lesson 2: Everyday Examples of Geometric Transformations


Module 2: Understanding Translations and Rotations

Section 1: Introduction to Translations
- Lesson 1: What Is a Translation?
  - Module 2, Section 1, Lesson 1: What Is a Translation?
- Lesson 2: Properties and Effects of Translations
  - Module 2, Section 1, Lesson 2: Properties and Effects of Translations

Section 2: Techniques for Translating Figures
- Lesson 1: Step-by-Step Process for Translating Shapes
  - Module 2, Section 2, Lesson 1: Step-by-Step Process for Translating Shapes
- Lesson 2: Plotting Translations on the Coordinate Plane
  - Module 2, Section 2, Lesson 2: Plotting Translations on the Coordinate Plane

Section 3: Introduction to Rotations
- Lesson 1: What Is a Rotation?
  - Module 2, Section 3, Lesson 1: What Is a Rotation?
- Lesson 2: Understanding Rotation Angles and Directions
  - Module 2, Section 3, Lesson 2: Understanding Rotation Angles and Directions

Section 4: Performing Rotations
- Lesson 1: Rotating Figures About the Origin
  - Module 2, Section 4, Lesson 1: Rotating Figures About the Origin
- Lesson 2: Identifying Center of Rotation in Figures
  - Module 2, Section 4, Lesson 2: Identifying Center of Rotation in Figures

Section 5: Comparing Translations and Rotations
- Lesson 1: Similarities and Differences Between Translations and Rotations
  - Module 2, Section 5, Lesson 1: Similarities and Differences Between Translations and Rotations
- Lesson 2: Combined Effects of Translation and Rotation
  - Module 2, Section 5, Lesson 2: Combined Effects of Translation and Rotation


Module 3: Reflections and Dilations

Section 1: Basics of Reflections
- Lesson 1: Definition and Examples of Reflections
  - Module 3, Section 1, Lesson 1: Definition and Examples of Reflections
- Lesson 2: Understanding the Line of Reflection
  - Module 3, Section 1, Lesson 2: Understanding the Line of Reflection

Section 2: Reflection Techniques
- Lesson 1: Creating Mirror Images Using Reflections
  - Module 3, Section 2, Lesson 1: Creating Mirror Images Using Reflections
- Lesson 2: Reflecting Figures on the Coordinate Plane
  - Module 3, Section 2, Lesson 2: Reflecting Figures on the Coordinate Plane

Section 3: Introduction to Dilations
- Lesson 1: What Is a Dilation?
  - Module 3, Section 3, Lesson 1: What Is a Dilation?
- Lesson 2: Understanding Scale Factors and Their Impact
  - Module 3, Section 3, Lesson 2: Understanding Scale Factors and Their Impact

Section 4: Performing Dilations
- Lesson 1: Step-by-Step Dilating Geometric Figures
  - Module 3, Section 4, Lesson 1: Step-by-Step Dilating Geometric Figures
- Lesson 2: Dilation on Coordinate Grids
  - Module 3, Section 4, Lesson 2: Dilation on Coordinate Grids

Section 5: Combining Reflections and Dilations
- Lesson 1: Analyzing the Effects of Multiple Transformations
  - Module 3, Section 5, Lesson 1: Analyzing the Effects of Multiple Transformations
- Lesson 2: Practical Exercises with Combined Transformations
  - Module 3, Section 5, Lesson 2: Practical Exercises with Combined Transformations


Module 4: Introduction to Symmetry in Shapes

Section 1: Basics of Symmetry
- Lesson 1: What Is Symmetry?
  - Module 4, Section 1, Lesson 1: What Is Symmetry?
- Lesson 2: Identifying Symmetry in Simple Geometric Figures
  - Module 4, Section 1, Lesson 2: Identifying Symmetry in Simple Geometric Figures

Section 2: Axial (Line) Symmetry
- Lesson 1: Understanding Line Symmetry
  - Module 4, Section 2, Lesson 1: Understanding Line Symmetry
- Lesson 2: Recognizing Axial Symmetry in Shapes
  - Module 4, Section 2, Lesson 2: Recognizing Axial Symmetry in Shapes

Section 3: Rotational Symmetry
- Lesson 1: Concepts and Examples of Rotational Symmetry
  - Module 4, Section 3, Lesson 1: Concepts and Examples of Rotational Symmetry
- Lesson 2: Determining the Order of Rotational Symmetry
  - Module 4, Section 3, Lesson 2: Determining the Order of Rotational Symmetry

Section 4: Symmetry in Nature and Art
- Lesson 1: Exploring Natural Examples of Symmetry
  - Module 4, Section 4, Lesson 1: Exploring Natural Examples of Symmetry
- Lesson 2: Symmetry in Artistic Patterns and Designs
  - Module 4, Section 4, Lesson 2: Symmetry in Artistic Patterns and Designs

Section 5: Symmetry in Problem Solving
- Lesson 1: Using Symmetry to Simplify Geometrical Problems
  - Module 4, Section 5, Lesson 1: Using Symmetry to Simplify Geometrical Problems
- Lesson 2: Real-Life Applications of Symmetry in Design and Analysis
  - Module 4, Section 5, Lesson 2: Real-Life Applications of Symmetry in Design and Analysis


Module 5: Basics of Group Theory in Geometry

Section 1: Introduction to Group Theory Concepts
- Lesson 1: Understanding the Idea of a Mathematical Group
  - Module 5, Section 1, Lesson 1: Understanding the Idea of a Mathematical Group
- Lesson 2: Importance of Group Theory in Geometry
  - Module 5, Section 1, Lesson 2: Importance of Group Theory in Geometry

Section 2: Exploring Transformation Groups
- Lesson 1: Defining Transformation Groups in Geometry
  - Module 5, Section 2, Lesson 1: Defining Transformation Groups in Geometry
- Lesson 2: Examples of Common Transformation Groups
  - Module 5, Section 2, Lesson 2: Examples of Common Transformation Groups

Section 3: Introduction to Symmetry Groups
- Lesson 1: What Are Symmetry Groups?
  - Module 5, Section 3, Lesson 1: What Are Symmetry Groups?
- Lesson 2: Classifying Symmetry Groups in Regular Shapes
  - Module 5, Section 3, Lesson 2: Classifying Symmetry Groups in Regular Shapes

Section 4: Applications of Group Theory in Transformation Geometry
- Lesson 1: How Group Theory Explains Transformation Properties
  - Module 5, Section 4, Lesson 1: How Group Theory Explains Transformation Properties
- Lesson 2: Exploring Symmetry Operations Through Group Concepts
  - Module 5, Section 4, Lesson 2: Exploring Symmetry Operations Through Group Concepts

Section 5: Review and Integration of Concepts
- Lesson 1: Summarizing the Fundamentals of Transformations and Symmetry
  - Module 5, Section 5, Lesson 1: Summarizing the Fundamentals of Transformations and Symmetry
- Lesson 2: Integrating Group Theory with Transformation Geometry
  - Module 5, Section 5, Lesson 2: Integrating Group Theory with Transformation Geometry