This 4-week project engages 11th-grade students in exploring the intricate mathematical concepts within Pascal’s Triangle through the lens of fractal patterns and combinatorial properties. By synthesizing critical thinking and problem-solving skills, students validate mathematical theorems and cultivate a deeper understanding of how these principles connect to broader real-world applications. Through a blend of technology, collaboration, and artistic expression, students will develop interactive simulations and visual demonstrations that articulate their discoveries, culminating in a Math and Art Symposium that encourages community interaction and appreciation. This approach ensures that students navigate and appreciate the depth of mathematical reasoning, linking algebraic expressions to geometric and artistic representations.

Students will explore and identify fractal and combinatorial patterns within Pascal's Triangle, providing algebraic proofs for their consistency and exploring their applications across various domains. They will develop critical thinking and problem-solving skills by simulating and visualizing these patterns using digital tools, engaging in collaborative analysis, and refining their approaches through peer critique. Through reflective journaling, students will document their learning journey, understanding complex mathematical concepts and expressing these visually and algebraically. Additionally, students will learn to communicate their findings effectively during the Math and Art Symposium, connecting mathematical theory with artistic expression.

Students will create interactive digital models using coding platforms like Scratch or Python to simulate and visualize the fractal patterns within Pascal’s Triangle. These models will allow users to manipulate parameters, observing how changes affect the patterns in real time. Additionally, students will develop visual artifacts such as posters or animations that include both the discovered fractal patterns and the algebraic proofs supporting their existence. These products will be presented at the 'Math and Art Symposium', bridging mathematical exploration with creative expression and community engagement.

Kick off the project with a hands-on 'Fractal Launch Expedition,' where students dive into Pascal's Triangle by using interactive software simulations to explore and manipulate fractal patterns like the Sierpinski Gasket. This initial experience is designed to spark curiosity and will be supplemented with discussions on the visual and mathematical significance of these patterns, setting the stage for deeper exploration and investigation. As students generate questions and hypotheses during this session, they'll start forming connections between the visual beauty of fractals and their combinatorial properties, laying the groundwork for the project's essential question.

Students will transform the classroom into a dynamic 'Math and Art Symposium' to showcase their discoveries. They will present their interactive digital models and visual representations of fractal patterns in Pascal’s Triangle through engaging posters and animations. Family members and community stakeholders are invited to interact with the simulations, exploring the mathematical concepts firsthand. The event encourages dialogue between attendees and students, prompting discussion on the real-world applications of the patterns and proofs explored. This culminating experience not only celebrates their mathematical insights but also bridges connections between math, art, and community learning.