Pascal's Patterns: Fractals and Fibonacci Fun!

11th Grade Project 4 weeks

Pascal's Patterns: Fractals and Fibonacci Fun!

Ari D

CCSS.Math.Content.HSG-SRT.B.4

CCSS.Math.Content.HSG-CO.C.10

CCSS.Math.Content.HSG-SRT.B.4

CCSS.Math.Content.HSG-CO.C.10

CCSS.Math.Content.HSA-APR.C.5

+ 1 more

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Purpose

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.

Learning goals

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.

Standards

[Common Core] CCSS.Math.Content.HSG-SRT.B.4 - Prove theorems about triangles.
[Common Core] CCSS.Math.Content.HSG-CO.C.10 - Prove theorems about triangles.
[Common Core] CCSS.Math.Content.HSG-SRT.B.4 - Prove theorems about triangles.
[Common Core] CCSS.Math.Content.HSG-CO.C.10 - Prove theorems about triangles.
[Common Core] CCSS.Math.Content.HSA-APR.C.5 - (+) 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.

Competencies

Critical Thinking & Problem Solving - Students consider a variety of innovative approaches to address and understand complex questions that are authentic and important to their communities.

Products

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.

Launch

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.

Exhibition

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.

Learning Journey

Question

Students will engage in collaborative brainstorming sessions to formulate specific, investigable questions that probe the fractal and combinatorial patterns within Pascal's Triangle, guiding their subsequent exploration and research.

Days 1 - 3

Pascal Fractal Encounter

Launch 60m

Question Quest Planning

Deliverable

Design

Students will collaboratively design simulations using mathematical software to systematically plan their investigation, ensuring methodology aligns with proving combinatorial and fractal properties in Pascal’s Triangle, while preparing for peer feedback sessions for critique and refinement.

Days 4 - 7

Investigation Design Blueprint

Deliverable

Fractal Background Review

Deliverable

Collect

Students will systematically collect data using computational tools to identify and record fractal patterns and combinatorial properties in Pascal’s Triangle, preparing for upcoming analysis through comprehensive note-taking and digital documentation, while leveraging peer feedback for data clarity.

Days 8 - 11

Data Methods Essentials Quiz

Assessment 40m

Analyze

Students will analyze collected data and insights to decode fractal and combinatorial patterns within Pascal’s Triangle, using peer feedback to refine their proofs and innovative models, while documenting reflections on their methodology and discoveries.

Days 12 - 16

Analyze Data Patterns

Deliverable

Draft Preliminary Findings

Deliverable

Conclude

Students will synthesize their mathematical discoveries by articulating conclusive statements, acknowledge the boundaries of their findings, reflect on peer feedback, and prepare for their final presentations at the 'Math and Art Symposium.'

Days 17 - 20

Investigation Reflection Journal

Deliverable

Fractal Pascal Symposium

Assessment 2m

Plan

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Pascal's Patterns: Fractals and Fibonacci Fun!

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