MATH CIRCLE ACTIVITY 9 PASCAL’S TRIANGLE HUNT

 MATH CIRCLE

ACTIVITY 9
PASCAL’S TRIANGLE HUNT

DATE: 27-11--2025 DAY: Thursday

Sub-topic:

Patterns in Numbers – Pascal’s Triangle and Its Mathematical Connections

Objective:

To enable students to explore patterns in Pascal’s Triangle, understand its relationship to binomial expansion, and discover connections with other number sequences such as Fibonacci numbers.

Purpose:

To help students recognize how mathematical structures like Pascal’s Triangle reveal deep interconnections among numbers, patterns, and algebraic concepts, fostering curiosity and analytical thinking.

Learning Outcomes:

By the end of this activity, students will be able to:

• Construct Pascal’s Triangle up to several rows using combinatorial logic.
  
  

• Identify patterns such as symmetry, odd-even coloring, and triangular number arrangements.
  
  

• Connect Pascal’s Triangle to Fibonacci numbers and binomial coefficients.
  
  

• Strengthen logical reasoning, pattern observation, and algebraic understanding.
  
  

Skills Developed:

🔹 Pattern Recognition
🔹 Algebraic Reasoning
🔹 Logical Analysis
🔹 Creative Mathematical Thinking

Competencies Involved in this Activity:

• Numeracy and Quantitative Aptitude: Understanding numerical growth and relationships.
  
  

• Analytical Thinking: Identifying and justifying number patterns and symmetries.
  
  

• Representation and Visualization: Constructing Pascal’s Triangle neatly and interpreting it visually.
  
  

• Reasoning and Proof: Explaining why the triangle relates to binomial expansion and Fibonacci sequence.
  
  

• Collaboration and Communication: Working in pairs or groups to discuss and validate observed patterns.

Procedure:

1. The teacher introduced Pascal’s Triangle and explained how each number is obtained by adding the two numbers directly above it.
   
   

2. Students constructed the triangle row by row on chart paper or in their notebooks, starting from the top (1).
   
   

3. The teacher guided students to observe patterns:
   
   

• The triangle’s symmetry on both sides.
  
  

• Diagonal patterns forming counting numbers, triangular numbers, and Fibonacci sequence.
  
  

• Odd-even number coloring to form fractal-like shapes (Sierpiński pattern).
  
  

4. Students identified the link between Pascal’s Triangle and Binomial Expansion (using (a+b)n.
   
   

5. As an extension, students created color-coded triangles highlighting specific patterns such as even numbers, Fibonacci diagonals, or powers of 2.
   
   

Teacher’s Observations:

• Students were deeply engaged in identifying patterns and making mathematical connections.
  
  

• The visual nature of the activity made abstract algebraic concepts easier to grasp.
  
  

• Many students independently recognized the connection to Fibonacci numbers and powers of 2.
  
  

• The collaborative setup fostered rich mathematical discussion and creativity.
  
  

Student’s Feedback / Reflections:

“Building Pascal’s Triangle was so much fun! I loved finding patterns like Fibonacci numbers and even the odd-even designs. It was exciting to see how everything in math connects. I now understand binomial expansions better too. This activity made me see math as a pattern-filled puzzle.”
— By ___________

 Students constructing Pascal’s Triangle during the Mathematics Circle Activity — exploring number patterns, binomial connections, and Fibonacci relationships through creative teamwork. ------------------------------------------- 🧮 Pascal's Triangle Math Circle Activity 9 Row 1: 1 Row 2: 1 1 Row 3: 1 2 1 Row 4: 1 3 3 1 Row 5: 1 4 6 4 1 Row 6: 1 5 10 10 5 1 Row 7: 1 6 15 20 15 6 1 Row 8: 1 7 21 35 35 21 7 1 Row 9: 1 8 28 56 70 56 28 8 1 Row10: 1 9 36 84 126 126 84 36 9 1 🌟 Key Observations - Each number = sum of two numbers above - Symmetric along vertical axis - Outer edges always 1 - Diagonals: counting numbers, triangular numbers, Fibonacci pattern 📘 Connection - Each row = coefficients of (a+b)^n - Example: (a+b)^4 = 1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4 💡 Extensions - Color odd numbers → Sierpiński triangle - Sum of each row = 2^n -------------------------------------------

🧮 Pascal’s Triangle

Math Circle Activity 9 — Pascal’s Triangle Hunt

1

1   1

1   2   1

1   3   3   1

1   4   6   4   1

1   5   10   10   5   1

1   6   15   20   15   6   1

1   7   21   35   35   21   7   1

1   8   28   56   70   56   28   8   1

1   9   36   84   126   126   84   36   9   1

🌟 Key Observations

• Each number = sum of the two numbers above it.
• The triangle is symmetric along the vertical axis.
• Outer edges are always 1.
• Diagonals show counting numbers, triangular numbers, and Fibonacci patterns.

📘 Mathematical Connection

Each row corresponds to coefficients of the binomial expansion (a+b)ⁿ.

Example: (a+b)⁴ = 1a⁴ + 4a³b + 6a²b² + 4ab³ + 1b⁴

💡 Extension Ideas

• Color all odd numbers to reveal a Sierpiński triangle pattern.
• Find powers of 2 in each row: Row 0 → 1, Row 1 → 2, Row 2 → 4, etc. (Sum of row = 2ⁿ).