MATHEMATICS SUBJECT ENRICHMENT ACTIVITY Class: VIII Chapter: We Distribute, Yet Things Multiply

 

 MATHEMATICS SUBJECT ENRICHMENT ACTIVITY

Class: VIII
Chapter: We Distribute, Yet Things Multiply
Activity Title: Coin Conjoin – Flipping Triangular Coin Patterns

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🔷 Topic

Triangular Numbers, Patterns, and Optimization of Moves

Coin Conjoin Arrange 10 coins in a triangle as shown in the figure below on the left. The task is to turn the triangle upside down by moving one coin at a time. How many moves are needed? What is the minimum number of moves? A triangle of 3 coins can be inverted (turned upside down) with a single move, and a triangle of 6 coins can be inverted by moving 2 coins. The 10-coin triangle can be flipped with just 3 moves; did you figure out how? Find out the minimum possible moves needed to flip the next bigger triangle having 15 coins. Try the same for bigger triangular numbers. Is there a simple way to calculate the minimum number of coin moves needed for any such triangular arrangement?

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🎯 Aim of the Activity

• To explore triangular number arrangements using coins

• To develop logical reasoning and spatial visualization

• To find the minimum number of moves required to transform a pattern

• To identify patterns and general rules for larger numbers

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🧰 Materials Required

• 10, 15 or more identical coins

• Flat surface or chart paper

• Pencil and notebook

• Ruler (optional)

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🧠 Prior Knowledge

• Counting numbers

• Understanding triangular arrangements

• Basic reasoning and pattern recognition

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🧩 Activity Description (Given Situation)

Coins are arranged in the shape of a triangle.
The challenge is to turn the triangle upside down by moving one coin at a time, using the minimum number of moves.

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🪜 Procedure / Steps

Step 1: Forming the Triangle

1. Arrange 10 coins in a triangular shape (4 rows: 1, 2, 3, 4).

2. Observe the orientation (point facing upward).

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Step 2: Understanding Smaller Cases

3. Try flipping:

   • Triangle of 3 coins → 1 move

   • Triangle of 6 coins → 2 moves

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Step 3: Flipping the 10-Coin Triangle

4. Identify coins that remain in the same position in the inverted triangle.

5. Move only the coins that do not overlap with the final shape.

6. Count each coin shift as one move.

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Step 4: Extending the Pattern

7. Repeat the activity for:

   • 15 coins

   • 21 coins (optional)

8. Record observations.

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📊 Observation Table

Number of Coins	Rows	Moves Needed	Coins That Stay	Coins Moved
3	2	1	2	1
6	3	2	4	2
10	4	3	7	3
15	5	4	11	4

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✅ Answers / Solutions

✔ Solution for 10 Coins

• Minimum number of moves required = 3

• Reason: 7 coins overlap in both triangles

• Only 3 coins need repositioning

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✔ Solution for 15 Coins

• Minimum number of moves required = 4

• Coins form 5 rows

• Overlapping coins = 11

• Coins to move = 15 − 11 = 4

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🔍 Key Pattern Observed

For a triangle with n rows:

🔑 Minimum number of moves = n − 1

Rows (n)	Coins	Minimum Moves
2	3	1
3	6	2
4	10	3
5	15	4

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💭 Reflections (Student Response)

• I learned that not all coins need to be moved.

• Patterns help reduce effort and save time.

• Mathematics helps us find the best solution, not just any solution.

• Logical thinking is more important than trial and error.

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🔥 Higher Order Thinking Skills (HOTS)

1. Why do some coins remain fixed during inversion?

2. Can you predict the answer without physically moving coins?

3. How is this activity related to triangular numbers?

4. What happens as the number of coins becomes very large?

5. Can you design a similar puzzle with square numbers?

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📌 Conclusion

The Coin Conjoin activity shows how simple distribution leads to multiplication of ideas.
By observing patterns, students discover a general mathematical rule that reduces effort and increases efficiency.

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🧮 Mathematical Link

• Triangular Numbers:

  $$T_n = \frac{n(n+1)}{2}$$

• Optimization & reasoning

• Visual geometry

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✨ Extension Activities

• Predict moves for 21 coins

• Draw before–after diagrams

• Convert the rule into a formula

• Create your own coin puzzle