Exploring Co-primes through Thread Art class 6 Maths Subject Enrichment Activity

 

Mathematics Subject Enrichment Activity

Class: 6
Chapter: Prime Time (Co-primes)
Textbook: NCERT Ganita Prakash
Page No.: 116

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Topic:

Exploring Co-primes through Thread Art

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Aim:

To understand the concept of co-prime numbers by creating artistic designs with pegs and thread gaps, and to explore how mathematics can be used in art.

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Materials Required:

• A geometry box (compass, ruler, protractor)

• Pencil, eraser, sharpener

• Colored pens/sketch pens

• Chart paper or notebook

• Scale and divider

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Procedure:

1. Draw a circle and mark equal divisions on its boundary to represent pegs (like clock positions).

2. Choose a thread-gap (example: every 4th peg).

3. Connect each peg to the peg that is the chosen gap away, continuing until the thread returns to the starting peg.

4. Repeat the activity with different numbers of pegs and different thread-gaps.

5. Observe when the thread covers all the pegs and when it does not.

6. Relate the results to the concept of co-primes:

   • If the number of pegs and the thread-gap are co-prime, the thread covers all pegs.

   • If they are not co-prime, the thread covers only a subset of pegs and forms smaller polygons.

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Observation Table:

Number of Pegs	Thread Gap	Co-prime? (Yes/No)	Pattern Formed
12	4	No	Triangle inside circle
13	3	Yes	Star covering all pegs
15	10	No	Pentagon inside circle
10	7	Yes	Star covering all pegs
14	6	No	Heptagon (7-sided figure)
8	3	Yes	Star covering all pegs

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Reflections:

• When the number of pegs and the thread-gap are co-prime, the thread visits every peg and creates a complete star-like design.

• When they are not co-prime, the thread skips some pegs and forms smaller closed shapes (like triangles, pentagons).

• This shows a direct relationship between co-primality and coverage in thread art.

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Conclusion:

This activity demonstrates how mathematics (co-primes) can be represented visually and artistically. It helps students discover the importance of co-primes in patterns, designs, and geometry.

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Extension / Higher-Order Thinking:

1. What happens if the number of pegs is a prime number and the thread-gap is less than it?
   → The pattern always covers all pegs, making a beautiful star.

2. How can this activity be extended to polygons instead of circles?

3. Can you think of real-life applications where co-primes are used in designs (e.g., rangoli, gear wheels, music rhythm cycles)?

Extension / Higher-Order Thinking (with Answers):

Q1. What happens if the number of pegs is a prime number and the thread-gap is less than it?
👉 Since a prime number has no factors except 1 and itself, any smaller number chosen as the thread-gap will always be co-prime with it.
🔹 This means the thread will cover all the pegs and form a complete star pattern.
Example:

• 13 pegs, thread-gap = 3 → covers all pegs, forms a 13-pointed star.

• 17 pegs, thread-gap = 5 → covers all pegs, forms a 17-pointed star.

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Q2. How can this activity be extended to polygons instead of circles?
👉 Instead of marking pegs around a circle, we can take regular polygons like a hexagon, octagon, or decagon.

• Place thread on the vertices of the polygon.

• Use different gaps (skip 2nd vertex, 3rd vertex, etc.).

• Patterns like stars (e.g., Star of David from hexagon, pentagram from pentagon) will appear.
  🔹 This connects co-primes to polygonal star figures (star polygons) studied in geometry.

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Q3. Can you think of real-life applications where co-primes are used in designs (e.g., rangoli, gear wheels, music rhythm cycles)?
👉 Applications:

1. Rangoli / Kolam Designs (Art): Traditional patterns often skip fixed points to create symmetric star shapes — same principle as co-prime art.

2. Gear Wheels (Engineering): If two gears have co-prime teeth counts, every tooth of one gear will eventually touch every tooth of the other — ensuring even wear.

3. Music Rhythm Cycles (Math + Music): In Indian classical music, rhythmic beats (tala) use cycles. If one cycle has 7 beats and another has 5 beats (co-prime), the combined pattern repeats only after 35 beats.

4. Cryptography (Math + Computer Science): Co-primes are used in secure coding (RSA algorithm) to generate keys.

5. Clock Design: Hands of a clock meet after specific intervals based on co-prime properties of 12 hours and 60 minutes.