Co-prime pairs, and why sharing no factor is what matters

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What to assume they know

Common factors: which jump sizes can land on a number: factors, common factors, and the hopping game
Common multiples: why the first "idli-vada" is 15: common multiples and the first collision of a pair
Prime and composite: the rectangle test for a number's factors: prime and composite, so the naming trap can be named
Marking equally spaced points around a circle and joining them with straight lines

What they should be able to do

State the definition: a pair sharing no factor above 1
Decide whether a given pair is co-prime by intersecting factor lists
Give a co-prime pair in which neither number is prime, and a non-co-prime pair in which one number is prime
Connect co-primality to the hiding game: a pair is safe exactly when it is co-prime, once a hop of 1 is banned
Predict whether a pair's first common multiple will be the product or less
Predict, for a circle of pegs and a fixed thread gap, whether the thread will reach every peg
Explain why a shared factor is what makes the thread skip pegs
Recognise that co-primality relates two numbers and says nothing about either one alone

Where it usually goes wrong

"Co-prime means both numbers are prime." The word invites it and the book kills it in its first example: 4 and 9 are co-prime and both are composite. Section 5 exists for this misconception alone.
"If one of the two is prime, the pair is co-prime." 3 and 9 share 3. A prime is co-prime to everything it does not divide, and to nothing it does.
"Co-prime is something a number is." Ask "is 9 co-prime?" and the question has no answer. It is co-prime to 4, not co-prime to 6. The property lives on the pair.
"Sharing a factor means sharing a prime factor, so I should check primes only." True, and it is exactly what §5.4 proves next — but it is not obvious yet, and asserting it here skips the argument the following topic is built on.
"The thread skips pegs when the gap is large." Size is irrelevant. Thirteen pegs with a gap of 3, which the book does state, reaches every peg; work out twenty-four pegs with a gap of 6 and the thread reaches four. That second pair is arithmetic done for the sake of the contrast, not a reading of the printed twenty-four-peg picture, whose gap the book never gives — see Notes. What decides it is whether the two numbers share anything.
"The first collision is always the two numbers multiplied." Carried over from Common multiples: why the first "idli-vada" is 15, where it was deliberately left open. This is the topic that closes it.

Questions to check understanding

State whether a given pair is co-prime and justify it with the shared factor, or with the empty overlap
Give a co-prime pair of composites
Given one number, list the numbers below 20 co-prime to it
Say whether a stated hiding place is safe under the game's rules
Predict whether a thread on a stated number of pegs with a stated gap will reach every peg
Given a pair, say whether the product is where their first collision lands
The appendix bound with this chapter answers the §5.3 questions on footer page 6

Examples worth working on the board

The hiding game with the new rule (§5.3, p.115). Two treasures, one hop length kept throughout, both must be landed on — and now a hop of 1 is forbidden, which is what makes the game winnable for the hider at all.
The two worked placements (p.115). 12 and 26 fails as a hiding place, because a hop of 2 reaches both. 4 and 9 succeeds: no allowed hop reaches both.
Four pairs to judge (p.115). 15 and 39; 4 and 15; 18 and 29; 20 and 55. Two of the four are safe. Give the pairs, not the verdicts; the appendix on footer page 6 records them.
The definition and the worked example (p.116). Checked against the printed page. The book states that the safe pairs are the ones with nothing dividing both but 1, names that co-prime, and immediately contrasts 15 with 39, which share 3, against 4 and 9, which share nothing.
Five more pairs (p.116). 18 and 35; 15 and 37; 30 and 415; 17 and 69; 81 and 18. Three are co-prime and two are not. The pair with 37 and the pair with 17 are the ones that teach the naming trap from the other side: a prime paired with a composite that it does not divide is a co-prime pair, and 81 with 18 is a pair of composites that is not.
The counting-game observation (p.116). A character reports that for some pairs the first collision landed exactly on the two numbers multiplied, and for others it came earlier, and the reader is asked to produce examples of each and to say how it relates to the pair being co-prime. This is the bridge back to Common multiples: why the first "idli-vada" is 15, and it should be resolved: when the pair shares a factor, the two rhythms can meet before the product; when they share nothing, they cannot.
The thread pictures (p.116). Four circles, checked against the printed page. The book states the setup for the first two only: twelve pegs with the thread tied every fourth peg, and thirteen pegs with the thread tied every third. The first draws a triangle touching only pegs 12, 4 and 8; the second draws a many-pointed star touching all thirteen pegs. The third circle has sixteen pegs and its thread touches only the eight even-numbered pegs; the fourth has twenty-four pegs and its thread touches only pegs 6, 12, 18 and 24, giving a diamond. The book does not print the gaps for the third and fourth — it asks the reader what they are. Redraw all four; the printed art is the book's own.
Four patterns to construct (p.117). Fifteen pegs with a gap of 10; ten pegs with a gap of 7; fourteen pegs with a gap of 6; eight pegs with a gap of 3. These are the book's own inputs. Two of the four reach every peg.

Figures to have open

Four numbered peg circles, redrawn: 12 pegs, 13 pegs, 16 pegs, 24 pegs, each with its chord set. The redraw must reproduce which pegs the thread reaches, since that is the mathematical content; the styling is free.
Two side-by-side factor lists with the overlap marked, reused from Common factors: which jump sizes can land on a number so the student recognises the picture.
A counting-game track carrying two rhythms, reused from Common multiples: why the first "idli-vada" is 15, for section 7.
No photograph or textbook data table is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 5 "Prime Time", §5.3 "Co-prime Numbers for Safekeeping Treasures", pp.115–117 — the hiding game with the new rule and the four pairs to judge (p.115), the definition, the five further pairs, the counting-game observation and the four thread pictures (p.116), and the four constructions (p.117)
Chapter Summary, p.128, for the book's printed wording of co-prime
Backward pointers inside the same chapter: §5.1, pp.107–110, for the two games this section reuses
Forward pointer inside the same chapter: §5.4, pp.117 and 120, where the factor-list test is replaced by a prime-factor test
Solutions appendix bound with this chapter file, footer page 6, for §5.3