Prime and composite: the rectangle test for a number's factors

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What to assume they know

Common factors: which jump sizes can land on a number: listing every factor of a number, in pairs
Rows-times-columns as a way of counting a rectangular array
Multiplication tables to 12, and division as the reverse of multiplication
Common multiples: why the first "idli-vada" is 15: multiples, so that "more than two factors" can be checked against a real list

What they should be able to do

Build every rectangular arrangement of a given number of objects, and read a factor pair off each one
State the classification the book uses: exactly two factors, or more than two
Name the first several primes and the first several composites without a table
Explain why 1 falls into neither class, using the factor count rather than a rule
Explain why 2 is the only even prime
Decide whether a given two-digit number is prime, and show the factor that settles it when it is not
Judge true-or-false claims about primes and composites and produce a counterexample when one is false
Recognise a number that is a product of three different primes

Where it usually goes wrong

"A prime is a number that cannot be divided." Every number can be divided by 1 and by itself. A prime is a number with no other divisors, and the book makes this a count — exactly two — rather than a prohibition.
"1 is prime, because you can only make one rectangle from it." You can, and that rectangle is one cell. But primality here is counted in factors, and 1 has one factor, not two. A prime's single layout is a strip whose two sides are different numbers, so it yields two factors; 1's single layout is square, so its two sides are the same number and it yields one. Counting layouts cannot tell those apart — the factor count decides.
"Odd means prime." 9, 15, 21 and 25 are odd and composite; the book's own composite list starts 4, 6, 8, 9. Meanwhile 2 is even and prime.
"Big numbers are more likely to be prime, because they have more room." The opposite. The larger a number, the more candidate factors it has had to survive — which is why the sieve in the companion topic thins out as it climbs.
"To test whether a number is prime you must try every smaller number." You need only test up to where the factor pairs cross over, because factors come in pairs and one member of each pair is at most the square root. This is an added argument, not the book's — say so — but it turns question 4 from a chore into a method.
"Primes have no factors." Printed as a true-or-false claim on p.115 for precisely this reason. They have two.

Questions to check understanding

Draw every rectangular layout of a given number of counters and list its factors
State whether a given number is prime or composite and justify it with a factor
Say why 1 belongs to neither class
Give a counterexample to a stated false claim about primes
Find twin primes within a stated range
Identify which of several numbers is a product of exactly three different primes
Explain why 2 is the only even prime
The appendix bound with this chapter answers the §5.2 questions on footer pages 4 and 5

Examples worth working on the board

The fruit-packing scene (§5.2, p.112). Two children grow figs and want to box them, one packing twelve to a box and the other seven. The question the section opens with is how many rectangular layouts each of them can make.
The printed layouts (p.112). Checked against the printed page. Three orange grids appear for twelve: a single column twelve cells tall, a block six rows tall and two columns wide, and a block four rows tall and three columns wide. One purple strip appears for seven: a single row of seven cells. The running text picks out the middle orange block and reads it as two columns of six. Redraw the grids; do not lift the artwork.
Reading a rectangle as a product. Rows across multiplied by columns down returns the number of objects, so each side length is a factor. Twelve's side lengths across the three printed layouts are 1 and 12, 2 and 6, 3 and 4 — the full factor list of 12, arrived at without dividing once.
The printed lists (p.112). First primes as printed: 2, 3, 5, 7, 11, 13, 17, 19. First composites as printed: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20. Give these as data.
The status of 1 (p.113). The book states that 1 has a single factor and puts it in neither class. Checked against the printed page. Do not soften this into "1 is a special prime" — the chapter is explicit, and §5.4 needs it, because a factorisation into primes would never terminate if 1 were allowed in.
Counting in a decade (p.113). How many primes, and how many composites, sit between 21 and 30.
A hand test (p.114, question 4). Which of 23, 51, 37 and 26 are prime. Two of these fail, and each fails for a different reason worth showing — one to an obvious small factor, one to a factor a student would not guess at first glance. Give the four numbers, not the verdicts.
Pairs adding to a multiple of five (p.114, question 5). Three pairs of primes, each member below 20, whose two members total a multiple of 5. Give the ceiling and the target; several pairs qualify, so this is a search with more than one right answer rather than a one-answer drill.
Primes built from the same digits (p.114, question 6). 13 and 31 are the printed example — two primes below 100 written with the same two digits in opposite orders. The reader hunts for the others. Worth pairing with the last-digit argument, since reversing a numeral moves a different digit into the deciding position.
Doubling a prime and adding one (p.115, question 12). The printed instance is 3, where twice the prime plus one is again prime; at least five more primes behaving the same way are wanted. Give the rule and the number of examples asked for. The search is short enough to run live, and it is the one item in the set that treats primes as objects with behaviour rather than a list to recite.
The true-or-false set (p.115, question 9). Five claims: that no prime ends in the digit 4; that a product of primes can itself be prime; that primes have no factors at all; that every even number is composite; and that after 2 and 3, a prime is never followed immediately by another prime. Each is a misconception in disguise and each deserves its own beat.
Product of three different primes (p.115, question 10). Candidates as printed: 45, 60, 91, 105 and 330. Exactly one of them is a product of three distinct primes; the trap is the candidate that has exactly three distinct prime factors but repeats one of them, so its product is not those three, and the other whose distinct primes run to four.
Three-digit numbers from 2, 4 and 5 (p.115, question 11). All six arrangements of the three digits, each to be tested for primality. Give the digits; the last-digit argument does the rest and previews §5.5.
Twin primes (p.114, question 8). Defined in the question itself, with 3 and 5, and 17 and 19, as the printed examples; the reader hunts for the rest below 100.

Figures to have open

The rectangular layouts of twelve and of seven. Redraw as unit-square grids; the printed art is the book's own. The redraw must keep the fact that seven admits only the single strip, because that contrast is the whole section.
A factor-pair ladder for a two-digit number showing pairs closing in from both ends and meeting in the middle. Standard schematic; the book does not print one, and it carries the section-9 method.
A branching diagram for the three-way classification. Standard schematic.
No photograph or textbook data table is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 5 "Prime Time", §5.2 "Prime Numbers", pp.112–115 — the packing problem and the two definitions (p.112), the status of 1 and the decade count (p.113), the first Figure it Out and the historical panel (p.114), and the remaining questions (p.115)
Chapter Summary, p.128, for the book's printed definitions of prime and composite
Forward pointer inside the same chapter: §5.4, p.118, where the primes become building blocks in practice
Solutions appendix bound with this chapter file, footer pages 4–5, for §5.2