Breaking a number down to primes, and why the result is unique

What to assume they know

Factor and multiple as they were used in Class 6 (Prime Time): a factor divides exactly, leaving nothing over
What a prime number is, and the Sieve of Eratosthenes as a way of listing primes — both are recalled here rather than built
Multiplying three or four small numbers together in any order and getting the same product
Factorise and regroup instead of multiplying head-on — regrouping a product by factorising its parts, which is the same manoeuvre applied to arithmetic

What they should be able to do

Say why writing out a number's factors one by one is unreliable, and name the two things that go wrong with it
State what a prime is, in the form the book states it
Produce the prime factorisation of a two- or three-digit number
Take one number apart along two different first splits and check that the same primes come out
State what the chapter claims about that sameness, and state that the chapter offers no proof of it
Give the factorisation of a prime
Read a circled factor-tree diagram: say how each circled entry is built from its two neighbours
Carry out the division method and collect the divisors and the final quotient into a factorisation

Where it usually goes wrong

"You have to start with the smallest prime, or you will get the wrong answer." This is the belief the 90 example exists to kill. The book deliberately offers 3 × 30 and 2 × 45 as legitimate first moves and shows one of them arriving. Any first split works; an explanation that drills "always divide by 2 first" is teaching a habit as if it were a requirement.
"Different splittings give different factorisations." The order the primes come out in changes. The primes themselves, and how many of each, do not. Keep those two things visibly separate.
"The chapter proves the factorisation is the same every time." It does not. It works one example, calls the result remarkable, and moves on. Say so. A student who thinks a single verified example is a proof has learnt the wrong lesson from a chapter that later spends a whole page on exactly this distinction (Part II, §3.1, p.52 and §3.3, p.58).
"A prime has no prime factorisation." It has a one-term one: itself. The book states this in a single line because it is the case that makes the general statement work without exceptions.
"Bigger numbers take longer to factorise." 121 is larger than 96 and splits into two primes against 96's six.
"The circled diagram and the ruled layout are two different methods." They are the same method with the circles rubbed out. Show the erasure.

Questions to check understanding

Write the prime factorisation for a stated two-, three- or four-digit number
Given a partly finished factor tree or division ladder, complete it
Given two different first splits of the same number, carry both down and compare the primes obtained
Decide whether a given product of primes equals a given number
State what a prime's own factorisation is
Given a number's factorisation, say how many prime factors it has counting repeats — the step that feeds the factor-listing work of Reading every factor of a number off its prime factorisation
Board-style items on this material are almost always one-mark "write the prime factorisation of …" prompts; the reasoning about why the answer does not depend on the route is examined, when it is examined at all, as a "give an example / explain" item

Examples worth working on the board

The pairs that broke the listing method (Part II, §3.1, p.49). 30 and 50; 28 and 42. The chapter names these two pairs as the ones that got cumbersome in the exercise just before, which is why it now changes technique. Use them as the motive, not as worked examples.
90, split three ways (Part II, §3.1, p.49). The book prints 90 = 3 × 3 × 2 × 5 as the finished factorisation, and names 3 × 30 and 2 × 45 as two other places you could have started. It then carries 3 × 30 through: 3 × 30 → 3 × 2 × 15 → 3 × 2 × 3 × 5.
The two circled diagrams (Part II, §3.1, p.49). Checked against the printed page. Two small trees, side by side. Left: 105 in a circle with 3 written to its left, 35 in a circle below it with 5 to its left, and a bare 7 at the bottom. Right: 30 in a circle with 2 to its left, 15 in a circle below with 3 to its left, and a bare 5 at the bottom. The rule the reader has to spot is stated on the next page: every circled entry equals whatever sits to its left multiplied by whatever sits beneath it. The book's own check of that rule: 30 = 2 × 15 and 15 = 3 × 5.
The collecting sweep (Part II, §3.1, p.50). Checked against the printed page. The 105 tree redrawn with a long blue curve running down the left-hand column and round the bottom, gathering 3, 5 and 7 and delivering 105 = 3 × 5 × 7. This one curve is the picture the explanation needs; it is what turns a diagram into an answer.
The bare division-method layout (Part II, §3.1, p.50). Checked against the printed page. The same two examples with the circles removed: divisor at the left, the number to its right, the quotient on the line beneath, the last quotient standing alone. 3 | 105, 5 | 35, 7 — and 2 | 30, 3 | 15, 5.
1200 (Part II, §3.1, p.50). The reader is asked to factorise 1200 by the division method, and the book sets against it the start of the other route, 1200 = 40 × 30 = 5 × 8 × 5 × 6.
Anshu's counterexample, borrowed forward (Part II, §3.1, p.51). 96 = 2 × 2 × 2 × 2 × 2 × 3 and 121 = 11 × 11. Useful here as evidence that the length of a factorisation is not fixed by the size of the number. Its full treatment belongs to Reading every factor of a number off its prime factorisation and Conjecture and generalisation: what mathematicians mean by those words.
The page-number rings (Part II, "It's Puzzle Time! — Mystery Colours!", p.66). Checked against the printed page, and this page barely extracts as text, so the printed page is the only source. Every page of the book carries the page number inside a coloured ring, and this closing page prints all one hundred rings for 1 to 100 in a ten-by-ten grid. The ring is cut into as many arcs as the number has prime factors counting repeats, and each arc is coloured by which prime it is: green for 2, magenta for 3, orange for 5, blue for 7, and red for any prime above 7. That larger prime is printed tiny in white on the red arc only where the red arc shares the ring with others; a prime's own ring is a single unlabelled red annulus, since the number sitting inside the ring already names it. Rings read off the printed page: 34 is green + red("17"); 35 is orange + blue; 36 is two green and two magenta; 37 is one red ring, unlabelled; 40 is three green and one orange; 45 is two magenta and one orange; 48 is four green and one magenta; 49 is two blue; 50 is one green and two orange; 1 is a single grey ring. The book prints no key — it asks the reader to decode the scheme. The decoding above is worked out here from the checked grid, so present it in the explanation as something to be discovered, not as something the book states.
The closing aside on the biggest known prime (Part II, p.65, below the SUMMARY box). Checked against the printed page. A pale green speech balloon, with a startled child drawn beside it asking how long it would take to write the number out: the biggest prime anyone had found runs to 4,10,24,320 digits and turned up on 12 October 2024. It is the natural last beat of section 10 — every ring on the next page is built out of primes, and this is how large one of them can get. The figure is grouped in the Indian system — 4 crore, 10 lakh, 24 thousand, 320 — which is 41,024,320 digits in the international grouping.

Figures to have open

A factor tree that can be drawn three times from three different first splits of 90 and end with the same leaves. Standard schematic; do not lift the book's circled art.
The ruled division layout, redrawable for 105, 30 and 1200. Standard schematic.
The ten-by-ten grid of coloured rings for 1 to 100. This one is the book's own page design and carries the section-10 payoff, so redraw it from the scheme described above rather than reproducing the printed page. The colour assignment must be checked against the printed page of p.66 before it is used.
No numbered figure exists to cite: this chapter prints no "Fig. 3.x" label anywhere on pp.47–66 (all twenty pages checked).

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part II, printed Chapter 3 "Finding Common Ground", §3.1 "The Greatest of All", p.49 — the paragraph on why listing factors is unreliable, and the two bold subheadings "Primes" and "Prime Factorisation"
Same section, "Procedure for Prime Factorisation", Part II, pp.49–50 — the two circled diagrams, the collecting sweep, the bare layout named as the division method, and the 1200 comparison
Same section, Part II, p.51 — the 96 and 121 counterexample, used here only as supporting evidence
Same chapter, SUMMARY, Part II, p.65, bullets 1 and 2, and the speech-balloon aside below the box giving the digit count of the biggest known prime
Same chapter, "It's Puzzle Time! — Mystery Colours!", Part II, p.66
Backward pointer: Class 6 Ganita Prakash, "Prime Time", for primes and the Sieve of Eratosthenes, which this section recalls rather than re-teaches
Forward pointer: Reading every factor of a number off its prime factorisation, which turns a factorisation into a complete list of factors