Conjecture and generalisation: what mathematicians mean by those words

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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What to assume they know

HCF: take the fewest occurrences of each prime — the HCF as the shared primes at their smaller counts
LCM: take the most occurrences of each prime — the LCM as every prime at its larger count
What HCF and LCM do for consecutive, even, and co-prime numbers — the patterns this chapter asks to be stated in general
Using a letter to stand for any number
Multiplying two numbers written as products of primes

What they should be able to do

Say what makes a claim a conjecture rather than an established fact
Refute a stated conjecture by producing one instance where it fails
Say what a general statement claims, and what would be needed to earn it
Explain why the LCM of two numbers can never exceed their product
Show, on a worked pair, that the product divided by the LCM is the HCF
State how a pair's HCF, its LCM and its product are related, and state it with the same caution the chapter uses
Follow the chapter's hint far enough to explain why the relation holds for two numbers
Test whether the relation survives three numbers, and report honestly what the test shows

Where it usually goes wrong

"A conjecture is a wrong guess." It is an unsettled claim. Some conjectures are true and unproved; Anshu's happens to be false. Nothing about the word says which. The chapter's second box is explicit that the reader should expect to make conjectures of their own, which would be strange advice if the word meant error.
"Lots of examples make a claim true." They make it worth trying to prove. The chapter works one pair through the HCF-times-LCM relation, sets three more for the reader, names the pattern it sees across all four, and still declines to state it as settled.
"One counterexample only weakens a claim." It ends it, as a general claim. The claim may survive in a narrower form — Anshu's holds only for powers of a single prime, and fails even for numbers built from the same primes — but the sentence as stated is finished.
"HCF × LCM = product holds for any number of numbers." It does not. This is the single most likely error an explanation will make here, because the two-number version is so tidy and because the chapter's own question invites the leap. 2, 3 and 4 settle it. Present the three-number case as an experiment with a surprising outcome, and never as a stated rule.
"The LCM could be bigger than the product if the numbers are big enough." It could not, ever: the product is already a common multiple, and the LCM is the least of them. Size is irrelevant, which is why the chapter does not let the reader stop at examples but goes on to ask for a reason or a proof.
"The chapter proves the relation." It gives a hint towards a proof and asks the reader for one. If the explanation supplies the argument — and it should — it must say that this is the explanation doing what the book asked, not the book speaking.

Questions to check understanding

Given a stated claim about numbers, decide whether it holds and produce either a reason or a failing case
Given two numbers, verify that the HCF times the LCM matches the product
Given two of {HCF, LCM, one number, the product}, find the missing one
Find two numbers meeting stated HCF and LCM conditions (Part II, p.63, question 4)
Explain why the LCM of two different primes has to be their product (Part II, p.64, question 10)
Explain, in words, why the relation holds — the "give a reason" item that this chapter's style makes likely
Competency-based papers phrase this as "state whether the following is always, sometimes or never true", which is precisely the distinction the topic teaches

Examples worth working on the board

Anshu's claim and its refutation (Part II, §3.1, p.51). The claim, the sentence disproving it and both factorisations all sit on p.51; p.52 carries only the side box naming conjecture and counterexample. The claim, paraphrased: a bigger number has a longer prime factorisation. The two numbers that end it: 96 = 2 × 2 × 2 × 2 × 2 × 3 and 121 = 11 × 11. This is the topic's opening exhibit, and its whole force is that the claim is reasonable — it holds for plenty of pairs — and still dead.
The two side boxes (Part II, §3.1, p.52). Checked against the printed page. Both are pale blue panels with a small owl-in-a-mortarboard character at the left. The upper one names conjecture and counterexample and applies both to Anshu. The lower one, further down the same page, tells the reader that their own conjectures will turn up while they work, and to reason about them and share them. Read together they set the topic's stance: conjecturing is not a mistake to be avoided, it is the normal first move.
The naming of generalisation (Part II, §3.3, p.58). The example carrying it: every pair in which one number divides the other has that number as their HCF. The chapter's move is to name the kind of claim this is before doing anything more with it.
The algebra (Part II, §3.3, pp.58–59). Inputs as printed: n and 5n, whose HCF is n. The teaching point is not the fact but the notation — one line with a letter replaces an unbounded list of checked pairs.
The size question (Part II, §3.3, p.62). Which of two quantities is bigger — a pair's LCM, or that same pair multiplied out? The book invites examples first and then presses on to a reason or a proof, and supplies a hint: the product is itself a common multiple. That examples-then-generalise order is the rhythm this whole topic is about, so keep it. Note the equality case, which the book does not raise: the two coincide exactly when the pair is co-prime.
105 and 95 (Part II, §3.3, p.62). Inputs as the book prints them: 105 = 3 × 5 × 7 and 95 = 5 × 19; the LCM as 3 × 5 × 7 × 19; the product in factored form as 3 × 5 × 5 × 7 × 19; and the observation, printed, that the product equals the LCM times 5. What the 5 is is deliberately not said yet.
The three test pairs (Part II, §3.3, p.62). 45 and 105; 275 and 352; 222 and 370. For each the reader works out the LCM, divides the product by it, and looks at what comes out. Do not pre-print the answers — the chapter does not.
The relation (Part II, §3.3, p.63). As printed: the HCF times the LCM equals the product of the two numbers. The wording around it matters and should be preserved in spirit: the chapter presents this as what the observations point towards, not as a rule handed down, and immediately asks for an explanation or a proof.
The proof hint (Part II, §3.3, p.63). The book's own outline, in paraphrase: sort the primes of the two numbers into those common to both and those in only one; then see where each kind ends up in the HCF, in the LCM, and in the product, and compare. That is enough for a Class 7 argument.
The three-number question (Part II, §3.3, p.63). The chapter asks whether the property still holds for three numbers and stops there. The answer, computed here and not printed anywhere in the book: it does not hold in general. Take 2, 3 and 4 — the HCF is 1, the LCM is 12, the product is 24, and 1 × 12 is not 24. It can still hold for particular triples: any three numbers no two of which share a factor give HCF 1 and LCM equal to the product. So the honest report is that the claim survives for some triples and fails for others, which is exactly what makes it a good exploration and a dangerous thing for an explanation to assert.

Figures to have open

Two rows of primes with a product row beneath, so that the LCM can be highlighted inside the product and the surplus ringed. Standard schematic; this is the topic's central figure.
A sorting figure for section 10: one bin for primes both numbers hold and one for primes only one holds, with counters flowing from the bins into the HCF, the LCM and the product.
A three-number comparison panel with editable inputs, for section 11.
The two owl-and-panel side boxes are the book's own page furniture; redraw the idea rather than reproducing the art.
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", pp.51–52 — Anshu's claim, the counterexample, and the two side boxes naming conjecture, verification and counterexample
Same chapter, §3.3 "Patterns, Properties, and a Pretty Procedure!", Part II, p.58 — the bold naming of general statement and generalisation
Same section, "Property Involving both the HCF and the LCM", Part II, pp.62–63 — the size question, 105 and 95, the three test pairs, the relation, the proof hint, and the three-number question
Same chapter, SUMMARY, Part II, p.65, the final bullet, which names the two words as things the chapter set out to teach
Same chapter, closing exercises, Part II, pp.63–64, questions 4 and 10
Backward pointers: Reading every factor of a number off its prime factorisation for the counterexample in its original context, and What HCF and LCM do for consecutive, even, and co-prime numbers for the general statements this vocabulary was invented to describe