The sign rule for division, and why it follows from multiplication

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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

Why a negative times a negative must be positive: the four sign cases for a product, and the token argument behind them
Division of whole numbers as the inverse of multiplication — knowing 6 × 7 = 42 gives 42 ÷ 7
Negative numbers on the number line, and adding and subtracting them: additive inverse and the notation −(−a)
Times tables to at least 12, and the ability to recognise a multiple

What they should be able to do

Restate any integer division as a multiplication with an unknown factor
Answer that unknown-factor question using a multiplication fact already known
State the sign of a quotient from the signs of dividend and divisor, in all four cases, and justify it from the matching multiplication
Read and use the chapter's three printed division identities, including the condition attached to them
Use dividend, divisor and quotient correctly
Explain why the answer to the unknown-factor question is unique when it exists
Recognise that reframing a division does not guarantee an integer answer, and say what the answer is when the divisor does not divide the dividend
Place Brahmagupta's fortune-and-debt statement alongside the modern sign rules and see them as the same content

Where it usually goes wrong

"Division needs its own sign rule, so that's eight rules in total." There are four facts, readable in two directions. The section's entire method is to convert every division into a multiplication before any sign question is asked.
"Because dividing makes things smaller, a negative divided by a negative should be negative." Size and sign are separate questions, as they were for products. (−100) ÷ (−4) is 25: positive, smaller in magnitude than the dividend, and — this is the half that does the work — larger in magnitude than the divisor. Dividing shrank the answer relative to one of its inputs and grew it relative to the other, so "dividing makes things smaller" is not even a statement about size, let alone about sign.
"The rules on p.39 use a and b, so a and b could be anything." Read the condition: on p.39 the letters stand for positive integers and the negatives are written as −a and −b. This is precisely where a student who has memorised the shape rather than the statement gets caught.
"Every integer division has an integer answer." Nothing in the section promises that, and item 11 on p.44 contains two that do not. The reframe only answers the question when the divisor actually divides the dividend; otherwise the hole has no integer filling.
"Dividing by zero gives zero." The chapter attaches b not equal to zero to its general statement and leaves it there. The reframe explains why: 0 ×? can never reach a non-zero number, so the question has no answer at all.
"Brahmagupta's lines are a quaint old version; the real rule is modern." They are the same rule, and the chapter says so. The interesting point is that fortune and debt gave the seventh century a reason to accept it, which is the same job the token bag does on p.30.

Questions to check understanding

Evaluate an integer division and state the sign rule that settled it
Fill a blank so that a quotient statement comes out true (p.39, two such items)
Given a product of two integers, write down the two divisions it answers
Evaluate a compound expression built from one product and one quotient (p.44, item 11)
State the sign of a quotient when both signs are given, without computing
Explain why a division question has no answer when the divisor is zero
Word problems that ask "how many bags / hours / rounds", where the answer must come out as an exact count (p.39, item 3(b) is exactly this shape)
Competency items typically hide the division inside a rate: a temperature falling so many degrees an hour, or a per-unit loss set against a total

Examples worth working on the board

(−100) ÷ 25 (Part II, p.38). Inputs: the division, its reframing as 25 ×? = −100, and the multiplication fact 25 × (−4) = −100 that answers it. The order matters — the section states the question, then the known product, then the conclusion. Keep the three beats separate.
(−100) ÷ (−4) (Part II, p.38). Inputs: the same dividend, the divisor now negative, reframed as (−4) ×? = −100, answered by (−4) × 25 = −100. Setting this beside the previous case is what makes the sign pattern visible; run them as a pair.
50 ÷ (−25) (Part II, p.38). Inputs: the known product (−25) × (−2) = 50, and the division it answers. Note that this one is presented in the reverse order — the product first, the division read off it.
The three printed identities (Part II, p.39). For positive integers a and b with b not zero, the chapter prints three statements: dividing by −b negates the quotient, dividing −a by b negates the quotient, and dividing −a by −b gives the plain quotient. Note carefully that the letters here stand for positive integers and the minus signs are written in explicitly. That is a different convention from §2.2's a and b, which range over all integers.
Brahmagupta's rules (Part II, p.35). Checked against the printed page. This passage is not in a box. It is a bold sub-heading, then ordinary running body text, then an indented four-line quotation, with no frame, rule or tint around any of it; a redraw must not invent one. The passage cites the Brāhmasphuṭasiddhānta (628 CE), chapter 18, verses 30–32, in which four short lines cover product and quotient together for the four sign combinations, using fortune (dhana) and debt (ṛṇa). Do not read the quoted lines out; state their content in the wording used here and give the source. The chapter's own claim about them — that this was the first such articulation — is the historical hook.
The SUMMARY's division bullet (Part II, p.45). Checked against the printed page. Beside the bullets sits a four-quadrant multiplication table, seven columns by seven rows, whose central row and central column are ringed in green and carry the factors. The central row prints −3, −2, −1, ×, 1, 2, 3 with signs. The central column prints 3, 2, 1, ×, 1, 2, 3 without signs — above the × the factor is positive and below it the factor is negative, and the reader has to supply that. Every entry in the top-left and bottom-right blocks is negative; every entry in the top-right and bottom-left blocks is positive. The table's numbers do survive text extraction, but they arrive out of order and without the green rings or the row-and-column geometry that gives them their meaning, so the printed page is what settles the layout. It works just as well read as a division table: pick an entry, divide by its row factor, land on its column factor.
Divisions that do not come out whole (Part II, p.44, item 11). Checked against the printed page. Item 11 asks for four values built from products and quotients. Worked through, 11(a) and 11(d) are integers; 11(b) is 32 ÷ 648 and 11(c) is (−300) ÷ (−1215), neither of which is an integer — they are 4/81 and 20/81. If the explanation uses item 11, it must not present all four as integers. This is a computed observation about the printed items, not a claim the chapter makes.
Division practice in the chapter (Part II, pp.39, 42). Inputs: 36 ÷ (−18) and (−46) ÷ (−23) on p.39; (−27) ÷ 9, 84 ÷ (−4) and (−56) ÷ (−2) on p.42; and two fill-the-blank quotient statements on p.39. All of these are exact.

Figures to have open

An equation frame with an empty box in one factor position, fillable — the central device of the whole topic. Standard schematic.
A 2 × 2 sign grid that can carry a multiplication fact and its matching division in each cell.
The four-quadrant multiplication table from p.45, redrawn at seven by seven, with tintable quadrants. The signs implied on the lower half of the central column must be made explicit in the redraw, since that is the one thing the printed table leaves to the reader.
A short timeline mark for 628 CE with the Brāhmasphuṭasiddhānta named. No portrait or manuscript image is needed or available; do not invent one.
No photograph or dataset from the textbook is required.

Where this sits in the book

NCERT Class 7 Mathematics, Ganita Prakash Part II, printed Chapter 2 "Operations with Integers", the unnumbered sub-heading "Division of Integers" inside §2.2, pp.38–39.
Part II, p.35, the passage under the bold sub-heading "Brahmagupta's Rules for Multiplication and Division of Positive and Negative Numbers", citing Brāhmasphuṭasiddhānta 18.30–32 (628 CE).
Part II, p.45, SUMMARY, second bullet and the accompanying four-quadrant table.
Part II, pp.39, 42 and 44, for the practice items named above.
Backward pointer: Why a negative times a negative must be positive, whose four sign cases this topic reads backwards.