Why a debt times a debt is a fortune

What to assume they know

• Debts and fortunes: negative numbers close subtraction — integers, and the debt-and-fortune reading of the two signs
• From śhūnyatā to śhūnya: turning nothing into a number you can compute with — the printed rule that any number multiplied by zero gives zero
• Multiplication as repeated addition, on whole numbers
• The distributive law on whole numbers, in the form a(b + c) = ab + ac
• That a number and its opposite add to zero

What they should be able to do

• State the two printed product rules for signed numbers and give the chapter's instances of each
• Explain a negative multiplier as the removal of repeated quantities, using the chapter's debt hint
• Compute products of integers with mixed and matching signs
• Derive the value of a product of two negatives from distributivity and the zero-product rule, and identify which printed rules the derivation uses
• Explain why no separate convention is needed once distributivity is required to hold
• Distinguish the sign rule for products from the sign rule for sums, and give a case where confusing them gives the wrong answer
• Connect the product rule to the subtraction rule met in the previous topic, and say why they are the same idea

Where it usually goes wrong

• "Two negatives make a positive — that holds for adding too." It does not: (−5) + (−4) = −9. This is the single commonest signed-arithmetic error and the four-cell comparison in section 9 exists to kill it.
• "The rule is a convention mathematicians agreed on." It is not available for agreement. Once distributivity is required and a × 0 = 0 holds, the value +12 is the only one left.
• "You cannot repeat something a negative number of times, so the rule is meaningless." Correct on the first clause, which is exactly why the chapter reinterprets the negative multiplier as removal rather than repetition. Naming the reinterpretation is the lesson.
• "Removing a debt gives you money." It leaves you better off by the amount of the debt without any cash changing hands. Students who think cash arrives get the arithmetic right and the meaning wrong, and then fail the explain-with-an-example question.
• "(−3) × (−4) is bigger than 12 because two negatives are involved." The magnitude is just 3 × 4. Only the sign is at issue, and separating magnitude from sign is the procedural habit worth drilling.
• "Distributivity is a fact about positive numbers and cannot be used here." The chapter states it for rational numbers, which include all the integers, at §3.4, p. 48. The argument is licensed by the book, though the book does not run it.
• "The debt story is just a mnemonic." It is a model, and it makes a testable prediction — that removing four debts of ₹3 leaves you exactly ₹12 up, not approximately.

Questions to check understanding

• Compute products of integers with mixed and matching signs
• State the rule for a product of two negatives and give a worked instance
• Explain with a debt example why multiplying two negatives gives a positive — the direct form of the chapter's Think and Reflect
• Justify the sign rule using distributivity and the zero-product rule, naming the rules used
• Spot the error in a worked solution that has applied the product sign rule to a sum
• Evaluate a mixed expression combining signed sums and signed products, where order matters
• Explain why subtracting a negative and multiplying by a negative are the same underlying move

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated inputs. The chapter prints no answers.

• Rule 4, the mixed-sign product (§3.3.1, p. 45). Printed as (−3) × 4 = −12, glossed by the chapter as taking on four debts of ₹3 for a total debt of ₹12. This is the case students accept without argument.
• Rule 5, the matching-sign product (§3.3.1, p. 45). Printed as (−3) × (−4) = 12, with no gloss attached to the rule itself.
• The chapter's own justification (Think and Reflect, §3.3.1 area, p. 46). Inputs, as the box supplies them: a negative number stands for a debt; multiplying by a negative stands for the removal of that debt; and the hint is that should another party cancel four debts of yours, each worth ₹3, you stand ₹12 better off. The box states the conclusion (−3) × (−4) = +12. Note: this is an argument, not a definition, and it is the chapter's only attempt at one — the numbered rule on p. 45 is bare.
• The distributivity argument (not in the book, built from printed rules). Inputs: distributivity, printed for rational numbers at §3.4, p. 48, law 4; and Brahmagupta's rule that anything times zero is zero, printed at §3.2.2, p. 44. The working: (−3) × [(−4) + 4] = (−3) × 0 = 0 by the zero rule; expanding the left side by distributivity gives (−3) × (−4) + (−3) × 4; and (−3) × 4 = −12 by rule 4. So (−3) × (−4) + (−12) = 0. Verified: the only value that satisfies this is +12. Emphasise what has happened: the answer was not chosen, it was cornered.
• Exercise instances (Exercise Set 3.2, Q3, p. 46). Q3(i) (−12) × 5 and Q3(ii) (−8) × (−7) are the two product items. Verified: −60 and 56. Q3(iii) 0 − (−14) and Q3(iv) (−20) ÷ 4 belong with Debts and fortunes: negative numbers close subtraction, though 3(iv) can be brought in here to show the sign rule surviving division. Verified: 14 and −5.
• Signs of sums against signs of products, side by side (the explanation's construction). Give it these four inputs and let it compute: (−5) + (−4), (−5) × (−4), (−5) + 4, (−5) × 4. Verified: −9, 20, −1, −20. Two negatives added give a negative; two negatives multiplied give a positive; the pattern differs between the two operations and this table is the cheapest way to show it.
• The tie to subtraction (Exercise Set 3.2, Q4, p. 46, and this topic). The identity the exercise supplies, 10 − (−5) = 15, is the same removal idea acting once instead of four times. Present them as one idea at two scales rather than as two rules.
• No figure is printed for this material. §3.3.1 and the p. 46 Think and Reflect box are set as text. Everything visual here has to be built.

Figures to have open

• A debt-slip prop that can be stacked and destroyed, carrying a running balance. Sections 1, 3 and 4 all use it. Standard schematic; the chapter prints no figure.
• A four-cell grid comparing sum and product for the same pair of signed inputs. Standard schematic, built from the values listed above.
• An annotated equation frame for the distributivity argument, with each step labelled by the printed rule that licenses it — one label pointing to §3.2.2, p. 44, one to §3.4, p. 48, one to rule 4 on p. 45. The provenance labels are the point; without them the derivation looks like sleight of hand.
• No textbook figure is required, because the chapter supplies none for this material.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics, printed Chapter 3 on the world of numbers. Section §3.3.1 carries the printed heading "The Arithmetic of Integers" and sits on p. 45; rules 4 and 5 of the five printed there are this topic's material.
• The Think and Reflect box arguing the case appears at the top of p. 46.
• Exercise Set 3.2, Q3(i) and Q3(ii), p. 46.
• Distributivity and commutativity for rational numbers, used by the second argument, are printed at §3.4, p. 48, law 4. The zero-product rule is at §3.2.2, p. 44.
• Chapter Summary, p. 67, first bullet, restates the matching-sign product rule in symbols.