Debts and fortunes: negative numbers close subtraction

What to assume they know

• One-to-one correspondence: counting without number words — that subtraction can leave the natural numbers, set up by Exercise Set 3.1, Q3
• From śhūnyatā to śhūnya: turning nothing into a number you can compute with — zero as a number with rules, defined as a − a
• Reading a number line, and that equal steps mean equal differences
• Addition and subtraction of whole numbers
• Everyday debt and profit, and that owing is the opposite of holding

What they should be able to do

• Explain why the natural numbers, and even the whole numbers, cannot answer every subtraction question
• State what integers are, and which three ingredients the chapter combines to form them
• Give the symbol for the integers and say where the letter comes from
• Match Brahmagupta's two named states to positive and negative numbers
• Locate positive and negative integers and zero on the number line, with zero as the boundary rather than an endpoint
• Apply the printed rules for adding integers of the same sign, and for subtracting zero from either kind
• Translate a sequence of financial events into a single integer expression and evaluate it
• Explain, using a debt, why removing a negative amount has the same effect as adding a positive one
• Solve a signed-quantity word problem involving temperature change

Where it usually goes wrong

• "Negative numbers are numbers less than nothing, which is impossible." They are numbers on the other side of a chosen origin. A temperature of −11 °C and a debt of ₹100 are both perfectly definite; the minus records a direction, not an impossibility.
• "−5 is smaller than −4 because 5 is bigger than 4." Position on the line settles it: −5 sits further left, so it is the smaller. Fig. 3.2 is the evidence.
• "The natural numbers were just missing some answers we later found." Inside the naturals, 3 − 5 has no answer at all. The answer did not exist and was not waiting to be discovered; the set was enlarged so that it would.
• "Adding two negatives should somehow move you towards zero." Rule 2 says it does not, and the debt gloss is why: borrowing more does not reduce what you owe.
• "Zero is where the negatives start." Zero is neither a debt nor a fortune in the chapter's scheme; Fig. 3.2 marks it as the point both arrows leave from. Treating zero as the first negative breaks the symmetry the figure is drawn to show.
• "Two of the trader's three events are losses, so he must end deeply in debt." He ends only ₹100 down. Signed arithmetic does not obey a majority vote, and this exercise is built to catch that.
• "Subtracting a negative is a rule you memorise." It is the removal of an obligation. Students who have the debt picture never misremember the sign.

Questions to check understanding

• Evaluate a subtraction whose result is negative, and mark it on a number line
• State what the integers consist of and give their symbol
• Match Brahmagupta's two named states to the two signs
• Temperature problems: a starting reading and a fall or rise through zero
• Translate a sequence of financial events into one signed expression, then evaluate — the board's standard two-mark form of Q2
• Explain, with a real-world debt, why cancelling an obligation raises a balance
• Order a mixed set of integers, including two negatives, from least to greatest
• Compute (−5) + (−4) and 0 − (−14) with reference to the printed rules

Examples worth working on the board

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

• The subtraction that fails (§3.3, p. 45). The chapter opens by contrasting 5 − 5 = 0 with 3 − 5, and prints the latter with an empty box for its answer. That empty box is the whole motivation. Verified: 3 − 5 = −2.
• Fig. 3.2 (p. 45), read from the printed page. A horizontal number line with arrowheads at both ends, ticked and labelled from −5 to 5. A red arrow above the left half is labelled with negative integers and debt, the Sanskrit Ṛiṇa in brackets; a blue arrow above the right half is labelled with fortunes, Dhana in brackets, and positive integers. A small filled dot sits on 0, and the label above it names zero and Śhūnya. The two arrows point outward from the middle, so the figure reads as two directions of travel from a single origin rather than as two separate lines.
• The printed integer rules that belong here (§3.3.1, p. 45, rules 1 to 3 of five). Rule 1: a fortune added to a fortune stays a fortune, with 5 + 4 = 9. Rule 2: a debt added to a debt stays a debt, with (−5) + (−4) = −9, glossed by the chapter as owing ₹5 and then borrowing ₹4 more. Rule 3: subtracting zero changes neither kind, with 7 − 0 = 7 and −6 − 0 = −6. Rules 4 and 5 concern products and belong to Why a debt times a debt is a fortune.
• Ladakh overnight (Exercise Set 3.2, Q1, p. 46). Inputs: in Ladakh's high-altitude desert the noon reading is 4 °C, and it falls 15 °C by midnight. Verified: the midnight reading is −11 °C. Worth drawing on the number line, because the fall crosses zero and the crossing is the point.
• The spice trader's week (Exercise Set 3.2, Q2, p. 46). Inputs, in order: a loan of ₹850, then a profit of ₹1,200 the next day, then a loss of ₹450 the following week. The exercise asks for this as one integer expression before it asks for a value. Verified: (−850) + 1200 + (−450) = −100, so the trader ends ₹100 in debt. The sign of the answer is the interesting part — two of the three events are losses and the profit is the largest single figure, so students who guess without computing usually guess wrong.
• Subtracting a negative (Exercise Set 3.2, Q4, p. 46). Inputs: the identity the exercise supplies as its example, 10 − (−5) = 15, and the instruction to explain it through a real debt. The reading that works: cancelling a ₹5 debt from a position of ₹10 leaves ₹15. Note that this is the additive twin of the product rule in the next topic, and the two should be taught as one idea seen twice.
• The division item (Exercise Set 3.2, Q3(iv), p. 46): (−20) ÷ 4. Give it as an input and flag as a check that §3.3.1 states rules for adding and multiplying integers only — division of integers appears in the exercise without a printed rule preceding it. The numbered list runs to five items and none of them is a division rule.
• Other Exercise Set 3.2 data belonging to the next topic: Q3(i) (−12) × 5, Q3(ii) (−8) × (−7). Q3(iii) 0 − (−14) can be used here as a rule-3 and subtract-a-negative composite.

Figures to have open

• Fig. 3.2 (p. 45) redrawn: one number line from −5 to 5, zero marked as a filled point, two outward arrows labelled with the debt and fortune names. This is the chapter's own figure and the argument of sections 3 to 5 depends on the single shared origin, so keep that feature.
• A vertical thermometer scaled through zero for the Ladakh problem. Standard schematic; the chapter prints no figure for the exercises.
• A two-column ledger graphic for section 3. Standard schematic.
• A debt-slip prop for section 10, so that "removing a debt" is an action rather than a phrase. Standard schematic.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics, printed Chapter 3 on the world of numbers. Section §3.3 carries the printed heading "Integers: Expanding the Horizon" and occupies p. 45, with Fig. 3.2 on the same page.
• §3.3.1, printed heading "The Arithmetic of Integers", p. 45 — rules 1 to 3 of the five printed there.
• Exercise Set 3.2, p. 46 — questions 1, 2, 3(iii), 3(iv) and 4 belong here; 3(i) and 3(ii) belong to Why a debt times a debt is a fortune.
• Backward pointer: the closure question that motivates this topic is Exercise Set 3.1, Q3, p. 43.
• Chapter Summary, p. 66, third bullet, restates the extension of the line to the left of 1 and the debt-and-fortune categorisation.