Brahmagupta's rules, and how long it took the world to accept them

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What to assume they know

Addition as movement: starting position plus movement gives target position and Subtraction as the movement that gets you from start to target: adding and subtracting signed numbers on a shaft or a line
The additive inverse, and how it turns every subtraction into an addition: the inverse, and turning a subtraction into an addition
Zero pairs: why a positive and a negative token cancel: tokens in two colours, which the historical rods echo
Credits, debits, and what a negative balance actually means: credits and debits, which is where the history starts
Reading dates written BCE and CE, and placing them in order

What they should be able to do

State where and in what activity negative numbers were first written down
Describe the Chinese rod convention and match it to the chapter's own tokens
Name the Indian works the chapter cites and place them on a timeline
State what was new in Brahmagupta's treatment, as distinct from earlier uses
Apply each of the five addition rules to an example of your own
Apply each of the five subtraction rules likewise
Justify any one of the rules by walking it on the building or on the number line
Recount the route the ideas took to the Arab world and to Europe, with centuries
Explain why a mathematician could still call these numbers absurd in the 1700s

Where it usually goes wrong

"Brahmagupta discovered negative numbers." The chapter is careful: the earliest uses it cites are older and are Chinese and Indian ledger practice. His contribution is the unified treatment.
"Ancient mathematics is a preface to the real thing." The rules on these two pages are the rules a Class 6 student is being examined on. Nothing has replaced them.
"Europe rejected these numbers because it had not heard of them." They had arrived by the thirteenth century and were still being called absurd in the eighteenth. The obstacle was not information.
"Zero was the Indian contribution and negatives came from elsewhere." The chapter's claim is that the general treatment of all three together — positive, negative and zero — is what Brahmagupta gave.
"The rules are a list to memorise." Each one can be walked in three seconds on the building, and the chapter asks the reader to do exactly that. An explanation that recites ten rules has inverted the section.
"A minus sign has always been written in front." The manuscript the chapter names put its mark after the number. Notation is a choice that was made, not a fact about the numbers.

Questions to check understanding

State which of Brahmagupta's rules a given calculation uses
Give your own example of a stated rule
Justify a stated rule using the number line
Place the works and dates named in the section in chronological order
State what was new in Brahmagupta's treatment
Say how many centuries separate 628 CE from the European acceptance the chapter describes
Explain why an ordinary calculation such as 2 – (– 3) is an instance of a rule rather than a special trick
The chapter sets two tasks in this section and the solutions block bound with this chapter file does not answer them; both are open, and that is deliberate

Examples worth working on the board

The Chinese work (p.266). The chapter names The Nine Chapters on Mathematical Art, gives its Chinese title in transliteration, dates its completion to the first century CE or the second, and says the two kinds of number were shown by rods of two colours — red and black. It then points out that the chapter's own tokens do the same job with green and red. Make that comparison; it is the one place where a Class 6 model is shown to be two thousand years old.
The Indian works and their dates (p.267). Kautilya's Arthaśhāstra, given as around 300 BCE, described credit and debit and recognised that a balance could go below zero. The Bakśhālī manuscript, given as around 300 CE, wrote a negative number by putting a special mark behind it instead of in front.
Brahmagupta (p.267). His work is named, and the year given is 628 CE. What the chapter credits him with is the first treatment of positives, negatives and zero on equal terms, with rules for all four basic operations.
The five addition rules and their printed examples (p.267). Two positives: 2 + 3. Two negatives: (– 2) + (– 3). One of each: – 5 + 3, and 2 + (– 3), and – 3 + 5. A number with its inverse: 2 + (– 2). A number with zero: – 2 + 0, and 0 + 0.
The five subtraction rules and their printed examples (p.268). Smaller from larger: 3 – 2. Larger from smaller: 2 – 3. Taking away a negative: 2 – (– 3). A number from itself: 2 – 2, and – 2 – (– 2). Zero cases: – 2 – 0, and 0 – 0, and 0 – (– 2). The third of these is the rule earned in The additive inverse, and how it turns every subtraction into an addition, arriving here as history.
The two tasks the chapter sets (p.268). Explain each rule using the building or the number line, and give your own example of each. The first is the explanation's section 8 and is the only place in §10.5 where the reader does mathematics rather than reads about it.
The transmission dates (p.268). To the Arab world by the ninth century; to Europe by the thirteenth.
Carnot (p.269). An eighteenth-century French mathematician, named in the text, who rejected these numbers in one word. The chapter's point is the gap: more than a thousand years after Brahmagupta, and the argument was still live.
No portrait, map or manuscript image is printed with this section. Checked against pp.266, 267 and 268: all three pages are set text, with no illustration of any kind. Every visual in this topic will be not in the book.

Figures to have open

One timeline carrying every date the section gives, built across sections 3, 5, 10 and 11 rather than shown complete at the start. The gap between 628 CE and the eighteenth century is the argument, and a timeline that appears finished on first sight throws it away.
A pair of rod bundles in two colours beside the chapter's tokens, for section 2.
A number written with its sign trailing it, for section 4.
The building or number line from earlier in the chapter, reused unchanged for section 8. Reuse matters here: the point is that the rules need no new picture.
No portrait or manuscript facsimile exists in the chapter, and none should be invented.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 10 "The Other Side of Zero", §10.5 "A Pinch of History", p.266 — the claim about Asia, accounting as the first context, the Chinese work with its date, and the rods
§10.5, p.267 — Kautilya and his work with its date, the manuscript with its date and its trailing sign, Brahmagupta and his work with its year, and the five rules for addition with their examples
§10.5, p.268 — the five rules for subtraction with their examples, the two tasks set to the reader, the modern name for the system, and the transmission dates
§10.5, p.269 — Carnot and his verdict, and the pointer forward to algebra
Chapter Summary, pp.269–270, where the addition rules are restated as the chapter's own summary of how to add
§10.1, p.255, and §10.2, p.258, for the two places earlier in the chapter where the third subtraction rule was earned rather than told
Solutions block bound with this chapter file, which carries no entry for §10.5