Reciprocals, and Brahmagupta's rule for dividing fractions

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What to assume they know

Restating a division as a missing-factor multiplication — a division rewritten as a multiplication with a missing factor, and the four worked cases §8.2 builds
A fraction of a fraction, and why the numerators and denominators multiply — Brahmagupta's multiplication rule in letters
Cancelling a shared factor above and below a bar (Cancelling common factors before multiplying, not after)
Reading a formula written with letters as a statement about every case at once
That the order of two factors does not change their product (Part I, §8.1, p.186)

What they should be able to do

State what makes one number the reciprocal of another, in terms of their product
Write down the reciprocal of a fraction, and of a whole number
Explain why turning a fraction upside down produces its reciprocal, rather than asserting it
Carry out a division of fractions in two named steps: find the divisor's reciprocal, then multiply
Write the division rule in letters, and recognise the chapter's two printed forms as the same rule
Attribute the general form of the multiplication and division rules to Brahmagupta's Brāhmasphuṭasiddhānta of 628 CE, and the reciprocal phrasing to Bhāskara II's Līlāvatī of 1150 CE
Place the chapter's account of how fraction arithmetic travelled — India, then Arab and African mathematicians, then Europe — on a rough timeline

Where it usually goes wrong

"Invert and multiply is a trick you just have to remember." The chapter reaches the answers first and the formula second, and the reason is that the inverted fraction is the only thing that cancels the divisor away to 1. Show the cancellation before the slogan.
**"The reciprocal is the fraction upside down."** That is how you compute it for a fraction. What it is is the number whose product with the original is 1 — which is why 5 has a reciprocal, 1/5, with nothing to turn over, and why the idea survives when the number is not written as a fraction at all.
"You flip the first fraction." The reciprocal taken is the divisor's. In 2/3 ÷ 3/5 it is 3/5 that becomes 5/3, and the printed callout labels which is which.
"Every number has a reciprocal." On pp.188–189 as checked, the chapter works only with fractions and whole numbers that have one, and does not raise the case of zero.
"Brahmagupta invented fractions." The chapter says the opposite on p.194: fractions were in general use in the Śhulbasūtra tradition from about 800 BCE. What is credited to Brahmagupta is the codification of the operations in essentially their modern general form.
"The two printed formulas are two different rules to learn." They differ only in which factor is written first, and p.186 has already established that this cannot matter.
"The history is decoration." It carries the topic's actual claim — that going from four worked cases to one line of letters is a mathematical achievement with a date attached.

Questions to check understanding

Write down the reciprocal of a given fraction, of a whole number, and of a mixed fraction
State the property that defines a reciprocal, and verify it for one pair
Divide two fractions using the two-step method, showing both steps
Evaluate a grid of divisions of mixed types — the printed item on p.196 is a three-row, four-column grid whose last two cells are left empty, so it supplies ten, including whole ÷ fraction, fraction ÷ whole, and mixed ÷ mixed
Convert a mixed fraction before dividing, and be marked on that step
Say who first stated these rules in general form, in which work, and when
Explain why multiplying by the reciprocal is the same as dividing

Examples worth working on the board

The worked case the rule is read off (Part I, §8.2, pp.188–189). Inputs: 2/3 ÷ 3/5. The chapter identifies 5/3 as the number that multiplies the divisor 3/5 to 1, names 5/3 the reciprocal of 3/5, then multiplies 5/3 by the dividend 2/3 to reach 10/9. Checked against p.188: the printed line carries Dividend and Divisor on leader arrows above and Quotient on a leader arrow below, and the word reciprocal is set in bold where it is defined.
The reciprocal, described before it is named (Part I, §8.2, p.188). The chapter's own description is positional: the number wanted has the divisor's denominator on top and the divisor's numerator underneath. Keeping the two apart is the whole argument of this topic.
The two printed forms (Part I, §8.2, p.189). Both appear, one under the other. The first writes the reciprocal first and the dividend second; the second writes the dividend first. The chapter presents the second as a rewriting of the first. That they are the same rule is exactly the order-independence established on p.186 — say so, because it is a chance to use an earlier result rather than assert a new one.
The worked instance under the formula (Part I, §8.2, p.189). Inputs: 2/3 ÷ 3/5 again, now run through the formula rather than through the missing-factor argument, giving 10/9 a second time. The point is that the two routes agree; do not present the formula as replacing the reasoning.
Brahmagupta's two verses (Part I, "A Pinch of History", p.195). The chapter paraphrases a multiplication rule and a division rule from the Brāhmasphuṭasiddhānta, giving 12.1.3 as the verse reference for the multiplication one, and paraphrases Bhāskara II's Līlāvatī verse 2.3.40 as the restatement in terms of the reciprocal.
Bhāskara I's visual (Part I, "A Pinch of History", p.195). Checked against the printed page. A square is drawn with dashed internal lines making 5 columns and 4 rows; the bottom-left cell is labelled 1/20, with 1/4 outside the left edge and 1/5 beneath the bottom edge, and the caption states that this shows 1/5 × 1/4 = 1/20. The chapter dates the commentary Āryabhaṭīyabhāṣhya to 629 CE and Aryabhata's own work to 499 CE. This is the same unit-square picture §8.1 used, attributed.
The chain of transmission (Part I, "A Pinch of History", pp.195–196). Inputs, all printed with the chapter's own hedges: Śhrīdharāchārya about 750 CE, Mahāvīrāchārya about 850 CE, Caturveda Pṛithūdakasvāmī about 860 CE, Bhāskara II about 1150 CE; then al-Hassâr of Morocco about 1192 CE; then transmission to Europe over the following centuries, becoming widespread there only around the seventeenth century.

Figures to have open

A multiplication line into which candidate numbers can be dropped and the product watched, for section 3. Standard schematic.
A pair of fractions that can visibly exchange numerator and denominator, for sections 4 and 5.
The labelled division callout from p.188, redrawn.
One timeline axis reused in sections 9 and 11, wide enough to span 800 BCE to the seventeenth century CE without misleading. Every date this topic marks on it — 499, 628, 629, 750, 850, 860, 1150, 1192 — is CE, and not one of them belongs on the BCE side of the axis. The BCE dates that would push the axis that far left are the Śhulbasūtra's 800 BCE (pp.190 and 194) and Umasvati's 150 BCE (p.194), so a broken or logarithmic axis will be needed if that earlier material is included.
A unit square subdivided 5 by 4 with one cell picked out, for section 10.
No photograph from the textbook is required. The chapter prints no portrait of any of the mathematicians it names — checked against pp.194, 195 and 196.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 8 "Working with Fractions", §8.2 "Division of Fractions" — the Discussion block that defines the reciprocal and derives the rule, pp.188–189
Same part, the unnumbered "A Pinch of History" block, pp.194–196, for the verses, the attributions and the transmission account. Note that this block is printed inside §8.3 even though its content belongs to §8.1 and §8.2
Same part, SUMMARY, p.198, fourth and fifth bullets
Same part, §8.1, p.186, for the order-independence the two printed forms rely on
Same part, Figure it Out, p.196, item 1