A pinch of history: where solving for an unknown came from

What to assume they know

• Isolating the unknown, step by step — solving an equation with the unknown on both sides
• Doing the same thing to both sides preserves equality — an operation applied to each side preserves the assertion
• Adding and subtracting negative numbers, and the idea of a debt as a negative amount
• Substituting a value back into an equation to check it
• Reading a two-column table and matching corresponding entries
• A rough sense of the CE timescale — that 499 comes before 628, and 628 well before 1150

What they should be able to do

• Explain what bījagaṇita names and why a seed is the metaphor chosen
• Place Aryabhata, Brahmagupta, Al-Khwarizmi and Bhāskarāchārya on one timeline with the dates the chapter prints
• Trace the route by which the word algebra reached modern English
• Read an expression written in the ancient Indian notation of the printed table and give its modern form, and go back the other way
• Set up and solve the horse-and-debt problem, and check the answer against both men's holdings
• Apply Brahmagupta's formula to an equation of the form Ax + B = Cx + D
• Verify the formula against an equation already solved the long way earlier in the chapter
• Identify the case in which the formula cannot be applied, and connect it to the no-solution exercise set earlier

Where it usually goes wrong

• "Algebra was invented in Europe and reached India later." The chapter runs the transmission the other way and dates it: Indian work in the 5th to 7th centuries, into Arabic in the 8th, into Latin in the 12th. Say the direction out loud.
• "Brahmagupta's formula is a different, cleverer method." It is the same method, run once in general instead of every time in particular. Deriving it from Ax + B = Cx + D — subtract Cx, subtract B, divide by A − C — is worth ninety seconds and turns the formula from a thing to memorise into a thing that had to be true.
• "The formula always works." It does not, and the chapter attaches no condition to it on p.184. When A = C there is nothing to divide by. If B and D also agree, every value of x works; if they do not, none does — and that second case is exactly the equation with no solution the reader was asked to build back in §7.2 (Part II, §7.2, p.172). Join those two moments up; the chapter leaves the join to the reader.
• "Ancient notation is just modern notation with odd symbols." Three real differences are printed on p.183: the marker came before the number rather than after; a negative was shown with a dot above rather than a sign in front; and the two sides were stacked one above the other rather than joined by an equals sign. It is the ancient cell of the third row that carries no equals sign — the modern-notation cell beside it does print one, so the row is a contrast, not an absence.
• "They only ever had one unknown." yā, kā, nī, pī and lo are printed as distinct symbols for distinct unknowns, most of them abbreviations of colour names. The convention is nearer to modern x, y, z than the strangeness of the glyphs suggests.
• "A debt is just a smaller amount of money." In Example 16 the debt of ₹100 enters the equation as a subtraction, and getting that sign right is the whole modelling step. Negative numbers are being used here for what they are for.
• "History is the part you can skip." The section is the chapter's answer to what all the solving was for. It is also examinable: the exercise block that follows it includes a problem from the Bakhśhāli Manuscript.

Questions to check understanding

• Write a given modern expression in the ancient notation of the printed table, and translate one the other way
• State what bīja, rū and al-jabr mean and where each word comes from
• Place the four named mathematicians in order with their dates
• Set up and solve a problem stated in the old sources — the horse problem, or the Bakhśhāli distribution problem, in which the second share is twice the first, the third three times the second and the fourth four times the third, and the four together come to 132 (Part II, the exercise block following §7.4, p.188, question 14)
• Apply Brahmagupta's formula to a given equation of the four-number form, and confirm the answer by solving it the long way
• Say when the formula cannot be used, and what that says about the equation
• The chapter's longest exercise block sits directly after this section (Part II, pp.185–189) and mixes routine solving with history-sourced word problems

Examples worth working on the board

Printed values are marked as such.

• The seed metaphor (Part II, §7.4, p.182). Printed: bīja means seed; a tree lies hidden inside a seed as an answer lies hidden inside an unknown, and working a problem is likened to coaxing that tree out, a step at a time. Use it as the opening image; it is the section's own frame and it does real work.
• The dates (Part II, §7.4, pp.182–184). Printed inputs, all four: Aryabhata 499 CE; Brahmagupta's Brāhmasphuṭasiddhānta 628 CE, with Chapter 18 named as the relevant one; Al-Khwarizmi's book around 825 CE; Bhāskarāchārya's Bījgaṇita 1150 CE, printed on p.183 in that spelling, without the second a. A fifth date is printed in the exercise block: a manuscript from Bakhśhāli, which the chapter dates to 300 CE (p.188). Note the ordering the chapter itself asserts — Aryabhata first proposed a systematic method for a single unknown, and Brahmagupta set it out.
• The transmission (Part II, §7.4, p.182). Printed: Indian mathematical ideas went into Arabic in the 8th century; Al-Khwarizmi lived in what is now Iraq; his title translates as calculation by restoring and balancing; the book reached Latin and Europe by the 12th century; al-jabr became algebra.
• The Mumford line (Part II, §7.4, p.182). The section quotes the Fields Medallist David Mumford calling Brahmagupta the key figure in algebra's creation. Attribute it to Mumford — it is a cited judgement, not the textbook's own assertion, and the distinction is worth making to a Class 7 audience.
• The notation table (Part II, §7.4, p.183). Checked against the printed page. Two columns headed Modern Notation and Ancient Indian Notation, three rows: 2x + 1 against yā 2 rū 1, with the note that in each term the marker came before the number; 2x − 8 against yā 2 rū 8 with a dot set above the 8, and the note that the dot marked a negative; and 3x + 4 = 2x + 8 against two stacked lines, yā 3 rū 4 over yā 2 rū 8, with the note that an equation's two sides were written one under the other. Reproduce the layout, including the overdot and the stacking.
• Example 16, the horses (Part II, §7.4, p.183). Printed inputs: one man has ₹300 and 6 horses; the second owns 10 horses but also owes ₹100; the two are equally rich; every horse costs the same. The chapter names the price x, writes 300 + 6x = 10x − 100, and works down to a printed answer of ₹100 per horse. The debt entering as a subtraction is the interesting modelling step; do not let it go past unremarked.
• The two model equations (Part II, §7.4, p.184). Printed: 5x + 4 = 3x + 8, and 3x − 6 = 2x + 4. The reader is asked whether some operation on the four numbers in each could deliver the answer directly.
• The general form and the formula (Part II, §7.4, p.184). Printed: Ax + B = Cx + D, with x = (D − B) ÷ (A − C). Note the order — D minus B on top, A minus C underneath. Getting either subtraction the wrong way round flips the sign, and both come out wrong together only by accident.
• The formula tested (Part II, §7.4, p.184). Printed: applied to 650m + 4000 = 500m + 5050 it gives m = (5050 − 4000) ÷ (650 − 500). This is Example 9's savings equation from p.176 with the terms of each side reordered. Set the two solutions side by side — seven printed lines of working across pp.176–177 against one line here. That comparison is the section's argument.
• The reader's turn (Part II, §7.4, p.184). Printed: 2x + 3 = 4x + 5, to be solved by the formula. Here A − C is negative, so the answer is negative.
• The closing claims (Part II, §7.4, pp.184–185). Printed: algebra lets you generalise patterns, in numbers as much as in shapes or in situations, and lets you justify a claim — the example given is why two odd numbers always add to an even one. The section ends by saying that ancient Indian mathematicians recognised the power of algebra and hoping the reader will too.

Figures to have open

• A timeline running from 300 CE to about 1200 CE with six marks on it, each carrying a name, a date and a place. Standard schematic; the dates are all printed in the chapter.
• A two-column notation table whose rows can be revealed and translated in either direction, including a numeral carrying an overdot and a pair of stacked expressions with no equals sign between them. Redraw it — the printed table's cells wrap and should not be reproduced as an image.
• A seed-to-tree growth movement for section 1.
• A balance or equalising visual for the horse problem, with a debt shown as a quantity removed rather than added.
• A side-by-side of a long worked solution against a one-line formula, for section 10.
• No photograph from the textbook is needed. The chapter prints no manuscript images, no portraits and no maps in this section — all 28 printed pages were checked.

Where this sits in the book

• NCERT Ganita Prakash, Class 7, Part II, printed Chapter 7 "Finding the Unknown", §7.4 "A Pinch of History", pp.182–185 — the naming of bījagaṇita and the seed metaphor, Brahmagupta and the Mumford judgement, the transmission through Arabic and Latin, and the symbols for unknowns (p.182); the colour names, rūpa and rū, the notation table and Example 16 (p.183); Aryabhata's attribution, the general form, Brahmagupta's formula and the closing claims about what algebra is for (p.184); the final line of the section (p.185)
• Same part, same chapter, the exercise block following §7.4, pp.185–189 — the Bakhśhāli Manuscript problem is question 14 on p.188
• Same part, same chapter, §7.2, p.172 — the no-solution exercise this brief joins to the A = C case
• Same part, same chapter, §7.2, pp.176–177 — Example 9, the savings equation the formula is tested on
• Sibling topic: Mind the Mistake, Mend the Mistake, the §7.3 error-spotting set, marked video: no