The oldest known tables of squares and cubes

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What to assume they know

What makes a number a perfect square — square numbers, and the area reading of a square
Square roots, and the prime-factor test for a perfect square — square roots and the radical symbol
Cube roots, and what successive differences expose — cube roots
Reading dates on both sides of the Common Era, and that BCE dates count backwards
What a lookup table is, from any subject

What they should be able to do

State when and where the earliest known lists of squares and cubes were made, and what they were written on
Explain why a table is useful when no efficient method exists, and what a table costs
Name the practical work the chapter says those tables served
Give the Sanskrit terms for the square power, the cube power and the fourth power, and say what each also meant outside mathematics
Explain why the same word served the figure and the power
Trace the word for root from mula through Arabic and Latin to the modern usage
Attribute the two statements the chapter quotes to Aryabhata and Brahmagupta with their dates
Argue that a term's ordinary meaning is evidence about a concept's origin

Where it usually goes wrong

"They had tables, so they had our methods." A table is what you build instead of a method. That is the whole reason the chapter's next sentence names the practical problems: somebody needed answers before anybody had an algorithm.
"'Root' is a metaphor European mathematicians invented." radix is a translation of a usage already old in India; the chapter traces the chain explicitly, and the modern word carries an Indian idea in Latin clothing.
**"varga just means the shape."** It carries the figure and the power at once, and Aryabhata's statement is quoted precisely to establish that.
"Powers above the third had no names until algebra." varga-varga names the fourth. A named fourth power implies people were composing operations long before symbolic notation.
"The history section is decoration you can cut." The words are the evidence. Cut them and the claim that these ideas came out of measuring land has nothing behind it.
"1700 BCE is when squares were invented." It is the oldest surviving list. What came before it did not survive, which is a different statement.
"Aryabhata is defining squaring." He is recording that one word covers two things.
"The Sanskrit words are just old names for the same things." mula meaning cause and origin is doing real work: the side is treated as what the area comes from, which is exactly the relation an inverse operation expresses.

Questions to check understanding

State when and where the earliest known lists of squares and cubes were made
Name the Sanskrit terms for the second, third and fourth powers
Explain why the operation of finding a side is called taking a root
Attribute a given statement to Aryabhata or Brahmagupta with the century
Explain what a table of squares was used for, and why a table rather than a rule
Short reasoning: what the double meaning of varga tells you about how the square power was first understood
Match Sanskrit, Arabic and Latin terms for root to their languages

Examples worth working on the board

The Babylonian lists (Part I p.15). The chapter's claim: the earliest known compilation of squares and cubes is Babylonian and dates to roughly 1700 BCE; it survives pressed into clay tablets; and its purpose was rapid recovery of square roots and cube roots (Part I pp.15–16).
What they were for (Part I p.16): problems of land measurement, of architectural design, and other work where geometric calculation was needed.
The dates the section gives, all printed: about 1700 BCE for the Babylonian lists; at least the third century BCE for varga, ghana and varga-varga in use in India; at least the first century BCE for mula; 499 CE for Aryabhata; 628 CE for Brahmagupta. Those five dates are the whole timeline and no others appear in the section.
**The two senses of *varga*** (Part I p.16): the drawn square with its area on one hand, the second power on the other. The two senses of ghana: the solid cube, and a number taken three times as a factor. varga-varga names the fourth power.
Aryabhata's statement (Part I p.16), paraphrased — do not read the printed translation aloud. He records that a four-sided figure whose sides are equal, together with the number giving its area, both carry the name varga, and that a product of two equal quantities carries it too. The chapter's own conclusion from this: the power's name came out of the drawn figure.
Why "root" (Part I p.16). mula meant the root of a plant, and also basis, cause and origin. It was taken up for the operation of extracting roots. varga-mula was the term for a square root, and ghana-mula for a cube root. The usage was then followed in Arabic with jidhr and in Latin with radix, each of which is that language's word for a plant root.
Brahmagupta's statement (Part I p.16), paraphrased: the pada of a krti is the quantity of which it is the square. pada itself means foot, basis, cause, origin.
The photograph (Part I p.15). A fragment of a brown clay tablet covered in wedge-shaped impressions, printed at the right of the page beside the opening of §1.3. Verified on the printed page. The chapter gives it no caption, no collection or museum credit, and does not say which tablet it is — checked on the page images of Part I pp.15 and 16.
A concrete hook for section 2, not in the book and not the chapter's. A table of squares to 60 fits on one clay tablet; the same tablet answers "what is 47 squared" and, read backwards, "what is the side of a field of area 2209". One artefact, both directions. That is what a table buys, and the cost is that it stops at its last row.

Figures to have open

A clay tablet with impressed rows, readable as a list. The chapter prints a photograph of a tablet fragment (Part I p.15) with no caption; if the printed image is not used, a schematic tablet with a visible list of squares carries section 1 better, because the real fragment's marks are not legible at video size.
A timeline carrying the five printed dates, from 1700 BCE to 628 CE. An added figure and the spine of section 9.
A word tree: mula at the root, varga-mula and ghana-mula growing from it, then jidhr and radix as branches into Arabic and Latin, with the plant-root meaning drawn in. An added figure and the one the thesis depends on.
A pair of panels showing a square figure labelled varga and a solid labelled ghana, each with the corresponding power written beside it. Standard schematic.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 1, §1.3 "A Pinch of History", Part I pp.15–16. The section begins in the lower third of Part I p.15, beside the tablet photograph, and runs to the "Figure it Out" heading part-way down Part I p.16.
The chapter quotes Aryabhata (499 CE) and Brahmagupta (628 CE) in translation; both quotations are short and neither is reproduced in this brief.
No exercise items attach to §1.3; the "Figure it Out" set that follows it on Part I pp.16–17 is cube material, covered by What makes a number a perfect cube, and the three-identical-groups test and Cube roots, and what successive differences expose.
The chapter-end puzzle page "Square Pairs!" (Part I p.18) is marked video: no in the spine and has no brief.