A stretched rope as compass and straightedge: the Śulba-Sūtra constructions

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What to assume they know

The perpendicular bisector, and the equidistance property that justifies it — the perpendicular bisector, and why equal distances from the two ends are the whole of it
Constructing a 90° angle at a chosen point on a line — the right angle at a chosen point
Knowing that a circle is the set of points one fixed distance from a centre
Comfort with the idea that the same geometry can be done at different scales

What they should be able to do

Name the Śulba-Sūtras as India's earliest surviving texts carrying these construction methods, and say what they were written for
Explain what a rope can do that makes it stand in for both a compass and a straight edge
Follow the rope construction of a perpendicular bisector step by step
Explain why doubling the rope over before tying it is the step that guarantees the result
Justify the rope construction using the congruence argument already met
Propose a rope method for a right angle at a marked point
Propose a rope method for bisecting an angle
Say why a construction argument is independent of the tool that carries it out

Where it usually goes wrong

"The rope is a picturesque substitute, but the real construction needs a compass." The reverse is nearer the truth here: the rope came first, and the compass is the tabletop version of it. What the argument needs is a fixed distance, and both tools supply it.
"A rope is too floppy to be exact." It is exact exactly when it is taut — which is why the chapter says twice, once for A and once for B, that both parts must be fully stretched. Slack is the only source of error, and it is visible.
"Śulba-Sūtras means a book of geometry." They are ritual construction manuals; the geometry is there because the altars had to be built to specification. Say what they were for before saying what they contain.
"These are the same as the compass steps, so there is nothing to prove." The rope method produces A and B by a different physical route, so the chapter sets a separate justification question for it (question 1, p.142).
"Ancient means approximate." The chapter's own word for what these texts contained is exact; the rope construction is not an estimate and does not become one at courtyard scale.
"One rope length gives one answer." Change the rope's length and A and B move up and down the same line — which is the p.139 observation about C and D again, in a new costume.

Questions to check understanding

Describe, in order, how a perpendicular bisector is built with a rope and two pegs
Explain why the loops at the rope's ends are excluded when the middle is marked
Justify that the line through the two marked positions really is the perpendicular bisector (Figure it Out question 1, Part II, p.142)
Devise a rope procedure for a right angle at a marked point of a line (Figure it Out question 2, Part II, p.142)
Devise a rope procedure for bisecting an angle (Figure it Out question 5, Part II, p.144)
Name the Śulba-Sūtras and state what kind of text they are
Heritage-and-mathematics items of this shape are a standing NCF-SE 2023 feature and are usually asked as "name the text / state what it contains / carry out the method"

Examples worth working on the board

What a rope does (Part II, §6.1, p.142). Two capabilities, stated by the chapter: swung about a fixed end it traces circles and arcs; pulled tight between two points it gives a straight line.
Fig. 6.4 (Part II, §6.1, p.142). Checked against the printed page. Three panels, of which only the lower two carry the thin red frame. The top one, unframed, shows two hands — one at each end of a doubled rope — against a pale blue disc, and is the folding step. The middle one is a patch of brown ground with a white segment lettered X at the left end and Y at the right, and the rope pulled up from both pegs to a hand at a point above, lettered A. The bottom panel repeats it downward: the same white XY, a dot lettered A left over above it, and the rope pulled down to a hand at a point below, lettered B.
The fold (Part II, §6.1, p.142). Input: the rope is doubled over so that its middle can be marked, and the chapter says explicitly that the rope taken into the two end loops is not counted. That exclusion is not a detail — it is what makes the two working halves equal.
The justification (Part II, §6.1, p.142). Inputs: with the rope taut, AX = AY, because both are the same half of one rope; likewise BX = BY. The chapter asks the reader to finish from there, and everything needed was settled on pp.137–138. Do not print the finished proof as the book's.
The attribution (Part II, §6.1, p.142). The chapter names this rope construction as coming from the Kātyāyana-Śulbasūtra and gives the reference 1.2. Show the citation; it is a good moment.
The two open prompts. A right angle at a marked point using a rope (question 2, p.142) and an angle bisector using a rope (question 5, p.144). Both are flagged Math Talk. Neither is answered in the book.

Figures to have open

A rope-and-peg construction on the ground, able to be shown moving so the middle of the rope can be pulled up to A and down to B while both halves stay taut. This is the topic's key image and must be redrawn rather than lifted from the book's illustration.
A compass and a rope side by side, both shown holding one distance while turning. Standard schematic.
A fire-altar ground plan. This is not printed in the chapter — the chapter names the altars in words only — so any plan shown must be built as a generic schematic and captioned as such.
The four-equal-lengths overlay linking the rope figure back to the compass construction. Built fresh; the book never sets the two figures together.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part II, printed Chapter 6 "Constructions and Tilings", §6.1 "Geometric Constructions", the unnumbered subheading "Construction Methods in Śulba-Sūtras", pp.141–142, including Fig. 6.4 and the attribution to Kātyāyana-Śulbasūtra 1.2 on p.142
Same part, same chapter, §6.1, "Figure it Out", questions 1 and 2, p.142
Same part, same chapter, §6.1, "Figure it Out", question 5, p.144
Same part, same chapter, §6.1, pp.137–138, for the congruence argument the rope construction is asked to reuse
Backward pointers: The perpendicular bisector, and the equidistance property that justifies it, Constructing a 90° angle at a chosen point on a line