Why a compass beats trial and error for building a triangle

What to assume they know

Measuring and drawing a segment of a stated length with a marked ruler
Class 6 Ganita Prakash, the chapter "Playing with Constructions": using a compass to place a point at a fixed distance from a given point
A circle as the set of points at a fixed distance from its centre, and the words centre and radius
Naming points with capital letters, and reading a labelled figure
The four angles at a crossing: vertically opposite and linear pairs — angles at a crossing, and the habit of arguing about a figure rather than only measuring it

What they should be able to do

Name the three vertices, three sides and three angles of a labelled triangle, and write the triangle's name in any vertex order
Explain why a ruler-only construction of a triangle needs repeated trials while a compass construction does not
State what the arc of radius 4 cm centred at A contains, and why that makes it the right tool
Justify, without measuring the finished figure, why the crossing point of two arcs is at the correct distance from both base vertices
Construct an equilateral triangle of a given sidelength using a compass
Construct a triangle from three given sidelengths, choosing a base and working through the four printed steps
Say why the full circles may be replaced by short arcs without weakening the argument
Distinguish equilateral from isosceles by the number of equal sidelengths

Where it usually goes wrong

"The compass is just a neater way of drawing." It is a different claim. A ruler mark asserts one distance; an arc asserts a whole set of points, so the crossing satisfies two conditions simultaneously. Section 7 must make the student able to say why no check is needed afterwards.
"A triangle has to sit on its longest side / on a horizontal side." The base is a choice made by the person constructing, not a property of the triangle. In the 4-5-6 construction the chapter puts the 4 cm side at the bottom purely for convenience.
"∆ABC and ∆BAC are different triangles." They are the same triangle named two ways; p.146 says the vertex order is free, and letters its examples out of order to prove it.
"Two arcs always cross, so any three lengths work." They do not, and the whole of The triangle inequality: which three lengths can close is about when they fail. Do not let the explanation imply that the method is universal — it should end pointing at the question, exactly as §7.2 does.
"Isosceles means exactly two equal sides, so an equilateral triangle is not isosceles." The chapter's phrasing is "two equal sides", not "exactly two", and it derives isosceles triangles from a circle's radii — where all radii are equal. Treat "exactly two" as a convention some books use and this one does not state either way.
"The dashed circle in the figure is part of the answer." The dashes record what the compass could reach. Only the crossing point is used.

Questions to check understanding

Construct a triangle of three given sidelengths and state which length you took as the base and why
Given a completed two-arc figure, explain why the crossing point is at the correct distance from each base vertex
Name the three angles of ∆PQR using the vertex letters
Say why a ruler-only construction of a 4-4-4 triangle may need several attempts
Given a circle with its centre marked, produce an isosceles triangle and justify which two sides are equal
From a list of sidelength triples, sort them into equilateral, isosceles and neither (the printed set on p.150 is exactly this task)
Board-style items on this material ask for the construction and the steps in words; the four printed steps on p.150 are the expected form of that answer

Examples worth working on the board

The opening gallery of triangles (Part I, p.146). Checked against the printed page. Four triangles are drawn, deliberately unalike: a tall thin one labelled ∆ABC, a wide flat one labelled ∆YZX, one labelled ∆WUV and one labelled ∆BAU. The labels are the point — the vertices are lettered out of alphabetical order on purpose, so that the freedom to name a triangle in any vertex order is shown rather than only asserted. Redraw four visibly different triangles and letter them the same way.
The failed ruler attempt (Part I, §7.1, p.147). Checked against the printed page. Base AB = 4 cm drawn solid; AC = 4 cm drawn solid; BC drawn as a dashed line annotated "4 cm?" with a question mark. Inputs: two of the three lengths can always be forced with a ruler; the third is left to chance. Do not show the third side coming out right.
The two-arc construction (Part I, §7.1, pp.147–148). Inputs: AB = 4 cm; arc of radius 4 cm centred A; arc of radius 4 cm centred B; C is where they cross. The printed figures draw the arcs as bold and the rest of each circle as a dashed loop, which is exactly the picture the argument needs — the dashes are the points the compass could have reached.
The 4-5-6 triangle (Part I, §7.2, pp.148–150). Inputs: base AB = 4 cm, AC = 5 cm, BC = 6 cm. Fig. 7.1 is the bare base. Fig. 7.2 adds the circle of radius 5 cm centred A. Fig. 7.3 adds the arc of radius 6 cm centred B.
The four printed steps (Part I, §7.2, p.150). Choose a base; arc from A at the second length; arc from B at the third length so it crosses the first; join. The book's own aside is worth keeping: full circles are not needed, only the piece near the crossing.
The five sidelength sets to build (Part I, §7.2, p.150). In centimetres: 4, 4, 6 · 3, 4, 5 · 1, 5, 5 · 4, 6, 8 · 3.5, 3.5, 3.5. Give these as inputs. Note: the list is not innocent — 3.5, 3.5, 3.5 is equilateral, 4, 4, 6 and 1, 5, 5 are isosceles, and one of the sets is a right triangle, though the chapter does not say so here.
The two circle puzzles (Part I, §7.2, pp.150–151). Checked against the printed page. The first is a single circle with its centre marked as a dot: joining the centre to any two points of the circle gives two equal radii, hence an isosceles triangle. The second shows two same-size circles with centres A and B, each passing through the other's centre, and then a third overlapping circle centred C.

Figures to have open

A compass, step by step: one arm pinned at a point, the other sweeping an arc. Standard schematic. The pinned arm must stay visibly fixed — that is the whole argument.
Four visibly different triangles for the opening, lettered out of alphabetical order as on p.146. Redraw; do not lift the book's artwork.
The failed ruler attempt with a dashed third side and a question mark, as on p.147.
The two-circle picture with the intersection marked, drawn so the circles can fade back to arcs. This is the load-bearing figure of the topic.
A single circle with its centre and two points on it, joined, for the isosceles construction on p.150.
No photograph, table or dataset from the textbook is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 7 "A Tale of Three Intersecting Lines" — unnumbered chapter opening, p.146 (what a triangle is made of, naming, the three angles, and the straight-line question)
Same part, §7.1 "Equilateral Triangles", pp.146–148 — the 4 cm construction, the ruler attempt on p.147, and Steps 1–3 on pp.147–148
Same part, §7.2 "Constructing a Triangle When its Sides are Given", pp.148–151 — the 4-5-6 construction with Fig. 7.1–7.3 on p.149, the justification and the four steps on p.150, the "Construct" list and the naming of isosceles triangles on p.150, and the two circle puzzles on pp.150–151
Same part, SUMMARY, p.171, first bullet — the chapter's own one-line statement that the compass simplifies this construction
Backward pointer: Class 6 Ganita Prakash, "Playing with Constructions", named on p.147 as the source of the method