Regular hexagons, and why the angles round a point must total 360°

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What to assume they know

Constructing 60° from an equilateral triangle, and the arches built on it — constructing 60° from an equilateral triangle
Bisecting an angle, and halving 90° to get 45° — bisecting an angle
Knowing that a full turn about a point measures 360°
Knowing that an equilateral triangle has three 60° angles
Recognising when two triangles are congruent, and what follows from it

What they should be able to do

State what makes a polygon regular, and name the regular 3-gon and 4-gon
Say why the chapter can reach a regular hexagon and defers the regular pentagon
Describe the six triangles a regular hexagon splits into from its centre
Explain the reversal: assume six congruent equilateral triangles and see what they make
Use the 360° criterion to decide whether a given angle fills a stated gap
State the criterion for fitting angles round a point in your own words
Explain why six 60° angles fit exactly, and why each long diagonal of the hexagon comes out straight
Construct a regular hexagon of a stated side length using 60° and 120°

Where it usually goes wrong

"A hexagon is regular if its six sides are equal." Equal sides are not enough — the definition on p.151 asks for equal angles too, and a student can easily draw an equal-sided hexagon that sags.
"Six triangles round a point always close up." They close up because 6 × 60 happens to be exactly 360. Change the triangle and the arithmetic fails. The gap puzzle is placed there to make that visible.
"If a piece looks like it fits, it fits." The 70° wedge looks plausible against a 50° gap and is 20° out. Do the sum.
"The pentagon just needs a cleverer compass trick." The chapter puts it off until triangles and five-sided figures are better understood, and leaves it to later years. Report the deferral honestly rather than implying it is impossible or easy.
"AOD is a straight line because it looks like one." It is straight because 60 + 60 + 60 = 180. The chapter asks for exactly this reason.
"120° is a new construction." It is what is left on the line beside a 60° angle. One construction, two angles.
"The 360° criterion is about hexagons." It is about angles at a point, and it is the same idea that will decide whether a shape can tile the plane three pages later. Signpost that link.

Questions to check understanding

State the two conditions for a polygon to be regular
Given a fan of angles round a point with one gap, compute the gap and decide whether a stated angle fits (the shape of the puzzle on p.152)
Explain why six congruent equilateral triangles close up round a point
Explain why the long diagonals of a regular hexagon are straight lines through the centre
Construct a regular hexagon of a stated sidelength using a compass and a scale — a 4 cm or 5 cm side has to be measured off, which the p.140 two-tool restriction expressly allows, and the p.152 task names a ruler without calling it unmarked (the tasks on p.152 and p.153)
Construct the 6-pointed star and decide whether its six point triangles are equilateral (the task on p.154)
Construct a six-pointed star set inside a hexagonal outline, finding its angles first (Figure it Out question 3, Part II, p.155 — its printed hint sends the reader to the angles, which is the argument already made for the p.154 star)
The 60°-angle bullet of the SUMMARY (Part II, p.163) is the statement the chapter expects back

Examples worth working on the board

The pentagon and hexagon attempt (Part II, §6.1, p.151). Checked against the printed page: a five-sided figure and a six-sided figure drawn side by side, each with tick marks on every side and an arc at every corner, so both are marked as equal-sided and equal-angled. The chapter flags this "Try This" and expects the attempt to be made before the reasoning starts.
Fig. 6.12 (Part II, §6.1, p.151). Checked against the printed page: a regular hexagon lettered A at the top, then B, C, D, E, F clockwise, with the centre lettered O and all three long diagonals drawn, cutting it into six triangles.
The hexagon's corner angle (Part II, §6.1, p.152). Inputs: each triangle contributes 60° at a hexagon corner, and two triangles meet at each corner. The chapter prints the sum and then asks the reader to say why.
The gap puzzle (Part II, §6.1, p.152). Checked against the printed page. Wedges are drawn fanning out from a point lettered O, their far corners lettered A, B, C, D, E, F, G, H, I going round, and each wedge is tagged with its angle: 40°, 60°, 50°, 30°, 40°, 90°. The remaining wedge, ∠AOI, is tagged only with a question mark, and a separate grey triangle carrying a 70° tag is drawn beneath with an arrow pointing into the gap. Worked through: the six tagged angles come to 310°, so the gap is 50°, and 70° is 20° too wide to go in. That conclusion is the point of the page and must not be softened.
Six equilateral corners (Part II, §6.1, p.152). Inputs: 60° per corner, six corners. The chapter states the product.
The three diagonals (Part II, §6.1, p.152). Input: the chapter asks the reader to account for AOD, for BOE and for COF each being straight. Three consecutive 60° angles is what settles it.
The 60°/120° pair (Part II, §6.1, p.152). Checked against the printed page: a line with a ray on it, 60° tagged on one side and 120° on the other, captioned so that getting the first gives you the second.
Two hexagons to build (Part II, §6.1, pp.152–153). The chapter asks for one of sidelength 4 cm on p.152 and one of sidelength 5 cm on p.153. Both are compass-and-ruler tasks. Use both; the second comes after the 60° construction has been given, so it should be quicker.
The 6-pointed star (Part II, §6.1, p.154). Checked against the printed page: the star is printed twice, once plain and once lettered — outer points A, B, C, D, E, F and inner hexagon corners G, H, I, J, K, L. The chapter states it has a rotational symmetry, hints that a hexagon is hiding inside, and asks whether the six point triangles ΔAGH, ΔBHI, ΔCIJ, ΔDJK, ΔELK and ΔFLG are equilateral. It does not print the answer. See Notes.

Figures to have open

A regular hexagon with its three long diagonals and centre, able to be shown moving so the six triangles can be lifted out and put back. This is the topic's key image.
The gap-angle fan: wedges of 40°, 60°, 50°, 30°, 40° and 90° round a point with one gap, and a separate 70° wedge that can be dragged towards it and refused. Redraw; the figure is what makes the point.
A running 360° total that fills as each wedge is added.
The 60°/120° line pair.
A hexagon being walked out side by side with a compass and a 120° angle, able to be shown moving.
The 6-pointed star, lettered as on p.154, with the inner hexagon highlightable.
No photograph, table or dataset from the textbook is needed.

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 "Regular Hexagons", p.151, and "Regular Hexagon and Equilateral Triangles", p.151, with Fig. 6.12
Same part, same chapter, §6.1, p.152 — the corner-angle argument, the definition of a degree from the complete angle, the gap-angle puzzle, the general criterion, the six-times-60 conclusion, the straight-diagonal question and the 4 cm hexagon
Same part, same chapter, §6.1, "Construction of a 60° angle", p.153, and the 5 cm hexagon
Same part, same chapter, §6.1, "6-Pointed Star", p.154
Same part, same chapter, §6.1, "Figure it Out", question 3, p.155 — a six-pointed star inside a hexagonal outline, set as a construction whose hint sends the reader to the angles
Same part, same chapter, SUMMARY, p.163, the 60°-angle bullet
Backward pointer: Constructing 60° from an equilateral triangle, and the arches built on it
Forward pointer: What it takes to cover a region with no gaps and no overlaps, where the same round-a-point criterion decides which regular polygons tile the plane