The degree: why a full turn was cut into 360 equal parts

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What to assume they know

Straight and right angles as the landmarks of a full turn: full turn, half turn, quarter turn; straight angle and right angle defined by halving
Comparing two angles without measuring either of them: the transparent circle laid on a vertex, and why it compares two angles
Halving a quantity, and simple fractions of a whole
Division facts up to 12, and the idea of a factor

What they should be able to do

Explain why comparison alone is not enough, and what assigning a number adds
Describe how one degree is produced by cutting a full turn into 360 equal parts
State the measure of an angle as a count of unit parts
Give the degree measures of a full turn, a straight angle and a right angle, and derive each from the one before by halving
Recount the chapter's historical material about 360 without overclaiming what is known
Explain the arithmetical reason 360 is convenient, and test it by division
Compute the angle produced when a circle is cut into a given number of equal parts
Build a paper protractor by repeated folding and label its creases
Define bisecting an angle, and identify the angle bisector of a folded angle
Relate the folded creases to the marks on a manufactured protractor

Where it usually goes wrong

"There are 360 degrees in a circle because of something about circles." The chapter is unusually direct here: the reason is not fully known, and the practical justification is arithmetical convenience. An explanation that invents a geometric necessity contradicts p.34.
"A degree is a length along the rim." It is an amount of turn. Corrected by the fact that a larger drawn circle placed on the same angle still gives the same count — the same argument as the arm-length case in Comparing two angles without measuring either of them.
"Measuring an angle needs a protractor." It needs a unit. The handmade protractor at §2.9, pp.37–40 is the chapter's demonstration that the tool is a convenience over the unit, not a source of it.
"Halving forever gives whole numbers." It does not, and the chapter goes to 22.5° and 67.5° and prints them with the decimal. Do not round them.
"Bisecting means cutting into two pieces." It means cutting into two equal pieces; the fold is what guarantees equality.
"360 is divisible by every number up to 10." It is not — 7 is the exception, and the chapter says so. An explanation that drops the exception states something false.

Questions to check understanding

Write the degree measure beside each of a set of equally cut circles
Given the number of equal parts a circle is cut into, compute the angle at the centre of one part
State the measures of a full turn, a straight angle and a right angle, and show how each follows from the one before
Explain, in one line, what one degree is
Explain why 360 is a convenient number of parts, naming the exception
Identify the angle bisector in a folded figure and justify why the two parts are equal
Fill-in items of the form "one eighth of a turn is ___ degrees"
Numerical items built on repeated halving: 360, 180, 90, 45, 22.5

Examples worth working on the board

Fig. 2.12 (§2.9, p.32). Two drawings side by side: a plain circle with one radius and a small turn arrow at the centre, and a circle whose whole interior is filled with radial lines, labelled round the rim from 0° (360°) through 10°, 20° and on to 350°. The second is a full-circle protractor and it is the first time the student sees 360 marks at once.
The 30-unit fan (§2.9, p.33). An angle drawn with its interior filled by closely spaced radial lines and captioned as 30 units. It contains thirty 1° parts, so its measure is 30°. Use it to make measure = count concrete.
The landmark chain (§2.9, p.33). Full turn 360°; half of that is the straight angle, 180°; half again is the right angle, 90°. The book prints the 180-unit and 90-unit fans beside plain figures of the same two angles.
The 360 divisibility check (§2.9, p.34). Divide 360 by each whole number from 1 to 10: 360, 180, 120, 90, 72, 60, then 7 fails, then 45, 40, 36. Also 360 ÷ 12 = 30 and 360 ÷ 24 = 15. What the chapter asserts is that no count below 360 makes all of those divisions come out whole, once the division by 7 is set aside; the explanation can check the divisions, and the minimality is worth stating as the book's claim rather than re-deriving.
The ten cut circles (§2.9, p.34). A row of five circles and a row of five more, cut into 1, 2, 3, 4, 5, 6, 8, 9, 10 and 12 equal parts. The exercise is to write the degree measure beside each. The answers in order are 360°, 180°, 120°, 90°, 72°, 60°, 45°, 40°, 36° and 30°.
Figs. 2.13 to 2.18, the folding sequence (§2.9, pp.37–39). Cut a circle; fold and cut to a semicircle and write 0° at one bottom corner and 180° at the other; fold the semicircle in half to reach 90° at the top; fold again to get 45° and hence 135°; fold once more to reach 22.5°. Each fold halves the previous angle, which is the same move that produced the right angle in the previous topic.
Fig. 2.19 and Fig. 2.20 (§2.9, p.39). The opened semicircle with creases labelled OA through OI, and the same semicircle carrying 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, 157.5° and 180°. Eight equal angles fill the straight angle, so each is 180 ÷ 8 = 22.5°. The book asks why; that division is the answer.
What the ninth fold would give. 22.5 ÷ 2 = 11.25°, which no ordinary protractor marks. Worth showing as the point at which paper stops being practical and manufacture takes over. This step is added here, not the book's.

Figures to have open

Fig. 2.12 in both parts, especially the full-circle protractor with its rim labelled from 0° (360°) round to 350°. Standard schematic but it must carry the numbers; a plain circle will not do the work.
The 30-unit fan, the 180-unit fan and the 90-unit fan (§2.9, p.33). These are the figures that make counting visible; redraw with visibly separate rays.
The row of ten cut circles (§2.9, p.34). Needed at ten, in the printed order.
Figs. 2.13 to 2.18 — the fold sequence. Redraw as a step-by-step sequence rather than as six stills.
Fig. 2.19 (creases lettered O A to O I) and Fig. 2.20 (the same semicircle with the nine degree labels). Both needed; the letters and the numbers are two views of one object and the section joins them.
No photograph or dataset is required.

Where this sits in the book

NCERT Class 6 Mathematics (Ganita Prakash), Chapter 2 "Lines and Angles": §2.9 Measuring Angles, pp.32–34 for the unit and the historical passage; pp.37–40 for the handmade protractor and the bisector
Named sub-headings inside §2.9 used here: Measures of different angles, p.33; A pinch of history, pp.33–34; Degree measures of different angles, p.34; Make your own Protractor!, p.37; Angle Bisector, p.40 — all printed without numbers, so cited by page
Teacher's Note on making a protractor before using a standard one, §2.9, p.40
Summary, p.54, for the settled statement of the degree
Historical reference the chapter itself gives: Rigveda 1.164.48