An angle of symmetry: turning a figure onto itself

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What to assume they know

A line of symmetry is a fold that makes the halves coincide: the fold test and what a line of symmetry is
An angle is an amount of turn, not a pair of drawn arms: an angle as an amount of turn about a fixed point
Straight and right angles as the landmarks of a full turn and The degree: why a full turn was cut into 360 equal parts: the right angle, the straight angle, and the 360 degrees of a full turn
Labelling the corners of a square with capital letters

What they should be able to do

Give an example of a figure with no line of symmetry that still returns to itself under a turn
Identify the centre of rotation of a given figure
State what an angle of symmetry is, and test a proposed angle against a figure
Explain why 360° is an angle of symmetry of every figure whatsoever
List all four angles of symmetry of the windmill and of the square
Track where each labelled corner of a square goes under a quarter, half and three-quarter turn
Decide, for a figure that returns only after a full turn, that it has no rotational symmetry, and say why the definition is written to exclude it

Where it usually goes wrong

"Symmetric means it has a mirror line." The windmill is the counterexample the chapter opens §9.2 with, and it is the reason the word symmetry has to be qualified from here on.
"Any turn will do if the figure looks roughly the same." The angle has to be exact. The dashed reference line on p.231 exists because the eye cannot tell a turned square from an untouched one, and a nearly-right angle would leave the figure visibly off.
"360° counting as an angle of symmetry is a trick." It is a consequence: a full turn returns every point of every figure to where it started. That is why the definition of rotational symmetry has to exclude it explicitly, and section 11 should make the exclusion feel necessary rather than fussy.
"A figure with only 360° has rotational symmetry of order one, so it counts." Not by this chapter's definition. The trapezium strip is stated to have none.
"Rotating a figure is the same as flipping it." A half turn and a mirror flip give the same answer for some figures and different answers for others; the trapezium strip is exactly a case where the flip works and the half turn does not. Keep them apart here — Line symmetry and rotational symmetry are independent of each other makes the separation its whole subject.
"The centre of rotation must be marked on the figure." It has to exist, not be printed. The chapter asks the student to say where to mark it for the square.

Questions to check understanding

Name the centre of rotation of a given figure
List all angles of symmetry of a given figure
Decide, with reason, whether a described figure has rotational symmetry
State where a labelled corner ends up after a stated turn
Explain why 360° appears in every such list
Give an example of a figure with rotational symmetry but no line of symmetry
Explain why a figure whose only angle of symmetry is 360° is said to have none

Examples worth working on the board

The paper windmill, §9.2, p.230. A green four-bladed paper windmill with a small red dot marking its centre. The chapter's claim about it is exact: it has no line of symmetry, and a turn of 90° about the red dot leaves it looking untouched. Build the explanation's first minute on the gap between those two facts.
The windmill's angles, §9.2, p.231. The chapter prints them: 90°, 180°, 270° and 360°, glossed as a quarter, a half, a three-quarter and a full turn — four in all. These are the book's own numbers and may be stated as such.
Nothing below 90°, §9.2, p.230. The chapter asks whether a turn of less than 90° works and answers flatly that it does not. Show the attempt failing; the four blades are what forbid it.
The labelled square, §9.2, p.231. Read off the printed page of p.231: the initial square carries A at the top left, B at the top right, C at the bottom right and D at the bottom left, with a dashed line running out to the right from the middle of the right-hand side. After the 90° turn the same square is printed with B at the top left, C at the top right, D at the bottom right and A at the bottom left, and the reference line now points upwards. The turn drawn is therefore anticlockwise. Follow the panels, not the sentence above them. The running text on the same page describes the quarter turn as taking A to where B was, B to where C was, C to where D was and D to where A was — that is a clockwise turn, and it is the reverse of what the four panels beneath it draw. The count of angles is unaffected, but every corner destination is, so an explanation that narrates the sentence over the pictures will contradict itself. Show the movement of the panels' direction and let the letters land where p.231 prints them.
The other three panels, §9.2, p.231. After 180°: C top left, D top right, A bottom right, B bottom left. After 270°: D top left, A top right, B bottom right, C bottom left. After 360° the square must be back exactly as it began. Printing slip to work around: the printed 360° panel labels its two bottom corners C on the left and D on the right, which is the reverse of the initial square. Draw the final panel identical to the first — that is what a full turn means — and do not copy the printed labels.
The square's count. Four angles of symmetry, the same four as the windmill. The chapter poses this as a question and prints the four rotated pictures rather than the number; derive the four.
The worked Example, §9.2, p.232. A strip shaped like a trapezium — two parallel horizontal edges, the lower one longer, the two slanting ends leaning inwards — with a dot at its centre. Turned 180° clockwise it becomes the same trapezium upside down, wider edge now on top, and it does not lie on the original. A second 180° restores it. So the only angle that works is 360°, and the chapter concludes that the strip has no rotational symmetry.
The rule that follows, Summary, p.241. A figure counts as having rotational symmetry only if it owns an angle of symmetry lying strictly between 0° and 360°. This is the sentence that makes the strip's verdict a definition rather than an opinion.
**The Game, p.241 — optional, and a place where a turn does the arguing rather than the measuring.** Printed just above the Summary. On a 6 by 6 grid the two players move alternately; a move is one line drawn across a neighbouring pair of cells, upright or flat; no two lines may sit on the same cell; whoever has no move left has lost. The chapter asks for a winning strategy and prints none. The second player replies to each move with its half-turn image about the centre of the grid. Because 6 by 6 has an even number of rows and of columns, that centre is a corner shared by four cells and no legal move can be its own image, so the reply never clashes with the move it answers, and the position is symmetric again at the end of every reply — from which the second player always has somewhere to go. Run it as an extra beat after section 8, once the half turn is established; the argument is added here, not the book's, so derive it.

Figures to have open

The four-bladed paper windmill with a marked centre. Standard schematic; movement of the quarter turn is essential and is the topic's single indispensable figure.
The labelled square in its four positions, with the dashed reference line. Redraw from p.231 — turning anticlockwise as the panels do, and with the 360° panel corrected, as noted above.
The trapezium strip with a centre dot, in its original and half-turned positions. Redraw from p.232.
A circle of turn marked at 90°, 180°, 270°, 360° for section 7. Standard schematic; the book does not print one.

Where this sits in the book

NCERT Class 6 Mathematics (Ganita Prakash), Chapter 9 "Symmetry", §9.2 Rotational Symmetry, p.230 — the windmill, the centre of rotation, and the two bold namings
§9.2, p.231 — the windmill's four angles, and the square in four positions with its reference line
§9.2, p.232 — the worked Example on the strip, and the verdict that it has no rotational symmetry
Summary, p.241 — the definition of rotational symmetry and of centre of rotation
Game, p.241 — the grid game whose winning strategy is a half turn, used here as an optional extra beat after section 8 (not a closing beat: sections 9-11 follow it)