Counting the angles of symmetry of polygons and radial-arm figures

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What to assume they know

An angle of symmetry: turning a figure onto itself: angle of symmetry, centre of rotation, and why 360° is always on the list
Common multiples: why the first "idli-vada" is 15 and Common factors: which jump sizes can land on a number: multiples and factors, and testing whether one whole number goes into another exactly
The degree: why a full turn was cut into 360 equal parts: the 360 degrees of a full turn, and why 360 was chosen
Division with a remainder, and writing a quotient as a mixed fraction

What they should be able to do

Count the angles of symmetry of a radial-arm figure and list them
Work out a figure's smallest angle of symmetry when it has a given number of equally spaced arms, deriving it as 360 shared among the arms
Explain why unequally spaced arms leave a figure with no rotational symmetry
Show that every angle of symmetry is a whole-number multiple of the smallest one
Explain why a smallest angle of symmetry measured in whole degrees must be a factor of 360
Decide whether a stated angle can be a figure's smallest angle of symmetry
Express a smallest angle as a mixed fraction when it is not a whole number
Use the exercise's phrase order of rotational symmetry for the count

Where it usually goes wrong

"More arms means more angles of symmetry." Only if they are equally spaced and alike. The three-arm figure on p.233 has three arms and no rotational symmetry at all; the point is on the same page as the counterexample.
"The angles of symmetry are 90°, 180°, 270° and 360° for everything." Those are the four-fold case. The chapter deliberately walks 4, then 2, then 3, then 5, 6 and 7 so that the list is seen to depend on the figure.
"Every smallest angle is a whole number of degrees." The seven-arm figure is the counterexample the chapter sets. Its smallest angle is 51 3/7 degrees, and the figure is no less symmetric for it.
"If 17° is not allowed, no odd number is." 45° is allowed, and so are 5°, 9° and 15°: any factor of 360 is allowed. The test is divisibility, not oddness.
"The angles of symmetry are whatever turns happen to work, in no order." They form a chain: the smallest one, then its double, then its triple, and so on up to a full turn. The chapter's three lists on p.236 exist to make that visible before it is argued for.
"Order and angle are the same thing." The order is a count of angles; an angle is a measure in degrees. A figure of order 4 has a smallest angle of 90°, not of 4.

Questions to check understanding

List the angles of symmetry of a given figure
Work out a figure's smallest angle of symmetry when it has a stated number of equally spaced arms
Given the smallest angle, list the rest
Given one angle of symmetry and the number of smaller ones, find the smallest
Decide whether a stated angle can be a smallest angle of symmetry, with reason
Express a smallest angle as a mixed fraction where it is not a whole number
Colour or shade a sectored circle to a stated number of angles of symmetry
Give the order of rotational symmetry of a printed figure

Examples worth working on the board

The four-arm figure, §9.2, p.232. Four straight bars reaching out from a centre along dotted lines, with the printed note that neighbouring dotted lines are 90° apart. Worked: 90°, 180°, 270°, 360° — four angles of symmetry. The chapter then asks whether the angles between the arms can be changed with the figure still having four angles of symmetry, and tells the student to try drawing it. Work the answer and it is no: four angles of symmetry make 90° the smallest, a 90° turn takes each arm to the next, so adjacent arms have to stay 90° apart. What may be changed is the shape of the arms, provided all four stay alike. The following question on the same page — how to modify the figure down to only two angles of symmetry — is the one the next bullet answers.
The two-angle version, §9.2, p.233. Read off the printed page of p.233: the same four-armed figure with the top arm bent to the right at its far end and the bottom arm bent to the left, the two side arms left straight. A half turn carries top onto bottom and left onto right; a quarter turn does not. So its angles of symmetry are 180° and 360° — two of them.
Three arms that fail, §9.2, p.233. A three-armed figure whose arms are not equally spaced. Only a full turn brings it back, so by the chapter's own definition it has no rotational symmetry. The chapter suggests tracing and cutting a copy and turning the copy over the original — a good device.
The argument, §9.2, p.234. Two rough three-arm diagrams side by side, the angles between the arms marked A, B and C on one and rotated on the other. For the turned copy to lie on the original, each marked angle must land on the next, so all three are equal. Three equal angles fill a full turn, so each is 360 ÷ 3 = 120°. The chapter prints this division. Then the angles of symmetry are 120°, 240° and 360° — three of them, shown in the four-panel strip at the foot of p.234, where colour is added only to make the turns visible.
Five and six arms, §9.2, p.235. Same division: 360 ÷ 5 = 72°, giving 72°, 144°, 216°, 288°, 360°; and 360 ÷ 6 = 60°, giving 60°, 120°, 180°, 240°, 300°, 360°.
Seven arms, §9.2, p.235. 360 ÷ 7 is not a whole number: 7 goes into 360 fifty-one times with 3 left over, so the smallest angle of symmetry is 51 3/7 degrees. The chapter asks for exactly this and for the mixed-fraction form. Note what it shows — the seven-arm figure exists and turns onto itself; it is only the degree measure that comes out untidy.
Q1, §9.2, p.235 — three figures with a marked point. Read off the printed page: (a) an equal-armed plus made of two crossing bars, so 90°, 180°, 270°, 360°; (b) an upright segment with a small square at each end, both squares on the same side of it, so a half turn moves a square to the wrong side and only 360° survives; (c) an H — two upright bars joined by a crossbar — so 180° and 360°.
The three printed lists, §9.2, p.236. For figures with exactly two angles: 180°, 360°. With exactly three: 120°, 240°, 360°. With exactly four: 90°, 180°, 270°, 360°. The chapter's own observation follows: each list is the multiples of the number that starts it.
Why the smallest angle divides 360, §9.2, pp.236–237. Printed as a True or False item, so the reasoning is the explanation's job. Run it as division with a remainder: keep adding the smallest angle to itself; every total so far is an angle of symmetry. If no total lands exactly on 360°, one of them overshoots, and going a full turn and then undoing the totals below it leaves a turn that works and is smaller than the smallest — which cannot be. So a total lands exactly on 360°, and the smallest angle goes into 360 exactly.
Q4, Q5 and Q6, §9.2, p.238. Q4: if the smallest is 60°, the rest are 120°, 180°, 240°, 300°, 360°. Q5: 60° is an angle of symmetry and exactly two angles of symmetry lie below it, so 60° is the third multiple of the smallest and the smallest is 20°. Q6: 45° can be a smallest angle because 45 goes into 360 eight times; 17° cannot, because 17 does not go into 360 at all.
Q1, §9.2, p.238 — the sector circle. A circle cut by radii into equal sectors; counted from that figure, there are twelve of them, so each is 30°. To colour it for three angles of symmetry, repeat a four-sector block three times, giving a smallest angle of 120°. For four angles, repeat a three-sector block four times, giving 90°. The counts you can reach altogether are exactly the numbers that go into 12: one, two, three, four, six and twelve.

Figures to have open

The four-arm radial figure of p.232 with its 90° marking. Redraw; must show.
The bent-arm figure of p.233. Redraw from the page — the bend directions are the content, and they do not extract from the text at all.
The unequally spaced three-arm figure of p.233, and the two rough diagrams of p.234 with the angles marked A, B and C.
The four-panel 120° strip at the foot of p.234, redrawn with colour used the same way — to mark which arm is which, not to decorate.
The three figures of Q1, p.235. Redraw; the position of the two small squares in (b) is the whole exercise.
The twelve-sector circle of p.238, blank, so the explanation can colour it live.
A dial marked 0° to 360° for section 10, with the multiples of the smallest angle stepping round it. 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 — the bold sub-heading on figures with radial arms, pp.232–235
§9.2, p.234 — the three-equal-angles argument and the printed division by 3
§9.2, p.235 — the five-, six- and seven-arm questions, and Figure it Out Q1–Q2
§9.2, p.236 — Q3 and the three lists of angles, with the multiples observation
§9.2, pp.236–237 — the True or False block, which straddles the page break: its heading and its 360° item are at the foot of p.236, its factor item at the head of p.237
§9.2, p.238 — Figure it Out Q1 (the sectored circle) and Q4, Q5, Q6
Chapter 5 "Prime Time", §5.1, for the multiple-and-factor vocabulary this topic leans on