Line symmetry and rotational symmetry are independent of each other

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What to assume they know

A line of symmetry is a fold that makes the halves coincide and Why a figure can have several lines of symmetry: the fold test, and counting a figure's lines of symmetry
An angle of symmetry: turning a figure onto itself: angle of symmetry, and why a figure whose only angle is a full turn is said to have no rotational symmetry
Counting the angles of symmetry of polygons and radial-arm figures: that the angles of symmetry are the multiples of the smallest one
Shapes come in sequences too, with rules of their own: the shape sequences of Chapter 1, Table 3

What they should be able to do

State the two tests separately and say what each one measures
Give a figure with turns but no mirror line, and one with a mirror line but no turn
Sort a set of figures into the four cases the chapter's closing statement names
Explain why every angle whatsoever is an angle of symmetry of a circle
Explain why every diameter of a circle is a line of symmetry
Say why the circle is the exception to the rule about a smallest angle
Count both the lines and the angles of symmetry of the same figure and compare the two numbers
Sketch, or say why you cannot sketch, a figure meeting a stated combination of the two properties

Where it usually goes wrong

"Symmetric means mirror-symmetric." The windmill kills this, and it is why the chapter spends a whole section on the distinction rather than a sentence.
"If a figure has rotational symmetry it must have a mirror line too." A slanted parallelogram is the counterexample the chapter asks for at Q3(c), p.238: a half turn works, and no fold does.
"If a figure has a mirror line it must have rotational symmetry." The trapezium strip of p.232 and the kite of Q3(d) both refuse.
"A circle has one line of symmetry — the horizontal one." It has one through every point of the rim. The move from a diameter to every diameter is the whole content of section 8.
"Every figure has a smallest angle of symmetry." The chapter says most do and the circle does not. Any angle you name, half of it also works — so there is no least one.
"The number of lines and the number of angles are always equal." Only when there is at least one line. The windmill has four angles and no lines at all.
"The counts are equal by coincidence." Section 11 gives the reason: fold across one mirror line and then across another, and the two folds together are a turn; fold and then turn, and that is a fold across some third line. So the mirrors and the turns match up one for one.

Questions to check understanding

State whether a described figure has reflection symmetry, rotational symmetry, both or neither
Sketch a figure meeting a stated combination, or say why none exists
Give the number of lines and the number of angles of symmetry of a named regular polygon
Find the lines and angles of symmetry of a building outline, an emblem or a traditional design
Explain why every diameter of a circle is a line of symmetry
Explain why a circle has no smallest angle of symmetry
Two-part items of the shape "does it have reflection symmetry? does it have rotational symmetry?" about the same picture — the form Q7 on p.239 takes

Examples worth working on the board

The four cases, with a figure for each. Turns but no mirror: the paper windmill of §9.2, p.230 — the chapter states it has no line of symmetry, and p.231 gives it four angles. Mirror but no turn: the trapezium strip of §9.2, p.232 — the chapter states it has no rotational symmetry; its upright mid-line is a line of symmetry, which the chapter does not say, so derive that. Both: the square, four lines and four angles. Neither: the photograph of clouds from the chapter opening, p.218.
Q3, §9.2, p.238 — four sketches to order. (a) a triangle having two or more fold lines and two or more angles of symmetry: only one with all three sides equal will do, and it gives three of each. (b) a triangle having just one fold line and no rotational symmetry: two equal sides and a third different. (c) a four-sided figure that turns onto itself but has no fold line at all: a slanted parallelogram, neither rectangle nor rhombus — a half turn works, no fold does. (d) a four-sided figure with a fold line and no rotational symmetry: a kite, or the trapezium of p.232. Parts (c) and (d) carry the printed Try This marker.
The circle, §9.2, p.237. Two claims, both the chapter's own. Turn a circle about its centre through any angle at all and it lands on itself, so every angle is an angle of symmetry. Take any point on the rim, join it to the centre and continue to the far side: that diameter is a line of symmetry, and so is every other. The consequence the chapter draws — printed on this same page, a few lines above the heading on the circle's symmetries — is that the circle has no smallest angle of symmetry; it is the stated exception.
Fan, flower and wheel, §9.2, p.237. Three everyday objects printed as further cases of rotational symmetry: a three-bladed ceiling fan, a flower, and a many-spoked wheel. Use them for the "look around you" beat; a three-bladed fan gives 120°, 240°, 360°.
Q7, §9.2, p.239 — the new Parliament Building. A photograph, and beside it the outline of what the picture shows. Measured off that outline: it is a six-sided figure got by cutting the three corners off a triangle with equal sides whose apex points downward, the three long edges within about two per cent of one another and set at 0°, 60° and 120°, the three corner cuts likewise alike. So three lines of symmetry, and angles of symmetry 120°, 240° and 360°.
Q8 and Q9, §9.2, p.239, with Chapter 1, §1.5, Table 3, p.10. The regular polygon sequence runs triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon. Lines of symmetry: 3, 4, 5, 6, 7, 8, 9, 10. Angles of symmetry: the same eight numbers again. Both questions therefore produce the counting numbers from 3, and the fact that they produce the same sequence is the observation section 11 explains.
Q10, §9.2, p.239 — the Koch Snowflake sequence. Read off the printed page of Chapter 1's Table 3 on p.10: the row prints five shapes, the first a plain triangle and the second a six-pointed star, the remaining three progressively more jagged versions of that star. So the first has 3 lines and 3 angles, and from the second onward the count is 6 and 6, because every later shape is built by treating all twelve edges of the star alike. Derive this rather than asserting it; the chapter prints no answer.
Q11, §9.2, p.239 — Ashoka Chakra. The wheel carries twenty-four spokes, so twenty-four lines of symmetry and twenty-four angles of symmetry, the smallest being 360 ÷ 24 = 15°. The count of spokes is a fact about the emblem, not a number the chapter prints; show the spokes being counted.
The Summary's last bullet, p.241. The chapter closes by naming three of the four cases explicitly: a figure may have a line of symmetry and no angle of symmetry, or angles and no lines, or both. This is the sentence the whole topic is built to earn.

Figures to have open

A two-by-two arrangement of the four cases, one figure in each cell. The single most important figure of the topic; the book does not print one and it must be built.
The windmill and the trapezium strip, redrawn from pp.230 and 232.
A circle with an arbitrary turn shown step by step, then with a family of diameters drawn in one at a time. Standard schematic.
The Parliament outline of p.239, redrawn as the corner-cut triangle with its three mirror lines added.
The regular polygon strip from Chapter 1's Table 3, each polygon carrying both counts.
The Ashoka Chakra with its spokes countable. Standard emblem art; the chapter prints a small version on p.239.
The Koch Snowflake row of Chapter 1's Table 3, p.10, redrawn so the first two terms are distinguishable.

Where this sits in the book

NCERT Class 6 Mathematics (Ganita Prakash), Chapter 9 "Symmetry", §9.2 Rotational Symmetry — the bold sub-heading on the symmetries of a circle, p.237
§9.2, p.237 — the fan, flower and wheel, and the remark that the circle is the exception to the smallest-angle rule
§9.2, p.238 — Figure it Out Q2 and Q3
§9.2, p.239 — Figure it Out Q7 to Q11
§9.2, pp.230 and 232 — the windmill and the strip, reused here as the two one-sided cases
Chapter 1, §1.5, Table 3, p.10 — the regular polygon and Koch Snowflake sequences the exercises point back to
Summary, p.241 — the closing statement that the two properties come apart