Why a figure can have several lines of symmetry

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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 exactly overlap rules out
Shapes come in sequences too, with rules of their own: the shape sequences of Chapter 1, and what regular means for a polygon — its sides all equal and its angles all equal
Knowing a square, a rectangle, a triangle and a hexagon by sight, and what a diagonal of a quadrilateral is

What they should be able to do

Find every line of symmetry of a square by folding, and say how many there are
Explain why a square's diagonal is a line of symmetry and a non-square rectangle's diagonal is not
State the number of lines of symmetry of a regular polygon with a given number of sides, and justify it
Draw triangles with exactly three, exactly one, and no lines of symmetry
Argue that no triangle has exactly two lines of symmetry
Produce figures with a curved boundary having a stated number of lines of symmetry
Count the lines of symmetry of a traditional design or a paper cut-out

Where it usually goes wrong

"If one diagonal is a line of symmetry, every diagonal is." The square and the rectangle sit on the same page of the book precisely to break this.
"A rectangle has four lines of symmetry, like a square." The two mid-lines work; neither diagonal does. Two, not four.
"More sides always means more symmetry." True only for regular polygons. An irregular hexagon can have none. The word regular is doing all the work in section 8.
"Turning the figure changes its lines of symmetry." The square standing on a corner (Q6a, p.226) still has four; only their directions on the page move.
"Exactly two lines of symmetry is impossible." It is impossible for a triangle, which is what Q9 asks. A rectangle has exactly two. Keep the scope of the impossibility explicit or students over-generalise it.
"Curved figures cannot have lines of symmetry." Q10 exists to refute this; the disc, the oval and the rosette are all fold-testable.

Questions to check understanding

State the number of lines of symmetry of a named regular polygon
Decide, with reason, whether a given diagonal is a line of symmetry
Draw a triangle to a stated number of lines of symmetry, or say it cannot exist
Trace a printed figure and draw all its lines of symmetry
Complete a partly drawn figure so that two given lines both become lines of symmetry (the pattern of §9.1, p.229, Q12)
Add two segments to a figure on a dot grid so the result has a line of symmetry (the pattern of §9.1, p.230, Q13)
Count the lines of symmetry in a kolam, a rangoli or a folded-paper cut-out

Examples worth working on the board

The square-folding sequence, §9.1, p.220. Four photographs of one sheet, labelled Fold 1 to Fold 4, followed by two more labelled Vertical Fold and Horizontal Fold. The chapter's instruction is to fold upright, then flat, then along one diagonal, then along the other, opening the paper each time. Show the movement of the creases accumulating on one square; the four creases together are the answer.
The square's count. The chapter asks how many lines of symmetry a square has and does not print the number. Derive it: a fold line of the square must carry corners onto corners, and there are exactly four lines that do — the two through opposite side midpoints and the two through opposite corners. So four.
The rectangle and its diagonal, §9.1, p.221. The book prints a plain rectangle with a dotted diagonal and asks the student to predict, then fold and check. A rectangle 8 cm by 3 cm. At the corner where the fold starts, one side is 8 cm and the other is 3 cm, so the long side cannot land on the short one. Fold and a strip hangs out.
The rectangle's own count. The upright and the flat mid-lines both work, so a non-square rectangle has two lines of symmetry. The chapter asks only about the diagonal; the count of two is a derivation to show, not a statement to attribute to the book.
Three figures printed together, §9.1, p.221. A red spiky closed outline of a snowflake kind; a yellow flower of identical petals about a centre; a green interlaced band forming a square with a rounded loop at each corner. The chapter asks the student to find all their fold lines and prints no answers. Do not quote counts for these three; count them live.
Q6, §9.1, pp.226–227. Three parts. (a) a square drawn resting on a corner, beside an eight-pointed star; (b) a triangle whose three sides are all equal and whose three angles are too; (c) a hexagon described the same way. Worked: the square has 4 whatever its orientation on the page, the eight-pointed star has 8, the triangle 3, the hexagon 6.
Q8, §9.2, p.239, with Chapter 1, §1.5, Table 3, p.10. The regular polygon sequence there runs triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon. Their fold-line counts are 3, 4, 5, 6, 7, 8, 9, 10 — the counting numbers from 3, which is the very sequence Chapter 1 attaches to the number of sides (§1.6, p.11).
Q9, §9.1, p.228. Draw a triangle with exactly one fold line, one with exactly three, one with none — then decide whether exactly two is possible. The argument to show: a fold line of a triangle forces two of its sides to be equal; a second, different fold line forces a second pair equal; two pairs among three sides overlap, so all three sides are equal, and then there are three fold lines, not two. Two is therefore impossible for a triangle — though not for figures generally, since a rectangle has exactly two.
Q10, §9.1, p.228. Figures with at least one curved edge and exactly one, exactly two, exactly four fold lines. Workable answers: a half-disc for one, an oval for two, a four-petal rosette for four.

Figures to have open

One square that accumulates four creases. Indispensable; must show.
A rectangle with a marked diagonal and a visible overhang after the fold. Standard schematic, best drawn to the 8 cm by 3 cm numbers.
The three multi-line figures of p.221 — spiky outline, petalled flower, interlaced square knot. Redraw from the printed page; their shapes are the exercise.
The regular polygon strip, triangle through decagon, taken from Chapter 1's Table 3 rather than invented, so the two chapters agree.
Three triangles: equal-sided, two-sides-equal, all-sides-different.
A half-disc, an oval and a four-petal rosette for the curved cases.

Where this sits in the book

NCERT Class 6 Mathematics (Ganita Prakash), Chapter 9 "Symmetry", §9.1 Line of Symmetry — the bold sub-heading that opens the material on several fold lines at once, pp.220–221
§9.1, p.221 — the three multi-line figures and the rectangle-diagonal question
§9.1, pp.226–230 — Figure it Out items Q5 to Q13
§9.2, p.239, Q8 — regular polygons, pointing back to Chapter 1, §1.5, Table 3, p.10, and the counting-number sequence of §1.6, p.11
Summary, p.241 — the statement that a figure may have several lines of symmetry