Fold, blot, cut, punch: making a figure that is symmetric by construction

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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 the definition of a line of symmetry
Symmetry as reflection: the fold line acts as a mirror: reflection — a point crossing the line to a partner
Why a figure can have several lines of symmetry: that a square has four lines of symmetry
Class 5 paper folding and ink-blot work, which the chapter says the student has already done

What they should be able to do

Explain why a fold made before a cut guarantees that the crease is a line of symmetry of the result
Predict the shape of the opened figure from a described fold and cut, before opening it
State how many holes a single punch makes after one fold and after two folds
Deduce, from the positions of holes in an opened square sheet, which line the paper was folded along
Given a figure's line or lines of symmetry and some of its holes, mark the remaining holes
Plan the folds and the one straight cut that will produce a stated hole
Assemble a design from square tiles to a specified number of lines of symmetry

Where it usually goes wrong

"You have to open the paper to know what you made." The whole point of folding first is that you do not. Sections 7 and 10 are the payoff.
"One punch makes one hole." Only in unfolded paper. The number of holes is the number of thicknesses, and that is what Q1(d) is testing.
"Two folds always give a hole in each corner." They give four holes, whose positions depend on where the punch went. Corner holes come from punching near the corner of the folded quarter.
"The fold has to be upright or flat." Q1(b) is a diagonal fold, and the diagonal of a square is a genuine line of symmetry — established in Why a figure can have several lines of symmetry.
"An ink blot is symmetric because ink spreads evenly." It is symmetric because of the press, not the ink. Unpressed, the same drops give nothing.
"Any four-sided hole in the middle is a square." The chapter's own note under Q5 warns against this: equal sides are not enough without equal angles.
"With two lines of symmetry I add one more copy." Two perpendicular lines turn one quarter into four. The tile grid on p.240 is the clearest picture of this, since three of its four blocks are blank.

Questions to check understanding

Predict and sketch the opened figure from a stated sequence of folds and cuts
State the number of holes a single punch produces after a stated number of folds
Identify the fold line from the positions of holes in an opened sheet
Complete a figure's holes given its line or lines of symmetry
Describe the folds and the single cut needed to obtain a stated hole
Make a design from a fixed number of tiles with exactly one, or exactly two, lines of symmetry
Justify why a four-sided hole is or is not a square

Examples worth working on the board

Ink Blot Devils, §9.1, p.222. Fold a sheet in half, open it, drop ink on one half only, press the halves together, open again. The chapter asks whether the result is symmetric and where its line of symmetry is. The answer: the crease, always, because every drop is pressed onto the point facing it. The chapter also asks whether there is a second fold line, and there generally is not — the blot is otherwise irregular.
Fold and cut, §9.1, p.223. Two pictures of a green sheet folded once with a dotted cutting line, and a hand with scissors. Students sketch the opened sheet. One cut across the crease leaves two matching notches; a cut that does not reach the crease leaves two separate holes.
The decorative strip, §9.1, p.223. A blue repeating design of eight-pointed stars with diamonds above and below, made by folding a rectangular strip several times and cutting once. Its fold lines are to be identified in the repeating design.
The Punching Game, §9.1, p.223. Fold a square sheet, punch holes through it, open it out. The printed illustration shows a folded sheet, the same sheet with a punched hole, and the opened sheet carrying two.
Q1, §9.1, p.224 — four opened sheets. Read from the printed page of p.224: (a) two holes side by side at the same height, placed symmetrically about the upright mid-line; (b) two holes close together near the top right, offset slantwise; (c) two holes one above the other, symmetric about the flat mid-line; (d) four holes, one near each corner. Worked: (a) an upright fold, (c) a flat fold, (b) a fold along a diagonal, (d) two folds — upright and flat, in either order, because a single punch has produced four holes.
The counting rule. One fold gives two thicknesses, so one punch gives two holes. Two folds give four thicknesses, so one punch gives four. Three folds give eight. The rule doubles each time and it is what makes Q1(d) answerable.
Q2, §9.1, p.224 — five figures. A square with a diagonal line, a rectangle with a flat line, a triangle with an upright line, and two circles each with a slanted line, each carrying one marked hole. Mark the partner hole in each. The circles are there so that the line of symmetry is a diameter and the partner is not in an obvious grid position.
Q4, §9.1, p.225 — four predictions. The picture-notation is fixed first: the upright fold under Q3 at the foot of p.224, the flat fold at the head of p.225. Q4 itself, with all four of its fold-and-cut sequences, is printed on p.225, and asks what the hole will look like before the paper is opened; the student cuts afterwards to check. Read off the printed page of p.225, sequence (d) runs: a plain yellow sheet, a rectangle noticeably taller than it is wide, not a square; the sheet folded in half into a tall strip; the strip with a rectangle marked off against each of its two upright edges and a scissors at work; and last, the folded strip after both cuts, its outline now waisted like a capital I. The panel the book stops at is the sheet still folded — the opened figure is the answer, and it is not printed.
Q5, §9.1, p.226. Two pink sheets, each with a square hole in the middle; in the first the hole sits square to the sheet, in the second it stands on a corner. Each has to be got by folding a few times and then making one straight cut. The printed note tells the student to check that the four-sided hole really is a square — equal sides and equal angles.
Playing with Tiles, §9.2, pp.239–240. An eight-by-eight grid, counted from the printed page of p.240, with the top left four-by-four block already filled with sixteen two-colour tiles and a red upright line and a red flat line drawn through the middle. Complete the grid so the whole figure has exactly two lines of symmetry: the two red lines determine the other three blocks by reflection. The follow-up asks for sixteen-tile figures with exactly one line and with exactly two.

Figures to have open

The ink-blot sequence, step by step. Standard schematic.
A folded sheet with a cutting line, and the opened result. Standard schematic.
The four opened squares of Q1, p.224, with the holes placed as printed. Must be redrawn from the page — the hole positions are the exercise.
The five figures of Q2, p.224, including both circles with slanted lines.
Q4's fold notation — the labelled upright-fold and flat-fold pictures of pp.224–225 — and at least sequence (d) worked through.
The two pink squares of Q5, p.226, one hole square to the sheet and one on its corner.
The eight-by-eight tile grid of p.240 with its filled top left block and the two red lines. Redraw; the colours and the placement carry the exercise.

Where this sits in the book

NCERT Class 6 Mathematics (Ganita Prakash), Chapter 9 "Symmetry", §9.1 Line of Symmetry — the sub-heading on generating shapes with lines of symmetry, p.222, and Ink Blot Devils on the same page
§9.1, p.223 — Paper Folding and Cutting, the decorative strip, and the Punching Game
§9.1, pp.224–226 — Figure it Out items Q1, Q2, Q3, Q4 and Q5
§9.2, pp.239–240 — Playing with Tiles, with the grid printed on p.240
Summary, p.241 — the settled statement of line of symmetry