Naming corners in order, and why a rotated square is still a square

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What to assume they know

The two properties that define a rectangle, and the one more a square needs — the two conditions for a rectangle and the two for a square
A point fixes a location; a segment is the shortest route between two points — a segment is named by its two endpoints
Reading a position off a grid of dots

What they should be able to do

Explain what makes an ordering of four corner labels a legitimate name
List the valid names of a given rectangle, and say why two given orderings are rejected
Decide, for a stated ordering, whether it names the figure shown
State what a rotation changes and what it leaves alone
Argue that a rotated square still satisfies both square conditions
Extend the same argument to a rotated rectangle without repeating the work
Judge whether a tilted figure on a dot grid is a square from the positions of its corners alone

Where it usually goes wrong

"ABCD just means the corners are called A, B, C and D." Then ABDC would name the same figure, and the chapter says it does not. The order carries information: consecutive letters are joined by a side.
"There is one correct name and the rest are wrong." There are eight, and the chapter prints all eight for one rectangle.
"You have to start at A and go clockwise." Any corner, either direction.
"A square standing on its corner is a diamond, not a square." Diamond is not a mathematical category. Run the two conditions: all sides equal, all angles 90°. Both still hold, so the figure is still a square.
"Turning it must change something." It changes which side is at the top. It changes no length and no angle — and lengths and angles are all the definition mentions.
"You can't tell on a dot grid without measuring." The chapter's own Think prompt asks precisely whether the corner positions settle it, and they do: two corners a fixed number of dots apart give a length you can compare without a ruler.
"A rotated rectangle needs its own separate proof." It does not, and the chapter says as much: the argument used for the square did not depend on the sides being equal.

Questions to check understanding

Write down all the names of this rectangle
Say why ABDC fails as a name for the rectangle ABCD
Which of PQSR, SPQR, RSPQ and QRSP fails to name the square shown? (the chapter's own p.193 item)
A square is turned through 45°. Is it still a square? Justify by the conditions, not by appearance
Draw three rotated squares on a dot grid with all corners on dots, and verify the conditions for each
Explain why the argument for a rotated square also settles the rotated rectangle
Given a tilted figure on a dot grid, decide whether it is a square without using any measuring instrument

Examples worth working on the board

The eight names of Fig. 8.4, p.193. For the rectangle labelled A top-left, B top-right, C bottom-right, D bottom-left, the chapter prints ABCD and then seven more. Going one way round: BCDA, CDAB, DABC. Going the other way: ADCB, DCBA, CBAD, BADC. Those are the four starting points taken in each of the two directions round the boundary.
The two rejects, p.193. ABDC and ACBD. In ABDC the step from B to D jumps across the figure rather than along a side; in ACBD the very first step does. That is the whole diagnosis, and it is worth making the explanation state it as a rule the student can apply rather than a list to memorise.
The count. Four letters can be arranged in 4 × 3 × 2 × 1 = 24 orders. Valid names are four starting corners times two directions, so 8. Sixteen of the 24 orderings name nothing. The chapter prints neither number; they are supplied here as inputs.
The multiple-choice square, p.193. A square is printed with S at the top left, P at the top right, R at the bottom left and Q at the bottom right — checked against p.193, since those labels are artwork. Going round it one way gives S, P, Q, R; the other way gives S, R, Q, P. The four options offered are PQSR, SPQR, RSPQ and QRSP. Three of them are rotations of one of those two walks; PQSR is not, because it steps from Q to S across the figure. PQSR is the odd one out.
The rotated square, p.193. The chapter prints a filled square upright and the same square turned so that it stands on a corner, then walks both square conditions again: sides still equal, angles still 90°. Checked against p.193 — both figures are artwork.
The rotated rectangle, p.194. Drawn on a slant at the top of the page, with the text noting that the same reasoning carries over. Checked against p.194.
The three Figure it Out tasks, p.194. Reproduce the Fig. 8.3 arrangement on dot paper with the four squares placed symmetrically; identify which of four tilted figures on a dot grid are squares; and draw at least three rotated squares and rectangles with every corner landing on a dot. The middle task is the one that carries this topic — see The two properties that define a rectangle, and the one more a square needs for the reading of those four figures.

Figures to have open

Fig. 8.4 with its four labels, and a movement of the boundary walk. Redraw.
The 24-ordering grid for section 4. Not in the book; an added device.
The printed multiple-choice square with S, P, R, Q in the positions given above. The label positions matter to the answer, so this must be drawn faithfully.
The upright square beside the same square turned onto its corner, plus a slanted rectangle. Redraw rather than reproduce the printed art.
A dot grid with a tilted square on it, dots visible and countable.
No photograph or dataset is required.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 8 "Playing with Constructions", §8.2 "Squares and Rectangles", p.193 — the list of names, the two rejects, the order-of-travel rule, the multiple-choice square, and the sub-heading "Rotated Squares and Rectangles" with the turned square beneath it
§8.2, p.194 — the rotated rectangle and the three Figure it Out tasks
§8.5, pp.203 and 207, where the naming convention is relied on: the diagonal written PR, and the equal-sides step written AD = BC