The rectangle and square formulas are shortcuts for the same addition

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What to assume they know

Perimeter as the distance all the way round: perimeter as the length of the whole way round
A point fixes a location; a segment is the shortest route between two points: naming a figure by its vertices, so that AB means a particular side
That a rectangle's opposite sides are equal and a square's four sides are all equal
Multiplication as repeated addition, and taking a common factor outside a bracket

What they should be able to do

Derive the rectangle rule by adding the four sides and collecting the repeats
Derive the square rule the same way and say which property of the square is doing the work
State both rules in words and use them on given measurements
Use a rule in reverse: find a side from a perimeter, or from a perimeter and one other side
Explain why a perimeter alone does not determine a rectangle's two sides
Recognise that a length of wire or string bent into a new shape keeps its length but changes its enclosed region
Explain why cutting a rectangle and rejoining the pieces changes the boundary length, and predict the direction of the change
Say when a formula stops applying and the definition must be used instead

Where it usually goes wrong

"Perimeter of a rectangle is length × breadth." The single most common wrong answer in this chapter, and it is a collision with the area formula two sections later. Pre-empt it here, while only one measurement is in play.
"2 × (length + breadth) means 2 × length + breadth." Show the bracket arriving from the doubling, so it is never an unexplained mark.
"A formula is something new to learn." Sections 3 and 6 exist to show the rules being built out of the definition. A student who can rebuild them does not need to remember them.
"Knowing the perimeter tells you the rectangle." It fixes length + breadth and nothing more; many rectangles share a perimeter. Section 9 should show two or three of them side by side. This is the doorway to Same area, many perimeters: why one does not determine the other.
"Cut a shape in two and the boundary lengths must still add to the original." Cutting makes two brand-new edges; rejoining hides some edges and exposes others. The four arrangements on p.136 have the same two pieces and different boundaries.
"Every closed figure has a perimeter formula." The L, the T and the offset pair on p.136 have none. You add the sides.
"A square is not a rectangle, so its rule is a different kind of rule." The square rule is the rectangle rule when the two measurements happen to be equal; showing 4 × s falling out of 2 × (s + s) is a thirty-second movement and settles the point.

Questions to check understanding

Compute a rectangle's or a square's perimeter from its measurements
Find the missing side from a perimeter
Bend a fixed length of wire or string into a different figure and find its side
Cost of fencing: perimeter × rate, sometimes × number of rounds
Decide whether a described quantity is a perimeter or an area before computing
Find the boundary length of a figure assembled from two rectangular pieces
Construct a figure with a stated perimeter from given pieces
Find the perimeter of a border inset by a stated margin from a rectangle whose sides the student measures themselves
The answer key appended to the cached PDF answers the p.132 set across its footer pages 1 and 2; it gives no answers for the Split-and-rejoin arrangements on p.136 and none for question 6 on p.149

Examples worth working on the board

Rectangle ABCD (pp.129–130). Printed as a plain outlined rectangle with A top-left, B top-right, C bottom-right, D bottom-left, 12 cm marked along the top and 8 cm down the left. The book's chain of working is worth working through line by line, because each line is one small idea: the four named sides; the pairing of equals; the two doublings; the common factor taken out; the numbers put in. Printed answer: 40 cm. The side panel that licenses the pairing states that a rectangle's opposite sides are always equal.
Debojeet's photo frame (p.130). A square frame for a photo, 1 m along each side, with coloured tape wanted all round it. It is drawn as a plain outlined square with 1 m marked above it. The book adds 1 m four times first, gets 4 m, and only then replaces the addition by 4 × 1 m. Keep that order — the multiplication has to be shown arriving as a consequence, not as a rule handed down.
The tablecloth (p.131). 3 m by 2 m, lace all round. Printed answer 10 m, worked through the rectangle rule.
Usha's rounds (p.131). A square park with 75 m sides, three rounds. Printed working: one round 300 m, three rounds 900 m. This is the square rule used inside a bigger computation.
Reverse questions, p.132. (1a) perimeter 14 cm, breadth 2 cm, length wanted; (1b) perimeter 20 cm, side of a square wanted; (1c) perimeter 12 m, length 3 m, breadth wanted; (2) a wire bent as a 5 cm by 3 cm rectangle, straightened, then bent as a square. Question 2 is the best single item in the section for the argument of this topic: the wire's length is fixed, so the two rules have to agree about it.
Cost and rope, p.132. (4) a 150 m by 120 m park fenced at ₹40 per metre; (6) a 230 m by 160 m field with three rounds of rope, drawn as a three-rail fence with a farmer carrying rope.
Split and rejoin (p.136). A paper chit 6 cm by 4 cm is cut into two equal pieces; the thumbnail shows the chit standing with 4 cm across and 6 cm down and the cut running the long way, so each piece is 6 cm by 2 cm. Arrangement a. lays the two pieces end to end, 6 cm and 6 cm side by side with 2 cm marked as the height, and the book states its perimeter is 28 cm. Three further arrangements are drawn and their boundaries asked for: b. an L, one piece lying flat with the other hanging down from its right-hand end, 2 cm marked at the foot; c. a T, one piece standing upright on the middle of the other, with 2 cm marked on each side of the upright; d. two upright pieces side by side, the right-hand one dropped by 3 cm relative to the left. The section closes by asking for an arrangement whose perimeter is 22 cm. Is the only value the book gives.
The border on your own page (p.149, question 6). The reader rules a rectangular border on a page of their own book — inset by 1 cm at top and at bottom, and by 1.5 cm at each side — and finds its perimeter. The book prints no page size, so the sides have to be measured first and then reduced by 2 cm in one direction and 3 cm in the other before the rule is applied. There is no single printed answer and the appended key skips the question, going straight from 5 to 7. That is a feature: like the estimate-then-measure task on p.134, the find-objects-around-you task on p.136 and the corridor and playground questions on p.141, it asks the student to supply their own measurements, and it makes the rule something they use rather than something they are tested on. Do not call it the only such item — it is one of several.

Figures to have open

Rectangle ABCD as printed: vertices lettered, 12 cm along the top, 8 cm down the side. Standard schematic; must carry the letters, because the book's working is written in them.
The square frame with all four sides labelled 1 m. Standard schematic.
The four Split-and-rejoin arrangements from p.136, drawn from the same two 6 cm by 2 cm pieces, with the hidden join edge visible as a dotted line in each. This one must match the book's four joins, since the exercise is to compare them. Redraw.
A strip showing several rectangles that share one perimeter, for section 9. Standard schematic; not in the book at this point.
No photograph or textbook data table is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 6 "Perimeter and Area", §6.1 "Perimeter", pp.129–132 and p.136 — the sub-headings "Perimeter of a rectangle" (pp.129–130), "Perimeter of a square" (p.130), the two worked examples (p.131), the Figure it Out set (p.132) and "Split and rejoin" (p.136)
§6.3, p.149, question 6 — the inset border, a perimeter item printed among the chapter's closing area exercises
Chapter Summary, p.150, for the book's closing statement of both rules, where it says width in place of §6.1's breadth
Companion topics Perimeter as the distance all the way round (the definition these rules compress) and Triangles and regular polygons: when equal sides let you multiply (the same collapse for triangles and regular polygons)
Solutions appendix bound into the cached PDF after the book's last printed page, footer pages 1 and 2, for the p.132 exercises