Constructing a rectangle from one side and a diagonal

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What to assume they know

Constructing a square or rectangle from its side lengths — constructing a rectangle from its two side lengths
Why a rectangle's diagonals are equal, and when they split the corners evenly — the diagonals, and what they do to the corners
Every curve in these figures is part of a circle — a circle is every point at a fixed distance from its centre
Working out the order in which a figure has to be drawn — a rough diagram is drawn before the construction
Reading and drawing an angle with a protractor — setting an angle of a stated size

What they should be able to do

Draw a rough diagram for a construction whose data are an angle and a side
Construct a rectangle in which a diagonal makes stated angles at the corners it runs between
Give two different ways of placing the last corner, each justified by a different rectangle property
Explain why trial and error is unsatisfactory even when it succeeds
Replace the search for one point by the drawing of a whole set of points
Construct a rectangle from one side and a diagonal
Say when such a rectangle cannot exist, and why
Check a completed construction against R1 and R2

Where it usually goes wrong

"You draw the diagonal first, because it is the longest." You cannot: at the start you know its length but neither of its endpoints. The rough diagram is what shows you that.
"Sliding the ruler around until it reads 7 cm is a perfectly good method." It gets an answer and teaches nothing, and it is unrepeatable. The chapter names it and then replaces it.
"The circle is an extra flourish." The circle is the answer to the question "where could B be?", and drawing it converts a search into a crossing.
"Drawing the whole circle is wasteful, so the method is clumsy." The chapter agrees, and prints the arc version immediately afterwards. Wanting less of the circle is the right instinct; wanting none of it is not.
"Any side and any diagonal give a rectangle." Only if the diagonal is longer than the side. Equal, and the "rectangle" collapses; shorter, and the circle misses the line entirely.
"A 45°/45° diagonal gives a special rectangle." It gives a square, and it gives one for a reason: equal halves at a corner force the two sides there to be equal.
"Once the fourth point is marked, the job is done." The chapter ends by asking for a check against R1 and R2, p.210.

Questions to check understanding

Construct a rectangle in which a diagonal makes 60° and 30° at the corners it runs between (worked at p.205)
Do the same for 50° and 40° (asked at p.211)
Do the same for 45° and 45°, and say what happens to the sides (asked at p.211)
Construct a rectangle with one side 5 cm and a diagonal 7 cm (worked at p.208)
Construct one with a side of 4 cm and a diagonal of 8 cm (asked at p.211)
Construct one with a side of 3 cm and a diagonal of 7 cm (asked at p.211)
Give two different ways of locating the fourth corner, and name the property each one uses
A side of 6 cm and a diagonal of 5 cm are given. What goes wrong?

Examples worth working on the board

The 60°/30° rectangle, pp.205–207. Rough diagram, checked against p.205: A bottom-left, B bottom-right, C top-right, D top-left, the diagonal AC drawn, 60° marked between AB and AC and 30° between AC and AD, and a right-angle mark at D. The construction runs: draw AB at any length and raise a perpendicular at B; set a 60° ray from A and let it meet that perpendicular — that point is C; draw the perpendicular to AB through A, which is the line D must sit on; then place D either by dropping a perpendicular to BC at C, or by taking BC in the compass and marking AD equal to it. Checked against pp.205, 206 and 207.
The leftover angle, p.206. Once the 60° is set, the chapter asks what the other part of the corner comes to. It is 30°, because the whole corner is a right angle.
Two ways to finish, p.207. Method 1 uses R2: all the angles are right angles, so a perpendicular at C finds D. Method 2 uses R1: opposite sides are equal, so AD can be copied from BC with a compass. Naming which property each method leans on is the point.
The 5 cm and 7 cm rectangle, pp.208–210. Rough diagram, checked against p.208: A top-left, B top-right, C bottom-right, D bottom-left, with DC = 5 cm along the bottom and the diagonal DB = 7 cm. The construction: draw DC = 5 cm; raise the perpendicular to DC at C and call it l; B must be on l and must also be 7 cm from D; draw the circle of radius 7 cm centred at D; B is where circle and line cross; then perpendiculars at D and at B meet at A. Checked against pp.208, 209 and 210.
The arc instead of the circle, p.210. Only the part of the circle near l was ever needed, so Method 2 draws an arc. Same point, less ink — and the same economy reappears in §8.6.
The other side. The chapter never asks for it. For information only: with a side of 5 cm and a diagonal of 7 cm, the other side comes out at just under 5 cm — about 4 cm 9 mm.
The four practice constructions, p.211. A diagonal making 50° and 40°; a diagonal making 45° and 45°, with the question of what the sides do; one side 4 cm with diagonal 8 cm; one side 3 cm with diagonal 7 cm. The 45°/45° case produces a square and is the payoff of Why a rectangle's diagonals are equal, and when they split the corners evenly.
When it cannot be done. The circle centred at D only reaches the line l if its radius exceeds DC. Given a side of 5 cm and a "diagonal" of 4 cm, the circle never gets there and no such rectangle exists. The chapter does not raise this; it is worth attention because it explains what the circle is doing.

Figures to have open

The two rough diagrams, drawn unscaled and visibly so.
The full 60°/30° construction sequence, four steps, with both finishing methods.
The circle-meets-line figure. This is the topic's key image and must show the circle large enough that the crossing reads as a crossing.
The arc version beside the circle version.
A failure case: a circle whose radius is too small to reach the line.
No photograph or dataset is required.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 8 "Playing with Constructions", §8.5 "Exploring Diagonals of Rectangles and Squares", pp.205–211 — the 60°/30° construction on pp.205–207, the side-and-diagonal construction on pp.208–210, and the four practice items on p.211
§8.1, pp.188–189, for the circle as the set of all points at a fixed distance
§8.2, p.193, for R1 and R2, which the two finishing methods rely on
Summary, p.216, for the statement that a rectangle can be constructed from one side and a diagonal