What the diagonals alone tell you about a quadrilateral

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What to assume they know

The carpenter's problem: how to be sure a frame really is rectangular — equal diagonals bisecting each other produce a rectangle, and the isosceles-triangle argument behind it
Properties of a rectangle, and why the square is the special case — the square's diagonals, equal and crossing at 90°
The parallelogram: everything that follows from "opposite sides parallel" — the parallelogram, whose diagonals halve each other but need not match in length
The rhombus, and what its diagonals do — the rhombus, whose diagonals halve each other at 90°
Constructing a segment of given length and setting two segments at a given angle
Reading a dot grid, and using it in place of a geoboard

What they should be able to do

Explain why fixing the two diagonals fixes the quadrilateral, and hence why diagonal conditions can serve as definitions
Set up the three-question table — equal, halving, perpendicular — and place the rectangle, square, parallelogram and rhombus in it
Predict the shape produced by two equal bands crossed at right angles through their midpoints, and justify the prediction
Predict what changes when one of those bands is lengthened equally at both ends, and say which of the three conditions was broken
Construct a quadrilateral from stated diagonal lengths and a stated crossing angle
Decide which diagonal conditions are enough to pin a shape down and which leave more than one possibility, and give a counter-example for the ones that do
Explain why the kite sits outside the table's neat rows, having only one of its diagonals halved

Where it usually goes wrong

"You need the sides to know the shape." You do not. Two crossed segments fix all four corners, so the diagonals carry the same information as the sides — differently packaged.
"Perpendicular diagonals mean a rhombus." The kite has them too. What the rhombus adds is that both diagonals are halved; in the kite only one is. This is exactly why the chapter sets it as a true-or-false item.
"Equal diagonals mean a rectangle." Only if they also halve each other. Equal diagonals alone are cheap.
"Stretching one diagonal ruins everything." It changes one condition. Both are still halved at the crossing, and the crossing is still square, so the result is still a named shape — a different one.
"The 2 cm is arbitrary." The number is arbitrary; the both ends is not. If the band were stretched at one end only, the crossing would no longer be at its middle and the answer would change again.
"A geoboard is a toy." Every claim it makes has already been proved earlier in the chapter. The board's job here is to make a proved statement testable in five seconds, which is the chapter's method from Part I p.88 applied.
"A square and a rhombus have the same diagonal description." The rhombus entry has no equality condition. Adding it produces the square, and that is the whole difference.

Questions to check understanding

Construct a quadrilateral from stated diagonal lengths and a stated crossing angle, and name what you get
Name the shape and justify it, given only a description of the diagonals
True or false with justification, on diagonal conditions — Part I pp.108–109, item 11(i), (iii) and (iv)
Identify the figure formed by two perpendicular diameters of a circle (Part I p.94, item 3)
Produce an exact right angle from two equal sticks and a thread, and explain why it is exact (Part I p.94, item 4)
Given a shape name, state the diagonal conditions that characterise it — the reverse direction, which students find much harder
Explain what changes in the shape when one diagonal is lengthened equally at both ends

Examples worth working on the board

The claim that makes the topic possible (Part I, §4.1, p.93). The chapter states, while finishing the square construction, that the ends of the diagonals settle the corners of the quadrilateral. Everything in the explanation rests on that sentence.
Geoboard, first arrangement (Part I, §4.5, p.103). Two rubber bands of the same length, laid perpendicular to each other, and then the four ends joined. The chapter's photograph shows the two bands crossing as an upright cross on a square pegboard. The question printed underneath asks what quadrilateral appears and demands a justification. Input only.
Geoboard, second arrangement (Part I, §4.5, p.103). Starting from the first, one of the two bands is stretched out by 2 cm at each end. The second photograph shows the cross with one arm longer. Again the question asks for the shape and a justification. Input only. The 2 cm at both ends is the load-bearing detail: the crossing point stays in the middle of the stretched band.
The diagonal facts already proved in the chapter, which the table is built from — hand all of these to the teacher with their pages: rectangle, diagonals equal and each halving the other (Part I p.90, Property 4); square, the same and crossing at 90° (Part I p.93, Property 4); parallelogram, diagonals halving each other (Part I p.99, Property 4), and not necessarily equal (Part I p.98); rhombus, diagonals halving each other, crossing at 90°, and halving the corner angles (Part I pp.101–102, Properties 4, 5 and 6); kite, one diagonal halving the other at right angles and halving two of the corner angles (Part I p.105, Property 1).
Construction tasks stated as diagonal data. Part I p.94, item 2: both diagonals 8 cm, halving each other, crossing at 30°, at 40°, at 90° and at 140°. Part I p.102, item 2: a parallelogram from diagonals 7 cm and 5 cm crossing at 140°. Part I p.102, item 3: a rhombus from diagonals 4 cm and 5 cm. Part I p.107, item 2: a kite from diagonals 6 cm and 8 cm. Part I p.108, item 6: a square from a 6 cm diagonal, with no protractor.
The circle version (Part I, §4.1, p.94, item 3). A circle centred O with two perpendicular diameters PL and AM; identify the quadrilateral APML. This is the same table entry arriving in different clothing — two diameters of one circle are automatically equal and automatically halved at the centre.
The true-or-false list (Part I, §4.6 exercise, pp.108–109, item 11). Seven statements, of which four are about diagonals or angles that this table settles: (i) diagonals equal and halving each other force a square; (iii) diagonals halving each other force a parallelogram; (iv) perpendicular diagonals force a rhombus; and, for contrast, (v) equal opposite angles force a parallelogram. Give all seven.
The two sticks (Part I, §4.1, p.94, item 4). With two sticks of the same length and a thread, and no paper, produce an exact right angle. The intended route runs through this table.

Figures to have open

A single master diagram used throughout: two crossing segments with the crossing point marked, the four half-lengths labelled, and the crossing angle marked. Every named shape in the explanation is this one figure with different marks switched on.
The two geoboard arrangements of Part I p.103. Redraw them as dot grids with two straight bands rather than reproducing the photographs; the second must show the extension at both ends.
The three-question table itself, filled in row by row, with an empty row for the kite that never gets a tick in the second column. This is an added figure. The chapter states each diagonal fact under the shape it belongs to, and its summary (Part I pp.109–110) repeats them the same way, as a bullet list per shape rather than as a grid.
A circle with two perpendicular diameters and the four ends joined (Part I p.94, item 3).
Side-by-side rhombus and kite with their diagonals marked, for section 10.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 4, "Quadrilaterals", §4.5 "Playing with Quadrilaterals", unnumbered bold subheading "Geoboard Activity", Part I p.103 — one page, two arrangements, two questions.
The statement that the ends of the diagonals settle the corners: Part I p.93.
The diagonal properties gathered from earlier sections: Part I p.90 (rectangle), Part I p.93 (square), Part I pp.98–99 (parallelogram), Part I pp.101–102 (rhombus), Part I p.105 (kite).
Construction tasks phrased in terms of diagonals: Part I p.94 items 2 and 3; Part I p.102 items 2 and 3; Part I p.107 item 2; Part I p.108 item 6.
The true-or-false list: Part I pp.108–109, item 11.
Chapter summary of diagonal properties: Part I pp.109–110.