Estimating the area of a shape with no formula

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What to assume they know

Area as a count of unit squares: area as a count of unit squares, and the rectangle rule as a fast way of counting
Perimeter as the distance all the way round: the same dot grid used for perimeter, and its straight and diagonal units
Halves of a whole, and adding halves to whole numbers

What they should be able to do

Explain why some closed shapes have no area rule and still have an area
Trace a shape onto squared paper and count its squares
Apply each of the book's four conventions to a part-covered square and say which way it errs
Argue why the four rules together give an estimate that is close rather than merely arbitrary
Say what would make an estimate closer, and why
Recognise when counting on a grid stops estimating and becomes exact
Find the exact area of a figure drawn on a dot grid with straight and diagonal edges
Report an area with its unit, and describe it as an estimate when it is one

Where it usually goes wrong

"Estimating means guessing." The chapter separates the two on the same page: the guess comes first with no method, the estimate second with four stated rules. Keep both words and keep them apart.
"An estimate is a wrong answer." It is an answer with a known way of being improved. A guess is not.
"Rule 3 is cheating, because you are counting space that is not there." So is rule 2, in the other direction. Show them as a matched pair or neither makes sense.
"Half a square should be rounded up like anything else over half." Exactly half is the one case where no rounding is needed, so the book takes it exactly.
"Counting squares always gives an estimate." Not when every corner of the figure sits on the grid. The four dot-grid figures have exact areas, and the Summary says so in its own words.
"The spiky shape must be smaller because it has all those points." The points are where students' intuition fails, which is why the book puts the guess before the method rather than after it.
"A shape with a bigger perimeter has a bigger area." Section 9 puts the two measurements of one figure side by side and they behave independently — see Same area, many perimeters: why one does not determine the other, which is built on that.
"You need a formula, so shapes like these cannot be done." They can, and every practical land, floor and playground measurement in the chapter is done this way.

Questions to check understanding

Estimate the area of an irregular closed shape drawn on squared paper
Apply the four conventions correctly to a marked boundary square
Find the exact area of a figure whose corners lie on grid points
Say whether a stated area is exact or an estimate, and why
Explain what would make an estimate closer
Split a rectilinear figure into rectangles and total the parts
Compare two shapes by area when neither has a formula
The answer key appended to the cached PDF gives the four dot-grid areas on its footer page 3 and gives no answer to the guess question above them

Examples worth working on the board

The two blobs (p.140). Two closed outlines are printed side by side and the reader is asked to guess which encloses more. Read off the printed page: shape a. is a smooth wandering curve drawn in green, shape b. is a spiky star-like outline drawn in red with about nine points. The book prints no answer and the appended answer key gives none. Do not resolve the guess when explaining it; the guess exists to make the reader want a method.
The four conventions (p.140). In the book's order: one whole small square of the sheet counts as 1 sq unit; a part smaller than half a square is ignored; a part larger than half counts as a full 1 sq unit; a part that is exactly half counts as one half. Show them as four decisions about the same boundary square, and make the direction of each error visible — rule 2 under-counts, rule 3 over-counts, rule 4 is exact.
The tracing method (p.140). The shape is traced onto transparent paper and the tracing laid over squared or graph paper. Worth showing, because it is what makes the method usable on something that is not already drawn on a grid.
The four dot-grid figures (p.140). The same four figures that appeared on p.135 for perimeter come back here to be measured for area. Reconstructed from the printed page and worked out independently by coordinates, they are: purple, a flag-like figure with two sloping edges, 4 sq units; pink, a heart-shaped nine-sided figure, 9 sq units; green, a wide cup with two triangular notches, 10 sq units; dark red, shaped like a capital N, 11 sq units. The same four values appear in the answer key appended to the cached PDF. All four have their vertices on the dots, so these are exact areas, not estimates — that is the point of section 8, and it is the bridge to the triangle work of §6.3.
The same figures round the edge (p.135). Their perimeters, written in the book's straight-and-diagonal shorthand, are 8s + 2d, 4s + 6d, 12s + 6d and 18s + 6d in the same order. I derived these from the printed page as well and they agree with the appended key. Section 9 should show area and perimeter of one figure together: two different numbers, two different units, one drawing.
Where estimation earns its keep (pp.138, 141, 150). Question 4 on p.138 splits a rectilinear outline into rectangles and gets an exact answer, because the boundary follows the grid. The two tasks on p.141 — the floor outside the corridor, and the school playground — are real regions where an estimate is the only honest answer. The Summary on p.150 keeps both possibilities in one sentence.
A closeness argument to run. Take one of the printed blobs, count it on the given grid, then count it again on a grid of half the spacing. The boundary squares shrink and the estimate tightens. This demonstration is added here; the book does not print a second grid.

Figures to have open

The two blob outlines from p.140, drawn as printed — one smooth, one spiky — so the guess is a real guess. Redraw; the shapes are the book's and the comparison depends on them.
A single boundary square shown four ways, one per convention, with the part covered shaded. This is the topic's central figure. Standard schematic.
The four dot-grid figures on their lattice, the same four the perimeter topic uses. They must be the same drawings in both videos.
A blob counted on two grids, coarse and fine, for the closeness argument. Standard schematic; not in the book.
No photograph or textbook data table is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 6 "Perimeter and Area", §6.2 "Area", p.140 — the two shapes and the guess, the tracing method, the four numbered conventions and the four dot-grid figures
§6.1, p.135, for the same four figures measured round the edge, in straight and diagonal units — see Perimeter as the distance all the way round
§6.2, p.138, question 4, and §6.2, p.141, for the corridor and playground tasks
Chapter Summary, p.150, for the book's own sentence about regions being estimated or determined exactly by breaking them into unit squares
Solutions appendix bound into the cached PDF after the book's last printed page, footer page 3