Why a triangle takes exactly half the rectangle around it

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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
Estimating the area of a shape with no formula: counting exactly on a grid when corners land on grid points
A point fixes a location; a segment is the shortest route between two points: naming a figure by its vertices, so that ABE means a particular triangle
What a diagonal of a rectangle is — the chapter's Teacher's Note asks the teacher to recall this before starting

What they should be able to do

Cut a rectangle along a diagonal and check that the two triangles coincide
State that each triangle from such a cut is half the rectangle, and say why
Find the area of a right-angled triangle drawn on a grid by halving its rectangle
Drop a line from a triangle's top corner to its base and name the two rectangles it creates
Show that the two half-rectangles add to half the enclosing rectangle
Explain why two triangles of very different shape can have identical areas
Find the area of a grid figure by cutting it into rectangles and triangles
Say what the chapter has and has not established about triangles in general

Where it usually goes wrong

"Half only works for a right-angled triangle." That is where the argument starts and not where it ends. Sections 7 to 9 exist to carry it to a triangle whose top corner is not above a corner.
"Sliding the top corner along the top edge changes the area." It does not. The rectangle stays the same and so does the half. This is the most valuable single idea in the section, and it is what the red-and-blue pair is printed to show.
"The taller-looking triangle has more area." The two triangles on p.143 look very different and measure the same. The book puts that observation in a character's mouth for a reason.
"The two pieces from a diagonal cut are roughly equal." They coincide. The cutting activity is there to replace roughly with exactly.
"You need to multiply the base by the height and halve it." The chapter never writes that, and a student who has followed the halving argument does not need it. If the explanation reaches for it, the section's own reasoning is lost.
"A triangle is half of any rectangle you draw around it." Only of the one built on the same base and reaching the same height. Draw a wrong rectangle once and let the count refute it.
"Triangle ABE has to be handled by a new rule because it is not right angled." It is handled by using the old rule twice. That is the entire argument.
"A figure that is not a rectangle and not a triangle cannot be measured." The p.144 exercise is five such figures, and every one falls to cutting.

Questions to check understanding

Find the area of a right-angled triangle drawn on a grid
Find the area of a triangle whose top corner is not above a corner of its rectangle
Show that two triangles on the same base between the same pair of parallel grid lines have equal areas
Find the area of a compound grid figure by cutting into rectangles and triangles
Justify, in words, why a triangle is half of its rectangle
Given a rectangle's area, state the area of a triangle cut from it by a diagonal
The answer key appended to the cached PDF answers the p.144 exercise on its footer page 4; it gives nothing for the blanks on pp.142–144, which are the reader's own to fill

Examples worth working on the board

The paper cut (p.142). Draw a rectangle, draw one diagonal, cut along it, and check whether the two triangles cover each other exactly. Then repeat with rectangles of other sizes, and with a square. The reader is given a ruled line to write the inference on and the book prints nothing on it. The repetition is the point — the section is teaching that a relationship worth believing is one that survives being tried again.
The two coloured figures (p.142). A blue rectangle with one diagonal drawn across it, and a yellow triangle with a line drawn from its top corner straight down to its base. The reader is asked whether the rectangle's area is more, less, or the same as the triangle's, and then to write the relationship down; both answer lines are printed blank. Measured on the printed page, the triangle's base is about twice the rectangle's length and the two are about equally tall, so the intended answer is that they match — but the book does not say so.
Rectangle ABCD on the grid (p.143). A is the bottom-left corner, B the bottom-right, C the top-right, D the top-left. Read off the printed page, the rectangle is 5 grid squares across and 4 down, so 20 square units. E sits on the top edge, 3 squares from D. F sits on the bottom edge directly below E. The left part AFED is shaded green and is 3 by 4; the right part FBCE is shaded yellow and is 2 by 4. The blue triangle is BAD and the red one is ABE.
The two triangles (p.143). Both come to 10 square units, which is half of 20. The book asks for both areas, prints blanks for them, then states in words that BAD is half of ABCD. For ABE it prints the chain in full: ABE is AEF plus BEF; AEF is half of AFED; BEF is half of BFEC; so ABE is half of AFED plus half of BFEC, which is half of the two together, which is half of ABCD. On the numbers: AFED is 12 and AEF is 6; BFEC is 8 and BEF is 4; 6 plus 4 is 10.
The speech bubbles (p.143). One character remarks that the two coloured triangles measure the same although they look nothing alike; another asks what happens with ABE; a third answers that there are two halves of two different rectangles. Those three lines are the argument's spine.
The blank conclusion (p.144). Two ruled lines are printed under the chain, for the reader's own statement, and the book writes nothing on them.
The p.144 exercise. Five figures on a grid, lettered a to e, to be found by cutting into rectangles and triangles. Read off the printed page and computed from their lattice coordinates: a is a four-sided figure 4 wide with left edge from row 0 to row 6 and right edge from row 1 to row 7, giving 24 square units; b is a similar slanted four-sided figure 4 wide, giving 30; c is a five-sided figure 6 wide with a peak above its top edge, giving 48; d is a five-sided figure 4 by 5 with a notch cut down into its top edge, giving 16; e is a four-sided figure with no line of symmetry — from its top corner, one corner lies 2 units left and 2 down, another 2 units right and 2 down, and the bottom corner lies 6 units down and 1 unit to the right — giving 12. The same five values appear in the answer key appended to the cached PDF.

Figures to have open

Rectangle ABCD exactly as printed on p.143: 5 by 4 grid, A at the bottom left, E three units along the top edge, F below it, the two parts shaded differently, the blue and red triangles drawn over them. This is the topic's central figure and the argument cannot be told without it. Redraw on the same grid.
The diagonal cut movement — one rectangle, one diagonal, two pieces superimposed. Standard schematic.
A version of the ABCD figure in which E can slide along the top edge while the count is displayed. An added extension; the book prints one position.
The five grid figures from p.144, with cutting lines shown step by step into each. Take the lattice positions from the Worked examples rather than tidying the shapes: figure e in particular is not symmetric, and drawing it as a kite would change the figure the exercise is about.
No photograph or textbook data table is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 6 "Perimeter and Area", §6.3 "Area of a Triangle", pp.142–144 — the cutting activity, the Teacher's Note and the two coloured figures (p.142), the grid figure with ABCD, E and F and the printed chain of reasoning (p.143), and the blank conclusion with the five-figure exercise (p.144)
Chapter Summary, p.150, for the book's closing sentence about breaking regions into rectangles and triangles
Companion topic Area as a count of unit squares for the rectangle rule this argument halves, and Estimating the area of a shape with no formula for exact counting on a grid
Solutions appendix bound into the cached PDF after the book's last printed page, footer page 4