Halving a square: the doubling construction run backwards

What to assume they know

• Doubling a square: the diagonal is the construction — doubling a square by building on its diagonal, and the triangle-counting argument that justifies it
• A square's diagonals are equal, bisect each other, and cross at right angles
• Midpoint of a segment, and that joining midpoints of adjacent sides of a square gives a segment cutting off a right-angled corner triangle
• Congruent triangles have equal area, and the two standard tests (three sides; two sides with the included angle)
• Area of a square is side × side, so a length halved gives an area quartered
• Reading a paper-folding instruction: a crease is a fold line, and a folded flap covers the region it lands on

What they should be able to do

• Construct, inside a given square, a square of exactly half its area, by joining the midpoints of the four sides
• Explain why that inner figure is a square and not merely a rhombus, using the angles its diagonals create
• Prove the halving by counting congruent triangles: four inside the tilted square, eight in the outer one
• Answer the chapter's own question about halving the sidelength, and state the count of small squares that fill the original
• Perform the paper fold in which four corner flaps turn inward on creases through the side midpoints, and use the flaps themselves as the area argument
• Identify the halving construction as the doubling construction with the roles of given and constructed square exchanged
• Extend the halving into a chain — half, quarter, eighth — and state the sidelength ratio at each step in words the chapter has supplied

Where it usually goes wrong

• "Half the side, half the area." The dominant error, and the chapter attacks it head-on with its own question. Draw the four quarter-squares inside the original and count them before saying anything else.
• "A square of half the area is half a square — cut it down the middle." Cutting a square in half gives a rectangle. The task is a square, and the only way to get one is to tilt it.
• "The inner figure is a diamond, not a square." It is a square, tilted. The chapter deliberately asks for the proof, so an explanation that draws it and moves on has skipped the content. Show the equal, perpendicular, mutually bisecting diagonals.
• "The creases go corner to corner." Folding on the diagonals gives a smaller triangle, not a square. The creases run midpoint to midpoint — four of them, each joining the midpoints of two adjacent sides. Two creases could not turn four corners in, and creasing along a midline would fold half the sheet over and never produce PQRS at all.
• "The flaps overlap in the middle, so the fold proves nothing." They meet at the centre and do not overlap. This is worth showing slowly in the figure, because if a student believes the flaps overlap then the whole area argument collapses for them.
• "Halving and doubling are two constructions to memorise." They are one picture with the labels swapped. If a student can say which square is given, they can produce either result from the same figure.
• "You can keep halving and reach zero." You can keep halving forever and never reach zero; each step is another factor of one half. Worth one sentence, since section 10 walks the chain.

Questions to check understanding

• Given a square, construct a square of half its area and name the four points you used
• Prove that the quadrilateral joining the midpoints of a square's sides is itself a square
• Given a square of stated area, give the area of the square on its side midpoints, and the area of the square with half the sidelength — two different answers from one square
• Fill in a table of side against area to show that halving a side quarters an area
• Explain, with a fold or a diagram, why the four corner triangles together equal the inner square
• Take a square of half the area and recover the original — that is, recognise the doubling construction when it is asked for in the other direction
• Dot-grid items: draw a square of area 2 on a grid whose unit square has area 1 (Part II, p.53, no.8)

Examples worth working on the board

Values marked derived are worked out here; the chapter answers none of its own questions in this section.

• The tilted square inside (Part II, §2.2, p.36, first figure). A solid outer square with a smaller square drawn inside it with a broken outline, standing on a corner, its four corners touching the middles of the outer square's four sides.
• The same figure with the midlines (Part II, §2.2, p.36, second figure). Now the inner square is drawn solid and the horizontal and vertical midlines of the outer square are drawn dotted through it, meeting at the centre. Those two lines cut the inner square into four pieces.
• The counting argument. Derived, because the chapter asks for it and does not supply it: the two midlines cut the inner square into 4 triangles; the four corner triangles of the outer square are copies of the same triangle, so the outer square holds 8 of them. Four out of eight is one half. Cross-check it against §2.1, where the same picture read outward gave 2 against 4.
• The chapter's three printed questions on the sidelength (Part II, §2.2, p.36): whether halving a side halves the area; if not, the reason; and the count of such smaller squares needed to cover the given one. Derived: no; because area is side × side, so halving the side takes the area to a quarter; and four of them fill it, in a 2-by-2 arrangement. This is the arithmetic counterpart of §2.1's 2 × 2 = 4, which the chapter did state — the two questions are the same question in opposite directions.
• The folding figures (Part II, §2.2, p.36, lower artwork). Two pictures of yellow paper. On the left, a plain square sheet. On the right, the same sheet with four creases, each joining the midpoints of two adjacent sides — together they are the four sides of PQRS — and the four corners turned in toward the centre, four blue arrows showing the direction of each fold, and the inner square's corners lettered R at the top, Q at the right, P at the bottom and S at the left. The horizontal S–Q and vertical R–P lines visible inside the folded figure are not creases: they are where the four folded flaps' free edges meet, i.e. the two diagonals of PQRS. The outer square's outline is drawn dashed to show where the paper used to reach. The letters were read off the printed page.
• The two questions carried onto the next page (Part II, p.37, top): why PQRS is a square, and why its area is half the sheet's. The chapter's instruction is to join QS and PR, work out the angles that appear, and use triangle congruence. Derived: QS and PR are the outer square's own midlines, they are equal, they cross at the centre at right angles and bisect each other, and a quadrilateral whose diagonals are equal, perpendicular and mutually bisecting is a square. That is the answer to the first question; the second follows from the four-against-eight count.
• The flap argument, which is the stronger one for video (derived — the chapter supplies the fold and not the reasoning): the four corner flaps land inside PQRS, they do not overlap, and they leave nothing uncovered. So the flaps together have the same area as PQRS. Flaps plus PQRS is the whole sheet, so PQRS is half the sheet. Nothing is measured anywhere in this argument.
• The chain, for section 10 (derived). Start from a square of side 1 and area 1. Halving gives area 1/2, then 1/4, then 1/8. The sidelengths run 1, then a length whose square is 1/2, then 1/2, then a length whose square is 1/8 — and every second square in the chain is the plain half-side square. The chapter does not build this chain; it builds the doubling chain (Part II p.35).
• Later exercises that live on this construction (Part II, p.53). No.7 asks for a square holding whatever area is left when a square of side 5 is taken away from one of side 7 — the subtractive cousin of doubling and halving, and it needs §2.4. No.8, marked Math Talk and Try This, asks which squares can be drawn with corners on a 5-by-5 dot grid, naming target areas of 2, 3, 4 and 5 square units; the tilted square of area 2 is exactly this section's construction applied to the 2-by-2 grid square.

Figures to have open

• The outer square with the tilted inner square on its side midpoints, with both midlines drawn, shaded so the four inner triangles and the four corner triangles are visibly the same triangle. This is the chapter's figure (Part II, §2.2, p.36) and sections 3 to 5 depend on it. Redraw as a schematic.
• A step-by-step paper fold: a square sheet, four creases, each joining the midpoints of two adjacent sides, four corners turning in, the corners lettered P, Q, R, S as the chapter letters them (Part II, §2.2, p.36). The letters must match the printed ones because the student will have the book open. This is worth showing as a movement.
• The original square containing four quarter-size squares in a 2-by-2 arrangement, to answer the chapter's own sidelength question. Standard schematic; not printed in the chapter.
• Optional for section 10: the halving chain drawn at true relative size.
• No photograph is needed. The chapter's paper illustrations can be redrawn as flat colour.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 2, "The Baudhāyana-Pythagoras Theorem", §2.2 "Halving a Square", Part II pp.36–37. §2.2 prints one bold unnumbered subheading, nameable but not citable by number: "Halving a Square Using Paper" (Part II p.36). The two follow-up questions about PQRS sit at the top of Part II p.37, above §2.3.
• Backward reference within the chapter: the construction and the midline device are §2.1's, at Part II pp.33–35.
• Related later exercises: Part II p.53, Figure it Out no.7 (a square from the difference of two squares) and no.8 (squares on a dot grid, marked Math Talk and Try This).
• The chapter's SUMMARY at Part II p.54 does not mention halving; it records the theorem, the isosceles relation, the two facts about √2, triples, and Fermat's Last Theorem. Checked on the printed page.