Doubling a square: the diagonal is the construction

What to assume they know

• A square's area is side × side, so scaling a side by a factor scales the area by that factor twice over
• A square's two diagonals are equal, cross at their midpoints, meet at right angles, and each one bisects the two corner angles it passes through
• Two triangles that agree in all three sides, or in two sides and the angle between them, are congruent, and congruent figures have equal area
• Cutting a figure into non-overlapping pieces and rearranging them leaves its area unchanged
• Reading a diagram in which some lines are drawn solid and others dotted, and knowing that a dotted line is still a real line of the construction

What they should be able to do

• Explain why doubling every side of a square multiplies its area by four rather than by two, and give the general reason in terms of side × side
• Carry out Baudhāyana's construction: given a square, draw its diagonal and build a square on that diagonal
• Draw the horizontal and vertical lines that make the doubling visible, and say why those lines pass through the new square's corners
• Count the congruent triangles in each of the two squares and use the counts 2 and 4 to justify the word "double" without measuring any length
• Justify the congruence of the four triangles rather than asserting it from the drawing
• Continue the construction to produce a chain of squares in which each has twice the area of the one before, and say how many triangles each holds
• Reproduce the doubling with paper: cut two identical squares into quadrant triangles and reassemble them into one square of twice the area
• State the same result as a fact about the diagonal, ready for §2.3 to convert into a length

Where it usually goes wrong

• "Double the area means double the side." The chapter opens on this guess precisely because it is the one students bring. It gives four times the area, and the chapter shows the four quadrants. Make the student count the quadrants before saying anything about diagonals.
• "Then double the area means side times 1.5, or thereabouts." No single familiar fraction works; the required side is a length that cannot be written as a fraction at all. That is §2.3–§2.4's business, but an explanation that lets the class hunt for a fraction now has set the later argument up properly.
• "The tilted square is smaller — it looks pinched." A square standing on its diagonal looks narrower on the page than the same square set upright. Tilting a figure changes nothing about its area. Rotate it until it is upright and the illusion dies.
• "The four triangles look the same, so they are." Looking the same in a diagram is not congruence. The chapter asks the student to explain the congruence: each triangle has two sides that are halves of the two equal diagonals of the broken tilted square built on the diagonal — not of the genuinely largest square in the figure — meeting at a right angle where those diagonals cross.
• "The extended sides pass through the corners because the drawing was made that way." They pass through because the original square's side lies along a diagonal of the new square, and a diagonal of a square runs from corner to corner. The chapter's hint puts it the other way round — via angle bisection — and either direction is fine as long as one is given.
• "Cutting and rearranging might lose a bit." Nothing is lost: the pieces are the same pieces. This is the whole reason the chapter argues by dissection rather than by arithmetic, and it is the habit the rest of the chapter needs.
• "This only works for a square." Section 9 should be honest here. The chapter's chain doubles squares; the argument used is specific to a square's diagonals being equal and perpendicular.

Questions to check understanding

• Given a square's area, state the area of the square built on its diagonal, and the area of the square with double the sidelength, and explain the difference
• Given a square, construct with ruler and compass a square of twice its area, and say which segment you used
• Justify, in writing, that the four triangles cut from the square on the diagonal are congruent
• Complete a chain: a square, the square on its diagonal, the square on that square's diagonal — and give the ratio of the first area to the third
• Count triangles in a supplied composite figure to compare two areas, with no lengths given at all
• Explain, to a student who has doubled every side, what went wrong and by how much
• Constructions beyond doubling: three times the area, five times the area (Part II, p.47, Figure it Out no.3, which cites Śulba-Sūtra Verse 1.10) — these need §2.4 as well, so set them after the whole chapter

Examples worth working on the board

Values marked derived are worked out here on the chapter's stated inputs. The chapter prints no answers to any question in §2.1 (Part II pp.33–35), which is this topic's range; elsewhere it does work examples of its own, in §2.3 and §2.7.

• The failed first guess (Part II, §2.1, p.33, two figures side by side). A plain square on the left; on the right the same square with every side doubled, and the larger square cut by one dashed horizontal and one dashed vertical line through its centre into four equal quadrants. Small tick marks on the outer edges mark where the original sidelength ends. The chapter states the factor as 2 × 2 = 4. Derived, and the general form: a square of side s has area s × s; a square of side 2s has area 2s × 2s = 4s². Doubling a length twice over is what area does.
• Baudhāyana's construction (Part II, §2.1, p.34, top figure — the prose that introduces it closes Part II p.33, which prints no figure after it). The original square drawn solid, its diagonal drawn, and a square built outward on that diagonal drawn with a broken outline. The same figure appears again at mid-page right on Part II p.34 with the original square's horizontal and vertical sides extended straight across the broken square; the extensions arrive exactly at the broken square's four corners, and they cut it into four triangles.
• The triangle count. The original square: 2 triangles. The square on the diagonal: 4 of the same triangle. The chapter states both counts. This — not any length — is its proof that the area doubles.
• The congruence figure (Part II, §2.1, p.34, lower artwork). The broken square is drawn inside a larger square, with the original square occupying its lower-left quarter, and two solid midlines drawn — one horizontal, one vertical, crossing at the centre (a square has two midlines, not four). The tick marks come in two styles, and both are load-bearing: double ticks on the four sides of the broken tilted square, and single ticks on each of the eight half-sides of the outer square — twelve ticked segments in all, counted on the printed page. The single ticks mark the triangles' legs equal and the double ticks mark their hypotenuses equal; that pairing is the chapter's evidence that the pieces match, and a redraw that keeps only one style loses the argument.
• The chain of squares (Part II, §2.1, p.35, top). Three composite figures in a row, each with an arrow down to the clean square it produces: first the small upright square, then the tilted square standing on its diagonal, then a larger upright square. The chapter gives the triangle counts as 2, 4 and 8. Derived: taking the first square's area as 1, the chain is 1, 2, 4, and the sidelengths are 1, √2, 2 — the explanation may show the third square as literally the doubled-side square from section 2, which is the reconciliation the chapter leaves for the student.
• Paper activity 1 (Part II, §2.1, p.35). Two identical paper squares, each drawn with both diagonals. Square 1's four quadrant triangles are numbered 1 (left), 2 (top), 3 (right), 4 (bottom). Identical Square 2's are numbered 5 (left), 6 (top), 7 (right), 8 (bottom), and in the artwork the four pieces of Square 2 are already drawn separated by white gaps. The instruction is to set pieces 5, 6, 7 and 8 around Square 1. Derived: the four loose triangles attach to Square 1's four sides, hypotenuse against side, and the outline that results is the square standing on Square 1's diagonal — the same figure as Part II p.34, now in paper.
• Paper activity 2 (Part II p.39, Figure it Out no.1, flagged Math Talk — printed inside §2.3, though its content belongs to §2.1's construction; §2.1 runs Part II pp.33–35 and §2.3 opens on Part II p.37). A second cutting, flagged Math Talk). A second cutting: two identical squares, each cut along one diagonal only, giving four right triangles, numbered 1 and 2 in the first square and 3 and 4 in the second. The item asks for a square of twice the area from those four pieces. Derived: set the four right angles at the centre and the four hypotenuses on the outside; each hypotenuse becomes one side of the new square. This is a different dissection from the first activity and lands on the same square, which is worth showing side by side.
• The forward pointer worth planting. A square of side 1 has area 1, so the square on its diagonal has area 2 and its side is a length whose square is 2. The chapter does not name that length until §2.3 (Part II, p.37). Section 9.

Figures to have open

• The original square with its diagonal and the square built on that diagonal, with the horizontal and vertical extensions drawn in and the four triangles shaded as one repeated shape. This is the chapter's central figure (Part II, §2.1, pp.33–34) and everything in sections 4 to 8 hangs on it. Redraw as a schematic; do not reproduce the printed art.
• The doubled-side square split into four quadrants, beside the original, at matching scale (Part II, §2.1, p.33). Standard schematic.
• The chain of three squares with their triangle counts (Part II, §2.1, p.35). Standard schematic. It must be drawn at true relative size or the doubling is not visible.
• The two paper dissections, numbered as the chapter numbers them (Part II p.35, inside §2.1, and Part II p.39, printed inside §2.3). Both are standard schematics; the piece numbers must match the printed ones because a student will have the book open.
• No photograph is needed anywhere in this topic.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 2, "The Baudhāyana-Pythagoras Theorem", §2.1 "Doubling a Square", Part II pp.33–35. Within §2.1 the chapter prints one bold unnumbered subheading, nameable but not citable by number: "Doubling a Square Using Paper" (Part II p.35).
• The chapter's own Verse 1.9 attribution for the doubling rule, in a tinted box at Part II p.33; the Verse 2.1 material used later for the general case is at Part II p.42.
• The second paper dissection is Figure it Out no.1 at Part II p.39, carrying the Math Talk marker.
• Forward pointer: the diagonal's length is named at Part II p.37 (§2.3), and the chapter's SUMMARY at Part II p.54 records the isosceles relation that follows from this construction.
• The area-triple and area-quintuple constructions are set at Part II p.47, Figure it Out no.3, attributed to Śulba-Sūtra Verse 1.10.