Area as a count of unit squares

What to assume they know

• Multiplication of two whole numbers, and division as the inverse of it
• What a square and a rectangle are, and that a rectangle's opposite sides are equal
• Perimeter as the total length around a closed figure
• That two figures are congruent when one can be laid exactly on the other
• Reading a measurement off a labelled diagram, including a label written inside a shape rather than beside it

What they should be able to do

• State what quantity is being counted when an area is reported, and name the square that is doing the counting
• Count the unit squares in a rectangle by rows and columns, and explain why that count equals the product of the two sidelengths
• Compute the area of a rectangle from two sidelengths, and recover a missing sidelength from an area and one side
• Write an area in both of the two forms the chapter uses, and say why the unit carries a square on it
• Argue that a rectangle's diagonal splits it into two pieces of equal area, and compute the area of either piece
• Propagate a chain of known areas and part-lengths through a figure made of several rectangles to find one unknown sidelength
• Find the area of a region built by adding or subtracting rectangles, including a border laid around an inner rectangle
• Explain why moving a piece of a region to a new position leaves the total area alone, and use that to replace a bent shape by a straight one
• Give two different divisions of a square into four parts of equal area, one of which uses no straight cut all the way across

Where it usually goes wrong

• "Area is a formula you apply." It is a count. The formula is a shortcut available for rectangles, and the rest of the chapter is a series of arguments for extending the shortcut to other shapes. Open with the count.
• "Cutting a shape up changes its area." The chapter's very first figure and its very last exercise both depend on the opposite. Say why: the material is conserved, so the count is.
• "Rearranging can create area — look, the square got bigger." The cardboard puzzle on Part II p.152 makes a bigger square out of the same four pieces, and a student who does not notice the hole will conclude that area is not conserved after all. Account for the hole explicitly.
• "cm² means 'centimetres, and then square the number'." It names a square whose side is 1 cm. Draw the square. This misreading is what produces the student error 1 m² = 100 cm². The chapter never converts between m² and cm² — its printed conversions are in² ↔ cm² (Part II p.170) and in² ↔ ft² and m² ↔ km² (Part II p.171) — so do not let the false equation read as something the book prints. The place where the squaring bites in print is the question "How many m² is a km²?" on Part II p.171.
• "You can only measure whole unit squares, so awkward shapes have no area." Fractional counts are in the definition on Part II p.149, and the triangle on Part II p.150 is the first place they are needed.
• "Half the rectangle needs half the unit squares to come out whole." The diagonal slices through squares. The count is still exactly half, because the two triangles are congruent, not because the squares cooperate.
• "Equal perimeter means equal area." The chapter's own two rectangles are 22 cm around apiece and 28 cm² against 24 cm² inside. This is the hinge into Why perimeter cannot stand in for area; raise it here and hand it over.
• "A missing sidelength needs a ruler." In the pinwheel figure every unknown is recovered by dividing an area by a length. Division is the tool.

Questions to check understanding

• Given two sidelengths, state the area with its unit; given an area and one side, state the other
• Given a figure built from several rectangles with some areas and some lengths marked, find a named missing length
• Given a rectangular border of stated width around a stated inner rectangle, find the border's area two ways — by subtraction and by breaking it into rectangles — and check the two agree
• Divide a square into four parts of equal area in a way that uses no straight cut across the whole square, and justify the equality
• Given a shape of constant width made of straight arms, find its area by straightening it, stating what happens at each corner
• Explain, in a sentence, what quantity a number in cm² is counting
• Decide whether a stated rearrangement of pieces could have changed the total area, and say why not — the competency-style item this section invites

Examples worth working on the board

Items marked printed are worked out on the page; items marked not in the book are worked out here on the chapter's inputs and must not be presented as something the chapter states.

• The square in four equal parts (Part II p.148, two line drawings). The first drawing is the obvious division: one vertical and one horizontal cut through the centre, giving four congruent smaller squares. The second drawing keeps the same four pieces but replaces each straight internal cut with a stepped, key-shaped boundary, so no piece is a square any more and no two are congruent. Printed: the chapter's reason is that each piece loses material along one edge and gains the same material along another, so if the loss and the gain match in size the four areas are still equal. The page states openly that there are infinitely many such divisions, and sets a Math Talk asking for more.
• The rangoli rectangles (Part II p.149). Two solid coloured rectangles: one 7 cm by 4 cm, one 8 cm by 3 cm. The question is which needs more powder if the colour is laid on evenly. The unit is fixed on the page as the square of side 1 cm. Printed: the first holds 7 × 4 = 28 unit squares, the second holds 8 × 3 = 24, so the first needs more. Not in the book: the two boundaries are the same length — 22 cm each — which is the fact section 3 should hold back until Why perimeter cannot stand in for area.
• The definition and the formula (Part II p.149). Printed: area is measured by finding how many unit squares match the region, and the count may come out fractional; for a rectangle that count is the product of the two sidelengths. Printed: the two ways of writing the result, 28 sq. cm and 28 cm².
• Half of a rectangle (Part II p.150, top). A 7 cm by 4 cm rectangle with one diagonal drawn. Printed: the diagonal gives two congruent triangles, so each is half the rectangle, and half the area fills half the unit squares; each triangle is ½ × 7 × 4 = 14 cm².
• The pinwheel of four rectangles (Part II p.150, Figure it Out 1(i)). Four rectangles arranged around a centre, each sharing part of a side with the next. The printed areas are 28 in², 21 in², 35 in² and 14 in². The printed lengths are: 4 in, the vertical drop from the top edge of the 28 in² rectangle to the top edge of the 21 in² rectangle; 3 in, the part of the 35 in² rectangle's width that lies to the left of the 28 in² rectangle; 7 in, the full width of the 21 in² rectangle; 2 in, the part of the 14 in² rectangle that hangs below the 35 in² rectangle. The unknown, marked with a question mark, is the bottom width of the 14 in² rectangle. Not in the book chain, which is the whole point of the item: 21 ÷ 7 = 3, so the 21 in² rectangle is 3 in tall; 3 + 4 = 7, so the 28 in² rectangle is 7 in tall; 28 ÷ 7 = 4, so it is 4 in wide; 4 + 3 = 7, so the 35 in² rectangle is 7 in wide; 35 ÷ 7 = 5, so it is 5 in tall; 5 + 2 = 7, so the 14 in² rectangle is 7 in tall; 14 ÷ 7 = 2, so the answer is 2 in. Every step is one division or one addition, and the figure is built so that the chain has exactly one order in which it can be walked.
• The second missing-lengths figure (Part II p.150, Figure it Out 1(ii)). A thick-outlined rectangle whose area is given as 50 m². Its upper part is stippled and marked 29 m², and a vertical arrow beside that upper part marks its height as 4 m; the rest of the thick-outlined rectangle, below, is left blank. Hard against its right-hand side, sharing the same top and bottom edges as the 29 m² part, sits a hatched rectangle marked 11 m². Three question marks ask for the width of the thick-outlined rectangle, the width of the hatched rectangle, and the height of the blank lower part. Not in the book, and a warning: as printed these do not come out in whole metres. 29 ÷ 4 = 7.25 m for the first width and 11 ÷ 4 = 2.75 m for the second are exact on either reading of the 50 m² label. The third length is not settled by the figure: if 50 m² is the whole thick-outlined rectangle, the blank lower part is 50 − 29 = 21 m² over a width of 7.25 m, giving 21 ÷ 7.25 ≈ 2.90 m; if 50 m² is the blank lower part alone, its height is 50 ÷ 7.25 ≈ 6.90 m. Re-measured on the printed page, the leader terminates on the left edge of the thick-outlined rectangle at a height that falls inside the lower blank band, and the internal divider between the stippled and blank parts is drawn as thick as the outer boundary — so the leader does not distinguish the two readings. Neither reading terminates, so the printed awkwardness is real either way. A teacher must not promise tidy answers here. The teaching value of the item is which label constrains which length, not the arithmetic.
• A border around a park (Part II p.151, Figure it Out 2). A rectangular park EFGH sits inside a larger rectangle ABCD, and the shaded strip between them is the path. Part (i) asks which measurements are needed and for a formula, with the printed hint that the three areas — inner park, path, outer rectangle — are related. Part (ii) asks whether knowing the path's width along each side is enough, with the printed hint to break the path into rectangles. Part (iii) shows three drawings in which the inner park sits in three different positions inside the outer rectangle, and asks whether the path's area changed. Not in the book: it did not, because the path is always the outer area minus the inner area and neither of those moved. That is additivity doing the work, and it is the cleanest instance of the section-10 point in the chapter.
• A crosspath (Part II p.151, Figure it Out 3, a Math Talk). A plot with sides 14 m and 12 m carrying a cross-shaped path — one strip across and one strip down. The student chooses the two strip widths and gives a formula. Not in the book: for widths p across and q down the path is 14p + 12q − pq, and the subtracted term is the square where the two strips overlap, counted twice if you are careless.
• The spiral tube (Part II p.152, Figure it Out 4). A tube of constant width 1 spiralling inward, with an opening at the top left and another near the centre. The marked lengths, reading outward in, are 20 across the top, 20 down the right, 20 along the bottom, 15 down the left, 15 across the next turn, 10 down and 10 across the turn after that, and 5 across and 5 down the innermost turn; the width 1 is marked twice, once at each opening. The printed hint offers a second drawing: an L-shaped tube of width 1 whose two arms measure 5 and 5, beside a straight tube of unknown length, and asks what length of straight tube has the same area. Not in the book: the L has area 9, not 10, because the corner square belongs to both arms and may only be counted once — so the equivalent straight tube is 9 long. That off-by-one at the corner is the entire content of the hint, and it is what makes straightening the whole spiral a reliable method rather than a hopeful one.
• Rearranging a square around a hole (Part II p.152, Figure it Out 6, a Math Talk). Two lines are drawn inside a square, perpendicular to each other and neither passing through the centre, cutting it into four pieces. The pieces are to be reassembled into a larger square with a hole in the middle. The chapter suggests cutting the shape out of cardboard or chart paper. Not in the book: no material was created — the four pieces total exactly what they always did, and what grew is the hole, whose area is precisely the difference between the two squares. Used carelessly this figure is the standard way of "proving" something false.

Figures to have open

• The two divisions of a square (Part II p.148). Both drawings are needed and the pairing is the argument, so redraw them as one figure with the four pieces colour-coded so a student can follow which piece became which. Schematic; do not reproduce the printed art.
• The two rangoli rectangles with a unit grid added (Part II p.149). The printed rectangles are solid blocks of colour with no grid drawn on them. The grid is the whole point of the section.
• The 7 by 4 rectangle with its diagonal (Part II p.150). Schematic, but keep the unit grid on it so the "squares get cut and it still works" point lands.
• The pinwheel of four rectangles (Part II p.150, item 1(i)). Must be redrawn with all four areas and all four part-lengths in place, because the item is unsolvable if any one label is dropped. This is the figure the explanation most needs.
• The three park positions (Part II p.151, item 2(iii)). Three panels, one area readout. Schematic.
• The spiral tube and its straightened equivalent (Part II p.152, item 4). Both the spiral with its nine marked lengths and the L-shaped hint tube. Schematic; the corner square should be shaded in the L so the count of 9 is visible rather than asserted.
• No photograph is needed. The printed rangoli artwork on Part II p.149 is decorative and carries no measurement.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 7, "Area", §7.1 "Rectangle and Squares", Part II pp.148–149, for the opening square-division question, the rangoli comparison, the definition of area by unit squares, and the rectangle formula.
• Part II p.150, top third of the page, for the diagonal of a rectangle and the 14 cm² triangle. The unnumbered bold subheading ("Why Can't Perimeter be a Measure of Area?") begins immediately below it and belongs to Why perimeter cannot stand in for area.
• Part II pp.150–152, the Figure it Out block that closes this stretch of §7.1: item 1 (missing sidelengths, two figures), item 2 (the park path, three parts), item 3 (the crosspath, a Math Talk), item 4 (the spiral tube with its hint), item 6 (the cardboard rearrangement, a Math Talk). Item 5, on doubling a square's sidelength, is handed to Units for real areas, and why converting between them squares the length factor.
• Part II p.150 points the student to grid paper bound at the end of the book. That material is the "LEARNING MATERIAL SHEETS" section on Part II pp.172–176, of which Part II p.175 is headed "Isometric Grid".
• Forward pointers inside this chapter: dissection is named on Part II p.161; the chapter SUMMARY is on Part II p.171 and states no rectangle formula, because the rectangle is treated as already known.