Reading (a + b)² off a partitioned square

What to assume they know

• Area of a rectangle as length times breadth, and of a square as side squared
• That a length can be split into two parts and the parts named separately
• Multiplying out a(b + c) by the distributive property
• What an identity claims, and that a claim about lengths is narrower than a claim about numbers (An identity holds for every value; an equation need not)
• Squaring a two-digit number by ordinary multiplication, so the shortcut has something to beat

What they should be able to do

• Draw a segment of length a + b as a segment of length a followed by one of length b, and mark all three lengths
• Construct a square on that segment and account for every one of its four pieces by area
• State the area of the whole square in two ways and read the identity (a + b)² = a² + 2ab + b² off the two statements
• Explain why the two rectangles are congruent and why there are two of them rather than one
• State the restriction under which the area argument is valid, and say what the chapter does next because of that restriction
• Use the identity to square a two-digit or three-digit number by splitting it at a round value
• Obtain (a − b)² = a² − 2ab + b² from (a + b)² = a² + 2ab + b² by replacing b with −b, and check the signs of all four terms
• Account for (a − b)² inside a square of side a by subtracting two rectangles, and explain where the +b² comes from
• Use (a − b)² to square a number that sits just below a round value

Where it usually goes wrong

• "a² + 2ab + b² has four terms because the square has four pieces, and that is the whole story." The count is a consequence, not a reason. The reason there are two ab pieces is that the horizontal cut and the vertical cut are made independently, so each of a and b pairs with the other exactly once.
• "The two ab rectangles are the same rectangle." They are congruent but differently oriented: one is b wide and a tall, the other a wide and b tall. The chapter's own artwork gives them the same fill, which is a hint about equal area, not about being one object.
• "The picture proves the identity for all numbers." The page itself limits the claim to lengths, and lengths are positive. The general claim is earned three paragraphs later by distributivity.
• "(a − b)² = a² − b²." This is the commonest error in the whole chapter, and Fig. 4.3 kills it visually: a² − b² is not even the right shape — removing a b-by-b corner from a square of side a leaves an L, not a square.
• "Replacing b by −b flips every sign." It flips one. a² has no b in it, and (−b)² is positive. Walk the three terms separately.
• "The +b² in (a − b)² is a fudge to make the numbers work." It is the overlap you would otherwise remove twice. Show the movement of the two removals and let the corner get taken away, then given back.
• "You can only use the identity when a and b are whole numbers." Exercise Set 4.1 puts fractions and reciprocals into both slots; nothing in the derivation cared.
• "Splitting 193 as 200 − 7 is harder than just multiplying." Have the explanation race the two methods once. The point of the identity here is not elegance, it is that two of the three pieces are trivial to compute.

Questions to check understanding

• Given a square partitioned into four pieces with two lengths marked, write the total area in two ways and hence state an identity
• Expand (a + b)² and (a − b)² for given algebraic a and b, including cases where a or b is a fraction or a reciprocal
• Evaluate a square such as 79² or 205² by choosing a convenient split, and say which of the two identities the split calls for
• Say what is wrong with (a − b)² = a² − b² and support it with a diagram
• Obtain one identity from another by a stated substitution, and justify each sign
• Label the pieces of a supplied partitioned square so that the drawing represents a given identity — the form of the Think and Reflect on p. 76

Examples worth working on the board

Inputs, not answers. Values marked Verified are worked out here; this book prints no answer key.

• Fig. 4.1 (§4.2, p. 69). A single horizontal segment, tick-marked once. The two parts below the line are labelled as a units and b units; the whole, above the line, is labelled as (a + b) units, with arrows spanning each. Read on the printed page — the labels are set in artwork and survive extraction only partially. This is the figure the whole chapter rests on and it contains no arithmetic at all.
• Fig. 4.2 (§4.2, pp. 69–70). A square of side (a + b), partitioned into four pieces by one horizontal and one vertical cut. Read off the printed page, because every interior label is artwork lettering: the top-left piece is the larger square, labelled a²; the bottom-right piece is the smaller square, labelled b²; the top-right and bottom-left pieces are both labelled ab. The top edge carries a then b, the left edge carries a then b, and both the top and the left of the whole figure are marked (a + b) units. The four pieces are drawn in three fills, not four: the a² square is yellow, the b² square is magenta, and both ab rectangles carry the same green — which is the figure quietly telling you they are the same area. Verified as a statement: the top-right rectangle is b wide and a tall, the bottom-left is a wide and b tall, so both have area ab and neither is a square; adding a² + ab + ab + b² recovers (a + b)².
• The restriction, printed (§4.2, p. 70). Immediately after reading the identity off the figure, the page states that what Fig. 4.2 has established holds for a and b that are lengths of line segments, and then asks what happens for numbers that are not lengths. The picture first, the limitation second, and only then the numerical trials and the distributive argument (which are the business of An identity holds for every value; an equation need not).
• The area argument turned into arithmetic (§4.2, p. 71, Example 4; Exercise Set 4.1 Q2, p. 72). 43² split as (40 + 3)². Then 64², 105², 205². Verified: 43² = 1600 + 240 + 9 = 1849; 64² = (60 + 4)² = 3600 + 480 + 16 = 4096; 105² = (100 + 5)² = 10000 + 1000 + 25 = 11025; 205² = (200 + 5)² = 40000 + 2000 + 25 = 42025. Draw 43² as an actual square of side 43 with the 40-cut in it, and the three terms become a 40 × 40 tile, two 40 × 3 strips and a 3 × 3 corner. That is the same picture, to scale.
• Replacing b by −b (§4.3, p. 73, Think and Reflect). The box asks what happens to (a + b)² = a² + 2ab + b² under that replacement, and the text on the same page gives (a − b)² = a² − 2ab + b². Inputs: perform the substitution term by term. Verified: a² is untouched because b does not appear in it; 2ab becomes 2a(−b) = −2ab; b² becomes (−b)² = +b², so only the middle term changes sign. That single-sign-change observation is the whole content of the exercise and it is worth slowing down, because students routinely flip the last sign too.
• Fig. 4.3 (§4.3, pp. 73–74). A square of side a, cut into three pieces — labels read off the printed page. The left column has width a − b and is cut in two: the upper piece is a square labelled (a − b)² and the lower piece is a rectangle labelled b(a − b). The right column runs the full height a with width b and is one rectangle labelled ab. The outer dimension arrows mark a across the top, subdivided as a − b then b, and a down the left side, subdivided as a − b then b. Three fills, one per piece.
• The printed derivation from Fig. 4.3 (§4.3, p. 74). The page states the four areas and then subtracts: (a − b)² = a² − ab − b(a − b), which it opens out to a² − ab − ba + b² and collects to a² − 2ab + b². The mechanism to narrate: you remove a full-height ab rectangle, then remove the bottom strip — but the bottom strip is only a − b wide, so it is b² less than ab. Subtracting ab twice would take away b² too much, and the +b² in the identity is putting it back.
• Squaring from above (§4.3, p. 73, Example 8; Exercise Set 4.2 Q2, p. 75). 29² as (30 − 1)². Then 79², 193², 299². Verified: 29² = 900 − 60 + 1 = 841; 79² = (80 − 1)² = 6400 − 160 + 1 = 6241; 193² = (200 − 7)² = 40000 − 2800 + 49 = 37249; 299² = (300 − 1)² = 90000 − 600 + 1 = 89401. Point worth making: the three-digit cases are the ones where the shortcut is obviously faster than long multiplication, and 193 needs b = 7, so the b² term is not negligible.
• A comparison the chapter sets up but does not draw. Fig. 4.2 partitions and adds; Fig. 4.3 partitions and subtracts. In Fig. 4.2 the four pieces are disjoint and fill the square. In Fig. 4.3 the three pieces are also disjoint and fill the square, but the identity is obtained by moving two of them to the other side of the equals sign. That difference is the reason the second argument needs the distributive property and the first does not.

Figures to have open

• Fig. 4.2, the square of side (a + b) with its four labelled pieces (p. 69). This is the chapter's own figure and the topic cannot be taught without it. Redraw as a clean schematic; keep the two ab rectangles in a shared fill, because that fill is doing pedagogical work.
• Fig. 4.1, the cut segment (p. 69). Trivial to redraw and worth one beat on its own, because it is where a and b are defined.
• Fig. 4.3, the square of side a in three pieces (pp. 73–74). The chapter's own figure. Redraw; the piece labels — (a − b)², b(a − b), ab — must be legible, and the dimension arrows subdividing both a-edges as (a − b) then b must be kept, since without them the figure is unreadable.
• A step-by-step overlay for section 9 showing the corner b² being subtracted twice and restored once. Standard schematic; the chapter has no such figure and the argument is the topic's payoff.
• A to-scale 43 × 43 square for section 6. Standard schematic.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics, printed Chapter 4, "Exploring Algebraic Identities": §4.2 Visualising Identities, pp. 69–72, including Fig. 4.1 and Fig. 4.2; and the (a − b)² material inside §4.3, pp. 73–74, including Fig. 4.3.
• Exercise Set 4.1 Q1–Q2, pp. 71–72; Exercise Set 4.2 Q2, p. 75.
• Think and Reflect, p. 73 — the b → −b prompt whose answer the running text supplies on the same page.
• Chapter summary, p. 90, second bullet, on visualising identities with geometrical models or tiles; the identity list is on p. 91.
• Companion topics: An identity holds for every value; an equation need not for why the picture is not enough, Recognising an expression as an identity in disguise for reading the same identity backwards as a factorisation, (a + b + c)² by substitution, and the square that proves it for the three-letter version of the same square.