(a + b)² and (a − b)²: why there is a middle term at all

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What to assume they know

Expanding a product of two two-term brackets (Multiplying two two-term expressions, and where the four terms come from)
Like terms, and combining them (Multiplying two two-term expressions, and where the four terms come from)
Area of a square and of a rectangle, and that a region's area is the total of the areas it is cut into
Exponent notation, including that a product raised to a power carries the power onto each factor
Integer sign rules, in particular that a negative squared is positive
Squares of the round numbers 10, 20, …, 100 and of the single digits

What they should be able to do

Compute the square of a two-digit number by splitting it into a round part and a small part and adding four areas
Expand (a + b)² by the distributive property and account for the middle term as two congruent rectangles
State Identity 1A and use it on numbers and on expressions
Decide when (a + b)² exceeds a² + b² and when it does not, in terms of the signs of a and b
Choose a split of a given number that makes both squares easy, and show that different splits give the same total
Expand the square of a two-term expression such as (6x + 5)², both by distributivity and by the identity, and compare the two routes
Build the square of a difference geometrically, by removing two overlapping strips from a larger square and restoring the doubly-removed corner
Derive Identity 1B from Identity 1A by substituting a negative, and say why only the middle term changes sign
Apply both identities to squares of numbers just above or just below a round number

Where it usually goes wrong

"The square of a sum is the sum of the squares." The single most durable error in school algebra, and the chapter's figure is the antidote: the two rectangles are drawn, tinted and labelled, and they are plainly not nothing. Run it numerically too — 60 squared plus 5 squared is not 65 squared, and the shortfall is exactly the two 300s.
"The middle term is ab." It is two copies of ab, because there are two rectangles and they are congruent. Get a student to point at both.
"The square of a difference is a² − b²." Two separate errors live here: the missing middle term, and the belief that the last term should be negative. Squaring a negative gives a positive, which is why only the cross term flips. The next topic in this chapter is where a² − b² actually comes from, and confusing the two is the reason to teach them in this order.
"The square of a sum always beats the sum of the squares." Not when the two numbers have opposite signs, and not when either is zero. The chapter asks this as a Math Talk question for exactly that reason.
"There is a right way to split a number before squaring." 65 can be cut at 60 and 5, at 30 and 35, or at 52 and 13. The total is fixed; only the difficulty changes. The chapter sets all three.
"You must remember both identities." 1B is 1A with the second letter negated, and the chapter shows the substitution explicitly. One identity plus the sign rules is enough — and the owl box says that even that is optional if you can expand.
"The square of 6x is 6x²." The printed working takes it to 36x² in a visible step. Do not skip it.
"The 55-square argument is faulty because it removes the corner twice." It does remove it twice, and that is the interesting part. The chapter narrates the double removal as a problem and then repairs it, which is a better lesson than a clean derivation would have been.

Questions to check understanding

Square a two-digit number by splitting it into a round part and a small part, showing the three contributions
Expand the square of a two-term expression, both by the identity and by the distributive property, and confirm the two agree
Given a square of a difference, expand it and state the sign of each term
Choose which of several candidate expressions matches a stated description, and say why the rejected ones fail
Decide whether the square of a sum exceeds the sum of the squares for stated values, and characterise when it does
Prove a general claim about the squares of even or odd numbers by writing the number in a suitable form and expanding
Write an expression for a tiled or paved region built from squares and uniform strips
Show that two expressions arising from one description are equal, by expanding

Examples worth working on the board

Inputs. Items marked "printed" are the chapter's own working; the rest the chapter leaves open.

The opening question (Part I, §6.2, p.145). Printed inputs: a square of sidelength 60 has area 3600 sq. units, a square of sidelength 5 has area 25 sq. units, and the question is whether those two facts can produce the area of a square of sidelength 65.
The 65-square figure (Part I, §6.2, p.145, artwork). A single tinted square with a dashed horizontal line and a dashed vertical line cutting it into four regions. The top edge is marked 60 and then 5; the left edge likewise. The four regions carry the labels 60² (top left, the large one), 60 × 5 (top right, set vertically), 5 × 60 (bottom left) and 5² (bottom right, the small one). All four labels and both edge markings are artwork lettering — read them from the printed page.
The chapter's own arithmetic (Part I, §6.2, p.145). Printed: 65 squared taken as 60 squared plus 5 squared plus twice the product of 60 and 5, and evaluated as 3600 + 25 + 600 = 4225 sq. units. Printed alongside: the same thing by distributivity, (60 + 5)(60 + 5) expanded to four products and then gathered.
The alternative splits (Part I, §6.2, p.145). Printed as a question: what if 65 squared is written with a split at 30 and 35, or at 52 and 13? The page asks for the figures to be drawn and the areas checked. It gives no answers. This is the best available section-6 material — three different cuts, one total.
The general square figure (Part I, §6.2, p.145, artwork). A tinted square cut by one dashed horizontal and one dashed vertical line into four regions labelled a², a × b (set vertically), b × a and b², with the top edge marked a then b and the left edge marked a then b. Beside it, the expansion of (a + b)(a + b) into four products with two curved arrows over the pairings, and then the collected form. The page notes that this was already done as Example 2 in §6.1.
Identity 1A, boxed (Part I, §6.2, p.145): the square of a + b equals a² + 2ab + b².
The Math Talk question (Part I, §6.2, p.145). Printed: for any two integers, is the square of the sum always greater than the sum of the squares, and if not, when is it greater? The answer hangs entirely on the sign of the doubled middle term, which is the whole reason the question is worth asking — but the chapter leaves it open.
Inputs for the identity on numbers (Part I, §6.2, p.145): 104 squared and 37 squared, with the chapter's own printed hint that each be decomposed into a sum or a difference of numbers whose squares are easy.
Inputs on expressions (Part I, §6.2, p.146): the squares of m + 3 and of 6 + p.
The two-column comparison (Part I, §6.2, p.146). Printed as a table with two headed columns, one for the distributive route and one for the identity route, both expanding the square of 6x + 5. The distributive column runs four lines and ends 36x² + 60x + 25; the identity column runs two lines and ends 36x² + 25 + 60x. Same expression, different order — worth pointing out, because students read the two orderings as two answers.
The owl box (Part I, §6.2, p.146). Printed advice: if the identity is hard to recall, expand with the distributive property instead and the same result appears. This is the chapter's own position on memorisation.
Input to be done both ways (Part I, §6.2, p.146): the square of 3j + 2k.
The 55-inside-60 figure (Part I, §6.2, p.146, artwork). A square of sidelength 60 with a square of sidelength 55 sitting in its upper left corner. The 55-square is tinted one colour; the two overlapping strips along the right and the bottom, together with the small corner square where they cross, are tinted another. The top edge is marked 55 then 5, the left edge 55 with the whole height marked 60. Four labels sit on leader lines below and to the right of the figure: 60 × 60, −5 × 60, −60 × 5 and +5 × 5. Every one of these is artwork lettering. The running text alongside narrates the argument as a question — the two strips remove the corner square twice, so what should be done about it — and then answers it.
The chapter's own arithmetic for the difference (Part I, §6.2, p.147). Printed: the square of 60 minus 5 taken as 3600 − 300 − 300 + 25, evaluated as 3025, and stated as the area of a square of sidelength 55.
The two derivations of Identity 1B (Part I, §6.2, p.147). Printed. First by distributivity: (a − b)(a − b) gives a² − ba − ab + b², then a² − 2ab + b². Then by substitution, with a printed hint that the square of a − b is the square of a + (−b), and the identity applied with the negative in place. Identity 1B is boxed on that page in the order a² + b² − 2ab; the chapter's SUMMARY on Part I p.157 gives the same identity in the order a² − 2ab + b². Both are the same expression; pick one.
Asked but not answered (Part I, §6.2, p.147): find the general expansion of the square of a difference geometrically, in the way the chapter did it for 55.
Inputs for 1B on numbers (Part I, §6.2, p.147): 99 squared and 58 squared.
Inputs to expand both ways (Part I, §6.2, p.147): the squares of b − 6, of −2a + 3, and of 7y − 3/(4z). The third has a fraction with a letter in its denominator; check it on the printed page before setting it, since the raised and lowered digits collapse in extraction.
Chapter-end inputs. Part I p.154 no. 1(i), 46 squared using Identity 1A, and no. 1(iii), 91 squared using Identity 1B; no. 2(iv), the square of 6x + 5y, and no. 2(v), the square of 2x − 1/2. Part I p.155 no. 3(i) offers six candidate expressions for "two more than a square number" — 2 + s, the square of s + 2, s² + 2, s² + 4, 2s² and 2²s — and asks which fit; the second of those is the distractor this topic exists to defuse. Part I p.155 no. 5(iii) makes two claims and asks whether they hold: that an even number's square is always a multiple of four, and that an odd number's square always exceeds some multiple of eight by exactly one. Part I p.156 no. 8 describes a procedure — add two numbers, multiply the total by half of itself — and asks for the expression and a proof that the result is half the square of the sum.
The Dhauli park (Part I, chapter-end "Figure it Out", p.156, no. 10, with a figure). Printed data: a park plan drawn as a tiled rectangle containing two equal square green plots, each of area g² sq. ft.; everything outside the plots is a walking path w ft. wide that has to be tiled. The figure carries five double-headed arrows — two vertical ones, each marked w ft., in the gap above the left plot and the gap below the right plot, and three unlabelled horizontal ones spanning the gap at the left edge, the gap between the two plots and the gap at the right edge — and the two plots are labelled with their areas. An expression for the tiled area is asked for. Hand over the figure and the two letters; the expression is the exercise.
The chapter prints no answers to any of its exercise items. Its own computed values in this section are 3600, 25, 600, 4225, and 3600 − 300 − 300 + 25 = 3025, plus the two expansions of the square of 6x + 5.

Figures to have open

The general (a + b)² square cut into two squares and two congruent rectangles, with the congruence made visible — same tint, same dimensions marked. This is the chapter's own figure (Part I, §6.2, p.145) and the argument of sections 3 and 4 cannot be made without it. Redraw as a schematic rather than reproducing the printed art.
The numbered version of the same figure for 65, with 60 and 5 on the edges and the four areas written in (Part I, §6.2, p.145).
The 55-inside-60 figure, with the two overlapping strips tinted for removal and the corner square tinted for restoration (Part I, §6.2, p.146). Essential; it is the whole content of sections 8 and 9, and it is where the chapter's second route to Identity 1B lives.
A three-panel comparison of the splits 60 + 5, 30 + 35 and 52 + 13 for the same square. Standard schematic; the chapter asks the student to draw these and draws none of them.
The Dhauli park plan: a rectangle holding two equal square plots with a uniform path around and between them, the path width and the plot side marked (Part I p.156). The chapter's own figure.
No photograph is needed.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 6, "We Distribute, Yet Things Multiply", §6.2 "Special Cases of the Distributive Property", Part I pp.145–147. The printed subheading covering this material is "Square of the Sum/Difference of Two Numbers", which opens §6.2 on Part I p.145.
Identity 1A is boxed on Part I p.145; Identity 1B on Part I p.147.
Example 2 of §6.1 (Part I p.141) is the expansion the chapter reuses rather than repeating; it is carried in Multiplying two two-term expressions, and where the four terms come from.
Chapter-end "Figure it Out", Part I p.154 nos. 1(i), 1(iii), 2(iv), 2(v); Part I p.155 nos. 3(i), 5(iii); Part I p.156 nos. 8 and 10.
The chapter's SUMMARY, Part I p.157, lists both identities among the special cases of the four-term expansion, and gives 1B with its middle term written before its last term.
Forward pointer: Identity 1C, and the two printed patterns that use both 1A and 1B, are in Why the product of a sum and the matching difference is a² − b², and the patterns that follow.