Multiplying two two-term expressions, and where the four terms come from

What to assume they know

• The distributive property, and expanding a single bracket (What happens to a product when you nudge one factor)
• Identity 1 and what makes an equality an identity (What happens to a product when you nudge one factor)
• Exponent notation for repeated multiplication of the same letter (Why the angles of any quadrilateral add to 360°)
• Multiplying and simplifying fractions
• Integer sign rules, including the product of two negatives
• Place value for two-digit numbers, for the long-multiplication comparison

What they should be able to do

• Expand a product of two two-term brackets and account for each of the four terms by naming the pair it came from
• Justify the four-term expansion by applying distributivity twice, treating a bracket as a single term at the first step
• Extend the same argument to a bracket with three or more terms and predict how many products appear before collecting
• Multiply a term with a fractional coefficient and a letter across a three-term bracket, and simplify each product using exponent notation
• Decide whether two terms are like terms, and explain why unlike terms cannot be combined
• Expand (a + b)(a + b) and identify why two of its four products merge
• Expand (a + b) against a three-term expression and collect the result into four terms
• Recognise ordinary two-digit long multiplication as this expansion applied to place values
• Continue the pattern of products of the form (a − b) against a lengthening sum, and state the next identity in the family

Where it usually goes wrong

• "(a + b)(c + d) is ac + bd." The two cross products are the ones students drop, and they are exactly the two blocks in the off-diagonal of the picture. Draw the rectangle and point at the missing area.
• "There is a mnemonic for the order of the four products." There is, and it is a trap: it only ever covers two terms against two, and it fails silently on the chapter's own no. 4, where a three-term bracket needs six products. The chapter never offers such a mnemonic. Teach the rule that counts nothing.
• "a²b and ab² are like terms — same letters." Same letters, different counts of each. They measure different things: one is a squarish slab, the other a different squarish slab. Example 3 depends on keeping them apart.
• "(3/2)a² and (3/10)a can be added because both have a in them." The chapter puts this question on the page precisely to answer it: no.
• "ba and ab are two different terms." Commutativity says otherwise, and Example 2 turns on it — that is where the doubled middle term is born.
• "(3a/2) × a is (3a/2)a and you cannot do better." You can: (3/2)(a × a) and then exponent notation. The chapter walks it line by line because students stall here.
• "Long multiplication is a separate algorithm you were taught in Class 4." It is this expansion with the brackets written as tens plus units. Once (10a + b)(10c + d) is expanded, the partial products of the school method are standing there labelled.
• "Distributivity is a modern convenience." The chapter's history note dates the first explicit statement to Brahmagupta and the implicit use back much further. The rule is old; only the letters are new.

Questions to check understanding

• Expand a product of two brackets and state which pair of terms produced a named term of the answer
• Expand a product where one bracket has three or more terms, and say how many products appear before collecting
• Expand a product involving a fractional coefficient and simplify every term
• Identify the like terms in a given expression and give the collected form
• Decide whether two given expressions are the same expression, by expanding both
• Fill the cells of a table window with expressions when one cell is given in letters
• Verify or refute a claim of the form "this expression is always a multiple of four" by expanding it
• Continue a family of identities one step further, and check the new member 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 four-block array (Part I, §6.1, p.139, artwork). Four dot-array blocks in a 2-by-2 arrangement labelled ab, an, mb, mn, with a rows and m rows down the left, b columns and n columns across the top, and the whole figure labelled (a + m)(b + n). All lettering sits inside the artwork; read it from the printed page. The page states the governing observation in words: the product is the total of every first-bracket term against every second-bracket term.
• Signs handled by the same identity (Part I, §6.1, p.140). Printed: the general (a + u)(b − v) expansion, ending ab + ub − av − uv, together with the page's remark that the signs follow from ordinary integer multiplication, and an owl box making the same point. Printed answers for the other two sign cases: (a − u)(b + v) gives ab − ub + av − uv; (a − u)(b − v) gives ab − ub − av + uv.
• Example 1 (Part I, §6.1, pp.140–141). Printed in full. Expand (3a/2)(a − b + 1/5). The three products are formed first, then simplified one at a time: (3a/2) × a becomes (3/2)(a × a) and then (3/2)a²; (3a/2) × b becomes (3/2)ab; (3a/2) × (1/5) becomes (3/10)a. Result: (3/2)a² − (3/2)ab + (3/10)a. The page then asks whether any two of those terms can merge, singling out (3/2)a² and (3/10)a, and states that no two terms share the same letters, so nothing further can be done. This is where the term like terms is introduced.
• Example 2 (Part I, §6.1, p.141). Printed in full. Expand (a + b)(a + b). The four products are a × a, b × a, ab and b × b, giving a² + ba + ab + b²; then ba and ab are the same term, so ba + ab is 2ab; result a² + 2ab + b². Keep this in the explanation — §6.2 later cites it rather than redoing it.
• Example 3 (Part I, §6.1, pp.141–142). Printed in full. Expand (a + b)(a² + 2ab + b²). Six products appear: a³ + a²b + 2a²b + 2ab² + ab² + b³. The page simplifies each product line by line, including a marginal "why?" beside a × a² = a³. Then a²b + 2a²b gives 3a²b and ab² + 2ab² gives 3ab², and the result is a³ + 3a²b + 3ab² + b³.
• A Pinch of History (Part I, §6.1, p.142). Printed content, paraphrased: the property was in use, without ever being stated, across five ancient traditions the page names — Egyptian, Mesopotamian, Greek, Chinese and Indian. Euclid's use of it is geometric; Āryabhaṭa's is algebraic; both are extensive and both leave it implicit. The first explicit statement is credited to Brahmagupta in the Brahmasphuṭasiddhānta, verse 12.55, under the name khaṇḍa-guṇanam, glossed by the chapter as multiplication by parts — the multiplier is broken into parts that add back to it, the other number is multiplied by each part, and the results are added, which for two parts is exactly (a + b)c = ac + bc. The very next verse, 12.56, gives a fast-multiplication method, which the chapter takes up in the following subsection.
• The multiplication grid (Part I, §6.1, p.142, "Figure it Out" no. 1). Printed figure: a 10-by-10 multiplication table with both the header row and the header column running 1 to 10, so the entries run 1 to 100. A 3-by-3 window is outlined in red over the rows headed 3, 4, 5 against the columns headed 5, 6, 7 — that is the entries 15, 18, 21 above 20, 24, 28 above 25, 30, 35. Beside the table, a panel shows those nine entries written as products: 3 × 5, 3 × 6, 3 × 7 above 4 × 5, 4 × 6, 4 × 7 above 5 × 5, 5 × 6, 5 × 7. Below that panel sits an empty 3-by-3 frame with pq written in its middle cell only. The task is to fill the other eight cells with expressions. Hand over the whole figure as data: the concrete window, the products panel and the letter frame. The eight expressions are the exercise.
• Products to expand, §6.1 "Figure it Out" no. 2 (Part I p.143): (3 + u)(v − 3); (2/3)(15 + 6a); (10a + b)(10c + d); (3 − x)(x − 6); (−5a + b)(c + d); (5 + z)(y + 9). The third of these is the long-multiplication case and deserves its own section — it is a two-digit number times a two-digit number written in letters.
• Products to expand, no. 4 (Part I p.143): (a + ab − 3b²)(4 + b), and (4y + 7)(y + 11z − 3). Both have a three-term bracket, so both test whether the student is counting terms or applying the rule.
• The family of identities, no. 5 (Part I p.143): expand (a − b)(a + b), then (a − b)(a² + ab + b²), then (a − b)(a³ + a²b + ab² + b³); the item asks for the pattern, for the next identity in the family, and for a check by expanding. The chapter returns to the first of these on Part I p.148. Hand over the three products; the pattern is the exercise.
• Further inputs from the chapter-end set: (p − 1)(p + 11) (Part I p.154, no. 2(i)); −(2y + 5)(3y + 4) (Part I p.154, no. 2(iii)) — the chapter's only product carrying a minus in front of the whole thing, and so the natural companion to section 4's sign work; (7p) × (3r) × (p + 2) (Part I p.154, no. 2(vi)); (k + 1)(k + 2) − (k + 3), claimed always to equal 2, and (2q + 1)(2q − 3), claimed to be a multiple of 4 (Part I p.155, no. 5(i) and 5(ii)); and the remainder item on Part I p.155, no. 6, where one number leaves 3 on division by 7 and another leaves 5, and the remainders of their sum, their difference and their product are asked for. The last is expansion put to work: write the two numbers as 7s + 3 and 7t + 5 and multiply out.
• The chapter prints no answers to any of its exercise items.

Figures to have open

• The four-block rectangle for two two-term brackets, each block labelled with its own product. The chapter's own figure (Part I, §6.1, p.139) and the spine of sections 1 and 3. Redraw as a schematic; do not reproduce the printed art.
• A three-by-two block grid for a three-term bracket against a two-term bracket, so section 5 can show six products in a picture rather than in a list. Standard schematic.
• The 10-by-10 multiplication table with a 3-by-3 window highlighted and the centre cell labelled (Part I p.142). This is the chapter's own figure and the exercise cannot be posed without it.
• A like-terms sorter: bricks whose shapes encode a²b, ab², a³ and b³, so merging is visibly a matter of matching shapes. Standard schematic.
• A side-by-side of an ordinary two-digit long multiplication and the letter expansion of the same product, with the four partial products connected by arrows. Standard schematic; it does not appear in the chapter, which supplies only the letter form as an exercise item.
• No photograph is needed. The history section can be carried by a plain timeline.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 6, "We Distribute, Yet Things Multiply", §6.1 "Some Properties of Multiplication", Part I pp.139–143. Identity 1 and its picture are on Part I p.139; the sign cases are on Part I p.140; Examples 1 to 3 run from Part I p.140 to p.142.
• The printed subheading "A Pinch of History" occupies the upper half of Part I p.142.
• §6.1 "Figure it Out", Part I pp.142–143, items 1, 2, 4 and 5. Item 3 belongs to What happens to a product when you nudge one factor and is not repeated here.
• Chapter-end "Figure it Out", Part I p.154 nos. 2(i), 2(iii) and 2(vi), and Part I p.155 nos. 5(i), 5(ii) and 6.
• The chapter's SUMMARY, Part I p.157, states the four-term expansion using the letters a, b, c, d rather than the shift letters used inside §6.1 — a detail worth matching to whichever form the class has been taught.
• Forward pointer: Example 2's result is the subject of (a + b)² and (a − b)²: why there is a middle term at all, and item 5(i) of the "Figure it Out" set is picked up in Why the product of a sum and the matching difference is a² − b², and the patterns that follow.