What happens to a product when you nudge one factor

What to assume they know

Multiplication as a rectangular array or as repeated addition, and its commutativity
Letters standing for numbers, and reading 5a and xy as products with the multiplication sign left out (The carpenter's problem: how to be sure a frame really is rectangular)
Integer multiplication, including the sign rules for negative factors (Body parts and tally marks: counting before numerals)
Adding and subtracting integers
That two expressions are the same expression when every substitution gives them the same value

What they should be able to do

Predict how a product changes when one factor rises by one, and justify the prediction from the distributive property rather than from two computations
Expand a(b + c) and (a + b)c, and say which step uses distributivity and which uses commutativity
Expand (a + 1)(b + 1) by treating one bracket as a single term, and identify the increase as a + b + 1
Expand (a + 1)(b − 1) and decide, for given values, whether the product rose or fell
State what an identity is, and explain why substituting a few values supports an identity without establishing it
Use Identity 1 with negative values of the shifts to recover every increase-and-decrease case from one statement
Find the condition under which two opposite shifts leave a product unchanged
Explain why the two diagonal products of a 2-by-2 block of calendar dates always differ by the same amount

Where it usually goes wrong

"Add one to a factor, add one to the product." The commonest error in the chapter's opening question. Adding one more copy of a adds a. Ask the class for a prediction before revealing anything, then run 23 × 27 and 23 × 28.
"Up by one and down by one must cancel." They cancel only when b is one more than a. Otherwise the change is b − a − 1, which the chapter shows can be either sign — and it explicitly asks for three cases where the product falls.
"(a + 1)(b + 1) is ab + 1." Distributing shows three extra pieces: a strip of a, a strip of b, and one loose unit. The four-block array is the cure; the loose corner is a single marker and students can point at it.
"Decreases need their own rule." They do not. The whole point of Identity 1 is that the shifts are integers, so a decrease is a negative shift, and the ordinary sign rules produce the right signs unaided. The chapter's owl box says so directly.
"Distributivity is an arithmetic habit that letters make dubious." The letters stand for numbers, so nothing is being extended. The chapter checks negative substitutions precisely to make this visible.
"Substituting three values proves an identity." It does not, and the chapter is careful: it defines an identity as an equality holding whenever the letters are replaced by numbers, and it derives every one of them. Substitution is how you catch a mistake, not how you finish an argument.
"The increase depends on which bracket you open first." It cannot; the page invites the student to redo (a + 1)(b + 1) the other way round and get the same four pieces in a different order.

Questions to check understanding

Given a product, state the change when one factor is shifted by a named amount, without computing either product
Expand a product of two two-term brackets and identify which term of the answer came from which pair of terms
Use Identity 1 with one or both shifts negative, and confirm the result by multiplying out directly
Decide, for stated factors, whether an up-and-down pair of shifts raises or lowers the product
Construct examples meeting a condition — for instance, pairs whose product survives a named pair of opposite shifts unchanged
Explain why a stated equality is an identity rather than an equation with a particular solution
Prove a numerical observation about a table or calendar block by labelling its entries with one letter and 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 product (Part I, §6.1, p.136). Printed: 23 × 27, followed by three questions — the gain when 23 rises by one, the gain when 27 rises by one, and the gain when both rise by one. The page then asks for a generalisation to any two numbers. Note the deliberate ordering: the chapter takes the second question first, because commutativity makes it the easy one.
The dot array for a(b + c) (Part I, §6.1, p.137, artwork). Two blocks of round markers side by side inside a single bracket-like frame. The left block is labelled ab, the right ac, the whole thing a(b + c). The row count is marked a rows down the left; the two column counts are marked b columns and c columns across the top. All four labels are lettering inside the artwork — read them from the printed page, not from extracted text.
The numeric instance (Part I, §6.1, p.137). Printed: with a = 23, b = 27 and c = 1, the identity gives 23 × 27 plus 23, and the page boxes the 23 and labels it as the increase.
The mirror-image derivation (Part I, §6.1, p.137). Printed in three lines: (a + b)c is turned round by commutativity, expanded by distributivity, and each product turned round again. Worth showing in full — it is the chapter's only place where commutativity is used twice to buy one result.
One factor up by one (Part I, §6.1, p.137). The display line expands a(b + 1) with the a boxed as the increase, so the increase is a and the line must read a(b + 1) = ab + a. A printed slip sits here: the operator before the boxed increase is set as a multiplication sign where the argument needs a plus. This is a printing error, not an extraction artefact — the printed page and the extracted text agree, and the controls are on the same line and the same page: the plus inside the bracket sets correctly, and so does the plus in the numeric instance eleven lines above. Anyone showing this page must correct the operator, exactly as on Part I p.148; reproducing the line as printed puts a false equation in front of the class.
Both factors up by one (Part I, §6.1, p.138). Printed, with the algebra on the left and the same steps in numbers on the right: (a + 1)(b + 1) is expanded by taking (a + 1) as a single term, giving (a + 1)b + (a + 1)1, and then ab + (b + a + 1), with the bracket boxed as the increase. Numeric column: (23 + 1)(27 + 1) becomes 23 × 27 plus the bracket 27 + 23 + 1. The page then asks what happens if (b + 1) is taken as the single term instead.
One up, one down (Part I, §6.1, p.138). Printed, again in two columns: (a + 1)(b − 1) becomes (a + 1)b − (a + 1)1, then ab + b − (a + 1), then ab + b − a − 1, with the last three terms boxed as the increase. Numeric column: 23 × 27 plus 27 minus 23 minus 1. The page asks whether the product always increases and sets the student to find three cases where it decreases.
The negative substitutions offered (Part I, §6.1, p.138). Printed as suggestions: a = −5 with b = 8, and a = −4 with b = −5. The page states that integers satisfy distributivity, writing it with the letters x, y, z, and concludes that every expression for an increase already holds for negative values.
The two printed examples of identities (Part I, §6.1, p.139): a(b + 8) equal to ab + 8a, and (a + 1)(b − 1) equal to ab + b − a − 1. These are the page's illustrations of the word, not exercises.
Identity 1, boxed (Part I, §6.1, p.139). Printed: (a + m)(b + n) equals ab + mb + an + mn, with the increase given as an + bm + mn. The box carries four curved arrows in two colour-coded pairs — two magenta above the line, from a to b and from a to n, and two blue below it, from m to b and from m to n. The terms on the right are tinted to match, and that colour coding is the part that makes the figure readable when it is redrawn.
The four-block array for Identity 1 (Part I, §6.1, p.139, artwork). Four dot-array blocks in a 2-by-2 arrangement, labelled ab, an, mb and mn, with a rows and m rows marked down the left and b columns and n columns across the top; the whole figure is labelled (a + m)(b + n). Again all lettering is inside the artwork.
Recovering the earlier case from Identity 1 (Part I, §6.1, p.140). Printed: writing (a + 1)(b − 1) as (a + 1)(b + (−1)) and taking the shifts to be 1 and −1 reproduces ab + b − a − 1 exactly.
The two shift tasks (Part I, §6.1, p.140). Set for the student: (i) one factor down by 2 while the other is up by 3; (ii) both down, one by 3 and the other by 4. The page also asks that the answers be checked by multiplying out without turning the subtractions into additions.
The general one-up-one-down expansion (Part I, §6.1, p.140). Printed: (a + u)(b − v) in three lines, ending ab + ub − av − uv, followed by the instruction to check that it matches Identity 1 with the second shift negative.
The other two sign cases (Part I, §6.1, p.140). Printed answers on the page: (a − u)(b + v) gives ab − ub + av − uv, and (a − u)(b − v) gives ab − ub − av + uv.
Unchanged products (Part I, §6.1, p.143, "Figure it Out" no. 3, a Math Talk item). Set for the student: find three pairs of numbers whose product is unchanged when one factor rises by 2 and the other falls by 4.
Calendar diagonals (Part I, chapter-end "Figure it Out", p.155, no. 4, a Math Talk item). Printed: a February calendar with columns headed for the seven days, the 1st falling on the last column of the first week and the month running to 28. A 2-by-2 block containing 4, 5, 11, 12 is outlined in red. The page works the two diagonal products of that block, 4 × 12 = 48 and 5 × 11 = 55, and asks for the same on other blocks, for the observation, and for the reason. Printed hint: label the block a and a + 1 on the top row, a + 7 and a + 8 beneath. Hand over the hint; do not hand over the difference.
The chapter prints no answers to any of its exercise items.

Figures to have open

The array for a(b + c): a rows of markers split into a block b wide and a block c wide, with all three labels. This is the chapter's own figure (Part I, §6.1, p.137) and section 2 depends on it. Redraw as a schematic rather than reproducing the printed art.
The four-block array for (a + m)(b + n), with the row and column counts marked and each block labelled (Part I, §6.1, p.139). Essential — it is the picture that makes the four terms inevitable rather than remembered.
A bar or strip diagram for ab, (a + 1)b and (a + 1)(b + 1) on one scale, so the added strips can be seen as strips and the corner as a single unit. Standard schematic.
A 2-by-2 calendar block with the four dates replaced by a, a + 1, a + 7, a + 8, and both diagonals drawn (Part I p.155). Standard schematic.
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", §6.1 "Some Properties of Multiplication", Part I pp.136–140. Within §6.1 the printed subheading covering this material is "Increments in Products", beginning on Part I p.136.
The chapter's own opening paragraphs, Part I p.136, set distributivity up as the property that links multiplication to addition and announce that the chapter will use it to describe multiplication patterns.
Identity 1 is boxed on Part I p.139; its visualisation is on the same page.
§6.1 "Figure it Out", Part I p.143, no. 3.
Chapter-end "Figure it Out", Part I p.155, no. 4.
The chapter's SUMMARY, Part I p.157, states the four-term expansion of two two-term brackets as the chapter's first result.