The dropped multiplication sign, and what it silently means

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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What to assume they know

A letter-number stands for any number, not one number — letter-numbers and algebraic expressions
Why every rule of arithmetic carries over to algebra — sum of terms, and why arithmetic rules survive the arrival of letters
Multiples and multiplication tables; the position of a term in a sequence
Multiplication is done before addition unless brackets say otherwise
Adding and subtracting negative numbers

What they should be able to do

Rewrite 4 × n in the chapter's shortened form and back again
State the order convention — the number is written first, the letter or letters after it
Find the value of an expression such as 7k or 5m + 3 for a given value of its letter-number, restoring the hidden multiplication sign before computing
Diagnose a wrong substitution: say which step went wrong and what the value should be
Show that two expressions are different by finding one value of the letter that separates them
Explain why 10y – 3 and 10(y – 3) are not the same, in terms of what the multiplication acts on
Say why a missing sign can only mean multiplication

Where it usually goes wrong

"4n is a two-digit thing, like 42." Card 2 of the p.87 block is exactly this error: 3d at d = 6 read as the digits 3 and 6 side by side. Card 4 is the same error with an addition behind it — 2r at r = 8 read as 28, then 1 added to reach the printed 29, where the value is 17.
"5u and 5 + u are two ways of writing the same thing." The chapter does not merely deny this; it sets up a test. One value of u on which the two disagree settles it.
"Since the sign is invisible, it happens last." The order of operations is untouched by the convention. 5m + 3 multiplies before it adds, exactly as 5 × m + 3 would. Card 3 on p.87 (3s – 2 claimed to be 15 at s = 7) is what happens when this slips: the 3 has been made to multiply the whole of s – 2 rather than s alone, and the value is 19.
"You can drop the plus sign too, if it is obvious." Nothing is written without a sign except a multiplication. That is the entire safety of the convention: seeing no sign, there is only one thing you can put back.
"Brackets are decoration once you know the values." 10y – 3 and 10(y – 3) contain the same three symbols and agree at no value of y at all: the first stands 27 above the second wherever you look, because the letter falls out of the difference between them.
"A minus in front of a bracket only affects the first thing inside." Card 9 on p.87 is this error wearing a disguise: subtracting (3 – n) when n is larger than 3 means subtracting a negative.

Questions to check understanding

Find the value of an expression such as 7k, 2r + 1 or 3(m + 1) for a given value of its letter-number
Given a completed substitution, say whether it is right; if not, explain the error and give the correct value — this is the exact shape of the p.87 block and of competency-based items on algebraic notation
Decide whether two given expressions are equal, and justify by finding a value that separates them
Write, in the shortened form, an expression given in words
Write the nth term of a simple sequence of multiples
Say what 10(y – 3) means in words, and how it differs from 10y – 3

Examples worth working on the board

The sequence (Part I, §4.3, p.86). 4, 8, 12, 16, 20, 24, 28, … The page identifies it as the multiples of 4 in increasing order, gives the third term as 4 × 3 and the twenty-ninth as 4 × 29, and then asks for the nth.
The shortening (Part I, §4.3, p.87). 4 × n becomes 4n. The page states the order convention in the same breath: the number is written first, then the letter or letters.
Two substitutions the page works (Part I, §4.3, p.87). 7k at k = 4 gives 7 × 4 = 28. 5m + 3 at m = 2 gives 5 × 2 + 3 = 13, and the page spells out that 5m is 5 × m before doing the arithmetic.
The nine substitutions put up for diagnosis (Part I, §4.3, p.87). Printed in a three-by-three block of rounded cards, numbered 1 to 9; checked against p.87. The cards do reach the text layer in reading order, so the values below can be checked against it as well as against the printed page — what the printed page settles is the grid, since nothing in the text layer says which card sits beside which. Each card gives values and a claimed result: (1) a = –4, claims 10 – a = 6; (2) d = 6, claims 3d = 36; (3) s = 7, claims 3s – 2 = 15; (4) r = 8, claims 2r + 1 = 29; (5) j = 5, claims 2j = 10; (6) m = –6, claims 3(m + 1) = 19; (7) f = 3, g = 1, claims 2f – 2g = 2; (8) t = 4, b = 3, claims 2t + b = 24; (9) h = 5, n = 6, claims h – (3 – n) = 4. The book prints no answers. Worked through for this brief, the correct values in the same order are 14, 18, 19, 17, 10, –15, 4, 11 and 8 — so card 5 is the only one already right, and an explanation that says "spot the mistakes" without saying that one card has none will teach the class to invent a mistake.
**5u against 5 + u (Example 10, Part I, §4.4, p.92).** Two four-armed diagrams, each with the expression in a centre box and four empty boxes at the corners, the arms labelled u = 11, u = 2, u = 8 and u = 5. The book fills in one box on each diagram — 10 on the 5u diagram and 7 on the 5 + u diagram, both for u = 2 — and leaves the other six to the student. Checked against p.92.
**10y – 3 against 10(y – 3) (Part I, §4.4, p.93).** The same two diagrams again, arms labelled y = 2, y = 0, y = 7 and y = 10, with one box filled on each: 17 and –10, both for y = 2. The page also glosses the two expressions in words — three less than ten times y, against ten times a number that is itself three less than y. Checked from p.93.

Figures to have open

The nine diagnostic cards from p.87, redrawn as nine labelled cards. The three-by-three arrangement matters: it invites comparison across the row.
The four-armed value diagram, drawn twice, for 5u and 5 + u. This is the chapter's own device for "test it at several values" and is reused on p.93, so it is worth building once and reusing.
A fading-× movement for 4 × n → 4n. Standard schematic; the book states the convention in words only.
No photograph or dataset is needed for this topic.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 4 "Expressions using Letter-Numbers", §4.3 Omission of the Multiplication Symbol in Algebraic Expressions, pp.86–87, including the unnumbered Mind the Mistake, Mend the Mistake block on p.87.
Same chapter, Part I, §4.4, pp.92–93: Example 10 and the Math Talk pair that follows it. They sit in §4.4 but they are notation arguments, so this topic carries them; the simplification work of §4.4 belongs to Collecting like terms, and what "simplest form" is for.
Backward pointer: Part I, §4.1, p.82 and p.83, where 2 × n and 4 × q are still written with their signs — the same expressions, before the convention.