Restating a division as a missing-factor multiplication

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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 fraction of a fraction, and why the numerators and denominators multiply — multiplying two fractions
Cancelling common factors before multiplying, not after — cancelling a shared factor above and below a bar
Whole-number division, and the fact that it undoes multiplication
A fraction times itself upside down giving 1, at least as something to notice
Reading a sentence with a blank in it as an equation to solve

What they should be able to do

Rewrite any division as a multiplication with one factor missing
Name the dividend, the divisor and the quotient in a written division
Find the number that multiplies a given fraction up to 1, by cancelling
Solve a division whose divisor is a fraction and whose dividend is 1
Extend that to a whole-number dividend by scaling the answer
Extend it again to a fractional dividend
Describe the two-move pattern the four worked divisions share
Explain why the missing-factor reading survives a fractional divisor when the sharing-into-groups reading does not

Where it usually goes wrong

"Dividing by a fraction needs a new rule." It needs no rule at all at this stage — it needs the question re-asked. The chapter deliberately reaches the answers before it states any formula, and the formula only arrives on p.189.
"Division means sharing into equal groups, so a fractional divisor is meaningless." Sharing into 2/3 of a group is meaningless; the missing-factor question is not. Show the sharing reading breaking, and the missing-factor reading carrying straight through — that is why the chapter opens §8.2 with the restatement rather than with an example.
"The blank is something you guess and then check." It is found, not guessed: you ask what would leave 1 behind after cancelling, and the fraction turned upside down is what does it.
"12 ÷ 4 and 4 ÷ 12 are the same because both are divisions." The callout on p.186 labels which number is which for a reason, and the labels are not interchangeable.
"To get 3 ÷ 2/3 you must start again from scratch." The chapter reuses the answer to 1 ÷ 2/3 and scales it. Make the reuse visible; it is the argument, not a shortcut.
"A quotient must be smaller than the dividend." 3 ÷ 2/3 comes out as 9/2, which is larger than 3.

Questions to check understanding

Rewrite a given division as a multiplication with a missing factor
Label the dividend, divisor and quotient in a written division
Find what multiplies a given fraction up to 1
Divide a whole number by a fraction, showing the missing-factor step
Divide a fraction by a fraction, showing the same step
Choose, from four candidate expressions, the one that answers a word problem — the printed item on p.196 offers exactly this for a length of lace cut into quarter-metre pieces, for a half-metre of ribbon shared between eight badges, and for five kilograms of flour used a sixth at a time
Explain why a division and its matching multiplication have the same content

Examples worth working on the board

12 ÷ 4 (Part I, §8.2, p.186). Inputs: 12 and 4. The chapter states the quotient 3, then immediately restates the problem as a multiplication with a blank: something times 4 must make 12. Checked against p.186: three speech-bubble labels sit around the printed line, Dividend over the 12, Divisor over the 4, Quotient under the 3.
1 ÷ 2/3 (Part I, §8.2, p.187). Inputs: 1 and 2/3. The chapter asks what multiplies 2/3 to give 1, observes that removing the 2 and the 3 would leave 1, and reads 3/2 off the page. Checked against p.187: the printed line shows 2/3 times a boxed 3/2, with the 2s and 3s struck through diagonally and an arrow from the box down to the word Answer.
3 ÷ 2/3 (Part I, §8.2, p.187). Inputs: 3 and 2/3. The chapter reuses the 3/2 just found and multiplies it by 3, reaching 9/2. The printed working boxes the whole of 3/2 × 3 and labels that the answer. This is the step that turns one special case into a method: solve for a product of 1, then scale.
1/5 ÷ 1/2 (Part I, §8.2, pp.187–188). Inputs: 1/5 and 1/2. The chapter works out that 2 multiplies 1/2 up to 1, then multiplies that 2 by 1/5, giving 2/5. Note the shape: the scaling factor is now itself a fraction, and the method does not notice.
2/3 ÷ 3/5 (Part I, §8.2, p.188). Inputs: 2/3 and 3/5. The chapter finds that 5/3 multiplies 3/5 up to 1, then multiplies 5/3 by 2/3, giving 10/9. This is the example the Discussion on the same page then generalises, so it is the one.
The order the chapter builds them in. Dividend 1 with a fraction divisor; then a whole-number dividend; then a fractional dividend with a unit-fraction divisor; then both fractional. Each step changes exactly one thing. Keeping that order is most of the pedagogy of §8.2.

Figures to have open

An equation line with an empty box that can be filled, cancelled through and then scaled — sections 3 to 9 are all this one object, and re-using it is what makes the shared pattern visible. Standard schematic.
The three-role callout for a division, redrawn.
Diagonal cancellation strokes that can be shown moving on, for section 6.
No photograph, data table or textbook figure is required. §8.2's opening pages carry no diagram beyond the labelled callout.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 8 "Working with Fractions", §8.2 "Division of Fractions", pp.186–188 — the 12 ÷ 4 restatement with its labelled callout (p.186) and the four worked divisions (pp.187–188)
Same part, §8.2, Discussion, pp.188–189, where these four are generalised — covered in Reciprocals, and Brahmagupta's rule for dividing fractions
Same part, §8.1, p.183, for the cancelling this topic relies on
Same part, Figure it Out, p.196, item 2, for the assessment shape