Why dividing can make a number bigger

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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

Reciprocals, and Brahmagupta's rule for dividing fractions — the reciprocal, and division rewritten as multiplication by it
Why multiplying can make a number smaller — when a product lands above or below each of its factors, and why 1 is the hinge
Comparing two fractions
Whole-number division, and the expectation it builds that quotients are small
That the reciprocal of a fraction below 1 is a number above 1, at least as something to check on examples

What they should be able to do

Predict whether a quotient will land above or below its dividend, from where the divisor sits relative to 1
Produce a division whose quotient exceeds its dividend, and one whose quotient falls below it
Explain the prediction by converting the division into a multiplication by the reciprocal
State the chapter's completed rule for division and say for which numbers it has been established
Read a division of the form "how many pieces of this size fit into that", and say why such a count can be large
Recognise that the chapter leaves the divisor-to-quotient comparison as an open question, and say why no rule in terms of the divisor alone can settle it
Choose, from candidate expressions, the division that answers a word problem

Where it usually goes wrong

"Division makes numbers smaller." 6 ÷ 1/4 = 24 is printed on p.189 to break exactly this. The habit comes from years of whole-number division, where the divisor is always at least 1.
"Dividing by a fraction makes the answer bigger." Not always: 3/2 is a fraction, and dividing by it makes things smaller. What matters is which side of 1 the divisor lies on, not whether it is written with a bar. This is the same correction the multiplication topic needed, and saying so out loud is the point of the whole topic.
"The quotient is always bigger than the divisor." 6 ÷ 3 gives 2, which is below 3. The chapter raises this comparison as a question and does not answer it; an explanation that answers it in one line will be wrong.
"There must be a rule for the divisor and the quotient too, in the same shape." There is not one in terms of the divisor alone. The demonstration has to hold the divisor still and move the dividend, because that is the number a divisor-only rule would have to ignore: 6 ÷ 3 = 2, which is below the divisor 3, while 12 ÷ 3 = 4, which is above the same divisor 3. Same divisor, opposite answer, so no rule phrased in the divisor alone can exist. It works the same way below 1: 1/8 ÷ 1/4 = 1/2 is above the divisor 1/4, but 1/100 ÷ 1/4 = 1/25 is below it. Present this as an added observation, offered as a reason the chapter left the question open. Do not argue it from a pair with two different divisors — 6 ÷ 3 beside 1/8 ÷ 1/4 is exactly consistent with a divisor-only rule and proves nothing.
"24 came out of nowhere." It came out of counting quarters in 6, which is what dividing by a quarter asks. Give the counting picture at least once.
"The rule for division is a second thing to memorise." It is the multiplication rule seen through the reciprocal, and the chapter's own instruction on p.190 is to fetch the earlier result rather than build a new one.
"The chapter states the division rule in §8.2." It states it in the SUMMARY on p.198. In §8.2 it is a question with a hint attached.

Questions to check understanding

Given a division, say without computing whether the quotient will exceed the dividend, and give the reason
Complete the chapter's own open statements about the quotient and the dividend
Divide a whole number by a unit fraction and interpret the answer as a count of pieces
Choose the expression that answers a word problem, from four candidates differing only in the operation and the order (the printed set on p.196)
Word problems of the "how many portions" kind — 5 kg of flour at a sixth of a kilogram per loaf is the chapter's own
Explain why dividing by a number below 1 has the same effect as multiplying by a number above 1
Judge the claim "dividing always makes a number smaller" and produce a counter-example

Examples worth working on the board

6 ÷ 3 (Part I, §8.2, p.189). Inputs: two whole numbers. The chapter states the quotient 2 and records that it falls below the dividend, writing the comparison out as an inequality. This is the baseline expectation the rest of the block overturns.
6 ÷ 1/4 (Part I, §8.2, p.189). Inputs: 6 and 1/4. The chapter states the quotient 24 and marks the reversal with an exclamation. Twenty-four is four times six.
1/8 ÷ 1/4 (Part I, §8.2, p.189). Inputs: two unit fractions. The quotient is 1/2, again above the dividend, and this time everything in sight is small. It rules out "the answer got big because 6 was big".
The two open questions (Part I, §8.2, pp.189–190). Printed, in this order: when is the quotient below the dividend and when above it; and whether divisor and quotient stand in any comparable relation. The chapter then tells the reader to use what they worked out for multiplication. It supplies no worked answer to either question on these pages — checked against pp.189 and 190.
The completed rule (Part I, SUMMARY, p.198, last bullet). A divisor between 0 and 1 puts the quotient above the dividend; a divisor above 1 puts it below. This is the only place in the chapter where the statement is printed in finished form; §8.2 itself leaves it as a question.
The reciprocal bridge. Not printed as an argument. Dividing by 1/4 is multiplying by 4; 4 sits above 1; and §8.1 (p.185) has already established what a factor above 1 does to a product. Run the three worked divisions through this once each.
The three word problems on p.196. Inputs only, no answers: 8 m of lace used a quarter-metre at a time, asking how many bags; half a metre of ribbon shared between 8 badges; 5 kg of flour used a sixth of a kilogram per loaf. Each offers four candidate expressions, of which one is the division that answers it. Two of the three are "how many pieces fit" and one is a sharing — which is what makes the set a discrimination task rather than a computation.

Figures to have open

One number axis, reused across sections 2, 3, 5 and 9, on which a dividend, a divisor and a quotient can all be placed. The whole topic is the movement of one mark relative to another, so the axis must not change between sections. Standard schematic.
A 6-unit bar that can be ticked into quarter-unit pieces and counted, for section 4. This picture is added here; the chapter prints no diagram in this block — checked against pp.189 and 190.
A pairing movement that turns "÷ 1/4" into "× 4", for section 8.
No photograph, table or textbook figure is required for this topic.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 8 "Working with Fractions", §8.2 — the unnumbered block "Dividend, Divisor and the Quotient", pp.189–190, with the three worked divisions and the two open questions
Same part, SUMMARY, p.198, last bullet, where the rule appears completed
Same part, §8.1, p.185, for the multiplication result this topic reuses, and §8.2, p.188, for the reciprocal
Same part, Figure it Out, p.196, item 2, for the discrimination task