A fraction of a fraction, and why the numerators and denominators multiply

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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 whole number times a fraction, read as repeated distance — a fraction times a whole number, and the two roles of multiplier and multiplicand
A fraction of a whole understood as: cut into equal parts, then take some
Area as the amount of surface a region covers, and the area of a rectangle as length times breadth for whole-number sides
Equivalent fractions, so that 6/20 and 3/10 can be recognised as one number
Reading a number written as a mixed fraction, such as one and a half

What they should be able to do

Represent a fraction as a shaded part of a unit square
Find a fraction of a fraction by cutting the shaded region again, in the other direction, and counting cells
Say why cutting one way and then the other produces exactly (rows × columns) equal pieces
State and use the rule for multiplying two fractions whose numerators are both 1, and explain the denominator from the picture
State the general rule for multiplying two fractions, and attribute it to Brahmagupta's Brāhmasphuṭasiddhānta of 628 CE
Read such a product as the area of a rectangle built on the two fractions
Rewrite a whole number as a fraction over 1 so that the same rule covers a whole-number factor
Explain why the picture method stops being practical for fractions such as 1/12 and 1/18, and what replaces it

Where it usually goes wrong

"'Of' means add, or means divide." A fraction of a fraction is a multiplication, and the unit square is what shows it: the second cut lands inside the first shading, not beside it.
"You multiply the denominators because that is the rule." The chapter never asserts it as a rule. Cutting into 5 rows and then into 4 columns produces 20 pieces because that is what crossing cuts do.
"The whole changes when you cut again." It does not, and Fig. 8.1 is drawn so that this is visible: the outer square is the same square before and after the second set of cuts, which is why the answer is read against 20 and not against 8.
"The multiplicand must be smaller than one whole." Fig. 8.2 is exactly the counter-case — the multiplicand is 3/2, drawn as a full square plus a half — and the same procedure runs unchanged.
"6/20 and 3/10 are different answers." They are one number written two ways. The chapter records both on p.179 without comment, and simplification becomes the subject of Cancelling common factors before multiplying, not after.
"Area is a formula for rectangles; this is fractions; they are different chapters." The chapter's point on p.180 runs the other way: the product of two fractions is an area, so a multiplication can always be drawn.
"A whole number cannot go into the fraction formula." Writing it over 1 is not a trick to make the formula apply; it is a true statement about the number, and the chapter uses it on p.182 to retire the separate case handled in A whole number times a fraction, read as repeated distance.

Questions to check understanding

Multiply two fractions by shading a unit square, and state the row and column counts used
Find a product by the formula and check it against a drawing
Given a shaded rows-and-columns picture, write down the multiplication it shows
Work out a rectangle's area when both its sides are fractions (a printed item on p.184 asks exactly this for sides of 3¾ ft and 9⅗ ft)
Multiply where one factor is a whole number, by rewriting it over 1
Explain in words why the denominators multiply
State who first set out the general multiplication rule, in which work, and in which year — the chapter gives all three on p.182 and again on p.195

Examples worth working on the board

Half an hour for the tortoise (Part I, §8.1, pp.177–178). Inputs: 1/4 km in 1 hour, 1/2 hour. The chapter shades 1/4 of a unit square, cuts that shading in half the other way, observes that the whole is now in eight pieces with one shaded, and records 1/2 × 1/4 = 1/8. Checked against pp.177–178: on p.177 the unit square is drawn in four horizontal strips with the top strip solid yellow; on p.178 the same square gains one vertical cut across the top strip, and the resulting left half is hatched.
The small table on p.177. Checked against the printed page. Two columns headed Hour and Distance, two rows: 1 against 1/4, and 1/2 against a question mark.
Fig. 8.1 (Part I, p.178). Checked against the printed page. A unit square cut into 5 rows and 4 columns, so 20 cells. The top two rows are yellow — that is the 2/5. The leftmost column of those two rows is hatched — that is the quarter of the 2/5, and it is 2 cells, so 2/20. Inputs: 2/5 km in 1 hour, 3/4 hour; the chapter finds the quarter-hour distance first (2/20) and then triples it, reaching 6/20 and then 3/10.
The p.179 companion picture. Checked against the printed page. The same 5-by-4 square with six of the twenty cells hatched — three of the four columns, across the top two rows.
Fig. 8.2 (Part I, p.179). Checked against the printed page, and the detail matters. The whole is a unit square cut into 2 rows and 4 columns, so 8 cells. Below it sits a half-square strip of 4 more cells, and the two together are bracketed as 3/2 — so the multiplicand is larger than one whole, which is why this example is set as a Math Talk. Quartering the 3/2 means taking the leftmost column, which is 3 cells; against a whole of 8 cells that is 3/8. Multiplying by 5 then gives 15/8. Inputs: 5/4 and 3/2.
Fig. 8.3 (Part I, p.180). Checked against the printed page. A unit square cut into 4 rows and 2 columns; the top-left cell is hatched and carries a 1/8 label on a leader line. Its sides are 1/2 unit and 1/4 unit, and eight copies of it fill the unit square. This is the figure that licenses reading any product of two fractions as an area.
1/12 × 1/18 (Part I, §8.1, pp.180–181). Inputs: the two fractions, and the chapter's own rule for which denominator becomes rows and which becomes columns — rows from the multiplicand, columns from the multiplier. Checked against p.181: the square is drawn with 18 rows and 12 columns, one corner cell shaded, with the two counts labelled outside the figure. The chapter then writes the unit-fraction formula in letters.
5/12 × 7/18 (Part I, §8.1, pp.181–182). Inputs: the two fractions. The chapter divides 7/18 into 12 equal parts, then takes 5 of the results, giving (5 × 7) over (12 × 18), which is 35/216. Checked against p.182: the same 18-by-12 square with a solid block 5 cells wide and 7 cells tall shaded, the 5 and the 7/18 labelled on the block's two sides.
The general formula and its attribution (Part I, §8.1, p.182). The chapter writes the rule with letters and credits Brahmagupta's Brāhmasphuṭasiddhānta, 628 CE. The same claim is made again, with the verse reference, in the history block on p.195.
Whole numbers folded in (Part I, §8.1, p.182). Two printed instances: 3 × 3/4 rewritten with 3 over 1, giving 9/4; and 3/5 × 4 rewritten with 4 over 1, giving 12/5.
Figure it Out, pp.180–181. Inputs only: four unit-fraction products to do by drawing — 1/3 × 1/5, 1/4 × 1/3, 1/5 × 1/2, 1/6 × 1/5 — then 1/12 × 1/18, which is the one the drawing cannot survive; and four further products — 2/3 × 4/5, 1/4 × 2/3, 3/5 × 1/2, 4/6 × 3/5.

Figures to have open

A unit square that can be subdivided into arbitrary rows and columns, with the row and column counts labelled outside it, and with two independent shadings (a solid one for the multiplicand, a hatched one for the part kept). Everything in sections 3, 5, 6, 10 and 11 is built from this one object. Standard schematic; redraw rather than lifting the printed figures.
A version of that square that can be extended below by a partial strip, so that a multiplicand greater than one whole can be drawn as in Fig. 8.2.
A rectangle of sides 1/2 and 1/4 that can be copied eight times to fill the unit square, for section 9.
An 18-by-12 subdivision fine enough that one cell is visibly tiny — the point of section 10 is that the drawing has become impractical.
No photograph or data table from the textbook is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 8 "Working with Fractions", §8.1 — the unnumbered blocks "Multiplying Two Fractions" (pp.177–180), "Connection between the Area of a Rectangle and Fraction Multiplication" (p.180), and "Multiplying Numerators and Denominators" (pp.181–182); Figs. 8.1, 8.2 and 8.3
Same part, §8.1, p.182, for the general formula and the attribution to Brahmagupta
Same part, the unnumbered "A Pinch of History" block, pp.194–196, for Brahmagupta's verse reference and for Bhāskara I's geometric reading of the same rule
Same part, SUMMARY, p.198, first bullet