Cancelling common factors before multiplying, not after

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What to assume they know

A fraction of a fraction, and why the numerators and denominators multiply — the rule that numerators multiply and denominators multiply
Equivalent fractions: dividing numerator and denominator by the same number leaves the fraction's value alone
Factors and common factors of whole numbers, and finding them by inspection
Lowest form of a fraction
Multiplying where one of the two numbers is whole (A whole number times a fraction, read as repeated distance), for the exercises that mix the two

What they should be able to do

Write a product of two fractions as one fraction whose numerator and denominator are each an unmultiplied product
Spot a common factor shared by a numerator on one side and a denominator on the other, and divide both by it
Carry out two separate cancellations in one product, using different factors
Justify cancelling by the equivalent-fraction rule rather than by "it is allowed in multiplication"
Say why cancelling first and simplifying afterwards give the same number, and what is gained by doing it first
Name the process the chapter names — cancelling the common factors — and give its Sanskrit name, apavartana
Recognise, in a word problem, that a "fraction of a quantity" is the same multiplication, and that units must be matched before the fractions are

Where it usually goes wrong

"Cancelling is a special rule for multiplication." It is the ordinary rule for rewriting one fraction, and it becomes usable here only after the product has been written as a single fraction. Show that step happening before any striking-through, or the licence disappears.
"You can cancel a numerator against a numerator." Dividing only the top by 12 changes the number. Both strokes have to land on opposite sides of the same bar.
"You can cancel across a plus sign the same way." Nothing in the chapter licenses that, and it is false; the two numbers being cancelled must be factors of the numerator and of the denominator, which is exactly what Brahmagupta's rule has just made them.
"Cancelling gives a different, smaller answer." It gives the same number in a smaller-looking form. Run 12/7 × 5/24 both ways once — 60/168 simplified, and the cancelled route — and land on 5/14 twice.
"You must find the largest common factor before you may cancel." The second worked example cancels 14 and then, separately, 5. Repeated small strokes are as valid as one big one, and easier to see.
"Cancelling is optional decoration." It is optional, and the chapter says so by offering it as an alternative. What it buys is that the answer arrives already in lowest form, which is what the exercises ask for.
"12/15 of 500 g and 3/20 of 4 kg can be compared as they stand." The units differ. Comparing the fractions alone answers a different question.

Questions to check understanding

Multiply two fractions and give the answer in lowest form (the printed instruction on p.182 is exactly this)
Show the cancellation you used, as working — boards give method marks for the struck-through pair
Find a fraction of a given quantity, where the quantity carries a unit
Multi-stage "fraction of the remainder" problems, where each later fraction is taken of what is left and the answer is wanted as a fraction of the original
Compare two quantities each given as a fraction of a different total
Work out a rectangle's area when its sides are given as mixed fractions
Explain why cancelling does not change the value of the product

Examples worth working on the board

12/7 × 5/24 (Part I, §8.1, pp.182–183). Inputs: the two fractions. The chapter writes the product as one fraction, rings the 12 in the numerator and the 24 in the denominator, notes that 12 divides both, and replaces them by 1 and 2. The printed working then reads 1 × 5 over 7 × 2, giving 5/14. Checked against p.183: the printed layout strikes each ringed number through and writes its replacement small, above the 12 and below the 24.
14/15 × 25/42 (Part I, §8.1, p.183). Inputs: the two fractions. Two independent cancellations are shown at once and colour-coded to keep them apart — the 14/42 pair ringed and struck in black, with black replacements, a small 1 above the 14 and a small 3 below the 42; the 25/15 pair ringed and struck in red, with red replacements, a small 5 above the 25 and a small 3 below the 15. The working reduces to 1 × 5 over 3 × 3, giving 5/9. Checked against p.183. The two colours are what make it visible on the page that the factor need not be the same one both times.
The reason printed for it (Part I, §8.1, p.182). The chapter's own justification is the equivalent-fraction rule: divide both parts of a fraction by a factor they share, and its value stays put.
A Pinch of History (Part I, p.183). Inputs: the Sanskrit term apavartana for reduction to lowest terms; the Jaina scholar Umasvati, dated in the chapter to about 150 CE; and the point being made, which is that the idea was familiar enough outside mathematics to serve as an image in a work of philosophy. The chapter names neither that work nor the image itself.
Figure it Out, pp.183–184. Inputs only, no answers:
- A tap fills 7/10 of a tank in an hour; how much is filled in 1/3 hour, in 2/3 hour, in 3/4 hour, in 7/10 hour; and, marked Try This, how long the tap must run to fill the tank. The last part is a division and points forward to §8.2.
- The government takes 1/6 of Somu's land; of what is left she gives half to Krishna and a third to Bora, keeping the rest. Three parts, each asked as a fraction of the original land — which is the trap: the halves and thirds are taken of the remainder, not of the whole.
- A rectangle with sides of 3¾ ft and 9⅗ ft, whose area is wanted.
- Four saplings in a row, 3/4 m between neighbours; the distance from the first to the last. The printed hint asks for a rough diagram, because the count of gaps is one fewer than the count of saplings.
- A comparison of two weights, one given as 12/15 of 500 grams and the other as 3/20 of 4 kg, asking which one is the heavier. The two are stated in different units; matching them is the first step and the whole point of the item.

Figures to have open

A fraction bar wide enough to hold two unmultiplied products, with ringing, strike-through and small replacement digits — the whole topic is carried by this one piece of notation. It must carry two ink colours, because the 14/15 × 25/42 line on p.183 sets one of its two cancellations in black and the other in red. Redraw; the printed rings, strokes and two-colour replacement digits are the book's own setting.
A rectangular plot that can be cut down in stages, for the Somu item: first 1/6 removed, then a half and a third of what remains.
A row of four saplings with three labelled gaps, for the off-by-one item.
Two weights on a balance with their units shown, for the final item.
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 block "Multiplication of Fractions — Simplifying to Lowest Form", pp.182–183, with both worked cancellations
Same part, the unnumbered "A Pinch of History" block on p.183, for apavartana and Umasvati
Same part, §8.1, Figure it Out, pp.183–184
Same part, SUMMARY, p.198, second bullet
Same part, §8.1, p.182, for the multiplication rule this topic operates on