Fractional relations between two quantities

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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, and reading a product as an area
Cancelling common factors before multiplying, not after — cancelling, needed to reduce the long products here
Reciprocals, and Brahmagupta's rule for dividing fractions — dividing by a fraction using the reciprocal
That the order of the factors in a product does not matter (Part I, §8.1, p.186)
Area of a triangle as half the area of the rectangle around it
Converting between units of mass, volume and currency

What they should be able to do

Turn a word problem into a division or a multiplication before computing anything
Express a part of a figure as a fraction of the whole by chaining the containment relations and multiplying along the chain
Compute the area of a region inside a square by successive fractions rather than by measuring
Evaluate a long product of fractions and reduce it to lowest form
Read a historical fraction problem, extract the chain, and interpret the answer in the units the problem uses
Convert between quantities linked by a ladder of exchange rates, by multiplying along the ladder
Say why the answer to a division can be a fraction even when the thing being counted is a whole object

Where it usually goes wrong

"You add the fractions along the chain." Three-quarters of one-eighth is a product. Adding gives a number larger than either, which the picture immediately refutes — the shaded sliver is smaller than the triangle it sits inside.
"3/4 of the triangle means 3/4 of the square." Every fraction in the chain is measured against a different whole, and only the last multiplication brings it back to the square. Say out loud, at each link, what the current whole is.
"The final answer should be roughly a quarter, since we started with a quarter." Each link shrinks it further. Landing on 3/32 is the point.
"375/2 bricks is a mistake." It is the chapter's printed answer, and it is the honest result of the division. Whether you can lay half a brick is a different question from whether the arithmetic is right.
"Four fountains, so add the times." Adding a day, half a day, a quarter and a fifth would give a longer time than the slowest fountain alone, which is absurd for taps running together. What adds is how many cisterns each fills in a day.
"The miser gave 1/1280 of a coin, so almost nothing." He gave exactly one coin — the smallest one in circulation. The unit conversion is what turns a fraction into an object, and it is the punchline.
"The exchange rates on p.194 are historical fact." The chapter says they varied and introduces its own set as an assumption. Keep the hedge; the assumed ladder does not even reconcile with the 1280 cowries the tale gives.
"A word problem needs the right formula." Every one of these needs the right chain, and the arithmetic afterwards is §8.1 and §8.2 unchanged.

Questions to check understanding

Find what fraction of a whole square a shaded region occupies, from a figure with no measurements on it (printed on p.193 as a pair of figures, and again on p.197 as item 10)
Evaluate a long product of fractions and reduce it to lowest form
Word problems combining a fraction and a division: a novel part-read, a car's petrol, a journey by train against by plane, a cake shared after part is eaten (items 5 to 8, all four on p.197)
Sum a set of divisions of 1 by unit fractions — the Pāṭīgaṇita item on p.197, attributed there to Sridharacharya in the ninth century CE
A branching diagram in which a group splits at each junction, asking what fraction reaches each destination (Fig. 8.7, p.198). The chapter prints no answer, and the branch structure has to be read off the artwork: the third branch point is a four-way split, so an answer that assumes every split is in two comes out wrong. The structure and the fractions it gives are set out under Worked examples and Notes
Extend a product pattern and state the generalisation — item 12 on p.198 builds products of terms of the form one minus a unit fraction and asks for a general statement. The pattern is that consecutive terms cancel and the product from one-half up to one minus one-over-n equals one-over-n; that reduction is working added here, not something the chapter prints
Choose the correct comparison for a product of two fractions each above 1 (item 9, p.197)

Examples worth working on the board

Example 3 (Part I, §8.3, p.190). Inputs: 1/4 litre of milk, 5 cups. The chapter writes the division 1/4 ÷ 5, restates it as a multiplication with the cup-quantity unknown, takes 1/5 as the reciprocal of 5, and reaches 1/20 litre per cup. Checked against p.190: a number line runs across the page with four interior ticks, bracketed above as the quarter-litre shared over five cups and below as the milk in one cup.
Example 4 (Part I, §8.3, pp.190–191). Inputs: an area of 7½ square units; square bricks of side 1/5 unit. The chapter computes the brick area as 1/25, rewrites 7½ as 15/2, and divides, reaching 375/2. The attribution is to Baudhāyana's Śhulbasūtra, dated in the chapter to about 800 BCE.
The half brick. The chapter leaves the count as 375/2 and does not remark on it. Checked against p.191. That is a formal answer to a physical question.
Example 5 (Part I, §8.3, p.191). Inputs: four fountains filling one cistern, taking one day, half a day, a quarter of a day and a fifth of a day. The chapter asks how many times each fills the cistern in a day, sets out four divisions of 1 by those times, states the total as 12, and gives the combined time as 1/12 day. Three of the four divisions are printed with blanks for the student to fill; checked against p.191. The structure worth naming is that the fills per day add, while the times do not.
Fig. 8.4 (Part I, p.192). Checked against the printed page. A square carrying both diagonals, the two mid-lines, and a further set of lines subdividing the top-right quarter, which is itself cut into four with one of its own diagonals. A narrow strip inside the top-right quarter is hatched in brown. This is the figure the whole block turns on.
Fig. 8.5 (Part I, p.192). Checked against the printed page. The same square, with the top-right quarter outlined in red so the first link of the chain is visible.
Fig. 8.6 (Part I, p.192). Checked against the printed page. Two panels: on the left the red square drawn on its own and enlarged, with the triangle inside it filled yellow; on the right the whole square again with the red quarter marked.
The chain (Part I, "Fractional Relations", pp.192–193). Inputs, in the chapter's order: the whole square counts as 1 square unit; the top-right square is a quarter of it; the yellow triangle is half of that quarter, so 1/2 × 1/4; the shaded region is three-quarters of the triangle, so 3/4 × 1/8. The chapter asks, without answering, why the shaded region is three-quarters of the triangle.
The two follow-up figures (Part I, p.193). Checked against the printed page. Two more squares with hatched regions, set as practice for the same chaining method, with no answers printed. The chapter says problems of this kind return in a later chapter, without naming which.
A Dramma-tic Donation (Part I, pp.193–194). Inputs: the chapter gives this as a translated problem, from the Līlāvatī of Bhāskarāchārya — Bhāskara II — dated 1150 CE. The gift is described as a chain of six fractions of a dramma — 1/5, 1/16, 1/4, 1/2, 2/3, 3/4 — and the chapter reassembles them as a single product, evaluating it to 6/7680 and reducing that to 1/1280. The tale's exchange rate is 1 dramma to 1280 cowrie shells, so the gift is one cowrie. Say what the joke is: the extravagant-sounding chain resolves to the smallest coin there was.
The coin ladder (Part I, p.194). Inputs, and the chapter's own framing matters. It lists four coin types — gold, silver, copper and cowrie shells — states plainly that exchange rates varied by region, period, economic conditions and the weight and purity of the coins, and only then offers a set of rates as an assumption: 1 gold dinar to 12 silver drammas, 1 silver dramma to 4 copper panas, 1 copper pana to 6 mashakas, 1 pana to 30 cowrie shells. It then works one rung — 1 copper pana as 1/48 of a gold dinar, shown as 1/12 × 1/4 — and leaves two rungs blank. Note the tension the chapter has set up: the assumed ladder and the tale's 1280 cowries per dramma do not agree, which is exactly why the chapter hedged.
Fig. 8.7 (Part I, p.198, exercise item 11). Checked against the figure's own region; whole-page scans the earlier pass used, the upper arms cannot be told apart. A single path enters at the bottom, four branch points are marked with red discs, five green bands reach the bar labelled Mango tree and two reach the bar labelled Sugarcane field. Counting the arms that leave each disc, from the bottom up: two, two, four, two — the third disc is the four-way one, sending two arms to the mango tree, one to the sugarcane field and one on to the fourth disc. Inputs: that structure, and the item's own condition that the group divides into equal shares at each branch point.

Figures to have open

A square that can be progressively outlined, enlarged and re-shaded: whole → top-right quarter → triangle inside it → shaded part of the triangle. Sections 5 to 9 are all this one object, and the zoom-in is what carries the argument. Redraw; Figs. 8.4, 8.5 and 8.6 are the book's own artwork.
A running label showing what the current whole is at each link — this is the single most useful piece of scaffolding in the topic.
A rectangular region tiled with small squares, for Example 4, with one partial brick visible at the edge.
A stacked bar of four fill-rates, for Example 5.
A coin ladder from gold dinar down to cowrie shell, with the exchange rates on the links and a visible marker that they are assumed. Do not present it as a historical chart.
No photograph from the textbook is required. The chapter prints no image of any coin — checked against pp.193 and 194.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 8 "Working with Fractions", §8.3 "Some Problems Involving Fractions", pp.190–191 — Examples 3, 4 and 5
Same part, the unnumbered block "Fractional Relations", pp.192–193, with Figs. 8.4, 8.5 and 8.6
Same part, the unnumbered block "A Dramma-tic Donation", pp.193–194
Same part, the unnumbered block "A Pinch of History", pp.194–196, for the Śhulbasūtra context Example 4 draws on
Same part, Figure it Out, pp.196–198, including Fig. 8.7
Same part, §8.1 p.186 and §8.2 p.189, for the two rules every computation here uses