Brahmagupta's method: you can only add units of the same size

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What to assume they know

Comparing two fractions by giving them a common fractional unit: converting a pair of fractions to a shared fractional unit
Equivalent fractions: many names, one number, and lowest terms: equivalent fractions, and reducing to lowest terms
Mixed fractions: naming what lies past 1: the mixed form, in which some of the answers land
Common multiples of two numbers, from Common multiples: why the first "idli-vada" is 15
Addition and subtraction of whole numbers, and reading a repeated sum as a product

What they should be able to do

Add and subtract two fractions that already share a fractional unit, and explain why only the counts change
Explain why fractions with different fractional units cannot be added as they stand
Convert a pair of fractions to a shared fractional unit and complete the addition
State the three-step method the book attributes to Brahmagupta, in order
Choose a workable shared denominator, and recognise that more than one choice works
Reduce a result to lowest terms, or rewrite it in the mixed form, when asked
Subtract fractions in both cases, and say why the same method covers both operations
Turn a worded situation into a fraction sum or difference and answer the question that was asked

Where it usually goes wrong

"Add the tops and add the bottoms." The single most common error in the topic. The strip model kills it on sight: adding the bottoms changes the size of the blocks, which nothing in the picture does. Show the wrong answer landing in the wrong place on a number line.
"Different denominators means the problem is impossible." It means one preparatory step. Say plainly that the addition itself never changes.
"You must use the product of the two bottom numbers." The p.179 pair shows otherwise, and the book uses the least common multiple there. Any common multiple works; the product is the one that always exists without thought.
"The answer must be in lowest terms." The book says if desired on p.179. Reducing is tidying, not part of the addition. Say which convention the explanation is using and keep to it.
"Subtraction is a different method." It is the same method with one sign changed, which is why the book introduces it as an extension rather than as new material.
"'Subtract A from B' means A minus B." The p.182 items are worded that way on purpose. Three of them will be got backwards by someone in every class.
"An answer bigger than 1 means a mistake." The four-sevenths and six-sevenths sum on p.177 is exactly such a case and the book prints it in both spellings.

Questions to check understanding

Add two or three fractions with the same bottom number
Add two fractions with different bottom numbers, showing the conversion
Subtract in both cases, including items worded as "subtract A from B"
Reduce a result to lowest terms, or rewrite it in the mixed form
Solve a worded problem involving a sum or difference of fractions, and answer the question that was asked rather than only computing
Say which shared denominator you chose and why another would also have worked
Spot and correct an added-the-bottoms error in a worked solution
The solutions appendix bound with this chapter answers the §7.8 exercises on its §7.8 pages

Examples worth working on the board

Meena and her brother (§7.8, p.175). Checked against the printed page. A slab is drawn as a bar of four blocks. Meena eats one half, her brother one quarter, and the question asks how much the two ate together. Three successive bars show the slab halved, then the untouched half halved again, then the eaten portion shaded as one piece, each with a hand-lettered caption. The working then rewrites the half as two quarters and collects. Inputs: the two amounts eaten, and the fact that a half is two quarters.
What is left (§7.8, p.175). The page closes by asking how much of the slab remains. Inputs: the total eaten and the whole slab. This is the section's first subtraction, and it arrives before subtraction has been introduced — worth keeping in that order.
Two fifths and one fifth (§7.8, p.176). Checked against the printed page. Each fraction is drawn as a five-block strip with the right number of blocks shaded red, and the sum is drawn as a third strip. The text counts shaded blocks and reports the count. Inputs: the two fractions and the strip model.
Four sevenths and six sevenths (§7.8, pp.176–177). Checked against the printed page. Same model with seven blocks. The counting gives a total that exceeds one whole, so the book writes the result twice — once as a single fraction and once in the mixed form — and a speech bubble states the same-unit rule in words. It then invites the reader to redo the sum on a number line. Inputs: the two fractions.
One quarter and one third (§7.8, pp.177–178). The first sum whose fractions do not share a unit. The book takes the product of the two bottom numbers as the shared denominator, rewrites both fractions, and adds. Inputs: the two fractions and the chosen shared denominator.
The attribution and the three steps (§7.8, p.178). The book names Brahmagupta and the year 628 CE as the first general statement of this method, points forward to §7.9, and then sets out three numbered steps: find a shared fractional unit using a common multiple of the bottom numbers; add the counts, keeping that bottom number; reduce if wanted. Give this its own section — it is the only place the procedure is written as a procedure.
Two thirds and one fifth (§7.8, pp.178–179). Here the book uses the least common multiple rather than the product, which for this pair happens to be the same number. Inputs: the two fractions and the shared denominator.
One sixth and one third (§7.8, p.179). Here the two differ: one bottom number already divides the other, so one fraction does not have to change at all, and the answer that comes out can be shortened. The book shows the reduction as a separate, optional step. Inputs: the two fractions. This is the pair that carries sections 8 and 9 and it should not be rushed.
The addition exercise (§7.8, p.179). Thirteen sums lettered a to m, plus two worded problems. The worded ones: a painter mixes two thirds of a litre of one colour with three quarters of a litre of another and asks for the total volume; and two people buy two fifths of a metre and three quarters of a metre of lace for a border whose full length is 1 metre, asking both for the total and whether it is enough. Inputs only — the second one has a yes/no question attached to the arithmetic.
Subtraction with a shared unit (§7.8, p.180). Checked against the printed page. Six sevenths is drawn as six shaded blocks of seven, four shaded blocks are pulled away in a second row, and the remainder is counted. Two speech panels label the removed parts and state that this works directly only because the units match. Inputs: the two fractions and the strip.
Three quick differences (§7.8, p.181). Inputs: five eighths less three eighths; seven ninths less five ninths; ten twenty-sevenths less one twenty-seventh. All share their unit, and two of the three answers can be shortened.
Subtraction with different units (§7.8, p.181). Three quarters less two thirds, converted to twelfths and subtracted, with two margin questions asking why each fraction was multiplied by the number it was multiplied by. The three-step subtraction procedure is then set out. Inputs: the two fractions.
The subtraction exercise (§7.8, p.182). Checked against the printed page. Four differences; three "subtract this from that" items whose wording reverses the usual order; and two worded problems — a girl whose school is seven tenths of a kilometre away and who rides half a kilometre before walking the rest, and two friends whose lap times are ten thirds and thirteen quarters of a minute, where the question asks who is faster and by how much. Inputs only.

Figures to have open

A block strip that can be divided into any number of equal blocks, shaded, slid together with a second strip, and had blocks removed. This is the workhorse figure and carries sections 1 to 5 and 10. Standard schematic; redraw rather than reproducing the printed slabs and bars on pp.175–180.
A number line for sections 5 and 9, on which a sum can be walked out step by step and on which reduction visibly moves nothing.
A side-by-side of two strips whose blocks do not match, for section 6. This is the picture that makes the obstacle real, and the textbook does not print one — it is added here.
A word-problem frame for section 12, as described in the visual treatment.
No photograph, table or data figure from the textbook is required.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 7 "Fractions", §7.8 "Addition and Subtraction of Fractions", pp.175–182 — Meena's chikki (p.175); the same-unit sums on strips (pp.176–177); the first different-unit sum, the attribution and the three steps (pp.177–178); two further sums and the addition exercise (pp.178–179); subtraction with a shared unit and the quick differences (pp.180–181); subtraction with different units and the closing exercise (pp.181–182)
§7.6, p.164 and p.172, for equivalent fractions and lowest terms
§7.9, pp.182–183, for the history the attribution points forward to
Chapter 5 "Prime Time", §5.1, for common multiples
Summary, p.186, for the printed one-line statements of the two methods
Solutions appendix bound with this chapter file, §7.8 pages