Taxicab numbers: why 1729 is the number it is

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What to assume they know

What makes a number a perfect cube, and the three-identical-groups test — perfect cubes, the notation n³, and the cube table through 20³
Adding four-digit numbers
What it means for a claim to hold for all numbers below a bound
Comfort with the idea that two different sums can reach the same total

What they should be able to do

Verify that 1729 splits into two positive cubes in two distinct ways
State precisely what Ramanujan claimed about 1729, including the word smallest
Distinguish a property a number has from a record it holds, and say why the second is harder to establish
Describe how a search for smaller counterexamples would be organised
Name a taxicab number and recognise the family the term describes
Search for the two decompositions of a further taxicab number given to you
Explain why finding one decomposition is easy and finding two is not

Where it usually goes wrong

"1729 is the only number that is a sum of two cubes in two ways." It is the smallest. The chapter names 4104 and 13832 on the same page, so the explanation can correct this immediately rather than leaving it standing.
"The story shows Ramanujan calculated very fast." It shows he already knew. A minimality claim is not something you compute in a doorway; it is something you have previously established and can recall.
"Every number is a sum of two cubes if you look hard enough." Most are not. The rarity is what makes two representations remarkable.
"1³ + 12³ and 12³ + 1³ are two different ways." Order is not a way. The chapter's two ways use genuinely different pairs of cubes.
"Interesting is a matter of opinion." Here it has been given a definition — a stated property, plus a record. That is what lets the claim be checked.
"Hardy was careless." He had no property in mind for 1729, which is the only sense in which a number is dull. The exchange is about who had looked, not about who was cleverer.
"Taxicab number means the number on a taxi." It names a family of numbers; the taxi is only how the first one was met.

Questions to check understanding

Verify that a given number splits into two cubes in two distinct ways
Find both decompositions of a taxicab number you are handed
State what makes 1729 the Hardy–Ramanujan Number, using the word smallest
Explain the difference between showing a number has a property and showing it is the smallest with that property
Given a target and a list of cubes, decide whether the target is a sum of two of them
Short-answer history: who the two mathematicians were and where the exchange took place

Examples worth working on the board

The story as printed (Part I p.13). Ramanujan was working with G. H. Hardy at the University of Cambridge and was ill in hospital. Hardy arrived by taxicab numbered 1729 and remarked that the number struck him as dull, hoping this was not a bad omen. Ramanujan disagreed on the spot and gave the reason. Paraphrase the exchange; do not read the printed sentences aloud.
The two decompositions, printed on the page: 1729 = 1³ + 12³, and 1729 = 9³ + 10³.
The cube values a student needs to see it: 1³ = 1, 9³ = 729, 10³ = 1000, 12³ = 1728. Note that 12³ is one of the blanks in the chapter's own cube table (Part I p.12) — the taxi's number is the value of a cell the student was asked to fill, plus one. That connection is not drawn on the page and is worth drawing.
The cubes the search needs, and the ones to withhold. The chapter's own decompositions force 1³ = 1, 9³ = 729, 10³ = 1000 and 12³ = 1728; those are printed as the taxicab fact and are yours to use. The rest of the run up to 12³. compute as the search proceeds, not display as a ready-made strip — 6³, 7³, 8³, 9³, 10³ and 12³ are blank cells in the chapter's cube table on Part I p.12, and a pre-filled strip does that exercise for the student. Beyond 12³ the cubes exceed 1729, which is what makes the search finite and showable.
How the search is organised — an added framing, not the chapter's. Fix the smaller cube, then ask whether the remainder is a cube. With cubes only up to 12³ in range, the whole search is a short table, and the point to land is that it terminates: this is a claim you can finish checking, unlike a claim about all numbers.
The Try This task (Part I p.13): two further taxicab numbers are named, 4104 and 13832, and the student is asked for the two ways each of them splits into a pair of positive cubes. Inputs only — the pairs are the exercise and must not be handed over.
Littlewood's remark (Part I p.13), reported in the chapter: that for Ramanujan every positive whole number was something like a personal acquaintance. Paraphrase it; the printed sentence is a quotation and is not read out.
The illustration (Part I p.13). A drawing at the middle right of the page: Ramanujan sitting up in a hospital bed with a tray across his lap, and Hardy standing in the doorway holding a cup. Verified on the printed page.

Figures to have open

A balance or pair of stacked-cube towers showing 1 + 1728 and 729 + 1000 both reaching the same level. An added figure and the one the topic cannot do without.
A strip of the cubes 1³ to 12³, filled in as the search proceeds rather than shown complete, and used for the decompositions and the search. The values come from the chapter's own table (Part I p.12), several of them as blanks the student fills.
The hospital-visit illustration (Part I p.13). If the printed art is not used, a simple two-figure schematic with the taxi number visible does the same job; the number on the cab is the only detail that matters.
No photograph is needed.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 1, §1.2 "Cubic Numbers", the subheading "Taxicab Numbers", Part I p.13. The subheading occupies the lower two-thirds of that page, ending before the next subheading on Part I p.14.
The Try This task on the next two taxicab numbers is set in the margin badge on the same page.
The cube values used here come from the chapter's own table at Part I p.12; see What makes a number a perfect cube, and the three-identical-groups test.
The chapter's history section (Part I pp.15–16) is a separate topic, The oldest known tables of squares and cubes, and does not mention Ramanujan.