What makes a number a perfect cube, and the three-identical-groups test

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What to assume they know

What makes a number a perfect square — square numbers, and the shape-to-number reading
Square roots, and the prime-factor test for a perfect square — prime factorisation, and the two-pile test for a square
Volume as a count of unit cubes, and what an edge of a cube is
Multiplying three numbers together, and multiplying two decimals to several places
Multiplying fractions, and multiplying negative numbers

What they should be able to do

Count the unit cubes in a cube of a given edge, by layers
State what a perfect cube is and produce the first several
Write and read the notation n³
Complete a table of cubes and describe patterns in it
Cube a fraction, a decimal and a negative number
State which digits can end a perfect cube, and contrast that with squares
Reason about how many zeros can end a perfect cube
Test a number for cubeness by dealing its prime factors into three matching groups, and read the edge off one group
Explain why three matching groups is the right condition — that is, why unique factorisation makes the test valid

Where it usually goes wrong

"A cube of side 2 holds 6 unit cubes." Six is the face count. The layer picture is the fastest correction: two floors of four.
"Cubing means multiplying by 3." The exponent counts how many times the number is used as a factor, not what it is multiplied by. 3³ and 3 × 3 differ, and so do 3³ and 9.
"Cubes obey the same last-digit restriction as squares." They do not. Every digit occurs as the last digit of some cube, which is exactly why the units-digit test that rejects squares is useless for cubes — and, later, why it becomes a stronger tool for cube roots. See Cube roots, and what successive differences expose.
"A cube can end in exactly two zeros." A factor of 10 in the number becomes a factor of 1000 in the cube, so trailing zeros arrive three at a time.
"Negative numbers cannot be cubed, the way they have no square root." (−6)³ = −216. Cubing preserves the sign; squaring destroys it. That single difference explains most of what follows in this module.
"Two piles worked for squares, so two piles work for cubes." The pile count is the exponent. Say it once, plainly, and the whole section holds together.
"64 is a square and a cube by coincidence." 64 has the prime 2 six times over, and 6 splits evenly into two groups and also into three. Numbers that are both are exactly the ones whose primes occur a multiple of six times.
"Cubes are about as common as squares." Between 10 and 26 there is not one, as the chapter says outright. The gaps grow far faster.

Questions to check understanding

Count the unit cubes in a cube of stated edge, by layers
Complete rows of a cube table and state a pattern
Cube a given fraction, decimal or negative number
Decide by prime factorisation whether a given number is a perfect cube
Find the least multiplier that turns a given number into a perfect cube
True-or-false claims about cubes, each requiring a stated reason — the form the chapter itself uses at Part I p.16 item 3
"Can a cube end in exactly two zeros?" and similar, answered by reasoning about factors rather than by search

Examples worth working on the board

The opening figures (Part I p.11). Beside the §1.2 text the page prints two line drawings: a plain unit cube, and a cube whose faces are ruled into a 2-by-2 grid, so it reads as 2 by 2 by 2. Verified on the printed page. The questions beneath ask how many 1 cm cubes fill a cube of edge 2 cm, then one of edge 3 cm. Inputs only.
The first three cubes (Part I p.12): 1 = 1 × 1 × 1; 8 = 2 × 2 × 2; 27 = 3 × 3 × 3. The chapter then asks whether 9 is a cube, answers no, and adds that nothing from 10 to 26 is one either — a useful and easily-missed statement about how sparse cubes are.
The 4-cube, counted by floors (Part I p.12): each layer holds 4 × 4 = 16 unit cubes, and with four layers stacked the count is 4 × 4 × 4 = 64. Then 5³ = 5 × 5 × 5 = 125. Two further margin drawings on that page show a cube ruled 3 by 3 by 3 and one ruled 4 by 4 by 4 (verified on the printed page).
The cube table (Part I p.12). Two columns, and the pre-filled cells are scattered rather than regular — read them off the printed page. Column 1 has slots 1³ to 10³ with 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64 and 5³ = 125 printed and 6³ to 10³ blank. Column 2 has slots 11³ to 20³ with 11³ = 1331, 13³ = 2197, 14³ = 2744, 17³ = 4913, 18³ = 5832 and 19³ = 6859 printed, and 12³, 15³, 16³ and 20³ blank. The scatter is worth preserving: it forces the student to compute rather than read down a column.
Cubes of things that are not whole numbers (Part I p.13). The chapter first names the squares (4/6)², (13.08)² and (−6)², then computes the cubes: (4/6)³ = (4/6) × (4/6) × (4/6) = 64/216; (13.08)³ = 13.08 × 13.08 × 13.08 = 2237.810112; (−6)³ = −6 × −6 × −6 = −216. Both fractions are left unreduced on the page. The text layer flattens the stacked fractions to "46" and "216 64"; the printed page sets them properly.
Questions the chapter poses and does not answer (Part I pp.12–13): what patterns the cube table shows; which last digits are possible for cubes; how many cubes have one digit, two digits and three digits; whether a cube can end in exactly two zeros, with an explanation. Hand these over as prompts.
The three-pile test on 3375 (Part I p.14): 3375 = 3 × 3 × 3 × 5 × 5 × 5. Dealt into three matching groups of (3 × 5), giving (3 × 5) × (3 × 5) × (3 × 5) = (3 × 5)³ = 15³. Read the other way as triplets: 3375 = (3 × 3 × 3) × (5 × 5 × 5) = 3³ × 5³.
When it fails (Part I p.14): 500 = 2 × 2 × 5 × 5 × 5. Three matching groups cannot be made, so 500 is not a perfect cube. The 2 appears twice, not three times — that is the whole reason, and it is worth saying rather than pointing.
§1.2 Figure it Out (Part I p.16), the items belonging here: item 2, the multiplier that turns 1323 into a cube; item 3, five true-or-false claims, each wanting a reason — that the cube of an odd number is even, that no perfect cube ends in 8, that a two-digit number's cube may have three digits, that a two-digit number's cube may run to seven digits or beyond, and that every cube carries an odd count of factors. Inputs only; every one of them is answerable from this topic's argument.
Where the two-digit claims bite. For item 3 parts (iii) and (iv) the explanation needs the bounds, not the verdicts: the smallest two-digit number is 10 and the largest is 99, and the chapter's own table supplies 10³ as a blank and 19³ as a printed value. Let the student close the argument.

Figures to have open

An exploded 4-cube: four square floors lifting apart and settling back. This is an added figure and it carries sections 2 and 3; the chapter's margin drawings on Part I pp.11–12 are the model.
The chapter's two-column cube table with its scattered pre-filled cells (Part I p.12). Preserve which cells were printed.
A three-tray dealing diagram for 3375, then the same primes regrouped as two triplets. An added figure; run it beside the two-tray version from Square roots, and the prime-factor test for a perfect square so the only difference is the tray count.
A number strip from 1 to 30 with 1, 8 and 27 marked, to make the sparseness visible. An added figure.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 1, §1.2 "Cubic Numbers", Part I pp.11–13 (the un-subheaded opening of the section, from the §1.2 heading part-way down Part I p.11 to the "Taxicab Numbers" subheading on Part I p.13).
The three-groups test is printed a page later, under the "Cube Roots" subheading at Part I p.14, where 3375 and 500 are worked. The spine assigns that test to this topic; Cube roots, and what successive differences expose takes the root notation, the factorisation table and the difference work from the same stretch of pages.
§1.2 "Figure it Out", Part I p.16, items 2 and 3.
Part I p.17 SUMMARY, fourth and sixth bullets.