Supercells: a number's status comes from its neighbours

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What to assume they know

The same children, different numbers: a number depends on what is being counted: a number can be a report about neighbours rather than a property of the thing itself
Comparing multi-digit whole numbers, up to five digits
Reading a row and a grid of cells, and the words left, right, above, below
Writing large numbers with commas in the Indian place-value grouping

What they should be able to do

State the rule that makes a cell a supercell, including what happens at an end
Mark the supercells in a given row or grid, and justify each mark
Explain why two supercells can never be adjacent
Deduce the greatest possible number of supercells in a row of n cells
Build a row that achieves that greatest number
Argue that whichever number is biggest must win its own cell
Argue that the smallest one never can, when no number is repeated
Apply the rule in two dimensions, where a cell has up to four neighbours
Fill a partly blank grid so that a given set of cells are exactly the supercells
Change one number by swapping two of its digits and predict the effect on the whole pattern

Where it usually goes wrong

"Supercell means big." It means locally biggest. In the first printed row 34 is a supercell and 694 in the row below it is not, and 694 is about twenty times the size. Show those two side by side.
"A cell at the end can't be a supercell, it's only half surrounded." The opposite — an end cell has one rival instead of two, so it is the easiest place to be a supercell. 34 and 198 both win at the right-hand end.
"You could fill a row so that every cell is a supercell." You cannot fill even two neighbouring cells that way, so at best every other cell. This is the proof the whole topic turns on and it should be argued, not asserted.
"You could fill a row so that no cell is a supercell." Not with distinct numbers: whichever number is largest beats everything it touches, so it wins its cell no matter where you put it.
"In the grid, the corners are the hard cases." In the grid a corner has two neighbours, an edge cell three and an inside cell four — so corners are the easiest. Students import the wrong intuition from chessboards.
"Shading in a textbook is always the complete answer." In Table 1 it is not necessarily so — see Notes. Teach the rule and let the rule decide.
"Swapping two digits changes only that cell." It changes every comparison that cell takes part in — up to four of them — which is why one swap on p.72 moves the supercell count from one to four.

Questions to check understanding

Mark all the supercells in a given row or grid
Fill a table so that a named set of cells come out as exactly the supercells
"How many supercells are possible in a table of n cells?" with the reasoning demanded, not just the number
Explain why whichever number is biggest must win its own cell
Explain why the smallest one never can
Build a table in which the runner-up number fails to win its cell
Build a table in which the runner-up fails but the second-smallest number wins
Change one digit, or swap two digits, and state the new supercell count
True/false items on adjacency: "two supercells can be next to each other"

Examples worth working on the board

The two opening rows (§3.2, p.57). Two separate eight-cell rows, each judged on its own:
- 43, 79, 75, 63, 10, 29, 28, 34 — shaded: 79, 29, 34
- 200, 577, 626, 345, 790, 694, 109, 198 — shaded: 626, 790, 198 The shading is colour in the printed book and does not extract. The book itself walks through three of these cells in prose: 626 beating 577 and 345, 200 losing to 577, and 198 winning at the end of the row against its single neighbour 109.
The row to be marked (§3.2, p.57, question 1). 6828, 670, 9435, 3780, 3708, 7308, 8000, 5583, 52. Nine cells, unshaded.
The partly filled row (§3.2, p.57, question 2). Nine cells; 5346 in cell 1, 1258 in cell 4, 9635 in cell 8; cells 2, 4 and 9 are shaded, so one of the shaded cells is the printed 1258 and the other two are blanks. Only 4-digit numbers may be used, and the shaded cells must come out as exactly the supercells. That makes the puzzle tighter than it first looks: 1258 has to beat cells 3 and 5, which pins both of them below 1258 and above 999; cell 2 has to beat 5346; and cell 9 has to beat 9635, which leaves it only the range 9636 to 9999.
Table 1 (§3.2, p.58). A 4 × 4 grid:
```
2430  7500  7350  9870
3115  4795  9124  9230
4580  8632  8280  3446
5785  1944  5805  6034
```
Shaded in the printed book: 7500, 9870, 8632, 6034. The book works one of them through in prose — 8632 against 4580, 8280, 4795 and 1944. See Notes for the fifth cell that meets the printed rule and is not shaded.
Table 2 (§3.2, p.58). A 4 × 4 grid to be completed; every entry is a 5-digit number using each of 1, 0, 6, 3 and 9 exactly once, and the shaded cells — and only those — must beat all their neighbours. Printed entries, by row:
```
(shaded)  96,301    36,109    (shaded)
_         13,609    60,319*   19,306
_         _         60,193    _
_         10,963*   _         (shaded)
```
where * marks a printed number that is itself shaded and _ is blank. Three follow-up blanks are printed on p.59: the biggest number in the finished grid, the smallest even one, and the smallest one above 50,000; then the instruction to put the commas in.
The closing grid (chapter-end Figure it Out, p.72, question 1). A 3 × 3 grid
```
16,200  39,344  29,765
23,609  62,871  45,306
19,381  50,319  38,408
```
with exactly one supercell, 62,871, shaded. The task is to swap two digits inside one of the numbers so that the grid ends up with four supercells.

Figures to have open

The two printed eight-cell rows with their shading (§3.2, p.57). The shading is the data; a monochrome reproduction destroys the exercise.
The nine-cell row from question 1 and the partly filled row from question 2 (§3.2, p.57). Both needed as blanks the explanation can fill.
Table 1 and Table 2 with their shading (§3.2, p.58). Table 2 must keep its blanks — a completed version is a different exercise.
The 3 × 3 grid from p.72 with 62,871 marked. Needed twice: before and after.
A schematic of one grid cell with four arrows to its neighbours, and of a corner cell with two. Standard schematic, does not need to come from the textbook.

Where this sits in the book

NCERT Class 6 Mathematics (Ganita Prakash), Chapter 3 "Number Play": §3.2 Supercells, pp.57–59
The opening table and the prose statement of the rule, p.57
The Figure it Out block, questions 1–5 on p.57 and questions 6–10 on p.58
The restatement of the rule for grids, with Table 1 and Table 2, p.58
The three fill-in lines about Table 2 and the comma instruction, p.59
The chapter-end Figure it Out block, question 1, p.72 — printed after §3.12 and carrying no section number of its own
Backward link: §3.1, pp.55–56 (The same children, different numbers: a number depends on what is being counted), where the same relational move is made with children instead of cells