Encoding a line-up as a sequence of numbers

What to assume they know

Comparing two lengths or heights directly, without measuring either
Counting a subset of a group ("how many of these are taller than that one")
Reading a position in a list as first, second, third … and knowing that position matters
Comparing two expressions by reasoning, not by evaluating: comparing two things by reasoning about them rather than by working out numbers first
Sorting a claim into always true / sometimes true / never true, and knowing that one counterexample settles "always"

What they should be able to do

State the rule that turns a height arrangement into a sequence of numbers, and apply it to a drawn line-up
Explain why the first person in the line always announces zero
Explain why the k-th person can never announce more than k − 1
Decide, given a candidate sequence, whether any arrangement could produce it
Reconstruct the relative order of a line-up from its sequence alone, and say why the reconstruction is forced rather than guessed
Classify statements about the code as always true, as only sometimes so, or as never true, and give a counterexample when the answer is "only sometimes"
Say what information the code keeps and what it discards, and why that is the point of the activity

Where it usually goes wrong

"Saying 0 means you are the tallest." The front of the line has nobody ahead of it, so it says 0 regardless of height. Both printed line-ups open with a 0 said by a child who is not the tallest. The implication runs one way only: tallest ⇒ 0, never 0 ⇒ tallest.
"The biggest number belongs to the shortest child." Only sometimes. The shortest child says a large number when standing near the back and says 0 when standing at the front — the same child, the same height, a different number. Section 1's two drawings of the same seven children are built to make this land.
"Any list of seven numbers could be somebody's line-up." No: 1, 0, 0, 0, 0, 0, 0 is impossible, because the first speaker has nobody ahead to count. The ceiling is the whole content of legality here.
"The numbers tell you how tall the children are." They tell you nothing about heights, only about order — which is exactly why the same seven children produce two different sequences on one page.
"You would have to guess to get the line-up back." Nothing is guessed. Each new number pins one child's place among those already standing, so the rebuild is forced from the front of the line to the back.
"A child standing in the middle cannot say 0." They can — if they are taller than everyone ahead of them. Statement (d) on p.128 is designed to be refuted, and the third printed line-up refutes it outright.

Questions to check understanding

Given a drawn arrangement, write the sequence it produces
Given a sequence, draw or describe an arrangement that produces it
Given a sequence, decide whether any arrangement could produce it, and say which entry rules it out
Say how large a number can be heard when n people line up, and justify it
Judge always / sometimes / never claims about the code, supplying a counterexample for each "sometimes"
Explain why two different arrangements of the same children give different sequences, and what that shows about what the code records
Competency-based items on this chapter tend to ask for reasoning in words rather than a computed value; the six statements on p.128 are the model

Examples worth working on the board

First line-up (Part I, p.127, upper figure). Checked against the printed page. Seven stick children on a ruled height chart, each with a speech balloon. Left to right the balloons read 0, 1, 0, 2, 2, 2, 1. The child at position 3 is the tallest in the group and is the second one to say 0.
Second line-up (Part I, p.127, lower figure). Checked against the printed page. The same seven children re-ordered; the balloons read 0, 0, 1, 2, 2, 5, 6. The two shortest children have been moved to the back, which is what pushes the last two numbers up to 5 and 6.
Third line-up (Part I, p.128). Checked against the printed page. Seven children on the same ruled chart with empty balloons — this is the one the student fills in. The heights drawn there give 0, 0, 1, 0, 3, 0, 3.
The six target sequences (Part I, §6.1, p.128, Figure it Out Q1). (a) 0, 1, 1, 2, 4, 1, 5 · (b) seven zeros · (c) 0, 1, 2, 3, 4, 5, 6 · (d) 0, 1, 0, 1, 0, 1, 0 · (e) 0 followed by six ones · (f) 0, 0, 0, 3, 3, 3, 3. Feed these in as they stand. Two of them are worth the explanation's time: (b) forces the children into increasing height from front to back, and (c) forces exactly the reverse. Every one of the six respects the position ceiling, which is why every one of them is achievable — that is the observation, not an accident.
The six statements (Part I, §6.1, p.128, Figure it Out Q2). Two of them turn on the same asymmetry: the tallest child always says 0, but a child who says 0 need not be the tallest — whoever stands at the front says 0 whatever their height. The last item asks how large a number can be heard when eight people line up, and the ceiling answers it directly.
The reconstruction (not in the book). Not printed as such; derive it. Add the children one position at a time. When the k-th child arrives you are told how many of the k − 1 already placed are taller, so their rank among those k is fixed, and there is no choice left to make. Run it on 0, 0, 1, 2, 2, 5, 6 and watch the printed second line-up come back.

Figures to have open

A ruled height chart with seven redrawn figures and speech balloons that can be re-ordered. The book's own stick figures are its artwork — redraw them; but keep the ruled horizontal lines, because they are what makes the height comparison readable without numbers.
An arrow overlay: from one child back to each taller child ahead of them.
A position strip showing the ceiling 0, 1, 2, 3, 4, 5, 6 under the seven places.
No table, photograph or dataset from the textbook is required.
The printed book supplies stick-figure cutouts at the back for this activity; a physical-manipulative shot is not needed, but the explanation may mention them.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 6 "Number Play", §6.1 "Numbers Tell us Things", pp.127–128 — the two balloon line-ups and the withheld rule (p.127), the rule itself, the blank third line-up and the two Figure it Out questions (p.128)
Same part, SUMMARY, p.145, first bullet, for the chapter's own one-line account of what this activity established
Backward pointer: the same children appear in the Class 6 Ganita Prakash under a different calling-out rule; the Class 7 text says so on p.127 but does not name the Class 6 chapter
Solutions appendix bound after p.145 in the cached PDF (not part of the printed book), consulted only to confirm the third line-up's numbers and the six verdicts