Using row and column sums to prove a grid cannot be filled

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What to assume they know

Odd and even as "can this be arranged in pairs": an argument that rules out every attempt at once, rather than reporting on the attempts that were made
Adding three one-digit numbers, and adding a short list of two-digit numbers
Reading a grid by row and by column, and the words row and column
1 + 2 + … + 9 = 45, or the willingness to add it once and remember it
Using a known total to recover a missing addend
Answering "could this possibly fit?" by estimating in stages: bounding a quantity above and below instead of computing it exactly

What they should be able to do

Read a grid whose row totals and column totals are printed around its edge, and say which cells each circled number controls
Fill a partly-given 3 × 3 grid from 1 to 9 using its row and column totals, choosing the tightest constraint to start from
State the smallest and the largest value a row or column total can take, and justify both
Show that the three row totals must add to 45, and that the three column totals must too, whatever the arrangement is
Use those two facts to decide that a proposed set of totals is unachievable, and name which fact it violates
Distinguish "I did not find a solution" from "no solution exists", and say what is needed to earn the second
Recognise when a set of totals admits more than one filling

Where it usually goes wrong

"The setter can choose any six totals they like." They cannot choose even five of them freely. Both triples are pinned to 45, so once two row totals and two column totals are chosen, the third row and the third column are forced — at most four of the six are free. Any puzzle whose six numbers do not respect that was never solvable.
"I tried for a long time and could not do it, so it is impossible." That is a report about the solver, not about the puzzle. The chapter's argument is what upgrades it: an unreachable total is unreachable for everyone, including the people who have not tried yet.
"A total of 26 is fine — the numbers only go up to 9, but three of them make plenty." The three largest available are 9, 8 and 7 and they make 24. Nothing in the grid can beat that, so 26 is out of reach before the grid is even drawn.
"Each row total is a separate condition, so more totals means more clues." More totals means more clues only until 45 is reached; after that the totals start repeating information. Section 9 is where that becomes visible.
"A set of totals that passes both checks must be solvable." Passing means the two arguments in this section have nothing more to say; it is not a guarantee. Practice grid B passes and has several fillings; other totals can pass and have none.
"If the totals work out, the filling is unique." Practice grid B has more than one filling. A well-posed puzzle and a puzzle with exactly one answer are different things, and this page contains an example of the difference.

Questions to check understanding

Complete a partly-filled grid from its row and column totals
Given six proposed totals, decide whether a filling can exist and name the fact that rules it out
Find the missing sixth total when five are given
Explain why the three row totals must come to 45
Say whether a set of totals has one filling or several, and justify the answer
Set a solvable puzzle of this kind for a classmate, which is exactly what p.134 asks the reader to do, and which forces the setter to use the 45 constraint
Explain the difference between failing to find a solution and showing none exists

Examples worth working on the board

The demonstration grid (Part I, §6.3, p.133). Checked against the printed page. Cells, reading across: 4 7 5 / 6 1 2 / 3 9 8. Ringed to the right of the rows: 16, 9, 20. Ringed below the columns: 13, 17, 15. The ringed numbers are printed but never explained — the reader is asked to work out what they are.
Practice grid A (Part I, §6.3, p.133). Checked against the printed page. Only two cells are given: 9 in the top-left and 5 in the bottom-right. Row totals 13, 14, 18; column totals 24, 9, 12. The good opening move is the row totalling 13 with a 9 already in it, because two cells are then squeezed into a very short list.
Practice grid B (Part I, §6.3, p.133). Checked against the printed page. Given cells: 4 in the middle row's left cell and 3 in the bottom-right. Row totals 24, 15, 6; column totals 12, 16, 17. Note: this grid has more than one filling — so do not present any one answer as the answer. The row totalling 6 is forced to hold 1, 2 and 3 and the row totalling 24 is forced to hold 7, 8 and 9; what is left over is genuinely a choice.
The impossible grid (Part I, §6.3, p.134). Checked against the printed page. One cell is given, a 6 in the middle row's right-hand cell. Ringed to the right: 5, 21, 19. Ringed below: 9, 11, 26.
The two extreme totals (Part I, §6.3, p.134). Inputs: the three smallest available numbers and the three largest. The chapter uses 1 + 2 + 3 and 9 + 8 + 7.
Kishor's and Vidya's observations (Part I, §6.3, p.134). Two named children report two patterns from the grids already solved: that all six ringed numbers together always come to 90, and that the three row totals on their own, or the three column totals on their own, always come to 45. The chapter then asks why — and that question is the spine of sections 9 and 10.
The 45 itself (Part I, §6.3, p.134). Printed as a total of 1 through 9, written out term by term.

Figures to have open

A 3 × 3 grid with editable cells and six ringed totals outside it, able to hold the demonstration grid, both practice grids and the impossible grid. This is the figure the whole topic runs on.
A movement that lifts three rows out of a grid and re-pools their nine cells, then repeats it for the columns. This is what makes the 45 argument visible rather than asserted.
A short number line marked at 6 and at 24, with the ringed values plotted on it.
The printed grids are simple ruled squares with numerals; redraw them rather than lifting the page art. No photograph or dataset is required.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 6 "Number Play", §6.3 "Some Explorations in Grids", pp.133–134 — the demonstration grid and the two practice grids (p.133), the impossible grid with the range argument, the two named observations, and the derivation of 45 (p.134)
Same part, SUMMARY, p.145, fourth bullet, for the chapter's own account of using row and column totals to settle impossibility
Forward pointer: the same page turns from impossibility to construction and reaches the definition of a magic square — Constructing magic squares, and where they came from
Solutions appendix bound after p.145 in the cached PDF (not part of the printed book); it supplies fillings for the two practice grids as images and adds that other fillings exist