Why adding the odd numbers gives the squares

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What to assume they know

Drawing a sequence makes its rule visible — square numbers are dots arranged in a square, and a drawing can carry a definition. Note that §1.4 does not reuse Table 2's two-row picture of an odd number; it draws each odd number afresh as a bent band
Every number sequence is a rule, not a list — Table 1's rows are rules
Mathematics is the search for patterns *and* for why they hold — the reason matters as much as the pattern
Adding a short run of small numbers accurately

What they should be able to do

Compute the running sums of the odd numbers and recognise the squares in them
State why checking six cases does not establish a claim about all cases
Cut a square array of dots into bands of 1, 3, 5, 7, … and say what each band is
Use the cut to explain why the running sums of the odd numbers are the squares, at every size of square
Say what the first 10 odd numbers come to, and then the first 100, without adding them up, and justify the shortcut
Recognise the up-and-down sums as a second route to the same squares
Slice a square array along its diagonals and connect that slicing to the up-and-down sums

Where it usually goes wrong

"They checked it up to 36, so it is proved." The book itself asks whether it lasts forever, which is exactly the admission that six lines have not settled it. The picture is what settles it.
"The picture is a nice way to remember the result." Reverse it. The result is what the picture forces. Remove the picture and the claim is a guess with six confirmations.
"Each odd number is a new row of dots." It is a row and a column and the corner. A student who draws only the row gets an even number of dots and the argument dies.
"Up-and-down is a different fact about squares." Same squares, same square, different cut. Showing both cuts on the identical 6-by-6 array is the cleanest way to make that land.
"Getting the first 100 odd numbers to add up means doing 100 additions." The picture replaces the addition with a single multiplication. That replacement is the payoff of having a reason rather than a pattern.
"1 is not an odd number here, it's just the start." It is the first odd number and the first bent band — a band of one dot, which is the 1-by-1 square.

Questions to check understanding

Find 1 + 3 + 5 + 7 + 9 + 11 + 13 without adding, and say how you knew
Add the first 20 odd numbers without doing twenty additions
Draw a picture explaining why the running sums of the odd numbers are squares
Explain why growing a square adds an odd number of dots
Complete the up-and-down sums and say which sequence they produce
Give a pictorial reason for the up-and-down sums
1 + 2 + … + 49 + 50 + 49 + … + 2 + 1 =?
The solutions appendix supplies 100 for the first shortcut question (page footered [2]), and on the next appendix page 10,000 for the second and 10,000 for the long up-and-down sum, together with a drawing for the "find a similar pictorial explanation" question — square lattices standing on a corner, labelled 1, then 1 + 2 + 1 = 4, then 1 + 2 + 3 + 2 + 1 = 9. Checked against the appendix pages footered [2] and [3].

Examples worth working on the board

The six running sums as printed (p.6). The book prints, in this order: 1; 1 + 3; 1 + 3 + 5; 1 + 3 + 5 + 7; 1 + 3 + 5 + 7 + 9; 1 + 3 + 5 + 7 + 9 + 11 — with the totals 1, 4, 9, 16, 25 and 36 set in a second colour so the squares stand out down the right-hand side. Checked against p.6.
The cut (p.7). A 6-by-6 array of 36 dots, cut by lines into six bent bands of 1, 3, 5, 7, 9 and 11 dots. Each band wraps around the corner of the square already built: one dot, then a band of three completing a 2-by-2, then five completing a 3-by-3, and so on. Checked against p.7 — the cutting lines are drawn and extract as nothing.
The general step. Growing an n-by-n square to an (n+1)-by-(n+1) square needs one new row of n, one new column of n, and one corner dot: 2n + 1 dots, which is odd, and which is the next odd number every time.
The two shortcut questions (p.7). How much the first 10 odd numbers come to, and then the first 100. Both are asked and neither is answered in the chapter; the solutions appendix bound with the same file gives 100 on the page footered [2] and 10,000 on the page footered [3].
Adding up and down (p.7). The book prints: 1; 1 + 2 + 1; 1 + 2 + 3 + 2 + 1; 1 + 2 + 3 + 4 + 3 + 2 + 1; 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1; 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 — totals 1, 4, 9, 16, 25, 36, again in a second colour. Same six squares, a different road.
The blank picture (p.8). Immediately after asking for a pictorial reason for the up-and-down sums, the book prints a 6-by-6 array of 36 dots with no cutting lines on it. Checked against p.8. That blank array is the student's workspace and should appear in the explanation as an invitation, not as a finished diagram.
The diagonal slicing. Slice the 6-by-6 array along one set of diagonals. The diagonals hold 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 dots. That is the up-and-down sum, and it totals 36. The chapter does not print this, but the solutions appendix does, in a rotated form: it draws square lattices standing on a corner, labelled 1, then 1 + 2 + 1 = 4, then 1 + 2 + 3 + 2 + 1 = 9 (page footered [3]). Standing the square on its corner and slicing it in horizontal rows is the same cut as slicing an upright square along its diagonals.
The long one (p.8). 1 + 2 + 3 + … + 99 + 100 + 99 + … + 3 + 2 + 1. Asked, not answered in the chapter; the appendix gives 10,000 on the page footered [3].

Figures to have open

The p.7 cut: a 6-by-6 dot array with six bent bands marked. This must be drawn faithfully — the bands wrap a corner, they are not stripes. Redraw rather than reproduce the printed art.
The p.8 uncut 36-dot array, identical in size and spacing to the p.7 one so the two read as the same square.
The same array sliced along diagonals, for section 11.
A growth panel: an n-by-n square gaining a row, a column and a corner dot, with the three parts coloured separately.
No photograph or dataset is required.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 1 "Patterns in Mathematics", §1.4 "Relations among Number Sequences", pp.6–8 — the running sums and the bold "A picture can explain it" passage on p.6, the cut drawing and the up-and-down sums on p.7, and the uncut array and Figure it Out on p.8
Back reference inside the same chapter: Table 2, §1.3, p.4, for the square arrays of dots that §1.4 goes on to cut. Table 2's odd-number row is a different arrangement and plays no part in the p.7 argument
Solutions appendix bound with this chapter file, pages footered [2] and [3], for §1.4