Integer grids, and why the total comes out the same every time

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What to assume they know

Addition as movement: starting position plus movement gives target position: adding signed numbers fluently
The additive inverse, and how it turns every subtraction into an addition: converting between addition and subtraction
Laying the integers out in order, and why −8 is less than −2: ordering integers, for the exercises that close the section
Why adding the odd numbers gives the squares: the habit of asking why a pattern holds, not only that it does

What they should be able to do

Compute the sum of a row and of a column in a bordered grid and compare them
State what the section's border sum is, and find it for a given grid
Complete a partly filled grid so that all four border sums agree
Say which grids are forced to a single answer and which admit many, and why
Carry out the strike-out procedure on a grid and total the circled numbers
Predict the total before playing, by identifying the row and column numbers the grid was built from
Explain why the procedure always selects one entry from each row and each column
Construct a grid of either kind to a required total
Evaluate a repeating token pattern by finding its period and its value per period

Where it usually goes wrong

"The grid is magic." It is constructed. The whole value of the section is replacing "it always works" with "here is what was done to it", and the book asks that question outright.
"The border sum is the sum of all the numbers round the edge." It is not: it is the total of each single outer row and each single outer column, and the four corner entries are counted in two of those totals apiece. Getting this wrong makes every exercise in the section come out wrong.
"If I can complete the grid, my answer is the answer." Two of the three puzzle grids have many completions. A student who finds one and thinks they have found the one has missed the section's second question.
"In the strike-out game, choosing bigger numbers gives a bigger total." It gives the same total. Let the class try to beat it and fail; that failure is the motivation for section 8.
"The strike-out total is the sum of one row plus one column." It is the sum of all four row numbers plus all four column numbers, because the procedure takes one entry from every row and every column. Demonstrating that is section 9.
"Counting years back across the start of the common era is plain subtraction." It is not, and the chapter prints the hint that says why: the year before 1 CE is 1 BCE, with no year 0 between them. A count that crosses that boundary comes out one year short unless you allow for it. A count that stays on one side of it is unaffected, which is why only one part of the question needs the hint.
"To find the value of a hundred tokens you must count a hundred tokens." Find the repeating block, find its value, and count the blocks. The block is what the question is testing.

Questions to check understanding

Find the border sum of a completed grid
Complete a partly filled grid to a required border sum
State whether a given grid can be completed in more than one way, and why
Play the strike-out procedure and give the total
Predict a strike-out grid's total without playing
Construct a grid of either kind to a stated total
Find the value of a repeating token string of stated length
List every integer strictly between two given integers, and complete a stated integer sequence
Given two dice whose faces carry both signs, say which sums between the smallest and the largest cannot be rolled
Find a year a stated number of years before or after a given one, across the BCE and CE boundary
Say whether a stated combination of signs — positive minus negative, negative plus negative, and so on — forces the sign of the result or leaves it open
The solutions block bound with this chapter file answers the p.263, p.265 and p.266 exercises on its footer pages 11–13

Examples worth working on the board

The first hollow grid (p.263). Top row 4, – 1, – 3. Middle row – 3, empty, 1. Bottom row – 1, – 1, 2. The book works all four border totals for this one and states what they come to.
The second hollow grid (p.263). Top row 5, – 3, – 5. Middle row 0, empty, – 5. Bottom row – 8, – 2, 7. The four totals are the first exercise.
Three grids to complete (p.264). Each is three by three with an empty centre. The first has – 10 top left, 9 bottom left and – 5 at middle right, and must reach a border sum of + 4. The second has 6 and 8 in the top row, – 5 at middle right and – 2 at bottom centre, and must reach – 2. The third has only 7 top left and – 5 at middle right, and must reach – 4. The book then asks which of these can be completed in more than one way. The second is the one that is forced; the other two are not.
The strike-out grid (p.264). Four rows: 3, 4, 0, 9; then – 2, – 1, – 5, 4; then 1, 2, – 2, 7; then – 7, – 6, – 10, – 1. The procedure is to circle any entry, rule out its whole row and column, and repeat until nothing is left.
The book's own play-through (p.264). It circles – 1, then 9, then – 7, then – 2, drawn as four successive pictures, and states the total those four come to.
The structure to be discovered (p.265). The strike-out grid on p.264 is built by adding a row number to a column number. Reading its top row as the column numbers, the four rows are that row unchanged, then the same row reduced by 5, by 2, and by 10.
Two more strike-out grids (p.265). The first has rows 7, 10, 13, 16; then – 2, 1, 4, 7; then – 11, – 8, – 5, – 2; then – 20, – 7, – 14, – 11. The second has rows – 11, – 10, – 9, – 8; then – 7, – 6, – 5, – 4; then – 3, – 2, – 1, 0; then 1, 2, 3, 4. Both Checked against p.265. See Notes on the second entry of the first grid's last row.
Three questions in the exercise set that closes §10.4 (pp.265–266). They sit among routine drill and carry ideas nothing else in this topic touches. Q3 gives two dice whose faces both carry – 1, 2, – 3, 4, – 5 and 6, states the extreme sums as – 10 and 12, and asks which values in between cannot be rolled — the chapter's one question about the shape of a whole set of attainable sums rather than about a single sum. Q5 asks for the years 150 and 2200 years before the present, and for the year 320 years after 680 BCE, with a printed hint that there was no year 0; the era scale is the integers with 0 deleted, so an off-by-one appears in exactly those counts that cross between BCE and CE and in no others. Q8 asks, for each combination of signs in an addition or a subtraction, whether the sign of the result is forced or could go either way — the empirical form of the rules that arrive named in Brahmagupta's rules, and how long it took the world to accept them.
The hundred-token string (p.266). Checked against the printed page. A single row of tokens is drawn, running three green, two red, three green, two red and so on, with a curl at the right-hand end showing it continues. The text states that the string holds a hundred tokens in all.
The chapter's closing game (p.271). Checked against the printed page. After the numbered sections end, the chapter prints a board game on a grid of ten rows of ten numbered cells — the upper five rows carrying + 1 to + 50, the lower five carrying – 1 to – 50, each row running the opposite way to the one before it, so the path snakes outward from the middle. 0 is not a cell in any row: it is drawn as a separate tab protruding from the left edge, between the + 1 to + 10 row and the – 1 to – 10 row. Both players start there. It is played with one die numbered + 1 to + 6 and another numbered – 1 to – 6, the two results to be added or subtracted in either order. It carries no section number and is not part of §10.4, but it is the natural extension activity for this topic.

Figures to have open

Two three-by-three grids with an empty centre cell, and three more partly filled ones. The empty cells must read as empty, not as zero.
A four-by-four grid that can be struck through row and column at a time, and whose cells can be re-formed as sums. This is the topic's central figure and it has to show.
A row of tokens in the two established colours, repeating and running off the right of the frame.
A board for the closing game: ten rows of ten numbered cells carrying + 1 to + 50 above and – 1 to – 50 below, with 0 drawn as a separate tab at the left edge between the + 1 to + 10 row and the – 1 to – 10 row, if the explanation takes the extension. Redraw; the printed board is the book's own art.
No photograph or textbook data table is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 10 "The Other Side of Zero", §10.4 "Explorations with Integers", p.263 — the named sub-heading on the hollow grid, the two grids, the four totals worked for the first, the naming of the border sum, and the first exercise
§10.4, p.264 — the three grids to complete, the questions on multiple completions and on making your own, the named sub-heading introducing the second puzzle, the four-by-four grid, the procedure, and the book's own play-through
§10.4, p.265 — the two further grids, the question about what makes them special, and the opening of the closing exercise set: integers between given pairs, the two-dice question, the eight signed computations, and the first two year questions with the printed hint about there being no year 0
§10.4, p.266 — the third year question, the sequences to complete, the integer cards, the sign-combination question, and the hundred-token string
"Integers: Snakes and Ladders", p.271 — the unnumbered closing game, offered as an extension
Solutions block bound with this chapter file, footer pages 11–13