Magic grids of integers

What to assume they know

• Why a negative times a negative must be positive: multiplying integers and the four sign cases
• That a product is unchanged by reordering its factors and by regrouping them — the argument here depends on both. The chapter has argued the reordering half by this point (Part II, p.35) but does not reach regrouping until Part II, pp.39–40, and Commutative, associative, and distributive over the integers comes after this topic in the series, so state the regrouping fact inline rather than sending the student forward for it
• Recognising multiples and common factors within a row of numbers
• The idea of a multiplication table as a rectangle of products of its headers, from earlier classes

What they should be able to do

• Play the grid game correctly: circle, strike out the whole row and column, repeat until nothing is left
• State how many numbers get circled in a four-by-four grid, and why exactly one comes from each row and one from each column
• Multiply four signed integers accurately and report the result
• Test the claim empirically by playing the grid several times with different choices
• Recover a pair of hidden row and column lists from a printed grid by comparing rows to one another
• Explain, using commutativity and associativity, why every legal play gives the same product
• Predict the sign of the answer by counting the negatives in the two hidden lists rather than by multiplying
• Construct a new grid of the same kind, and say what freedom the constructor has

Where it usually goes wrong

• "It works because the numbers were chosen carefully." Half right, and the wrong half is the interesting one. The numbers were chosen so that the grid is a product table; after that, the arrangement does all the work. The book's Try This question puts exactly this fork to the reader.
• "You have to circle them in the order shown." The printed panels barely show an order at all — three of the four rings are already there in Round 1. Any order gives the same answer, and demonstrating that is the chapter's own instruction on p.38.
• "There must be some special diagonal." The worked play is not a diagonal — its cells sit in columns 4, 2, 3, 1 as the rows go down. The condition is one per row and one per column, nothing more.
• "Different cells, so different factors, so different answers." The factors really are different numbers. But collected together they are always the same eight hidden numbers multiplied out, just paired up differently — and pairing is not something multiplication can detect.
• "The sign depends on how many negative cells you circle." Tempting, and false as stated: different plays circle different counts of negative cells. The invariant is the count of negatives in the two hidden lists. Show a play with one negative cell beside a play with three, both landing on the same negative answer.
• "This is just a trick, not mathematics." The trick is a working demonstration of commutativity and associativity. Without those two laws the claim would be false.

Questions to check understanding

• Play the grid and report the product; play it again differently and compare
• Given a partly filled product grid, complete the missing cells
• Given a grid, recover a possible pair of row and column lists
• Predict the sign of the final product without multiplying, and justify the prediction
• Construct a four-by-four grid of this kind with a stated target product
• Explain which property of multiplication makes the answer independent of the play, and say where it is used
• Multiply four signed integers accurately — the arithmetic hidden inside the puzzle is the examinable part, and appears in that plainer form on pp.42 and 44
• Puzzle-shaped items of this type are competency-based rather than recall-based; the marks sit on the justification, not on the number

Examples worth working on the board

• The printed grid (Part II, p.37). Checked against the printed page. Four rows of four, set in pink cells:

     8    −4    12    −6
   −28    14   −42    21
    12    −6    18    −9
    20   −10    30   −15

These sixteen values are the single most important input in the topic; check them against the printed page before showing anything.

• The instruction flow chart (Part II, p.37). Checked against the printed page. Three boxes to the right of the grid, joined top to bottom by an arrow with a return loop from the third back to the second: circle a number; strike out that number's row and column; circle any number not yet struck out. The loop is the part that matters, and it is drawn, not written.
• The chapter's worked play (Part II, p.37). Checked against the printed page, and the four panels compared. Four small copies of the grid headed Round 1 to Round 4 — but they are not a progression, whatever the headings promise. Round 1 already carries three rings, −6 (top row, fourth column), 14 (second row, second column) and 20 (fourth row, first column), each with its row and its column already struck. Rounds 2 and 3 repeat Round 1 with nothing added. Only Round 4 adds a fourth ring, on 18 (third row, third column). So the four circled cells are the ones named, and the only pick order the page supports puts 18 last, not third; there is no panel on the page showing one ring, or two.
• A second, different play. Supply a contrasting legal set from the same grid — for instance 8, 21, −6, 30 (first row first column; second row fourth column; third row second column; fourth row third column) — so the explanation can land on the same answer twice by different routes. Any set with one cell from each row and each column will do; there are twenty-four of them.
• The hidden lists (the explanation's reconstruction). Every cell of the printed grid is the product of a row number from 2, −7, 3, 5 and a column number from 4, −2, 6, −3. Check a few cells — the third row is 3 times the column list, the second row is −7 times it — and let the pattern land before the general claim is made. These two lists are nowhere printed in the chapter; they are recovered by looking at the grid.
• The freedom in the construction (the explanation's analysis). The split into two lists is not unique: multiplying every row number by some k and dividing every column number by the same k rebuilds the identical grid. For this particular grid the only other whole-number split is the one that negates both lists. Either way the two products multiply back to the same thing, which is exactly why the ambiguity does not disturb the argument.
• The sign, without multiplying. Three of the eight hidden numbers carry a minus — one in the row list, two in the column list. An odd count makes the answer negative. Offer this as a prediction to be made before any arithmetic and then confirmed by it.
• The second grid (Part II, p.38). Checked against the printed page: the sixteen values on p.38 are identical to those on p.37, redrawn in a slightly narrower table. The reader is asked to run the game again on the grid printed below the instruction — and that second grid repeats the first one exactly. Say so plainly rather than implying a fresh puzzle.
• The open question (Part II, "Try This", p.38). Printed beside the second grid: what makes these grids special — is it the numbers, the arrangement, or both — and can the reader make more of them? The chapter prints no answer, and Part II has no answer key. This question is the explanation's thesis.

Figures to have open

• A four-by-four grid of integers that can be filled from the printed values, can show rings on chosen cells, and can strike out whole rows and columns. Standard schematic; the printed grid's pink cells are decoration and need not be copied.
• A three-box instruction flow chart with a return loop, mirroring p.37.
• A bordered table with a row of headers down the left and across the top, so the grid can be rebuilt as an outer product. This figure does not exist in the book — it is the explanation's explanatory addition.
• Grid thumbnails for a round-by-round play, with one, two, three and four rings. The book's four panels do not supply them.
• No photograph or dataset from the textbook is needed.

Where this sits in the book

• NCERT Class 7 Mathematics, Ganita Prakash Part II, printed Chapter 2 "Operations with Integers", the unnumbered sub-heading "A Magic Grid of Integers" inside §2.2, pp.37–38, including the "Try This" prompt on p.38.
• Part II, p.35, for commutativity, and p.40, for associativity — the two properties the explanation rests on; both are also restated in the SUMMARY, p.45.
• Part II, p.33, for the sign cases used in the sign prediction.
• Sibling topic Commutative, associative, and distributive over the integers, which establishes the two laws this topic spends.