Algebra grids: every row is an equation, and the shapes are the unknowns

What to assume they know

• Writing an unknown as a letter and reading a picture-symbol as an unknown
• Adding like terms, so that two copies of the same unknown become twice it
• Solving a one-step equation by dividing both sides
• Solving a two-step equation: subtract from both sides, then divide
• Substituting a known value into a second equation and solving what remains
• Checking a solution by putting it back into every equation it came from

What they should be able to do

• Read a row of an algebra grid as an equation and write it out in letters
• Identify the row that can be solved on its own, and say what makes it solvable
• Solve for one shape's value by division, and justify the division
• Substitute a known shape into another row and solve for the remaining shape
• Check a proposed pair of values against every row of the grid
• Complete a row whose total is left blank, and explain why that blank is not an extra unknown
• Recognise two rows that carry the same information despite being drawn differently, and say why no work is needed on the second
• Recognise a row that pins nothing down, and say what it would need in order to

Where it usually goes wrong

• "The shapes and colours mean something." They are names. A blue square is an unknown called "blue square"; nothing about squareness or blueness enters the arithmetic. Replacing every shape with a letter, once, settles this permanently.
• "A shape might mean different things in different rows." Then no row would constrain any other and the puzzle would be unsolvable. That the same shape holds the same value throughout is the assumption doing all the work, and it deserves to be said aloud.
• "Two rows cannot settle two unknowns." They can, provided they say different things. The reader's grids do exactly this, and the third row in each is a check rather than a necessity.
• "The blank total is a third thing to find." It is fixed the moment the shapes are known. Students treat it as an unknown and then complain there are too many.
• "You must work the rows in the order they are printed." Start wherever the arithmetic is easiest. In the chapter's worked grid that happens to be row 1; in the reader's first grid there is no such row and you must combine two.
• "Row 3 is more work because the shapes are in a different order." It is no work at all. Recognising this is a genuine algebraic insight, not laziness.
• "Every grid has one answer." The empty fourth row of the second grid does not, and could not. A row constrains its shapes only if it has shapes in it.
• "Guess-and-check is as good as solving." It finds these two answers quickly and teaches nothing transferable. The point of the section is that the picture is a system of equations, and the method scales to grids guessing cannot reach.

Questions to check understanding

• Write out each row of a given grid as an equation
• Solve a grid that contains a single-shape row, stating which row you began with and why
• Solve a grid that contains no single-shape row
• Fill in a blank total and justify it without recomputing the whole row
• Given values for the shapes, complete every blank in a grid
• Construct a grid of your own with two shapes and a unique answer
• Explain why a row of empty cells cannot be completed
• Given three rows, say whether the third adds any information to the first two

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated inputs. No answer to any exercise the chapter sets the reader is printed anywhere in Part II pp.135–147, and Part II has no answer-key appendix. (The chapter does print worked answers to its own demonstrations — the 291 decode resolved to 25th December on Part II p.137, the apex-10 and apex-60 pyramids shown completed on Part II pp.138–139, the sum-36 block given as 5, 6, 12, 13 on Part II p.141, and the worked algebra grid resolving to 9 and 5 on Part II p.142 — so do not say the chapter prints no answers at all.) All three grids below were read off the printed page and confirmed on the printed page, because the shapes are artwork and do not extract.

• The rule of the puzzle (Part II §6.4, p.142, under the printed unnumbered subheading "Algebra Grids"). Drawn shapes stand for numbers, and the number in the last column of a row is the total of the shapes to its left in that row.
• The grid the chapter works (Part II §6.4, p.142). Two rows, three shape cells each, plus a totals column.
  • Row 1: blue square, blue square, blue square — total 27.
  • Row 2: red circle, red circle, blue square — total 19. The printed working: three squares totalling 27 gives the square as 9; then two circles plus 9 make 19, so twice the circle plus 9 is 19, and the circle is 5. Note the chapter's own step of writing "twice the circle" — that collecting move is the one a student is most likely to skip.
• The first grid set for the reader (Part II §6.4, p.142). Three rows, three shape cells each, plus a totals column. Two shapes are used, the same blue square and red circle as above.
  • Row 1: square, square, circle — total 27.
  • Row 2: circle, circle, square — total 21.
  • Row 3: circle, square, circle — total left blank. Verified: twice the square plus the circle is 27, and twice the circle plus the square is 21. Doubling the first and taking the second from it leaves three times the square as 33, so the square is 11 and the circle is 5. Both rows check: 22 plus 5 is 27, and 10 plus 11 is 21. The blank total is 21, and there are two ways to get there — add 5, 11 and 5, or notice that row 3 holds the same three shapes as row 2 in a different order.
• The second grid set for the reader (Part II §6.4, p.142). Four rows and four columns; two shapes are used, a blue circle and a purple diamond.
  • Row 1: circle, diamond, diamond — total 18.
  • Row 2: diamond, circle, circle — total 15.
  • Row 3: diamond, circle, circle — total left blank.
  • Row 4: all four cells empty, the totals cell included. Verified: the circle plus twice the diamond is 18, and twice the circle plus the diamond is 15. Doubling the second and taking the first from it leaves three times the circle as 12, so the circle is 4 and the diamond is 7. Both rows check: 4 plus 14 is 18, and 8 plus 7 is 15. Row 3 repeats row 2's shapes exactly, so its total is 15 with no work at all.
• The fourth row of that grid is empty in every cell, totals cell included. Verified on the printed page. With no shapes in it there is nothing to compute: any total at all could be written there by choosing shapes to match. This is the best thing on the page for section 11, because it shows what an equation needs before it says anything.
• The two shortcuts worth naming. Verified: first, a row whose cells are all one shape is the only row you can solve without help, and it is always the place to begin if one exists. Second, two rows holding the same multiset of shapes must carry the same total, however the shapes are arranged, because addition does not care about order. Both of the reader's grids contain such a pair, which is unlikely to be an accident.
• What happens with no single-shape row. Verified: the reader's first grid has no such row, so the opening move has to be different — take a multiple of one row and subtract another. That is the step the chapter's worked example never needed, and it is the real difficulty of the exercise.

Figures to have open

• The three grids of Part II p.142, redrawn. Two visually distinct shapes per grid are enough; do not rely on colour alone, since the printed grids distinguish a blue square from a red circle in the first two and a blue circle from a purple diamond in the third, and a colour-blind viewer needs the outline to differ.
• A side-by-side of one grid and the same grid with every shape replaced by a letter. Standard schematic, and the single most useful figure here — it is the whole thesis in one frame.
• A bracket-and-divide annotation for the single-shape row. Standard schematic.
• The empty fourth row shown with three candidate shape-fillings, each giving a different total. Standard schematic; not in the chapter, and needed for section 11 to land.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 6, "Algebra Play", §6.4 "Fun with Grids", under the printed unnumbered subheading "Algebra Grids", Part II p.142. The subheading can be named but has no section number.
• The worked grid and its step-by-step solution occupy the middle of Part II p.142, with the grid on the left and the working set out to its right. The two grids for the reader sit immediately below, side by side.
• The Calendar Magic material higher up the same page is a different topic and is covered by Calendar magic: the position in the grid is the whole secret; §6.5 "The Largest Product" begins at the foot of Part II p.142 and is covered by Arranging given digits to make the largest product.
• The chapter's SUMMARY (Part II p.147) lists grids among the things algebra was applied to here.