Algebra tiles: factorising by rebuilding the rectangle

What to assume they know

• Area of a rectangle as the product of its two sides, including when the sides are expressions (Reading (a + b)² off a partitioned square)
• Multiplying two brackets by distributivity and collecting like terms
• That a product of two linear expressions has an x² term, an x term and a constant term (Univariate polynomials and what degree names)
• Reading a rectangular array as a multiplication
• What a factor is, and that factorising is the reverse of multiplying (Recognising an expression as an identity in disguise)

What they should be able to do

• Compute the area of a rectangle whose sides are x + 3 and x + 4 both by distributivity and by counting tiles, and check the two agree
• Identify the three kinds of tile in the chapter's model and state the area of each
• Lay out a product of two linear expressions as a rectangle and read the four regions off it
• Explain why the x-tiles must divide into two groups, one along each side
• Explain why the number of unit tiles is forced once the two groups are chosen
• Show that an alternative split of the middle term fails to close the rectangle, and say exactly what goes wrong
• Read the factorisation off a completed rectangle by naming its two sides
• Generalise the tile layout to (x + a)(x + b)
• Build the tile picture when the leading coefficient is not 1, as for (2x + 3)(3x + 1)
• Complete the general product (px + a)(qx + b) and confirm it by distributivity

Where it usually goes wrong

• "You can split the middle term any way you like." Then the tiles do not close. Run 1 + 6 and 2 + 5 and let the unit tiles run out or spill. The rectangle is the referee.
• "The 12 unit tiles are just a count." They are a 3 by 4 array, and their two dimensions are the two numbers in the split. This is the single most important thing the picture says.
• "The x-tile is a square." It is one unit by x units. If it is drawn nearly square the whole model becomes unreadable, so keep x visibly longer than 1 in every drawing.
• "The x²-tile is the x-tile squared in the sense of being twice as big." Its side is x, so its area is x × x. In a to-scale drawing with x = 4 it is four times the area of an x-tile, not twice.
• "Tiles are a crutch for people who cannot do algebra." The tiles supply the reason the two arithmetic conditions come as a pair; the algebra of §4.6 supplies the speed. Neither replaces the other.
• "With 6x² at the front you need six separate rectangles." You need one rectangle with six x²-tiles in a 2 by 3 block, because 6 = 2 × 3 and those two numbers are the leading coefficients of the two factors.
• "Tiles work for every quadratic." They need whole-number, positive tile counts. Nothing in §4.5 or §4.6 supplies a negative tile, so an expression like x² − 5x + 6 cannot be laid out as printed even though §4.6 factorises it symbolically without trouble.
• "Fig. 4.8's eleven x-tiles are eleven of the same thing." Nine of them arise one way and two another. Colour or group them differently.

Questions to check understanding

• Given two linear expressions, lay out the tile rectangle and write the product
• Given a quadratic with positive whole-number coefficients, arrange tiles so that the factors are visible, and state them
• Explain, for a stated wrong split of the middle term, exactly why the rectangle cannot be completed
• Complete a general product such as (px + a)(qx + b) and verify by distributivity
• Given a count of tiles of each kind, write the expression and find the dimensions of the rectangle they form
• Find possible length and breadth for a rectangle whose area is given as a quadratic expression — the form of End-of-Chapter Q5 and Q9, pp. 89–90

Examples worth working on the board

Inputs, not answers. Values marked Verified are worked out here or an added count on the printed page; this book prints no answer key.

• The opening product (§4.5, p. 78). A rectangle with sides x + 3 and x + 4. Its area is (x + 3)(x + 4), which the page multiplies out by distributivity as x² + 3x + 4x + 12 and collects. Verified: x² + 7x + 12. Note that the page keeps 3x + 4x visible for one line before collecting — that intermediate line is the whole hinge of §4.5 and §4.6.
• Fig. 4.7 (§4.5, p. 79), counted on the printed page because tile labels are artwork lettering. The layout: one large square tile marked x² in the top-left; three tall tiles marked x standing in a column to its right; four wide tiles marked x lying in a row beneath it; and twelve small tiles marked 1 filling the bottom-right corner in an arrangement of four rows of three. The outer dimensions are marked x + 3 across the top and x + 4 down the left side. Verified as a statement: 1 × x² + 7 × x + 12 × 1 recovers the expression, and the outer rectangle's two sides are the two factors.
• The two readings of one figure (§4.5, p. 79). The page states explicitly that the same picture gives both the multiplication (x + 3)(x + 4) = x² + 7x + 12 and the factorisation x² + 7x + 12 = (x + 3)(x + 4). One drawing, read outwards or inwards. This is the section's cleanest idea.
• The failing split (§4.5, p. 79, Think and Reflect). The box asks whether a similar rectangle can be formed if 7x is split as 2x + 5x, and invites other possibilities. No answer is printed. Verified by an added count: with two x-tiles in the right-hand column and five along the bottom, the corner they enclose is 2 wide by 5 tall, so it holds ten unit tiles — but the expression supplies twelve, so two are left over with nowhere to go. Every split of 7 should be tried: 1 + 6 needs 6 units, 2 + 5 needs 10, 3 + 4 needs 12. Only the last matches, and it matches because 3 × 4 = 12.
• Two more layouts to build (§4.5, p. 79, second Think and Reflect). Find the product of x + 2 and x + 3 with tiles; and lay out x² + 11x + 30 so that its factors become visible. Verified: (x + 2)(x + 3) = x² + 5x + 6, needing one x²-tile, five x-tiles split as 2 and 3, and six unit tiles in a 2 by 3 array; x² + 11x + 30 splits as 5 and 6 with a 5 by 6 array of thirty units, giving (x + 5)(x + 6).
• Generalising (§4.5, p. 80, Think and Reflect). The box supplies two solved cases — (x + 3)(x + 4) = x² + 7x + 12 and (x + 6)(x + 7) = x² + 13x + 42 — and asks for the pattern for (x + a)(x + b). Verified: 6 + 7 = 13 and 6 × 7 = 42, so the second case fits; the general form is x² + (a + b)x + ab. This is the statement that §4.6 then works with, and it also appears in the identity list on p. 91.
• Fig. 4.8 (§4.5, p. 80), counted on the printed page. The rectangle represents (2x + 3) × (3x + 1), with 2x + 3 marked across the top and 3x + 1 down the left. The layout: six x²-tiles in a block two across and three down; nine x-tiles filling the right-hand strip, three in each of three rows; two x-tiles lying along the bottom; and three unit tiles in the bottom-right corner. Verified: the total is 6x² + 11x + 3, and (2x + 3)(3x + 1) = 6x² + 2x + 9x + 3 = 6x² + 11x + 3, so the counts agree. What makes this figure worth its own beat: the x-term now comes from two differently shaped contributions — nine tiles from three unit-columns beside three x-rows, and two tiles from two x-columns beside one unit-row — so the "split the middle term" idea is no longer a split into two similar halves.
• The general blank (§4.5, p. 80). The page prints (px + a)(qx + b) = (___)x² + (___)x + ___ and asks for the three entries, then says to check by distributivity. Verified: pq, then pb + aq, then ab. Cross-check with Fig. 4.8: p = 2, q = 3, a = 3, b = 1 gives 6, then 2 × 1 + 3 × 3 = 11, then 3 — the counted tiles.
• Example 17, Saira's rectangle (§4.8, p. 87, continued on p. 88). Saira lays out one square of side x, eight rectangular strips each x units by 1 unit, and fifteen squares of side 1 unit, and arranges them all into one bigger rectangle. The page adds the areas — x², then 8x, then 15 — to get x² + 8x + 15, and then factorises to recover the sides. Verified: the split is 3 and 5, since 3 + 5 = 8 and 3 × 5 = 15, so the rectangle is (x + 5) by (x + 3). The page closes by telling the reader to actually draw Saira's rectangle from the pieces, which is why this example belongs with the tiles even though it is printed in §4.8.

Figures to have open

• Fig. 4.7, the tile rectangle for x² + 7x + 12 (p. 79). The chapter's own figure and the topic's centre. Redraw to scale with x noticeably longer than 1, keep the three tile types visually distinct, and keep the outer x + 3 and x + 4 dimension labels.
• Fig. 4.8, the tile rectangle for (2x + 3)(3x + 1) (p. 80). The chapter's own figure. Redraw; the 2 by 3 block of x²-tiles must be legible as a block.
• A movement of the failed 2x + 5x layout. Standard schematic, not in the book — the chapter poses the question and prints no picture of the failure.
• A drawing of Saira's rectangle from the stated pieces (p. 87). The chapter explicitly asks for it and does not print it, so this one must be built.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics, printed Chapter 4, "Exploring Algebraic Identities": §4.5 "Factorisation Using Algebra Tiles", pp. 78–80, including Fig. 4.7 and Fig. 4.8.
• Think and Reflect boxes on p. 79 (two of them) and p. 80.
• Example 17, printed in §4.8 on pp. 87–88, which is a tile problem in prose.
• Chapter summary, p. 90, second and fourth bullets, both of which name algebra tiles; the identity list on p. 91 carries both x² + (a + b)x + ab = (x + a)(x + b) and its two-coefficient counterpart, acx² + (ad + bc)x + bd = (ax + b)(cx + d).
• Companion topic: Splitting the middle term once the tiles come away, which does the same work without the tiles.