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Factorisation using algebra tiles · 4 marks

Algebra tiles can be used to represent products and find factors.
1. Figure out the product of x + 2 and x + 3 using algebra tiles.
2. Lay out algebra tiles for x2 + 11x + 30 in such a way that you will see its factors.

  1. 1. Figure out the product of x + 2 and x + 3 using algebra tiles.
  2. 2. Lay out algebra tiles for x2 + 11x + 30 in such a way that you will see its factors.
Answer: 1. (x + 2)(x + 3) = x2 + 5x + 6 (one x2-tile, 2 + 3 = 5 x-tiles, 2 × 3 = 6 unit tiles). 2. Put 5 x-tiles beside the x2-tile, 6 below, and the 30 unit tiles in a 5 × 6 block: the rectangle has sides x + 5 and x + 6, so x2 + 11x + 30 = (x + 5)(x + 6).

Step-by-step solution

Idea: A rectangle made of tiles has area = length × breadth. The x2-tile is an x × x square, an x-tile is x × 1, and a unit tile is 1 × 1. Building a rectangle with sides (x + p) and (x + q) always uses one x2-tile, p + q x-tiles and p × q unit tiles.

x + 2x + 3x²xxx11x11x11(x + 2)(x + 3) = x² + 5x + 6x + 5x + 6x²xxxxxx11111x11111x11111x11111x11111x11111x² + 11x + 30 = (x + 5)(x + 6)

1. Figure out the product of x + 2 and x + 3 using algebra tiles.

  1. Make a rectangle with top side x + 2 (an x length and 2 units) and left side x + 3 (an x length and 3 units).½ mark
  2. Fill it: the x × x corner is one x2-tile; the 2-unit strip beside it holds 2 x-tiles; the 3-unit strip below it holds 3 x-tiles; the 2 × 3 corner holds 6 unit tiles (see the left diagram).½ mark
  3. Count: 1 x2-tile, 2 + 3 = 5 x-tiles, 6 unit tiles.½ mark
  4. So (x + 2)(x + 3) = x2 + 5x + 6. Check with distributivity: x2 + 3x + 2x + 6 = x2 + 5x + 6 ✓.½ mark
x2 + 5x + 6

2. Lay out algebra tiles for x2 + 11x + 30 in such a way that you will see its factors.

  1. Tiles: one x2-tile, 11 x-tiles, 30 unit tiles. The 11 x-tiles must be split into p beside and q below, with p + q = 11 and the corner p × q = 30.½ mark
  2. Try pairs adding to 11: 1 × 10 = 10, 2 × 9 = 18, 3 × 8 = 24, 4 × 7 = 28, 5 × 6 = 30 ✓.½ mark
  3. Place 5 x-tiles beside the x2-tile and 6 below it, and fill the corner with the 30 unit tiles in 5 columns and 6 rows (right diagram). Every tile is used and the shape is a rectangle.½ mark
  4. Its sides are x + 5 and x + 6, so x2 + 11x + 30 = (x + 5)(x + 6).½ mark
(x + 5)(x + 6)
1. (x + 2)(x + 3) = x² + 5x + 6. 2. x² + 11x + 30 = (x + 5)(x + 6), seen as an (x + 5) by (x + 6) rectangle of tiles.

Check: Put x = 10: (12)(13) = 156 = 100 + 50 + 6 ✓; (15)(16) = 240 = 100 + 110 + 30 ✓.

Answer to write in the exam

1.

Tiles: one x2 tile, 2 + 3 = 5 x tiles, 2 × 3 = 6 unit tiles

(x + 2)(x + 3) = x2 + 3x + 2x + 6

∴ (x + 2)(x + 3) = x2 + 5x + 6

2.

11x = 5x + 6x, since 5 + 6 = 11 and 5 × 6 = 30

Tiles: one x2 tile, 5 x tiles beside it, 6 x tiles below it, 30 unit tiles in a 5 × 6 block

Sides of the rectangle: (x + 5) and (x + 6)

∴ x2 + 11x + 30 = (x + 5)(x + 6)

Common mistakes that cost marks

  • Counting the x-tiles as 2 × 3 = 6 instead of 2 + 3 = 5. The x-tiles add; it is the unit tiles that multiply.
  • Putting all 11 x-tiles along one side. Then the corner is 0 units wide and the 30 unit tiles cannot fit; split them 5 beside and 6 below.
  • Choosing a pair with product 30 but the wrong sum, such as 3 and 10 (sum 13) or 2 and 15 (sum 17).

How this can come in the exam

MCQ (1 mark)

A rectangle made of algebra tiles uses one x2-tile, 7 x-tiles and 10 unit tiles. Its sides are

  1. x + 1 and x + 10
  2. x + 2 and x + 5
  3. x + 3 and x + 4
  4. x + 7 and x + 10
Show answer

(B) x + 2 and x + 5
Need sum 7 and product 10: 2 and 5. (x + 2)(x + 5) = x2 + 7x + 10.

Assertion–Reason (1 mark)

Assertion (A): The tiles for (x + 2)(x + 3) include 6 unit tiles.
Reason (R): The unit tiles fill a corner that is 2 units by 3 units.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
2 × 3 = 6, so R explains A.

Try one yourself

Factorise x2 + 10x + 24 by thinking about algebra tiles.

Show answer

Need sum 10, product 24: 4 and 6. The rectangle is (x + 4) by (x + 6), so x2 + 10x + 24 = (x + 4)(x + 6).

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