Factorisation with area tiles: extending a courtyard
Extend a square courtyard on two sides and see how (x + a)(x + b) breaks into x², x strips and unit squares.
Core concept
(x + a)(x + b) = x² + (a + b)x + ab, so factorising means finding two numbers with the right sum and product
Topic: Algebraic identities and factorisation
A family has a square courtyard x m long on each side, so its area: How much flat space a shape covers, measured in square units like cm². All key words is x². They extend it by a m on one side and b m on the other. The new courtyard is (x + a) long and (x + b) wide, so its area is (x + a)(x + b). Laid out as tiles, the floor is one x² square, a strips of x, b strips of x and a × b small 1 m² squares: x² + (a + b)x + ab. Factorising goes the other way: to write x² + 5x + 6 as a product, find two numbers that add to 5 and multiply to 6.
What you will learn
- Expand (x + a)(x + b) using an area model.
- Use the identity (x + a)² = x² + 2ax + a².
- Factorise x² + px + q by finding two numbers with sum p and product q.
Before you start
- Area of a rectangle; adding and multiplying small numbers.
Materials
- Algebra tiles cut from card
Equipment
- A measuring tape
Procedure
- Drag the green right edge to extend the courtyard by a, and the red top edge to extend it by b.
- Read the expression and count the tiles of each colour.
- Make a square courtyard again, and a courtyard extended on one side only.
- Lay out the tiles for x² + 5x + 6 and for x² + 7x + 12, and record each factorisation.
How this simulation is simplified
- Lengths are whole metres; x is the side of the original square courtyard.