Exploring Algebraic Identities · Lesson 5 of 8
Factorisation Using Algebra Tiles
“Move a few tiles around and the factors practically introduce themselves.”
• Understand x²-tiles, x-tiles and unit tiles. • Visualise multiplication of two linear expressions. • Understand why the middle term must split correctly. • Generalise (x+a)(x+b). • Generalise products of the form (px+a)(qx+b).
Algebra tiles give a geometric meaning to algebraic expressions. An x²-tile is a square with area x², an x-tile is a rectangle with area x, and a unit tile is a small square with area 1. If these pieces form a rectangle, the side lengths of that rectangle reveal factors of the expression.
| Tile | Dimensions | Area |
|---|---|---|
| x²-tile | x × x | x² |
| x-tile | x × 1 | x |
| Unit tile | 1 × 1 | 1 |
Visualising (x+3)(x+4)
A rectangle with sides x+3 and x+4 has area (x+3)(x+4). When partitioned, it contains one x² region, three x-rectangles along one side, four x-rectangles along the other side, and a 3×4 block of unit squares.
Why 7x Must Split as 3x+4x
The 3x and 4x parts are not arbitrary. They come directly from the two side additions, 3 and 4. Their sum is 7, matching the coefficient of x, and their product is 12, matching the unit-tile corner.
If 7x were split as 2x+5x, the rectangular corner would need 2×5=10 unit tiles, not 12. So that split cannot represent the same area.
Factorisation Is the Reverse Process
Starting with x²+7x+12, if the tiles can be rearranged into a rectangle of dimensions x+3 and x+4, the factors are visible immediately.
Generalising the Pattern
The coefficient of x is a+b because the two rectangular strips have areas ax and bx. The constant term is ab because the corner rectangle has dimensions a and b.
General Linear Factors
Problem
Expand the product.
- 1.Multiply 2x by 3x to get 6x².
- 2.Multiply 2x by 1 to get 2x.
- 3.Multiply 3 by 3x to get 9x.
- 4.Multiply 3 by 1 to get 3.
- 5.Combine the x-terms: 2x+9x=11x.
- 6.Result: 6x²+11x+3.
Practice Problems
- Use the algebra-tile idea to expand (x+2)(x+3).
- Arrange x²+11x+30 as a rectangle and identify the factors.
- Explain why splitting 7x as 2x+5x does not work for x²+7x+12.
- Expand (x+6)(x+7).
- Expand (2x+3)(3x+1).
- Verify (px+a)(qx+b)=pqx²+(pb+aq)x+ab using distributivity.
Key Takeaways
• Algebra tiles connect algebra with area. • Products of linear expressions can be visualised as rectangles. • The middle-term split must satisfy both a sum and a product condition. • Factorisation reverses multiplication. • (x+a)(x+b)=x²+(a+b)x+ab. • (px+a)(qx+b)=pqx²+(pb+aq)x+ab.
Next, we factor quadratics without tiles by splitting the middle term algebraically.