medium difficulty5 min

Algebra-Tiles Visualisation

Algebra-tiles visualisation uses squares and rectangles to represent algebraic areas and understand identities geometrically.

In one line: One big square, four pieces, two middle rectangles.

Practice This Concept

Understand it

For (a+b)², imagine a large square with side a+b. It can be split into one a by a square, two a by b rectangles, and one b by b square. The total area becomes a²+2ab+b², so the identity is seen through area.

Example

A square of side x+3 has area x²+3x+3x+9, which simplifies to x²+6x+9.

Formula and rule

Area model for (a+b)²: total area = a²+ab+ab+b² = a²+2ab+b².

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1For side x+4, split the square into x and 4 on both sides.
2Areas are x², 4x, 4x, and 16, so total area is x²+8x+16.

Method you can reuse

Mark the full side, split each side into the two parts, label every region by length times breadth, then add all areas.

Sketch it in your answer

Draw a large square of side a+b, split both length and breadth into a and b, then label four regions as a², ab, ab, and b².

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why are there two ab rectangles in the area model of (a+b)²?

Think first, then show answer

There are two ab rectangles because the square side is split in both directions into a and b, giving one a by b rectangle and one b by a rectangle.

Common wrong answer

Writing x²+4x+16 for (x+4)².

Why it loses marks: It counts only one rectangle of area 4x.

Don't mix this up

Students may draw only one ab rectangle and write a²+ab+b² instead of a²+2ab+b².

Exam tip

In a diagram for (a+b)², look for two identical rectangles of area ab.