(a + b)² and (a − b)² with square tiles
Build a square of side a + b from four tiles and see why (a + b)² = a² + 2ab + b². Then cut strips from a square of side a to see (a − b)² = a² − 2ab + b².
Core concept
(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b², which can be seen as areas of squares and rectangles
Topic: Algebraic expressions
(a + b)² means a square whose side is a + b. Cut it along the lines at a: it splits into a square a × a (a²), a square b × b (b²) and two rectangles a × b (2ab). So (a + b)² = a² + 2ab + b². A common mistake is to write a² + b², forgetting the two rectangles. For (a − b)², start with a square of side a and cut a strip b wide off one side and one off the other. Both strips covered the small b × b corner, so it was taken away twice and must be added back: (a − b)² = a² − 2ab + b².
What you will learn
- Expand (a + b)² and (a − b)².
- Explain each identity with areas of squares and rectangles.
- Use the identities to square numbers quickly.
Before you start
- Know how to find the area of a square and a rectangle.
Materials
- Squared paper and scissors
Equipment
- None
Procedure
- Record (a + b)² with a = 3 and b = 2.
- Record (a − b)² with a = 5 and b = 2.
- Record three or more different values of a.
How this simulation is simplified
- a and b are lengths in the same unit.