Algebraic Identities – Cut and Rearrange
MathematicsAlgebraAges 13–14
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Algebra
Illustrates (a + b)², (a − b)² and a² − b² using the areas of squares and rectangles. Students drag the handles to change a and b, then press Split/Merge to watch the pieces separate or reassemble: a² − b² is cut into two pieces and rearranged into a (a + b)(a − b) rectangle. The formula with the current numbers updates as the shape changes.
Lesson: Difference of squares. Square of a sum or a difference
What it shows
Algebraic identities can be proved with areas. A square of side a + b splits into a square a², a square b² and two a × b rectangles, which shows why (a + b)² = a² + 2ab + b² and not a² + b². For (a − b)², the strips removed from a square of side a explain the term −2ab. For a² − b², a square b² is cut from the corner of a square a², and the L-shaped remainder is cut into two rectangles that rearrange into an (a + b) × (a − b) rectangle, which is the factorisation of a difference of two squares.
How to use
Choose an identity with the buttons (a + b)², (a − b)² or a² − b², then drag the handles or change Length a and Length b. With the area labels showing, ask students to add up the pieces and match them with the numerical formula below the figure. Press Split / Merge to separate the pieces and put them back together, especially for a² − b², where the rearranged rectangle makes the factorisation visible.
Parameters you can change
- Length a 2–10 units
- Length b 0.5–9 units
- Identity (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², a² − b² = (a + b)(a − b)
- Show area labels on pieces
Questions to explore
- Why is (a + b)² larger than a² + b² when a and b are positive, and by how much?
- Set a = 9 and b = 4. What are the sides of the rearranged rectangle, and what is 9² − 4²?
- What is the total area of the two ab rectangles, and where does it appear in the formula?
