Expanding Brackets and (a + b)²: An Area Model Lesson

To expand brackets, multiply every term in one bracket by every term in the other. For a square: (a + b)² = a² + 2ab + b². A square of side a + b, cut into four pieces, shows where the 2ab comes from, the term students most often forget.

Learning objectives

• Expand single and double brackets.
• Explain (a + b)² = a² + 2ab + b² with an area model.
• Recognise the special products (a − b)² and (a + b)(a − b).

The opening question

"Is (3 + 4)² equal to 3² + 4²?" The left side is 49, the right side 25. The missing 24 is 2 × 3 × 4: the 2ab. Numbers first, then letters.

The lesson, click by click

1. Click 0: a square of side a + b.
2. Click 1: cut it into four pieces: an a × a square, a b × b square and two a × b rectangles.
3. Click 2: add the areas: a² + 2ab + b².
4. Click 3: the algebra: (a + b)(a + b) = a² + ab + ba + b².
5. Click 4: the other special products: (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b².

Common mistakes to watch for

• (a + b)² = a² + b². The most common algebra mistake of all. Test it with numbers.
• Missing a product. (x + 2)(x + 5) has four products: x², 5x, 2x and 10.
• Signs. (x − 3)² = x² − 6x + 9: the last term is +9.
• Single brackets. 3(x + 4) = 3x + 12, not 3x + 4.

Check questions

1. Expand (x + 3)². (x² + 6x + 9.)
2. Expand (2x − 1)². (4x² − 4x + 1.)
3. Work out 101² in your head. ((100 + 1)² = 10,000 + 200 + 1 = 10,201.)

Building it in Tracé

1. Type each line of the expansion as an equation and reveal one per click; colour the 2ab term in yellow when it appears.
2. Draw the area model with a Custom animation: two squares and two rectangles as polygon … fill shapes, each appearing on its own click.
3. Present it as a "predict then reveal": ask for (3 + 4)² before showing the general rule.

Questions

More: Animated math slides for teachers · How to create math slides, step by step