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Maths

Expanding Double Brackets

Overview

Students learn to multiply out double brackets systematically, connecting the algebraic process to area models and building the foundation for factorising quadratics.

Learning Objective
Students expand the product of two linear brackets using FOIL or the grid method, collect like terms, and recognise special cases including perfect squares and difference of two squares.

Resources needed

  • Mini whiteboards
  • Squared paper (for grid method)

Lesson stages

0 / 7 done
  1. 1 Expand 3(x+2) = 3x+6. 'Every term inside the bracket is multiplied by the outside term.' Quick fire: expand 5(x−3), 2(3x+1), x(x+4). Establish fluency before double brackets.
  2. 2 Expand (x+3)(x+4). Draw a 2×2 grid. Top: x, +4. Side: x, +3. Fill in cells: x², 4x, 3x, 12. Sum: x²+7x+12. 'The grid ensures we don't miss any terms.' Students use grid for three examples.
  3. 3 FOIL: First, Outside, Inside, Last. (x+3)(x+4): First: x×x=x². Outside: x×4=4x. Inside: 3×x=3x. Last: 3×4=12. Total: x²+7x+12. Same answer as grid. Students choose their preferred method.
  4. 4 (x−5)(x+2). FOIL: x²+2x−5x−10 = x²−3x−10. Key: be careful with signs when multiplying. Students practise six examples with mixed signs.
  5. 5 (x+3)² = (x+3)(x+3) = x²+6x+9. Pattern: (a+b)² = a²+2ab+b². Students expand (x+5)², (x−4)², (2x+1)². Identify the pattern — 'middle term is twice the product of the two terms.'
  6. 6 (x+4)(x−4) = x²−4x+4x−16 = x²−16. The middle terms cancel. (a+b)(a−b) = a²−b². Students spot and apply the shortcut for three examples.
  7. 7 Display: (x+3)(x+2) = x²+5 (missing middle term). (x−2)² = x²−4 (missing middle term). (x+4)(x−4) = x²+16 (wrong sign). Students identify and correct.

Tap a step to mark it as done.

Variations

  • Extend to expanding three brackets
  • Expand (2x+3)(3x−1) with non-unit leading coefficients
  • Link to finding the area of a rectangle with algebraic dimensions
More information

Display: expand, bracket, coefficient, like terms, collect. FOIL acronym displayed with each letter's meaning. Use 'every term by every term' as a simple summary.

Use the grid method exclusively before introducing FOIL. Provide expansion grids with headers already filled. Focus on a=1 cases only initially.

Are all four products found? Are like terms collected? Are signs handled correctly for negative terms? Do students recognise the special cases?

Grid drawn on mini whiteboards. No squared paper needed.

The most common error is (x+3)² = x²+9. Students 'square each term' rather than multiplying out the brackets. Show explicitly: (x+3)(x+3) — you cannot skip this step.

Activity details
Secondary Medium 40 minutes No prep Solo, Pairs Indoor
Topics
Algebra Brackets Expanding Quadratics