Math // Algebra

Expanding Double Brackets

Expand products of two or more binomials, e.g., (x + 3)(x - 2) = x² + x - 6, using the grid method or FOIL; simplify the result by collecting like terms

Ages
13 to 14
Type
Method
Builds on
2 topics
Leads to
0 topics
Standards
1 code

Quick check

Can the learner multiply out two brackets like (x + 3)(x − 2) and collect the terms to get x² + x − 6?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Expand the product of two binomials using a systematic method (grid or FOIL)
  • Collect like terms after expansion to simplify the result
  • Expand products involving negative terms correctly

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

2 topics
  • Collecting Like TermsRequired

    Algebra // Ages 11 to 12

    Expanding double brackets requires collecting like terms to simplify the result

  • Expanding Single BracketsRequired

    Algebra // Ages 11 to 13

    Expanding double brackets extends single-bracket expansion using FOIL/grid method

You are here

Expanding Double Brackets

Ages 13 to 14 // Method

Leads to

0 topics

Nothing in the catalogue builds on this topic yet. Browse Algebra

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Expanding two binomials applies the distributive law twice: every term in the first bracket multiplies every term in the second, so (x + 3)(x − 2) produces FOUR products (x², −2x, +3x, −6) that collect to x² + x − 6. It generalises the grid method for multiplying two-digit numbers — (x + 2)(x + 3) mirrors 12 × 13 as (10 + 2)(10 + 3). The grid/area model keeps all four products visible, which helps against the two most common errors: dropping the middle cross-products (writing x² − 6) and squaring each term of (a + 3)² to get a² + 9 instead of a² + 6a + 9. Directed-number care is essential for the signs; collecting like terms is a required final step.

Common misconceptions

  • Multiplies only the first terms and the last terms, MISSING the two middle cross-products, e.g. (x + 3)(x − 2) → x² − 6
  • Squares each term of a squared bracket: (a + 3)² → a² + 9 (or (a + b)² → a² + b²), forgetting the 2ab middle term
  • Sign errors with negative terms — applies a false 'same sign in the middle' rule, or drops the minus on a cross-product
  • Miscollects or mis-multiplies the middle/constant terms (e.g. 3b + 4b handled wrongly, or 3 × 4 written as 7)
  • Leaves the answer uncollected as x² + 3x − 2x − 6 instead of simplifying to x² + x − 6

Useful representations

  • 2×2 multiplication grid / area model (four cells, one per product)
  • Algebra tiles arranged as a rectangle
  • The numerical grid analogue, e.g. (10 + 2)(10 + 3) for 12 × 13
  • Substitution check (put x = 1 into both forms)

Teaching practices

  • Use the grid / area model so each of the four products occupies its own cell — nothing can be forgotten — and connect it to numerical grid multiplication (12 × 13, 24 × 36)
  • Say explicitly 'the product of two binomials has FOUR terms before collecting', so a missing middle term is noticed
  • Treat (a + 3)² as (a + 3)(a + 3), never squaring each term separately
  • Insist on directed-number care for negative terms (+3 × −2 = −6); reject the false 'keep the same sign in the middle' rule
  • Write all four products first, then collect the like middle terms — collecting is the final step, not optional
  • Check an expansion by substituting a number, and link back to factorising as the inverse ('if the product is this, what were the two brackets?')
  • Avoid teaching FOIL as a rule divorced from the distributive law

Activities to try

  1. 1

    Build the expansion on a 2×2 grid

    Why it works: A product of two binomials has FOUR products before collecting; the grid gives every product its own cell so none is forgotten, and it is the same idea as grid multiplication of two-digit numbers.

    1. Draw a 2×2 grid. Write the terms of the first bracket down the left of the rows (e.g. x and +3) and the terms of the second across the top of the columns (e.g. x and −2).
    2. Fill each of the four cells with the product of its row term and its column term, writing the sign explicitly: x², −2x, +3x, −6. Keep a running tally of how many of the four cells you have completed.
    3. Identify the two middle like-terms (−2x and +3x) and collect them (−2x + 3x = +x) to write the simplified answer x² + x − 6.
    4. Connect it to numbers: set out 12 × 13 as (10 + 2)(10 + 3) on the same grid and check the four cells add to 156 (100 + 30 + 20 + 6).
  2. 2

    Square a bracket without squaring each term

    Why it works: Squaring a bracket means multiplying it by itself, which still produces a middle term; (a + 3)² = a² + 6a + 9, not a² + 9, and substitution confirms which form is correct.

    1. Rewrite (a + 3)² as (a + 3)(a + 3) before doing anything else.
    2. Grid it: the four cells give a², +3a, +3a, +9. Notice the two identical middle cells.
    3. Collect the middle terms (3a + 3a = 6a) to get a² + 6a + 9.
    4. Check numerically with a = 2: (a + 3)² = 5² = 25, while a² + 9 = 13. They disagree, so a² + 9 is wrong; a² + 6a + 9 = 4 + 12 + 9 = 25 matches.
  3. 3

    Link expanding to factorising and check by substitution

    Why it works: Expanding and factorising are inverse processes; a substitution check catches errors, and some expansions (like (p+4)(p−4)) have middle terms that cancel rather than being absent.

    1. After expanding (x + 3)(x − 2) = x² + x − 6, substitute x = 1 into both forms and confirm they match (4 × (−1) = −4; 1 + 1 − 6 = −4).
    2. Run it backwards: given x² + x − 6, ask 'what two brackets multiply to give this?' and recover (x + 3)(x − 2).
    3. Try a difference-of-squares case (p + 4)(p − 4): expand to p² − 16 and notice the middle terms (+4p and −4p) cancel.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
The national curriculum in England: Key stages 1 and 2 framework document
KS3.Maths.Alg.4d

Math // Algebra

More in Algebra

All 25 topics