The Distributive Property: Expanding Brackets with the Area Model
Why is true, seen as the area of a rectangle; how to expand brackets with negatives without slipping; and how the same idea powers mental multiplication and factorising.
How would you work out in your head? Most people split into , multiply each part by , and add: . That everyday trick is the distributive property.
See it as area
A rectangle wide and long can be measured all at once, , or in two pieces, plus . The area is the same either way.
Splitting both sides by place value: .
This area model is the same picture used for long multiplication, and it carries straight over to algebra, where some of the side lengths are letters.
Expanding brackets
To expand a bracket, multiply the term outside by every term inside.
The 4 multiplies both terms in the bracket.
Negatives outside the bracket
A minus sign in front of a bracket multiplies every term inside by a negative, so every sign inside flips.
Expand and simplify .
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Distribute each bracket. The second is multiplied by .
, not . The negative multiplies both terms, and a negative times a negative is positive.
Two brackets
With two brackets, every term in the first multiplies every term in the second. The area model keeps track so nothing is missed.
Four products: each term on the left times each on the right.
. Write it as and you get : the middle term is easy to forget.
Running it backwards
Read right to left, is factorising: pulling a common factor out of every term. .
Check any expansion or factorisation by substituting a small number, like , into both forms. They must give the same value.
Common questions
- What is the distributive property in simple terms?
Multiplying a total is the same as multiplying each part and adding the results: . For example .
- Does the distributive property work with subtraction?
Yes. , because subtracting is the same as adding .
The The Distributive Property lesson has worked examples and unlimited practice that adapts to you.