Distributive Property Worksheets

Two directions

Expand gives you a(bx ± c) and wants abx ± ac. Factor gives you the expanded form and wants the bracket back. Mixed chooses per problem.

Sixteen questions by default, answer key optional.

Expanding is the easier half

Multiply everything inside the bracket by the thing outside it. Both terms, which is where the marks go: 3(2x + 5) is 6x + 15, and the pupil who writes 6x + 5 has multiplied one term and forgotten the other.

That error is so common it has its own arrow diagram in most textbooks, and the arrows are worth drawing until they are unnecessary.

Factoring is the direction that matters later

Harder, because you have to find the common factor rather than being handed it, and it is the one that unlocks quadratics, algebraic fractions and simplification generally.

The method: look at the numbers, find their highest common factor, take it outside, and write what is left inside. 12x + 18 has a common factor of 6, giving 6(2x + 3).

The near-miss is taking out a factor that is not the highest. 2(6x + 9) is a correct factorisation of the same expression and it is not fully factorised, and it will lose marks. Check whether the bracket has anything left in common — here 6 and 9 share a 3, so the job was unfinished.

Check by expanding

Every factorisation can be verified in about five seconds by multiplying it back out. If you do not get the original expression, something is wrong.

Teach that as automatic. It converts factoring from a skill with no feedback into one where a pupil knows immediately whether they are right, which is worth more than any amount of marking.

Why it is called distributive

Because the multiplication distributes across the addition, and the property is worth naming because it is the thing that makes algebra work at all.

It also explains mental arithmetic that pupils already do. 7 × 23 is hard; 7 × 20 plus 7 × 3 is easy, and it is the same property. Making that connection is a good way in for a class that finds the algebra abstract.

Level

Around ages eleven to fourteen for the basic form, revisited whenever quadratics arrive.

Related: factors and multiples, since factoring is a highest-common-factor problem wearing an algebra coat, and quadratic equations, which is where it pays off.