Distributive property worksheets

One idea, three directions

The distributive property says that a number multiplied by a sum can be multiplied into each part of the sum: a(b + c) = ab + ac. Children usually meet it first as an arithmetic shortcut, then use it to expand brackets with letters, and finally run it backward to factor. This generator covers all three, so one page can follow a student from sixth-grade numbers into seventh-grade algebra.

The sheet is aimed at grades 6 and 7. It helps to have worked through evaluating expressions first, because the same bracket habits apply. It leads straight into multi-step equations, where distributing is the first move.

Choosing a format

The Distributive form menu decides what each line looks like:

• With numbers: 6(4 + 3). Each problem is printed as a(b + c) = a × b + a × c = and the student finishes the arithmetic. The middle step is given, so this form teaches the pattern. The multiplier runs from 2 to 9 and the numbers inside from 1 to 12. This version uses one wide column.
• With variables: 3(x + 5). The student writes the expanded expression, such as 3x + 15. The letter varies between x, y, n and a. This is the best starting point for most students beginning algebra.
• Factor it back out. The line shows an expression like 12y + 18 and the student fills in the greatest common factor outside the brackets, with the bracket contents already printed. The generator makes sure the two terms share only that one factor, so the answer is unambiguous.

About three problems in ten use subtraction inside the bracket instead of addition. That is deliberate: a minus sign is where most expansion errors happen. In the numeric form a subtraction may even produce a negative total, for example 3(2 − 9) = −21.

Set Skill to "Distributive property" to use these options. The Level menu only affects the order of operations sheet. A sensible sequence is numbers, then variables with a mix of plus and minus, then factoring, spending a few days on each.

Worked example: expand 8(y − 3), then factor 12y + 18

Expanding.

1. Multiply the outside number by the first term: 8 × y = 8y.
2. Multiply it by the second term: 8 × 3 = 24.
3. Keep the sign that was inside the bracket: 8y − 24.

Factoring 12y + 18.

1. List what both terms divide by. 12 and 18 are both divisible by 2, 3 and 6.
2. Take the greatest, which is 6.
3. Divide each term by 6: 12y ÷ 6 = 2y and 18 ÷ 6 = 3.
4. Write 6(2y + 3).

Check by expanding again: 6 × 2y = 12y and 6 × 3 = 18, so the original comes back. Teach this check early. It turns every factoring line into a self-marking exercise before the key comes out.

What to look for when marking

• Only the first term multiplied. Writing 8y − 3 for 8(y − 3). The key's second term will be much larger than the student's.
• Sign dropped. Writing 8y + 24 for a subtraction problem.
• Partial factoring. In 12y + 18, writing 3 instead of 6. The brackets would then still hold terms that share a factor, which is a prompt to look again.
• Wrong operation when factoring. Subtracting the factor instead of dividing by it.

For a classroom warm-up, ask students to expand a numeric line both ways: compute 7(10 + 2) as 7 × 12, then as 7 × 10 + 7 × 2. Getting 84 twice is a quiet proof that the property works.

Sources

1. Common Core State Standards: Grade 6 Expressions and Equations (6.EE.A.3, apply the properties of operations to generate equivalent expressions) thecorestandards.org
2. Common Core State Standards: Grade 7 Expressions and Equations (7.EE.A.1, add, subtract, factor and expand linear expressions) thecorestandards.org