Key takeaway: The distributive property says a factor outside parentheses multiplies every term inside, so a(b + c) = ab + ac, which lets you split hard products into easy ones.
Grade 66.EE.A.3Expressions and Equations
Definition of Distributive Property
The distributive property connects multiplication with addition. When a number multiplies a sum, you can multiply it by each addend separately and then add the results, and the total is the same.
It works like splitting a rectangle into two smaller rectangles. A rectangle 7 units tall and 38 units wide has the same area as one 7 by 30 plus one 7 by 8. The property also works in reverse, which is called factoring out a common factor, and it holds for subtraction as well. Mental math uses it all the time: instead of a long column, you add two quick products, and in algebra it lets you remove parentheses from an expression.
Formula and notation
For any numbers \(a\), \(b\) and \(c\):
\[a(b + c) = ab + ac \qquad a(b – c) = ab – ac\]
Read left to right to expand, right to left to factor. With variables, \(6(2x + 5) = 12x + 30\).
Key facts
- The factor outside must multiply every term inside the parentheses, not only the first.
- Breaking one factor into tens and ones makes mental multiplication easier.
- Reading the property backward factors out a common factor, such as \(15x + 9 = 3(5x + 3)\).
- The area of a split rectangle is a picture of the property.
Distributive Property example
Use the distributive property to find \(7 \times 38\), then expand \(6(2x + 5)\).
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Split 38 as \(30 + 8\): \(7 \times 38 = 7(30 + 8)\).
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Distribute: \(7 \times 30 + 7 \times 8 = 210 + 56 = 266\).
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For the expression, multiply 6 by each term: \(6 \times 2x = 12x\) and \(6 \times 5 = 30\).
\(7 \times 38 = \) 266 and \(6(2x + 5) = \) \(12x + 30\).
Common mistake
Watch out
Rosa expands \(6(2x + 5)\) as \(12x + 5\).
Why it is wrong
She multiplied only the first term. The 6 must reach both terms inside the parentheses, so \(6 \times 5 = 30\) also appears: \(12x + 30\). Test with \(x = 1\): \(6(2 + 5) = 42\), and \(12 + 30 = 42\), while \(12 + 5 = 17\) fails.
Related terms and lessons
- Grade 6 · Greatest Common Factor and Least Common MultipleLessonWorksheetQuiz
- Grade 6 · Writing and Evaluating Algebraic ExpressionsLessonWorksheetQuiz
- Grade 7 · Equivalent ExpressionsLessonWorksheetQuiz
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