How it works
A student can recite a(b + c) = ab + ac and still not see why it is true. This distributive property graphic organizer sets the letters beside a number example and an area model, so the picture explains the rule.
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Why the area model works
A rectangle 3 high and 9 wide has an area of 27. Split the 9 into 4 and 5 and it becomes two rectangles with areas 12 and 15, which add to 27. The whole is 3 × (4 + 5); the two parts are 3 × 4 + 3 × 5.
The same rule runs backward
Expanding multiplies each term inside the parentheses by the number outside. Factoring undoes it: find the greatest common factor of the terms, write it outside, and divide each term by it to see what stays inside.
- A negative outside the parentheses changes the sign of every term inside: −2(4 − y) is −8 + 2y.
- To check an answer, put the same number in place of the letter in both forms. Different results mean a slip.
Questions and answers
What is the distributive property?
Multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. With letters, a(b + c) = ab + ac. It holds for subtraction too: a(b − c) = ab − ac.
What is an example of the distributive property?
3 × (4 + 5) = 3 × 4 + 3 × 5. The left side is 3 × 9, which is 27. The right side is 12 + 15, which is also 27. With a variable, 3(x + 4) = 3x + 12.
What grade is the distributive property taught?
The Common Core standards introduce it in grade 3 as a way to multiply: 8 × 7 as 8 × 5 plus 8 × 2. Students apply it to expressions with variables in grade 6 and to negative and fractional coefficients in grade 7.