Distributive Property

Background

We've found that addition and multiplication are both commutative (a + b = b + a and the same for multiplication) and associative (a + (b + c) = (a + b) + c and the same for multiplication). Are there any more properties we can explore? Yup.

Question

Does 2 x (4 + 3) = 8 + 6?

Answer

Yes - they both equal 14. We can show that from the Distributive Property.

Analysis

A form of multiplication that happens commonly is to have a number, like the 2, multiplying across the terms in a bracket, just like we're seeing with the question 2 x (4 + 3).

We can work our question using our usual procedure of the Order of Operations, and when we do that, we do the bracket first:

2 x (4 + 3) = 2 x 7 = 14

Awesome! But can we work this a different way? And the answer is yes - if we look at the Factors and Factoring entry, where we "factored" out the 2 in order to get from 8 + 6 to 2 x (4 + 3), we can also go the other way:

2 x (4 + 3) = 8 + 6 = 14

This ability to perform the multiplication first is called the Distributive Property and it can be expressed this way:

a x (b + c) = a x b + a x c

Ok - so how does this work?

When we did 2 x (4 + 3) = 2 x 7 = 14, we were looking for the number of times we'd move 2 to the right on a number line:



When we did it the other way, 2 x (4 + 3) = 8 + 6 = 14, what we're in effect doing is making two smaller jumps to have it add up to the big jump:



There are other ways we can use this property and we'll explore those in future entries.

Vocabulary used:

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Operations:

• Multiplication
• Order of Operations (PEMDAS)

Associated Operations:

• Factors and Factoring

Properties:

• Associative Property

Where might we go?

FOIL