Why is the Distributive Law true?

Remember back in sixth grade or so, when the teacher said, "Okay, here's the deal. a(b + c) = ab + ac." Maybe not. However, this is called the Distributive Law, or the Distributive Property. My class had to do a whole project on it, so it really got jammed into our heads. However, in the midst of this big project, I always was thinking in the back of my head "Why is this true?" And since the proof is so incredibly simple, I had to share it.

First off, let me describe the Distributive Property in further detail. I've used it in the past without thoroughly going over it, so that must get done now. Take this problem:

8(5 + 3) =

We could do it this way:

8(5 + 3)
8(8)
64

Or, we could use the distributive property. This means we distribute the 8 to the 5 and the 3 to get the answer.

8(5 + 3)
8(5) + 8(3)
40 + 24
64

This doesn't seem useful right now, but let's try it with variables and you'll see. Try simplifying the following.

3(x + 4)

We can't combine the x and 4 because they are not "like terms." We can't add apples and oranges, so we can't add x and 4. However, we can use the distributive property.

3(x + 4)
3(x) + 3(4)
3x + 12

And there you go. You may be wondering what I was at that moment; why does it work? I'll bet you can prove it algebraically, geometrically, and every other way, but we're going to prove this logically. Let's take a story problem:

I have eight bags of fruit, each with five apples and three pears inside. How much fruit do I have in all?

One way we could do it is figure out how much fruit is in one bag. To do it, we would add the five apples to the three pears to get eight pieces of fruit. By multiplying by the eight bags, we get sixty-four.

We could also total up the number of apples and the number of pears. Eight bags times five apples is forty apples, and three pears times the eight bags is twenty-four pears. 40 + 24 = 64, which is the same answer as before.

If you think about it, this second method is the exact same thing as the distributive property; we multiplied each term in the parentheses by the number of bags, which is known as the "factor," then added those two totals together. It makes perfect sense, it just takes that little story problem to understand it. This property is so important in Algebra, from FOIL to combining like terms, from radicals to point-slope form. It is endless what you can do with the Distributive Property. Sometime leading up to the problem of the weeks, I will post about something else you can use this property for, which will most likely come in handy in the problems!

For the majority of February, we will be putting together mathematics and technology, and using a calculator to graph equations, scatter plots, and even play, what my friends call "games" on a calculator. Except they throw coins and dice, not live birds that have explosives inside of them...