The Distributive Property

U01_L4_T2_we1 The Distributive Property

Rewrite the expression 4 times,
and then in parentheses

we have 8 plus 3, using the
distributive law of

multiplication over addition.

Then simplify the expression.

So let's just try to solve
this or evaluate this

expression, then we'll talk
a little bit about the

distributive law of
multiplication over addition,

usually just called the
distributive law.

So we have 4 times
8 plus 8 plus 3.

Now there's two ways to do it.

Normally, when you have
parentheses, your inclination

is, well, let me just evaluate
what's in the parentheses

first and then worry about
what's outside of the

parentheses, and we can do
that fairly easily here.

We can evaluate what
8 plus 3 is.

8 plus 3 is 11.

So if we do that-- let me do
that in this direction.

So if we do that, we get 4
times, and in parentheses we

have an 11.

8 plus 3 is 11, and then this
is going to be equal to--

well, 4 times 11 is just
44, so you can

evaluate it that way.

But they want us to use the
distributive law of

multiplication.

We did not use the distributive
law just now.

We just evaluated
the expression.

We used the parentheses first,
then multiplied by 4.

In the distributive law, we
multiply by 4 first. And it's

called the distributive law
because you distribute the 4,

and we're going to think
about what that means.

So in the distributive law, what
this will become, it'll

become 4 times 8 plus 4 times
3, and we're going to think

about why that is in a second.

So this is going to be equal to
4 times 8 plus 4 times 3.

A lot of people's first instinct
is just to multiply

the 4 times the 8, but no!

You have to distribute the 4.

You have to multiply it times
the 8 and times the 3.

This is right here.

This is the distributive
property in action right here.

Distributive property
in action.

And then when you evaluate it--
and I'm going to show you

in kind of a visual way
why this works.

But then when you evaluate it,
4 times 8-- I'll do this in a

different color-- 4 times 8 is
32, and then so we have 32

plus 4 times 3.

4 times 3 is 12 and 32 plus
12 is equal to 44.

That is also equal to 44, so
you can get it either way.

But when they want us to use
the distributive law, you'd

distribute the 4 first.
Now let's think

about why that happens.

Let's visualize just
what 8 plus 3 is.

Let me draw eight
of something.

So one, two, three,
four, five, six,

seven, eight, right?

And then we're going to add to
that three of something, of

maybe the same thing.

One, two, three.

So you can imagine this is what
we have inside of the

parentheses.

We have 8 circles
plus 3 circles.

Now, when we're multiplying this
whole thing, this whole

thing times 4, what
does that mean?

Well, that means we're just
going to add this to itself

four times.

Let me do that with
a copy and paste.

Copy and paste.

Let me copy and then
let me paste.

There you go.

That's two.

That's one, two, three, and then
we have four, and we're

going to add them
all together.

So this is literally what?

Four times, right?

Let me go back to the
drawing tool.

We have it one, two, three, four
times this expression,

which is 8 plus 3.

Now, what is this
thing over here?

If you were to count all of this
stuff, you would get 44.

But what is this thing
over here?

Well, that's 8 added to
itself four times.

You could imagine you're
adding all of these.

So what's 8 added to
itself four times?

That is 4 times 8.

So this is 4 times 8,
and what is this

over here in the orange?

We have one, two, three,
four times.

Well, each time we have three.

So it's 4 times this
right here.

This right here is 4 times 3.

So you see why the distributive
property works.

If you do 4 times 8 plus 3, you
have to multiply-- when

you, I guess you could imagine,
duplicate the thing

four times, both the 8 and the
3 is getting duplicated four

times or it's being added to
itself four times, and that's

why we distribute the 4.