Introduction to the Distributive Property

>> Professor Perez: Hey!
This is Professor Perez again.
Today we're going to look an introduction to the distributive property.
Of course, we've got to get Charlie out here.
He better be ready!
Hey, Charlie, you ready to go?
>> Charlie: Ya...
>> Professor Perez: All right, here we go.
Right here.
Introduction to the distributive property.
So, what the distributive property is, is what we're going to do,
is we're going to distribute a number across an addition or a subtraction using multiplication.
Let me give you an example here.
2 times 5 plus 3.
Notice the 5 plus 3 is in parenthesis.
Before we get to the distributive property,
what we could do is add the two numbers in the parenthesis.
You'll see when we get to the Order of Operations,
that that is what we're supposed to do first.
Okay, that's coming up soon.
So watch. We take what's in the parenthesis, 5 plus 3.
What's 5 plus 3, Charlie?
>> Charlie: 8.
>> Professor Perez: 8.
Don't forget we have a 2.
Now what's the operation, Charlie?
It's 2 outside of a parenthesis.
There is no operation written down, so it's assumed to be what?
>> Charlie: Multiplication.
>> Professor Perez: Multiplication.
And so, 2 times 8 is what, Charlie?
>> Charlie: 16.
>> Professor Perez: 16, same as 8 times 2.
Now, same problem, but now we're going to demonstrate the distributive property.
Notice there's a 2 outside the parenthesis
that 2 outside the parenthesis means you've got to multiply.
But, you don't have to add what's in the parenthesis first, you can add 5 plus 3,
but what we're going to do is we're going to distribute the 2
to the 5 and to the 3 by multiplication.
Watch, Charlie.
We first start with 2 times 5.
So we'll bring that down.
We did 2 times 5.
And now, in the parenthesis our operation was addition, so we'll bring that down
and now we take the 2 and multiply to the 3 and that's 2 times 3.
Notice the 2 was distributed to each the 5 and to the 3 and operation is multiplication.
And now, what's 2 times 5, Charlie?
>> Charlie: 10.
>> Professor Perez: 10, same as 5 times 2.
And what's 2 times 3?
>> Charlie: 6.
>> Professor Perez: 6, same as 3 times 2.
And anyway, what's 10 plus 6, Charlie?
>> Charlie: 16.
>> Professor Perez: 16 and notice the two answers are exactly the same.
16. Now, of course, most of you are probably saying, "Well,
I'm just going to do the parenthesis first."
Yeah, but you can't do that all the time, you'll see.
Because I know you're asking, "When am I ever going to use the distributive property?"
Well, you'll see very soon.
Now, same problem, Charlie.
2 times 5 plus 3.
Now somebody mentioned this technique to me so I'm going to go ahead and show it.
2 times 5 plus 3 basically means you have two 5 plus 3's a being added together,
which is true, okay?
And 5 plus 3 plus 5 plus 3, you can add in any order you want.
Remember adding numbers?
You can add in any order you want and we can just reorder it
as 5 plus 5 plus 3 plus 3 which is 10 plus 6.
Which is again, 16.
Another way of looking at what's 2 times 5 plus 3 in parenthesis, which is fine, okay?
Now, here's the example we haven't discussed variables yet in this class,
but we're going to demonstrate it right now with this distributive property.
Now, you cannot add x plus 3 in the parenthesis.
Some people think that x plus 3 is 3x.
No it's not 3x.
It is 3x if you want to repeat this class, Charlie.
>> Charlie: What?
Huh?
>> Professor Perez: Okay, x plus 3, you cannot add those two together.
3x actually means 3 times x, we'll get to those later.
But, x plus 3 you can't add.
So, are you stuck?
No. You can apply the distributive property.
You can distribute the 2 to both the x and to the 3.
So, 2 times x is written 2x.
2x means 2 times x. And our operation is addition
in the parenthesis so we'll bring that down.
And we'll take 2 times 3 which is what, Charlie?
>> Charlie: 6.
>> Professor Perez: 6, okay.
You cannon add 2x plus 6 and that is your answer.
So, you took 2 times the parenthesis x plus 3,
and applied the distributive property and then end up with 2x plus 6.
So, we'll be dealing with that a little later in the semester and you'll be dealing
with that a lot in Beginning Algebra which is the next class.
Okay, now, let's look at a subtraction in the parenthesis, Charlie.
Let's do the parenthesis first.
What's 7 subtract 3?
>> Charlie: 4.
>> Professor Perez: 4 and you're multiplying by 2 which does give you 8 okay?
Now let's apply the distributive property.
What do you, Charlie?
Distribute 2 times 7 is...okay...and your operation is subtraction so bring that down.
And then what?
>> Charlie: 2 times 3.
>> Professor Perez: Okay, 2 times 3, that's right.
All right, now, we did 2 times 7 is what, Charlie?
>> Charlie: 14.
>> Professor Perez: Okay, bring down your subtraction, 2 times 3 is?
>> Charlie: 6.
>> Professor Perez: 6, and what's 14 subtract 6?
>> Charlie: 8.
>> Professor Perez: 8, same answer, okay.
Now we'll go to this next one.
Here we're going to distributive across a subtraction and an addition.
Same process.
We take 2 times 7, bring down your operation which is a subtraction,
and then we have 2 times 3 which is...going to be 6.
Bring down our addition and then we have 2 times 2, right?
Okay, so let's do 2 times 7 is what, Charlie?
>> Charlie: 14.
>> Professor Perez: Subtract 2 times 3
>> Charlie: 6.
>> Professor Perez: Add 2 times 2.
>> Charlie: 4.
>> Professor Perez: Very nice, okay.
Now remember, you've got to work left to right.
14 subtract 6 is what, Charlie?
>> Charlie: 8?
>> Professor Perez: 8.
We've still got to add the 4 and what do we get?
>> Charlie: 12.
>> Professor Perez: Very nice there, Charlie, yes.
Now, the problem could have been...uh...the answer could have been gotten
by first doing the operations in the parenthesis.
7 subtract 3 is 4, and 4 plus 2 is 6, and 2 times 6 is 12.
Yes, that would be faster, but we're trying
to demonstrate the distributive property, how it can be used.
Okay, well, let's do 2 times 43.
Well, let's think about how do we do this.
Well, first, let's break the 43 in the expanded form, okay?
2 times 40 plus 3 and if we distribute the 2 through, what's 2 times 40, Charlie?
>> Charlie: 80.
>> Professor Perez: 80.
And we have an addition, and 2 times 3 is?
>> Charlie: 6.
>> Professor Perez: 6 and that gives you 86.
So this is another way of looking at 2 times 43.
Well, a lot of us tend to want to use this vertical format.
What you are soon going to see is this vertical format, the reason it works,
is because you're using the distributive property.
It's exactly what you're doing, watch.
43 times 2.
The first thing you are taught to do is do what, Charlie?
>> Charlie: 2 times 3.
>> Professor Perez: 2 times 3 which is 6, yes.
See? It's the distributive property.
2 multiplied by 3.
And then you take the what?
>> Charlie: 2 times 4.
>> Professor Perez: Go diagonally and go 2 times 4, which is 8 and you bring it down.
Notice you put the 8 in the tens place because you have 8 tens,
which is actually 80 and so 86 is just 80 plus 6.
That's your vertical format.
Watch, let's do a more complicated one.
Let's do 6 times 134.
Don't get scared!
What we're going to do is write 134 in expanded form.
In expanded form Charlie, what's 134?
>> Charlie: 100 plus...
>> Professor Perez: That's right.
100 plus 30 plus 4, okay, now, apply the distributive property.
What do we get, Charlie?
What do we do first?
>> Charlie: 6 times 100.
>> Professor Perez: 6 times 100 is 600.
Bring down our operation.
What's next?
>> Charlie: 6 times 30.
>> Professor Perez: 6 times 30 which is what, Charlie?
>> Charlie: 180.
>> Professor Perez: 180, because 6 times 3 is 18 and 6 times 30 is 180.
Okay, Charlie, 6 times 4 is what?
>> Charlie: 24.
>> Professor Perez: 24, okay.
So we work left to right, 600 plus 180 is 780 plus 24 which is 804.
that is the answer.
Now, Let's do the vertical format.
Now, we're going to a vertical format without carrying over.
You'll see what I mean.
Watch, we first start with 6 times 4 which is what, Charlie?
>> Charlie: 24.
>> Professor Perez: 24, that's the distributive property.
And then we do 6 times 3 which is 18.
Now notice, the 3 was in the tens place, so the 18, the 8 is written in the tens place
and we bring down a place holder, 180.
And then we go 6 times 1 which is 6.
It's in the hundreds place, we have 6 hundreds and we put in our place holders there
and you add them all together and the first column is what, Charlie?
>> Charlie: 4.
>> Professor Perez: 4.
Okay, and then we have the 2 and 8 is 10.
Put your 0, carry your 1, and 1 plus 1 plus 6 is 8.
It's 804. That's without the carry over.
Well, let's finish this lecture by doing the carry over here.
Okay, here we go Charlie, pay attention.
6 times 4 is 24.
Put the 4, carry the 2.
6 times 3 is 18, right?
Add the 2, is 20, put your 0 down, carry the 2, and 6 times 1 is 6 plus 2 is 804.
So there you go.
That's the vertical format.
But these vertical formats are using the distributive property, so there you go.
That's your introduction to the distributive property.
So, we're going to work on multiplication more, in the future, anyway,
keep up with your homework and we'll see you all again soon.