Let's now think about different ways
to represent a mixed number.
And let's say that our mixed number is 2 and 1/8.
Actually, let's make it a little bit more interesting.
Let's make it 2 and 1/4.
So let's first think about the whole number part, the 2.
Well, the 2 is literally two wholes.
You could literally view that if you want.
Right here we've drawn each whole.
We've cut it up into sections of 8, so it literally is 8/8.
So let me just do it like this.
So the 2 is this whole region right over here, that's 1.
So this right over here is 1.
And then this right over here is 2, 2 wholes,
so let me paint that in.
So that is 2 wholes.
And then I have 1/4.
So this last piece, this last whole,
is divided into 8 sections.
So let me divide it into fourths first.
So that's one 1/4, 2/4, and 3/4.
So we want one of those four to be
filled in-- one of those four in orange.
So one of those four to be filled in, just like that.
You might notice that I filled in two of the eighths,
and that's because 1/4 and 2/8 is the same thing.
So there I've represented this mixed number, 2 and 1/4.
Let's see how we can decompose this.
So let's get our grids back.
So how else could we do it?
And I'm just going to throw a bunch of fractions up there
and see what I get.
So the first thing I'm going to throw out is 1/2.
So how would I represent 1/2 here?
Well, if I take one of these wholes
and I put it into two sections right over here,
1/2 would be this section right over there.
So let me color that in.
So we have 1/2.
So I'm first going to add 1/2, which is the same thing as 4/8.
And you see that I just filled in four out of the eight
sections, which is exactly half of this first whole.
So we're making some progress.
Now let's throw in 3/8.
So what would 3/8 look like?
Each of these boxes are literally an 1/8
and I could fill it in however I want, but let
me just put this as 1, 2, and 3.
And then let's fill in plus another 8/8.
Now, what's 8/8?
Well, 8/8 is a whole, and I'll do that over here.
I still haven't filled this one in yet,
but I'll fill in this one right over here.
So let's do that.
So 8/8-- so that's 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8,
and 8/8, and it's a whole.
So I have a whole whole here, so that's 8/8.
I want to make this one a whole, because I want to get to 2,
so let me put in a 1/8 there.
So plus 1/8, well, that's going to be this one right over here,
so that's my 1/8.
And then let's add another 2/8, plus another 2/8.
Well, this is in eighths right over here,
so 2/8 is going to be two of these.
And notice, you see that the 2/8 is the same thing as 1/4.
If you took this 1/4 and split it into two,
so you have two times as many pieces, it becomes 2/8.
And you see that if 1 times 2 is 2, 4 times 2 is 8.
So that 1/4 is the same thing as 2/8.
You see the 8/8 is the same thing as a whole.
Now, you see, you could make another whole out of 1/2,
plus 3/8, plus 1/8, and they add up to a whole.
And just to make sense of why that worked,
1/2 is the same thing as 4/8-- because you
see that, we filled in the 4/8-- then you have 3/8,
and then you have 1/8.
And if you add all of these together, 4/8 plus 3/8 plus
1/8, you are going to get, in terms
of eighths, 4/8 plus 3/8 plus 1/8 is going to be 8/8.
4 plus 3 plus 1 is 8, so you get 8/8 which is this entire whole.
So hopefully that helps give you a visual understanding
of what we're doing when we're adding and decomposing
these fractions a little bit more.