Decomposing a mixed number

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 holes. You could literally view that if you want. Right here we've drawn each hole. 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 holes, so let me paint that in. So that is 2 holes. And then I have 1/4. So this last piece, this last hole, 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 holes 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 hole. 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 hole 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.