Decomposing a fraction visually

So let's think about all of the different ways that we can represent 7/9. So let's just visualize 7/9. So here I have 9 equal sections. And 7/9 you could represent as 7 of those equal sections. So let me get myself a bigger thing to draw with, so that I can fill this in fast. Actually, I don't like how that looks. I'm going to use the paint brush. So here we go. So that's 1, 2, 3, 4-- you know where this is going-- 5, 6, and 7. So that's one way of representing 7/9. We already know that. That's not too interesting. But let's see if we can represent 7/9 as the sum of other fractions. So let's imagine maybe we can represent it as-- let's do it as 2/9. Let me use a different brush here. So let's represent it as 2/9. 2/9 plus-- I don't know, let's see, maybe 3/9. But that doesn't quite get us to 7/9 yet. 2/9 plus 3/9 is going to get us to 5/9. So we're going to need 2 more. So it's going to be plus another 2/9. So what would this look like? So let's just draw another grid here. So this is going to look like-- and I'll try to do it right below it, so that we can see how they match up. So we have 2/9, this 2/9 right over here. Well, we have each of these as a ninth. We have 9 equal sections. So we're going to get 1 and 2. And then we're going to add 3 more ninths. So 1, 2, 3. So we add 3/9 right over there and then 2 more ninths, 1 and 2. So notice, when I added 2/9 to 3/9 to 2/9, this equals 7/9. And we know that when we add a bunch of fractions like this that have the same denominator, we can just add the numerator. And this is why. This is 2/9 plus 3/9 times 2/9 is going to give me 7/9. Let's do this one more time. This is actually a lot of fun. So let me draw my grid again. And then let's see what we can do. So let me get my pen tool out. Let me make sure my ink isn't too thick. Well, this is fine. And let's add a couple of ninths here. So let's add first 1/9. And I'm going to draw out all the 9's in blue. And let's add 2/9. And then we could add-- I don't know, maybe we could add-- let me give some space here so we can add more. And maybe we could add 3/9. And then we could add, let's see-- actually, let me just write this out. I'm going to try to add four fractions here. So let's say add first 1/9 and see where that gets us. So 1/9 is going to get us right over here. So that's 1/9. So let's say we add 2/9 to that. I've got my little paint brush going on. So that's 1 and 2 more, 2/9. So that still doesn't get us there. This gives us a total of 3/9. 1 plus 2 is 3-- 3/9. So let's add 4/9. And I'll do that in this blue color. So 4/9. That's different enough. So let's see where this gets us. Actually, well, why not? So 4/9. And so that's going to get us 1, 2, 3, 4. So that looks like it got us all the way, because 1 plus 2 plus 4 is going to give us 7-- 7/9. So what could we put here? Well, we could say 0/9. Why not? So we could call this one right over here 0/9. And how would we visualize that? Well, we're saying none of these. No ninths right over here. So this is 1/9 plus 2/9 plus 4/9 is equal to 7/9. So these are all different ways to decompose the exact same fraction.