Fractions and whole numbers
So let's say that this circle-looking thing represents a whole. We've already seen a couple of situations. We said, hey, look, we could divide this into two equal pieces. And if we shade in one of them-- so one of these equal pieces-- that would be 1/2. And then if we shade in two of them, this would be 2/2. So let me color that in a little bit better. So this right over here, I've divided into two equal pieces. And I'm shading in two of these equal pieces. So what fraction of the whole do I have shaded in? Well, we've seen this multiple times. This is 2/2 of the entire figure. And we see we've shaded in the entire figure. So this is equal to 1 whole. And we could do that. We don't have to just split it into two equal parts. We could split it into three equal parts. So let's say we were to split it into three equal parts-- let me do that, my best attempt to draw three equal parts. Three equal parts looks like a Mercedes symbol. And that's my best. I could draw a little bit better job of that. Let me be clear that I'm trying to make them equal. So that's three equal parts. And then if we were to actually shade in the three equal parts-- so that would be one of the three, so that's 1/3, 2/3, and 3/3. So once again, 3/3 is equal to 1 whole. Now, what if we were to do something, on some level, even simpler? What if we just take our whole and we divide it into only one equal section? Well, I've already done that. This is one equal section right over here. And then I were to select that one equal section-- so let me color it in. So I have one section, and I'm going to shade it in. So what fraction of the whole do I now have shaded in? Well, I had one equal section to begin with. And I shaded in that one equal section. So I have 1/1 of this shaded in. And this is also clearly an entire whole. So this is also equal to a whole. And I think you see a pattern here. 2/2, 3/3, or 1/1-- these all represent the exact same value. They all represent a whole. And you would also see this if you were to draw a number line. So this is 0. This is 1. We could keep on going. Well, this is 1/1. If I were to say, well, between 0 and 1, I just have 1, I divide it into one equal chunk, well, that's just this whole thing right over here. And if I were to move one of those equal chunks, I would get to 1. If I divide it into two equal chunks and if I make two jumps-- one, two jumps-- I still end up at 1. If I divide it into three equal sections-- so let's say one, two, three equal sections-- and I make three jumps-- one, two, three-- I end up at 1 again. So 2/2, 3/3, 1/1, or 1 over 1-- these all are different ways of representing the number 1, or 1 whole.