Fractions and whole numbers
Let's say that this circle right over here represents one whole. And we've divided this circle into one, two, three, four, five equal sections. So each of these sections represents 1/5 of the circle. And we've seen this already. 1/5, 1/5, 1/5, and 1/5. And if we were to then color in some of this-- So, let's say we were to color in three of these sections. So, that's one of the sections right over there, another one right over here. So, we've colored in two of the fifths and then three of the fifths. Notice, you put those three 1/5's together, how much have we now shaded in? We've shaded in 3/5 of the whole. So the fraction that's actually shaded in now is 3/5. 3/5 is what's shaded in. Now let's do something in some ways a little bit simpler but also in some ways kind of interesting. Let's start with the whole again. So, once again, this is one whole. Let me label it. One whole. And instead of dividing it into five equal sections, I'm just going to divide it into one equal section. So, if I were to shade this in just like that-- so I'm shading in my one whole, my one equal section, how many of the equal sections are now shaded in? Well, just to remind ourselves, there is one equal section, and I have shaded in exactly one of those one equal section. I've shaded in the whole thing. Or I could say that 1/1-- which you'll never hear someone actually say-- is shaded in, or I could say that the whole thing is shaded in. So this is equal to one whole. So that's a whole. That's interesting. And I want you to keep in mind, remember, look. We literally have one, two, three fifths, and we literally call that 3/5. Now, this is one whole. Now, what happens if we were to do this multiple times? Let me copy and paste that. So now I have another one whole and then another one whole right over here. So now, in total, how many wholes do I have? Well, I have three. One, two, three wholes. And I've actually shaded in three holes. So this one right over here is equal to-- Let me make sure I label it right. This right over here is equal to, if I were to take the combination, this is equal to 3. 3 holes. Or if I were to think of it in terms of numbers, just a number line, literally, this would represent the number 3. But what's another way I could represent it? Notice when I took a 1/5, another 1/5, and another 1/5, I could call that 3/5. So now if I take one 1/1, another 1/1, and another 1/1, well, I should be able to call this 3/1, or 3 firsts, or however you want to call it. So I could call this 3 firsts. So this is interesting. Now we're seeing where the top number on a fraction is larger than the bottom one. But another way of thinking about this fraction symbol is that it's division. So you could view this as 3 divided by 1 is equal to 3. Or you could say, well, look, 1 over 1 is a whole, and I now have three of them, so this is equal to 3 wholes. So 3/1 is the same thing as the number 3. And let me make sure, let me emphasize that. Let me draw this on a number line. So once again, let me go all the way to 3. So 0, 1, 2, and 3. So one whole gets us exactly one jump on the number line. So this right over here, that gets us to 1/1. We do another jump, now we've gotten 2 firsts, I guess you could say. We've essentially taken two of these jumps. Each jump is 1/1. Now we are at 2/1, which is the same thing as 2. You take another jump, and we essentially get to, or we do get to 3/1, which is the exact same thing as 3.