Visualizing equivalent fractions

Let's think about what fraction of this grid is actually shaded in pink. So the first thing we want to think about is how many equal sections do we have here? Well, this is a 1, 2, 3, 4, 5 by 1, 2, 3 grid. So there's 15 sections here. You could also count it-- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. So there are 15 equal sections here. And how many of those equal sections are actually shaded in this kind of pinkish color? Well, We have 1, 2, 3, 4, 5, 6. So it's 6/15 is shaded in. But I want to simplify this more. I have a feeling that there's some equivalent fractions that represent the exact same thing as 6/15. And to get a sense of that, let me redraw this a little bit, where I still shade in six of these rectangles, but I'll shade them a little bit in one chunk. So let me throw in another grid right over here, and let me attempt to shade in the rectangles as fast as possible. So that is 1-- 1 rectangle. I'll even make my thing even bigger. All right, 1 rectangle, 2 rectangles, 3 rectangles-- halfway there-- 4 rectangles, 5 rectangles shaded in and now 6 rectangles shaded in. So this right over here, what I just did, this is still 6 rectangles of the 15 rectangles are shaded in. So this is still 6/15. These are representing the same thing. But how can I simplify this even more? Well, when you look at it numerically, you see that both 6 and 15 are divisible by 3. In fact, their greatest common factor is 3. So what happens if we divide the numerator and denominator by 3? If we do the same thing to the numerator and the denominator, we're not going to be changing the value of the fraction. So let's divide the numerator by 3 and divide the denominator by 3. And what do we get? We get 2 over 5. Now how does this make sense in the context of this diagram right here? Well, we started off with 6 shaded in. You divide by 3, you have 2 shaded in. So you're essentially saying, hey, let's group these into sections of 3. So let's say that this right over here is one section of 3. This is one section of 3 right over here. So that's one section of 3. And then this is another section of 3 right over here. And so you have two sections of 3. And actually let me color it in a little bit better. So you have two sections of 3. And if you were to combine them, it looks just like this. Notice this is covering the exact same area as your 6 smaller ones did. And then how many equal sections of this size do you have on this entire thing? Well, you have 5 equal sections. Because this is one section of 3 right over here, this is another section of 3. And then this is another section of 3. So notice, you're covering the exact same area of the original thing. You're covering 2 out of the 5 equal sections. So 2/5 and 6/15 are equivalent fractions. So if you want to say what fraction of this is covered in the simplest form, you would say 2/5.