 So, this is sort of a video to help you visualize what it means, two break lines into segments and what fractions are. And fractions are basically parts of a whole, right, that we talked about in the previous video. So, we're in series four of the language of mathematics and we've been talking about fractions, ratios, units, how to do unit conversion stuff, right? And one thing that we did, we talked about the difference between ratios and fractions. And what I want to do is sort of go a little deeper into the fractions and just really sedify the concept of fractions because a lot of people have a hard time dealing with fractions, which is basically what we've talked about in the past is, you know, the definition of what rational numbers are. So if you're having a hard time, if you don't know how to deal with fractions, how to visualize fractions, that means you don't know how to deal with rational numbers, which is a serious problem because most of the numbers you're going to encounter, many of the numbers you're going to encounter in real life, are rational. They're fractions. If you can't deal with rational numbers, then what that means is you're stuck in the integer realm, right? Which means basically you're stuck in positive and negative whole numbers, which is a very, very small world, right? So what we're going to do right now is just visualize what fractions mean. And the way we're going to do it is we're going to break a line into pieces and we're going to break it into even pieces and we're going to break it into odd pieces. And this will be extremely useful, this visualization when you're dealing with the coordinate system, right? Cartesian coordinate system because, you know, you have an x-axis and you have a y-axis. And all it is, is those axes are rational numbers or the real number set. You can have irrational numbers too. But one thing you do have to do when you're putting points on a Cartesian coordinate system is visualize where that point is going to land, right? So you have to have at least a rudimentary understanding of how to visualize where fractions end up, okay? So what we're going to do right now is break a line into even pieces. So let's assume we want to break this, you know, stand. It's where the park. This is what we call a concession stand window, right? So let's say we want to break this concession stand window into eight different pieces, right? Let's say it was big enough to have eight different lines in. You want to have eight even pieces, segments of this concession window, right? So what we're going to do is just we're going to go from this end of the window to this end of the window, right? And let's say we want to break it into eight different pieces, right? So that's us visualizing what one eighth is going to look like or three eighths, multiples of an eighths, right? What they're going to look like on a number line. So let's draw our line. And what we're going to do, we're just going to go straight down from here. So let's say we want to break this line into even number of pieces. And the even number that we picked was eight. So we're going to break this line into eight different pieces. The way you do this is if it's an even number, right? So if we have one eighth, we want to break this thing into an eighth of a piece, right? If it's an even number in the denominator, the way we do it is we look at the line and we cut it down the middle, okay? So we take a look at the line and cut it exactly down the middle. If we're dividing it into even number of pieces, right? So one, two, three, four. So I'm going to cut this thing right here. So we just cut this thing into even pieces, right? So we want it eight from here all the way to here to the end. And what we did was we cut it straight down the middle because we wanted to break it into even pieces. And what happens is now what we have to do is put four pieces on this side of the line and put four pieces on this side of the line because we want it to be even, right? We want to break it into even number of pieces. So we took one eighth, right? We took a line, we cut it into a half, right? Now what we have to do is take each half and cut it into four pieces. And four again is an even number, right? So what we do, we go to this half and again we cut it in the middle and we do the same thing on this half. We go to this half and we cut it down the middle because we want four pieces on either side, right? So we're going to go on this one and we're going to cut it in half and we're going to go to this one and cut it in half. So what we have right now, we took one line initially. On the first cut, we cut it into a half, right? We cut it into two pieces. Then we took each piece and cut those into half. So what we have right now, we got four pieces going across, four hopefully even pieces going across, right? So we took one piece, right? And we cut it in half. That means we have two even pieces from this one piece now. And then we took each half, right? And cut those into half. That means we have four pieces going across, four even pieces going across. So we broke this window down to four even pieces, right? We wanted to break it down to eighth, right? Now we took this, cut it into half, cut it into quarters. Now what we need to do is we need to have two pieces in each segment now, right? Well, two again is even. So what we're going to do, we're going to go to each piece and cut that in half, right? That way we have one, two pieces in this half or in this segment, two pieces in this segment, two pieces in every segment. And what we're going to end up with, we're going to end up with eight even pieces. So what we did was break this line into eighths, right? So we're going to go to here, cut it in half. Go to here, do the same thing, cut it in half. Go to this, cut it in half. Go to this and cut it in half. So what we have right now is a line, right? That's been cut into eight different pieces, right? One, two, three, four, five, six, seven, eight, right? Now the way this works is if we have a coordinate system, if we have a number that we want to put on here, if this happened to be zero, right? Then three eighths would be one, two, three eighths. We'd be right here, right? Because we took, we went down three of the eight pieces, right? Three parts of the whole, right? If you had seven eighths, you'd go one, two, three, four, five, six, seven eighths, right? If you had a number that was, you know, beyond between zero and one, let's say we had a number like five and three eighths, right? Then all would happen is this guy would be your five, right? This was your zero, now it's your five. And five and three eighths would be five and one, two, three eighths, right? Five and seven eighths, you'd be here. Five and five eighths, one, two, three, four, oh, five over here, right? So five and five eighths would be here. This would be five and a half, right? If it's even, if you can reduce your fraction, you end up reducing your fraction. And if you're going in the other direction, in the x-axis, if you're going towards the negative, right? All that would happen is this would be your zero, if you're starting from zero and you're going towards the negative, that would be your negative one. So negative three eighths, you would go from this end, right? Zero, one, two, three. That would be negative three eighths. Negative seven eighths, negative one. Negative five eighths, right? If you were going beyond zero and negative one, right? If you were going from negative five to negative six, you do the same thing. This side would be your negative five, that would be your negative six. And if you wanted negative five and three eighths, you would go negative five, one, two, three. That's three parts of an eight piece line, right? So this would be negative five and three eighths. Negative five and five eighths, right? Negative five and a half because this would be negative four, negative five and four eighths. Four over eight is a half. So if you can reduce it, you do reduce it, right? So if you had, for example, if you go back from zero to one, if you had, you wanted to go down to go to two eighths, two eighths is just a quarter, right? So that would mean this is your two eighths, right? One, two. But this is also into one, two, three, four even pieces, right? So this would be one out of four even pieces. I hope that's clear. That's just basically dealing with your fraction, knowing how to reduce fractions. And that's how you cut a line into even pieces. Let's go take a look at how we cut a line into odd pieces, okay? So let's say what we wanted to do is break this next concession stand into 11 even pieces, right? What we'll do is we'll draw the line again and have it going from one end to the other end. But the process of breaking it into odd pieces is a little different than breaking into even pieces. So let's say we wanna break this into 11 different pieces, right? We're gonna have 11th happening along the line. What we end up doing is we look at the line and we find the middle, but we don't cut it in the middle. We offset it a little bit, okay, on either side. Because it's an odd break, we can't just go down the middle. Going down the middle, cutting something into half is even, right? So we can't cut it in half. We go to the middle and we offset it a little bit. The offset depends on how many more pieces you need to break the thing into, right? Depends on how large or how small your fraction is, the denominator in the fraction, right? If we're gonna cut this into 11 pieces, we're gonna go in the middle. We're gonna offset the break a little bit. The amount we're gonna offset it depends on how much we have to fit on either side of that. Because right in the middle, when we offset, that's one out of the 11 pieces. So what do we got left if we're taking one out of the fraction we've already got one piece out? We need five on either side, right? So what we end up doing is we have to estimate the break to be enough to fit five pieces on either side on either half that we have right now or either segment that we have right now that's gonna make it look even, okay? So what we're gonna do, I'm gonna approximate the middle for this to be around here, right? So the middle is here and what I'm gonna do is I'm gonna mark it, put a little thing there but I'm not gonna break it. This is just a marker for me to visualize it. And then what I'm gonna do is I'm gonna break it over here a little bit and break it on this side a little bit enough so I can fit five pieces on this side and five pieces on that side, okay? So if that's the middle, I'm gonna break it. I'm gonna break it. So here's the middle and what I'm gonna do, I'm gonna break it here and I'm gonna break it here. Okay and this is one, right? Out of the 11 pieces that I wanted, right? Now what I have to do is I'm gonna have to fit five on this side and five on that side. Now five happens to be even as well, right? So I can't break this part in half and then continue to break it in half. I have to go down the middle again, make a little mark and offset a little bit enough so I could fit two on this side and two on that side because that's gonna be one in the middle there, right? So I've taken one out of the five that I need so I need two on either side now, right? So I'm gonna go in the middle again. So I'm gonna assume my middle is around here and I'm gonna offset enough so I could put two even pieces here and two even pieces on this side, right? So if that's the middle and break it here, there I'm gonna break a little bit higher. And hopefully this segment is equal to this segment, right? They're approximately the same and what I need now is two on this side and two on the other side there, right? So what I'm gonna do because two is even, all I have to do is just break it in half, right? I have to break this segment in half and I have to break that segment in half and then I'm gonna do the same thing over here. Find the middle offset, break each one in half, right? So I'm breaking this guy in half. I'm breaking this guy in half, right? I'm gonna go with this side. I'm gonna pick the middle, right? The middle, I have to stand out away from it to pick it so I'm assuming my middle is around here and I'm gonna offset this the same amount. I'm gonna try to gonna pick the same type of segment here, right? So I'm gonna put my line there and there, right? Now I need two pieces here and I need two pieces here, two is even. So I'm gonna break each one of those in half, right? And this is our line cut into 11 pieces. So basically what we did was we marked off the center. We took one out of the equation and we made it even break on either side. Well, the same break on either side and if it's odd, again, we go to the middle, offset, break and break enough so we can have whatever pieces we need on either side again and we continue the thing up. And if we add up all of these, right? If we're starting at zero, that's one, two, three, four, five, six, seven, eight, nine, 10, 11, we're at one. So this thing's broken down to 11 pieces. And over here, you can tell that, you know, my breaks aren't the best, right? They're not exact. Over here, I'm offset a little bit. Over there, I'm offset a little bit. But this is sort of for us to do a visualization, right? If we needed to break this thing exactly into 11 pieces, we'd get a tape measure out and measure it out and divide it by 11, right? And then we'd start off at one end and just go down, know where the breaks are and just do the breaks, right? So if, just like the even, what do you call it? The even lines that we broke, right? If this was zero, this was one, then if we wanted to go to three 11s, we would go one, two, three. That's where three 11s would be. Seven 11s, one, two, three, four, five, six, seven 11s, right there, right? So if this line segment was beyond the boundary of zero and one, if it was between five and six, if it was five and three 11s, we'd go, that would be our five and then we'd go one, two, three. That's five and three 11s. One, two, three, four, five, six, seven, five and seven 11s, right? Because that would be our six. If these were negative, if this was zero to negative one and we wanted to go to negative three 11s, this would be zero, one, two, three. That would be negative three 11s. Negative seven 11s, one, two, three, four, five, six, seven, negative seven 11s, right? If this was negative five to negative six, negative five and three 11s, negative five, one, two, three 11s, negative five and seven 11s, negative five and 10 11s, whatever it is, right? So this is the way you break a line into odd pieces. Now the only thing you have to be careful with is sometimes when you're breaking a line, you're required to break a line to find out where you are on a line segment on a coordinate system. The break is even at first and then it goes odd and vice versa, I think vice versa anyway. So for example, if you're breaking a line into 10 pieces, right, 10 is even, you go down the middle, cut it in the middle, but in the other two segments, you need five on this side and five on that side. That's odd. So what you end up doing is you go in the middle offset and then you need two and two, right? So it's not necessarily if it starts out as being even, it's all even breaks or if it starts out as being odd, it's all odd breaks. What you need to do is think about what the break segments need and put in your ticks accordingly. And this is sort of just a basic visualization of what fractions look like. And it's really important to be visualized, to be able to visualize this and to be able to have a good grasp of what a fraction is, what breaking lines into segments is, which is basically parts of a whole, right? And that's about it. I guess I'll see you guys in the next video. Keep this in mind when we're doing the Cartesian coordinate system and we're graphing, because we end up dealing with a lot of rational numbers. I'll see you guys in the next video. Bye for now. And as an anecdote to get you started on why this is important, a student told me this story once where her dad's, her father's partner, when he was having interviews to hire people for a specific job, the first question he would ask someone is what's one eighth? And surprisingly, a lot of the time, people had no concept of one eighth was, right? Did we get the first question wrong in an interview? They wouldn't be able to answer that question. And if you've ever gone for a job interview, the first question you're asking in an interview is usually one of the most important questions you're gonna be asked, right? So they catch you off guard right off the bat. They ask you a question that they wanna answer for because they wanna know what type of person you are. And if you go into an interview and the person asking you, interviewing you, ask the question, what's one eighth? And you don't know what one eighth is. That says a lot about you, right? That says that your understanding of the world is limited. You don't know what rational numbers are. It means you don't know how to deal with fractions or ratios. And it means you're limited to the integers. So really important for you to be able to visualize what fractions are, visualize what it means to break lines, break things into pieces, visualize where something is placed on a coordinate system, visualize what ratios are, okay? And that's about it. For those of you that know what this is, that know how this works, more power to you because you took the time to learn it. And for those of you that don't know how this works, learn this. Super, super important. Okay.