 One of the ways to interpret division is we are splitting a number into an equal number of parts. For example, dividing 6 by 3 is 2 because if we split 6 into 3 equal pieces, each piece is 2. Perfect. How do we make sense of this when dividing by a non-integer such as 1.3 or 0.7? What does it mean to split 6 into 0.7? So 3 into 2 pieces or 1 point converted to fractions. You can look at it in the integer form or into 0.7 or 1.3 equal parts. Does this way of interpreting division just not make sense when dealing with non-integers? I would say division to me is fractions. Think about it this way. Let's take a look. Let's say you're doing this. For your example, you're dividing 6 by 3 or 2. Dividing 6 by 3 is 2 because 3 pieces. If you do it visually, if you got 6, let's do it by numbers. So if you're doing 6 divided by 3. The division, if you're going to do this, convert to multiplication. 6 times and this is 3 over 1 and when you're doing this, you flip this and it becomes 1 over 3. So 6 times 1 over 3 and 6 is just over 1. This turns into 6 divided by 3 which is equal to 2. So you're taking 6, breaking it into 3 pieces that have 2 each. So let's deal with a fraction. Let's say you're taking 6 and dividing it by 0.7. So 0.7 convert to a fraction. So this becomes 6 divided by 0.7 as a fraction is 7 over 10. Now how do you do division? You do division by going 6. You change it to multiplication and you flip this. So it becomes 10 over 7. Now multiply this out. Well top multiplies top, bottom multiplies bottom. So this is 60 divided into 7 pieces. So what you're doing, if you're going to think about it as integers, convert it into integers. The top into an integer and the bottom into an integer. So what 6 divided by 0.7 is means really as a direct comparison, sort of something that you can wrap your head around visually, you're taking 60 and divide it into 7 equal pieces. And whatever this turns out into, well this turns out into, we can do the multiplication with the division. Oh sorry, we're not going to do the multiplication with the division. Let's do the division, let's do long division here. 7, 60, right? 7 doesn't go into 6, it goes into 60 how many times? It goes into it 8 times. 8 times 7 is 56, subtract you get 4. Well now you add your 0, you add your decimal, bring the 0 down, 7 into 40 is 5. So that's 35, this becomes 5, you add another 0, bring it down, 7 into 50 is 7 times, which is 49, so that becomes 1, bring down another 0, and so on and so forth, right? So oh yeah, for sure, you can convert it into what do you call it, mixed number, right? But as a decimal it would be, as a mixed number this would be, thank you Graham by the way. So 7 goes into 8, 7 goes into 68 times, and what's left over, 4 over 7, right? If we didn't continue this it would be 4 over 7, right? So it would be 4 over 7, right? As a decimal it would be 8.571 dot dot dot, whatever it is, okay? That's the way you can visualize it if you want. If you want to visualize it in terms of integers, right? Because this makes it an integer, does that make sense? And dividing by any number, which is on a number scale, right? If this is 0, this is negative 1, and this is 1, open circle doesn't include that, open circle doesn't include that. If you divide any number, any number by anything between which is greater than, let's say you're dividing any number by x, and x is greater than negative 1 but less than 1, so if x is between negative 1 and 1, if you divide any number by any number that is between negative 1 and 1, it makes the number bigger, right? Which is sort of a weird thing. So one of the first weird things that students encounter when they're doing division. It's like, wait a second, you're dividing this by a number. Why is 6, why is 6 now bigger? Because you're dividing by 0.7 and 0.7 it would be here, 0.7. So if you divide any number by a number that is between negative 1 and 1, the number gets bigger, right? Why? Because if you convert that number, that decimal, there's decimals beyond us, but people refer to this decimal, this number to a fraction, it's really, if you're dividing by it, you flip it, so the 10 or the 100 or whatever it is is on the top, and the other number is on the bottom, so the number is getting bigger, right? Does that help? I know I didn't do the visual part, but it just doesn't make sense when you're going into this level, okay? For me it's easier to look at it in terms of fractions, 0.7 can go into 6, more times than 1 can go into 6, nice way of putting a graph, 0.7 can go into 6, right? More times than 1 can go into 6, that's a great way of sending a graph, I'm jacking that, I'm going to start using that, thank you for that by the way, and gang, keep this in mind, I don't give a rat's ass how good any teacher you think is, right? Another perspective can always introduce you to a different way of looking at a problem, right? So, if you're taking a course with someone, if you're trying to learn something and you're having a hard time learning it either from that student or from that book or from that video, don't think the concept is too difficult to understand, find yourself a new instructor maybe, a new booktory, or a new video to look at, okay? There's always another perspective that will all of a sudden make you go, aha, that makes sense, I get it, right? To me, I've never thought about it this way, as Graham said, 0.7 can go into 6, more times than 1 can go into 6, brilliant, right, brilliant, thank you for that, Graham, right? Joe, how does this translate to the real world? How does that translate into the real world? If I have 6 items and I'm asked to divide them by 0.7, how can I do that? Because the answer will be 0.87, so I end up with more coins than the number I originally have. Real world, Graham, can you think of a real world version of this? I mean, there's places you have this, but I'm trying to think of a simple way, a good question, by the way, percentages, percentages, 0.7 percent, 0.7 percent, 0.7 percent, 0.6 percent, yeah, yeah, I'm trying to think of something right now that's more less percentage-based, 70 percent of, yeah, that's the reverse of it, that's the reverse of it, so what Graham is saying, Joe, by the way, is this, so what you can do is reverse engineer it, right? You could say, hey, what's 70 percent? Or let's do a question that way, that question that way will be better, right? What if you were going to buy something, right? Let's say you walked into a store, right? And they said, sale, sale, right? Here's your item, right? And they have a tag on there saying sale price, sale price is $6, right? Sale price is $6. Let me write this more clearly so you see it better. I'm going to erase this, let's make it better. That's a nice example, actually, Graham. So they say sale price, sale price, $6, right, sale price, 70 percent off, 70 percent off, and it's $6. What if you wanted to find the original price? You would tell yourself, hey, the original price times the percent off, 0.7 has to equal 6, so you go X is equal to 6 divided by 0.7, and that's going to be 8.57. So that's the original price. Why would you want to do this? What if you walked into another store and they said sale price, sale price, 65 percent off, right? And it's $6.25, right? Well what was the original price of this one, right? We could do a comparison or something like this, right? I know that's probably not a tangible one because it involves percentages. Eagles and cycling. Think chocolate or polyurethane today, no, I don't think you would. Word is out he may, but who knows, maybe, no, I don't think so. Cheryl, especially if you're a visual thinker, especially if you're a visual thinker,