Multiplication as scaling with fractions

We have three expressions here. This is 2/3 times 7/8. The second expression is 8/7 times 2/3. This third expression is 5 times 2 over 3 times 5. And what I want you to do is pause this video right now and think about which of these expressions is the largest, which one is in the middle in terms of value, and which one is the smallest. And I want you to think about it without actually doing the calculation. If you could just look at them and figure out which of these is the largest, which of these is the smallest, and which of these is in the middle. So pause the video now. Now, you might have taken a shot at it. And I'll give you a little bit of a hint in case you had trouble with it. All of these involve multiplying something by 2/3. And you see a 2/3 here. You see a 2/3 here. And it might not be as obvious, but you also see a 2/3 here. And let me rewrite that to make it a little bit clearer. So this first expression could be rewritten as 7/8 times 2/3. This second expression here could be written as-- well, it's already written as 8/7 times 2/3. And then, this last expression, we could write it as, in the numerator, 5 times 2. And then in the denominator, it's over 5 times 3. 5 times 3, which is of course the same thing as 5/5 times 2/3. So you see, all three of these expressions involve something times 2/3. Now, looking at it this way, does it become easier to pick out which of these are the largest, which of these are the smallest, and which of these are someplace in between? I encourage you to pause it again if you haven't thought about it yet. So let's visualize each of these expressions by first trying to visualize 2/3. So let's say the height of what I am drawing right now, let's say the height of this bar right over here is 2/3. So this right over here represents 2/3. The height here is 2/3. So first, let's think about what this one on the right here represents. This is 5/5 times 2/3. Well, what's 5/5? 5/5 is the same thing as 1. This is literally just 1 times 2/3. This whole expression is the same thing as 1 times 2/3, or really, just 2/3. So this, the height here, 2/3, this is the same thing as this thing over here. This is going to be equal to-- this could also be viewed as 5 times 2 over 3 times 5, which was this first expression right over here. Now, let's think about what these would look like. So this is 7/8 times 2/3. So it's less than 8/8 times 2/3. It's less than 1 times 2/3. So we're going to scale 2/3 down. This is going to be less than 2/3. It's going to be 7/8 of 2/3. So this one right over here would look something like this. Let me see if I can draw it. Yeah, it would look something like this. If the yellow height is 2/3, then this right over here, then this height right over here-- let me make it clear. This height right over here would be 7/8 times 2/3. Likewise, let's look at this one right over here. Let's look at this one in the middle, 8/7 times 2/3. Well, 8/7 is bigger than 7/7. It's more than 1. This is more than 2/3. This is 1 and 1/7 times 2/3. So it's going to be the same height as 2/3 plus another 1/7. So it's going to look something like this. It's going to look something like this. So its height-- now we scaled the 2/3 up because 8/7 is greater than 1. So this right over here, this height is going to be 8/7 times 2/3. So the way that you could have spotted which of these is the largest and which of these is the smallest is to say, well, how are they scaling 2/3? This one right over here, you're essentially multiplying 2/3 by 1. So you're just going to get 2/3. You're not scaling it up, or you aren't scaling it down. This one right over here, you're scaling 2/3 down. You are multiplying it by something less than 1. If you multiply it by something less than 1, then you're going to be scaling it down. I should say, a positive number or a number between 0 and 1-- less than 1-- then you're going to be scaling it down. So this thing is scaled down. It's going to be the smallest. And here, you're multiplying the 2/3 times a number bigger than 1, by 1 and 1/7. So you're going to scale it up. So this expression is the largest, 8/7 times 2/3. The smallest is 2/3 times 7/8. And this one right over here is in between.