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## Multiplication as scaling

## Video transcript

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.