Fractions as division by power of 10

WEBVTT
Kind: captions
Language: en

00:00:00.940 --> 00:00:03.350
So I have six different
fractions written out here,

00:00:03.350 --> 00:00:05.970
and what I want you to do is
pause this video right now

00:00:05.970 --> 00:00:09.920
and try to rewrite each of
these fractions as a decimal.

00:00:09.920 --> 00:00:11.690
So I'm assuming you've
given a go at it,

00:00:11.690 --> 00:00:13.530
so let's go through
each of these.

00:00:13.530 --> 00:00:19.530
So we know that when we
say 34/10, or 34 over 10,

00:00:19.530 --> 00:00:26.910
that this could be interpreted
literally as 34 divided by 10.

00:00:26.910 --> 00:00:28.804
And so if you
start with 34-- let

00:00:28.804 --> 00:00:30.470
me put a little decimal
right over here.

00:00:30.470 --> 00:00:32.803
It will be clear that there
is an implied decimal there,

00:00:32.803 --> 00:00:35.500
so we could just keep
adding zeroes if we want.

00:00:35.500 --> 00:00:37.810
So when we divide
by 10, we're going

00:00:37.810 --> 00:00:41.292
to move the decimal one
space over to the left.

00:00:41.292 --> 00:00:43.000
And you should always
do a reality check.

00:00:43.000 --> 00:00:45.083
If you forget, hey, do I
move it over to the left,

00:00:45.083 --> 00:00:46.620
or do I do it over to the right?

00:00:46.620 --> 00:00:47.870
You should do a reality check.

00:00:47.870 --> 00:00:49.520
Look, if I'm dividing
by 10, should I

00:00:49.520 --> 00:00:51.340
get a smaller value
or a larger value?

00:00:51.340 --> 00:00:53.750
Well, clearly, if you're
dividing it into 10 groups,

00:00:53.750 --> 00:00:55.820
you should get a smaller value.

00:00:55.820 --> 00:00:58.840
So when you take your
decimal place to the left,

00:00:58.840 --> 00:01:00.400
you're going to get
a smaller value.

00:01:00.400 --> 00:01:06.440
34 is going to become 3.4.

00:01:06.440 --> 00:01:07.750
And that makes complete sense.

00:01:07.750 --> 00:01:10.160
If you were to
multiply 3.4 by 10,

00:01:10.160 --> 00:01:11.870
you would move the
decimal to the right,

00:01:11.870 --> 00:01:14.966
and you would get to 34.

00:01:14.966 --> 00:01:16.090
Let's think about this one.

00:01:16.090 --> 00:01:18.575
So this one right
over here is 3.4.

00:01:18.575 --> 00:01:20.510
Now let's think about 7/10.

00:01:20.510 --> 00:01:23.930
So the exact same idea.

00:01:23.930 --> 00:01:27.650
This is equal to
7 divided by 10.

00:01:30.430 --> 00:01:33.451
This fraction, horizontal
line symbol thing

00:01:33.451 --> 00:01:35.200
could literally be
viewed as "divided by."

00:01:35.200 --> 00:01:38.930
This could be read
as 7 divided by 10.

00:01:38.930 --> 00:01:44.170
So when you divide by 10, you
move the decimal one space over

00:01:44.170 --> 00:01:45.530
to the left.

00:01:45.530 --> 00:01:48.260
So we're going to
get to point 7.

00:01:48.260 --> 00:01:50.572
And just to be clear, we
could write this as 0.7.

00:01:50.572 --> 00:01:52.760
It's sometimes dangerous
to just write the decimal

00:01:52.760 --> 00:01:54.350
without the zero out front.

00:01:54.350 --> 00:02:02.130
So 7/10, or 7 divided by
10, can be rewritten as 0.7.

00:02:02.130 --> 00:02:05.590
Now let's try 53/100.

00:02:05.590 --> 00:02:07.829
Well, we'll start with our 53.

00:02:07.829 --> 00:02:09.370
Once again, this
could be interpreted

00:02:09.370 --> 00:02:14.510
as 53 divided by 100.

00:02:14.510 --> 00:02:16.730
So there's an implied decimal
point right over here.

00:02:16.730 --> 00:02:20.000
Now, if we're dividing by
100, that's dividing by 10

00:02:20.000 --> 00:02:21.790
and then dividing by 10 again.

00:02:21.790 --> 00:02:24.220
This is dividing by 10 times 10.

00:02:24.220 --> 00:02:26.780
So we're going to divide
by 10, and then we're

00:02:26.780 --> 00:02:28.170
going to divide by 100.

00:02:28.170 --> 00:02:30.620
So the decimal is going
to land right over there.

00:02:30.620 --> 00:02:36.910
So that's going
to get us to 0.53.

00:02:36.910 --> 00:02:39.840
Now let's tackle
2 divided by 100.

00:02:39.840 --> 00:02:44.336
So once again, this could be
rewritten as 2 divided by 100.

00:02:44.336 --> 00:02:46.930
2/100 is the same thing
as 2 divided by 100.

00:02:46.930 --> 00:02:50.600
If we start with a 2, put our
decimal point right there.

00:02:50.600 --> 00:02:52.463
So we're going to
divide by 10, and we're

00:02:52.463 --> 00:02:53.430
going to do that twice.

00:02:53.430 --> 00:02:54.930
So we're going to
divide by 10 once.

00:02:54.930 --> 00:02:55.825
That would be 2/10.

00:02:55.825 --> 00:02:57.565
That puts our decimal there.

00:02:57.565 --> 00:02:59.100
But we have to
divide by 10 again.

00:02:59.100 --> 00:03:00.840
You might say, hey, look,
there's nothing here.

00:03:00.840 --> 00:03:02.590
Well, you could just
throw a zero on here,

00:03:02.590 --> 00:03:04.530
just so that you
do something there,

00:03:04.530 --> 00:03:09.439
and then you have moved your
decimal over to the left twice.

00:03:09.439 --> 00:03:10.980
Remember, every time
you're doing it,

00:03:10.980 --> 00:03:11.965
you're dividing by 10.

00:03:11.965 --> 00:03:13.470
You're dividing by 10.

00:03:13.470 --> 00:03:15.140
Then you're dividing
by 10 again.

00:03:15.140 --> 00:03:17.830
Dividing by 10 two times, that's
the same thing as dividing

00:03:17.830 --> 00:03:21.130
by 100, because 10
times 10 is 100.

00:03:21.130 --> 00:03:27.384
So this is going to be 0.02.

00:03:27.384 --> 00:03:29.925
Now we've already seen that, if
you were to try to read this,

00:03:29.925 --> 00:03:32.320
this 2 is in the
hundredths place.

00:03:32.320 --> 00:03:35.067
So you would literally
read this as 2 hundredths.

00:03:35.067 --> 00:03:36.400
And we see that right over here.

00:03:36.400 --> 00:03:38.840
We've literally represented
it as 2 hundredths right

00:03:38.840 --> 00:03:42.040
over there-- 2/100.

00:03:42.040 --> 00:03:44.310
So now we have 1,098
divided by 100.

00:03:44.310 --> 00:03:45.800
Well, same drill.

00:03:45.800 --> 00:03:48.020
Or 1,098 hundredths I could say.

00:03:48.020 --> 00:03:52.710
That's the same thing
as 1,098 divided by 100.

00:03:52.710 --> 00:03:55.390
So we could start with 1,098.

00:03:55.390 --> 00:03:57.275
The decimal is implicitly
right over here,

00:03:57.275 --> 00:04:00.110
but we need to move it two
spaces to the left-- one, two--

00:04:00.110 --> 00:04:01.940
because we're dividing by 100.

00:04:01.940 --> 00:04:05.460
That gets us to 10.98.

00:04:05.460 --> 00:04:06.930
And always do a reality check.

00:04:06.930 --> 00:04:07.980
Look, it makes sense.

00:04:07.980 --> 00:04:10.390
We got to a significantly
smaller value

00:04:10.390 --> 00:04:12.910
when we divided by 100.

00:04:12.910 --> 00:04:16.579
Now we'll just divide
9,967 by 1,000.

00:04:16.579 --> 00:04:21.750
Well, 9.967, decimal
point implicitly there.

00:04:21.750 --> 00:04:24.880
Now, 1,000 is 10
times 10 times 10.

00:04:24.880 --> 00:04:26.590
So if you're dividing
by 1,000, that's

00:04:26.590 --> 00:04:28.920
dividing by 10 three times.

00:04:28.920 --> 00:04:32.220
So we'll divide by 10
once, divide by 10 twice,

00:04:32.220 --> 00:04:34.500
divide by 10 three times.

00:04:34.500 --> 00:04:38.520
We get to 9.967.

00:04:38.520 --> 00:04:44.370
So another way of writing
9,967 thousandths, well,

00:04:44.370 --> 00:04:47.946
that's just going to be 9.967.

00:04:47.946 --> 00:04:53.460
Or I could literally read
it as 9 and 967 thousands.

00:04:53.460 --> 00:04:54.600
And that also makes sense.

00:04:54.600 --> 00:04:58.540
I mean, you could break
this up into 9,000 plus 967.

00:04:58.540 --> 00:05:02.450
And so 9,000 thousandths
is going to be 9.

00:05:02.450 --> 00:05:06.002
And then you're going to be
left with the 967 thousandths.


Powers of 10: patterns

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.460
So let's think about what
happens when we multiply by 10,

00:00:03.460 --> 00:00:06.050
and see if we can see
some type of a pattern.

00:00:06.050 --> 00:00:12.030
So let's just start with 1
and multiply that 1 by 10.

00:00:12.030 --> 00:00:15.460
And this, of course,
is equal to 10.

00:00:15.460 --> 00:00:16.780
You knew that already.

00:00:16.780 --> 00:00:18.940
But let's multiply by 10 again.

00:00:18.940 --> 00:00:22.260
So let's do 1 times 10.

00:00:22.260 --> 00:00:24.570
And now let's multiply
by another 10.

00:00:24.570 --> 00:00:26.050
Let's multiply by another 10.

00:00:26.050 --> 00:00:27.880
So what's this going to be?

00:00:27.880 --> 00:00:30.800
This is going to be 1 times
10, which is 10, and then that

00:00:30.800 --> 00:00:32.020
10 times 10.

00:00:32.020 --> 00:00:36.820
So it's literally 10 10s or 100.

00:00:36.820 --> 00:00:38.850
Now let's multiply by 10 again.

00:00:38.850 --> 00:00:49.180
So we're going to have 1 times
10 times 10 times 10 times 10.

00:00:49.180 --> 00:00:51.160
What is this going
to be equal to?

00:00:51.160 --> 00:00:54.602
Well we already know that
1 times 10 times 10 is 100,

00:00:54.602 --> 00:00:56.810
and then we're going to
multiply that times 10 again.

00:00:56.810 --> 00:00:59.460
So we're going to literally
have 10 100s which

00:00:59.460 --> 00:01:03.650
we know is the exact
same thing as 1,000.

00:01:03.650 --> 00:01:06.230
So what's the pattern here?

00:01:06.230 --> 00:01:11.050
Well here we multiplied by
10 once and we have one zero.

00:01:11.050 --> 00:01:15.500
Here we multiplied by 10
twice and we have two zeroes.

00:01:15.500 --> 00:01:18.400
Here we multiplied
by 10 three times

00:01:18.400 --> 00:01:21.407
and we have three
zeroes in our answer.

00:01:21.407 --> 00:01:22.990
Let's do it again,
although this looks

00:01:22.990 --> 00:01:24.650
like a pretty clear pattern.

00:01:24.650 --> 00:01:26.670
Let's multiply by 10 four times.

00:01:26.670 --> 00:01:28.150
So we'll start with a 1.

00:01:28.150 --> 00:01:31.770
We'll multiply by one
10, that's our first 10,

00:01:31.770 --> 00:01:36.690
then our second 10, then
our third 10, and now

00:01:36.690 --> 00:01:38.220
our fourth 10.

00:01:38.220 --> 00:01:40.860
Well what's this
going to be equal to?

00:01:40.860 --> 00:01:47.380
Well it's going to be
this thing, I should say,

00:01:47.380 --> 00:01:52.800
times 10 or 1,000 times
10, which is 1,000s.

00:01:52.800 --> 00:01:56.470
So let me write that 10 1,000s.

00:01:56.470 --> 00:01:58.790
So once again, we
started with a one,

00:01:58.790 --> 00:02:04.165
we multiplied by 10 four
times, and we got 10,000.

00:02:04.165 --> 00:02:07.012
One followed by one,
two, three, four zeros.

00:02:07.012 --> 00:02:07.970
Let me make that clear.

00:02:07.970 --> 00:02:08.740
We start with a 1.

00:02:08.740 --> 00:02:14.200
We multiplied by 10 one,
two, three, four times,

00:02:14.200 --> 00:02:20.860
and our product is one followed
by one, two, three, four zeros.

00:02:20.860 --> 00:02:22.410
So I think you see
the pattern here.

00:02:22.410 --> 00:02:24.243
But there might be
something you're craving.

00:02:24.243 --> 00:02:26.600
You're like, hey
Sal, this is a pain

00:02:26.600 --> 00:02:28.680
to do this repeated
multiplication.

00:02:28.680 --> 00:02:30.570
I wish there was some
type of operation

00:02:30.570 --> 00:02:32.390
that just did it for us.

00:02:32.390 --> 00:02:34.339
The same way that
back in the day

00:02:34.339 --> 00:02:36.130
when you were doing
repeated addition where

00:02:36.130 --> 00:02:42.140
if you were saying 10 plus
10 plus 10 plus 10, you said,

00:02:42.140 --> 00:02:45.764
hey, this is the same
thing as 4 times 10.

00:02:45.764 --> 00:02:48.180
There should be another thing
for repeated multiplication.

00:02:48.180 --> 00:02:53.010
If I have 1 times 10
times 10 times 10 times

00:02:53.010 --> 00:02:56.559
10, which is really the same
thing as taking four 10s

00:02:56.559 --> 00:02:58.350
and multiplying them
together because the 0

00:02:58.350 --> 00:02:59.870
doesn't change the value.

00:02:59.870 --> 00:03:04.222
So 10 times 10
times 10 times 10.

00:03:04.222 --> 00:03:05.680
You're saying, hey,
there should be

00:03:05.680 --> 00:03:08.980
some type of shorthand for
this, and you're right.

00:03:08.980 --> 00:03:11.130
There is some type
of shorthand for this

00:03:11.130 --> 00:03:13.030
and it's called an exponent.

00:03:13.030 --> 00:03:16.090
So the same way if you take
four 10s and add them up,

00:03:16.090 --> 00:03:18.400
that's the same
thing as 4 times 10.

00:03:18.400 --> 00:03:21.790
If you take four 10s
and take their product,

00:03:21.790 --> 00:03:27.860
this is the same thing as
10 to the fourth power.

00:03:27.860 --> 00:03:31.700
10 to the fourth power
literally means taking four 10s

00:03:31.700 --> 00:03:33.290
and multiplying them together.

00:03:33.290 --> 00:03:35.590
Or you could view it
as starting with a 1

00:03:35.590 --> 00:03:38.030
and multiplying
by 10 four times.

00:03:38.030 --> 00:03:42.981
So this is also equal to
10 to the fourth power.

00:03:42.981 --> 00:03:44.730
This right over here,
we started with a 1.

00:03:44.730 --> 00:03:46.635
We multiplied by 10
three times, which

00:03:46.635 --> 00:03:48.300
is the same thing
as taking three 10s

00:03:48.300 --> 00:03:50.180
and multiplying them together.

00:03:50.180 --> 00:03:53.280
This is 10 to the third power.

00:03:53.280 --> 00:03:56.780
This right over
here is 10 squared,

00:03:56.780 --> 00:03:58.860
or 10 to the second power.

00:03:58.860 --> 00:04:01.484
And this right over
here, we just have one 10

00:04:01.484 --> 00:04:02.150
right over here.

00:04:02.150 --> 00:04:04.710
We just started with the one
and we multiplied by 10 once.

00:04:04.710 --> 00:04:08.340
This is 10 to the first power.

00:04:08.340 --> 00:04:10.760
And just so you know a little
bit of the terminology,

00:04:10.760 --> 00:04:13.280
here 10 is our base.

00:04:13.280 --> 00:04:15.180
The base is 10.

00:04:15.180 --> 00:04:18.040
And this 4 that we
have right over here,

00:04:18.040 --> 00:04:22.320
this is the exponent.

00:04:22.320 --> 00:04:26.520
So given that notation,
let's just do one practice.

00:04:26.520 --> 00:04:30.040
Let's think about what--
I'll do new colors-- what

00:04:30.040 --> 00:04:32.510
10 to the sixth power is.

00:04:32.510 --> 00:04:35.094
And I encourage you to pause
the video and think about it.

00:04:35.094 --> 00:04:36.760
Well there's two ways
to think about it.

00:04:36.760 --> 00:04:45.940
You could view this as six 10s--
so 10, 10, 10, 10, 10, and 10

00:04:45.940 --> 00:04:47.990
and multiplying them together.

00:04:47.990 --> 00:04:51.360
So we multiply
them all together.

00:04:51.360 --> 00:04:53.900
Or you could view it
as starting with a 1

00:04:53.900 --> 00:04:56.540
and multiplying it
by 10 six times.

00:04:56.540 --> 00:04:59.060
But either way, what
are you going to get?

00:04:59.060 --> 00:05:03.180
Well if you're multiplying
1 by 10 six times,

00:05:03.180 --> 00:05:05.710
you're going to end up with
1 followed by six zeroes.

00:05:05.710 --> 00:05:07.070
We already saw this pattern.

00:05:07.070 --> 00:05:11.490
So it's going to be 1 followed
by one, two, three, four, five,

00:05:11.490 --> 00:05:15.310
six zeros, or one million.

00:05:15.310 --> 00:05:19.950
So 10 to the sixth power
is the same thing as 1

00:05:19.950 --> 00:05:22.360
being multiplied
by 10 six times.

00:05:22.360 --> 00:05:25.760
We're taking six 10s and
multiplying them together.

00:05:25.760 --> 00:05:29.440
And that's going to be equal to
1 followed by six zeros, which

00:05:29.440 --> 00:05:33.230
is this pattern we
saw or one million.


Using fractions as division to create mixed numbers

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:04.600
We've already seen that
a fraction like 2/9

00:00:04.600 --> 00:00:08.884
can be interpreted
as 2 divided by 9.

00:00:08.884 --> 00:00:10.300
Let me do the 2
in the same color.

00:00:13.400 --> 00:00:15.920
That if we have a
fraction, it could

00:00:15.920 --> 00:00:20.080
be interpreted as the numerator
divided by the denominator.

00:00:20.080 --> 00:00:22.710
And this leads to all sorts
of interesting conclusions,

00:00:22.710 --> 00:00:25.180
some of which we've already
seen, and some of which

00:00:25.180 --> 00:00:26.750
are a little bit new.

00:00:26.750 --> 00:00:35.780
So, for example, if I
had the fraction 7/7,

00:00:35.780 --> 00:00:39.790
this can be interpreted
now as 7 divided by 7,

00:00:39.790 --> 00:00:43.270
as our numerator divided
by our denominator.

00:00:43.270 --> 00:00:47.070
And 7 divided by 7 is
of course equal to 1.

00:00:47.070 --> 00:00:49.350
And this is consistent with
what we've already seen.

00:00:49.350 --> 00:00:53.190
7/7 would get us to
a whole, and a whole

00:00:53.190 --> 00:00:56.260
is the exact same thing as one.

00:00:56.260 --> 00:00:58.880
But we could do things a little
bit more interesting as well.

00:00:58.880 --> 00:01:06.580
We could take something
like 18/6 and realize, wow.

00:01:06.580 --> 00:01:12.610
This is the same thing
as 18 divided by 6,

00:01:12.610 --> 00:01:15.350
which we know is equal to 3.

00:01:17.694 --> 00:01:19.360
And we should do a
little reality check.

00:01:19.360 --> 00:01:25.410
Does this make sense, that
18/6 should be equal to 3?

00:01:25.410 --> 00:01:26.560
Well, we could rewrite it.

00:01:26.560 --> 00:01:27.830
We could rewrite 18/6.

00:01:30.770 --> 00:01:34.140
Let me make the sixths
that same orange color.

00:01:34.140 --> 00:01:36.480
That's going to
be the same thing.

00:01:36.480 --> 00:01:40.320
18 is 6 plus 6 plus 6.

00:01:40.320 --> 00:01:47.070
And then all of that over 6,
and then that's the same thing.

00:01:47.070 --> 00:01:54.950
That's the same thing as
6/6 plus 6/6 plus 6/6.

00:01:54.950 --> 00:01:59.420
And I could make this
right over here in orange.

00:01:59.420 --> 00:02:02.840
And we've already seen, or we've
seen many, many videos ago,

00:02:02.840 --> 00:02:06.990
that 6/6, just like 7/7, these
are each equal to a whole.

00:02:06.990 --> 00:02:08.330
These are each equal to 1.

00:02:08.330 --> 00:02:10.370
And we can now view
this as 6 divided by 6,

00:02:10.370 --> 00:02:12.050
which is the same thing as 1.

00:02:12.050 --> 00:02:17.490
So this is 1 plus 1 plus 1,
which is, of course, equal

00:02:17.490 --> 00:02:18.519
to 3.

00:02:18.519 --> 00:02:20.560
But this starts to raise
an interesting question.

00:02:20.560 --> 00:02:26.220
This all worked out just fine
because 18 is a multiple of 6.

00:02:26.220 --> 00:02:28.640
6 divides evenly into 18.

00:02:28.640 --> 00:02:31.710
But what happens if we
start having fractions where

00:02:31.710 --> 00:02:35.620
the denominator does not divide
evenly into the numerator?

00:02:35.620 --> 00:02:43.588
Let's say we have a
fraction like 23 over 6.

00:02:46.354 --> 00:02:48.020
Well, we know that
we can interpret this

00:02:48.020 --> 00:02:49.852
as 23 divided by 6.

00:02:53.850 --> 00:02:57.230
And if we actually divide
23 by 6-- let's do that.

00:02:57.230 --> 00:02:59.665
So we divide 6 into 23.

00:03:03.390 --> 00:03:06.770
We know 6 goes into
23 three times.

00:03:06.770 --> 00:03:09.290
3 times 6 is 18.

00:03:09.290 --> 00:03:13.900
And then when you subtract, you
end up with a remainder of 5.

00:03:13.900 --> 00:03:23.717
So we might say, hey, 23 divided
by 6 is equal to 5 remainder 3.

00:03:28.020 --> 00:03:29.446
But that's not that satisfying.

00:03:29.446 --> 00:03:30.820
What do I do with
this remainder?

00:03:30.820 --> 00:03:32.153
This really isn't a number here.

00:03:32.153 --> 00:03:35.130
This is just saying that
we're going five times,

00:03:35.130 --> 00:03:37.750
and then we have a
little bit left over.

00:03:37.750 --> 00:03:41.640
What we can do now is
manipulate this a little bit

00:03:41.640 --> 00:03:46.720
so that we can realize that this
is a number, and in particular,

00:03:46.720 --> 00:03:48.480
a mixed number.

00:03:48.480 --> 00:03:51.560
So for example, we could
start with a 23 over 6,

00:03:51.560 --> 00:03:53.280
and we could divide
it into-- or we

00:03:53.280 --> 00:03:56.570
could decompose the
numerator into one part that

00:03:56.570 --> 00:03:59.810
is divisible by 6,
evenly divisible by 6,

00:03:59.810 --> 00:04:01.300
and the remainder.

00:04:01.300 --> 00:04:11.750
So for example, 23 over 6 we
can rewrite as 18 plus 5 over 6.

00:04:11.750 --> 00:04:15.070
Notice I decomposed the
23 into one part that

00:04:15.070 --> 00:04:19.200
is a multiple of 6, and
it's the largest multiple

00:04:19.200 --> 00:04:21.209
that essentially
fits into 23, or that

00:04:21.209 --> 00:04:25.060
is less than or equal to
23, and then the remainder.

00:04:25.060 --> 00:04:27.310
When you divide 6 into 23,
you get a remainder of 5.

00:04:27.310 --> 00:04:28.940
You could view it
as, I divided it

00:04:28.940 --> 00:04:31.297
into the remainder
and everything else.

00:04:31.297 --> 00:04:32.880
And the reason why
this is interesting

00:04:32.880 --> 00:04:40.170
is because we know that this is
going to be equal to 18 over 6

00:04:40.170 --> 00:04:44.060
plus 5 over 6.

00:04:44.060 --> 00:04:45.830
Well, we already
know that 18/6 is

00:04:45.830 --> 00:04:49.220
the same thing as 18
divided by 6, or 3.

00:04:49.220 --> 00:04:51.560
So this is the same thing as 3.

00:04:51.560 --> 00:04:55.020
So we know that 23 over 6,
which is the same thing as 18

00:04:55.020 --> 00:04:59.320
plus 5 over 6, is the
same thing as 3 plus 5/6.

00:04:59.320 --> 00:05:01.500
Or, if we want to write
it as a mixed number,

00:05:01.500 --> 00:05:07.700
we could write it as 3 and 5/6.


Understanding fractions as division

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:03.300
When we were first exposed to
multiplication and division,

00:00:03.300 --> 00:00:05.602
we saw that they had an
inverse relationship.

00:00:05.602 --> 00:00:07.060
Or another way of
thinking about it

00:00:07.060 --> 00:00:09.160
is that they can
undo each other.

00:00:09.160 --> 00:00:15.450
So for example, if I had 2 times
4, one interpretation of this

00:00:15.450 --> 00:00:18.030
is I could have
four groups of 2.

00:00:18.030 --> 00:00:23.320
So that is one group of 2, two
groups of 2, three groups of 2,

00:00:23.320 --> 00:00:25.110
and four groups of 2.

00:00:25.110 --> 00:00:29.540
And we learned many, many
videos ago that this, of course,

00:00:29.540 --> 00:00:31.410
is going to be equal to 8.

00:00:31.410 --> 00:00:34.960
Well, we could express a very
similar idea with division.

00:00:34.960 --> 00:00:37.370
We could start with 8 things.

00:00:37.370 --> 00:00:42.650
So let's start with one, two,
three, four, five, six, seven,

00:00:42.650 --> 00:00:44.320
eight things.

00:00:44.320 --> 00:00:46.430
So now we're going
to start with the 8.

00:00:46.430 --> 00:00:48.800
And we could say, well,
let's try to divide that

00:00:48.800 --> 00:00:52.730
into four groups,
four equal groups.

00:00:52.730 --> 00:00:56.420
Well, that's one equal
group, two equal groups,

00:00:56.420 --> 00:00:59.790
three equal groups,
and four equal groups.

00:00:59.790 --> 00:01:01.930
And we see when we
start with 8 divide it

00:01:01.930 --> 00:01:03.860
into four equal
groups, each group

00:01:03.860 --> 00:01:06.530
is going to have
2 objects in it.

00:01:06.530 --> 00:01:08.150
So you probably see
the relationship.

00:01:08.150 --> 00:01:09.565
2 times 4 is 8.

00:01:09.565 --> 00:01:11.980
8 divided by 4 is 2.

00:01:11.980 --> 00:01:15.770
And actually, if we did 8
divided by 2, we would get 4.

00:01:15.770 --> 00:01:17.540
And this is generally true.

00:01:17.540 --> 00:01:20.270
If I have something
times something else is

00:01:20.270 --> 00:01:24.240
equal to whatever their product
is, if you take the product

00:01:24.240 --> 00:01:26.010
and divide by one of
those two numbers,

00:01:26.010 --> 00:01:27.760
you'll get the other one.

00:01:27.760 --> 00:01:30.460
And that idea
applies to fractions.

00:01:30.460 --> 00:01:32.790
It actually makes a lot
of sense with fractions.

00:01:32.790 --> 00:01:38.840
So for example, let's say
that we started off with 1/3

00:01:38.840 --> 00:01:41.990
and we wanted to
multiply that times 3.

00:01:44.580 --> 00:01:47.110
Well, there's a couple of
ways we could visualize it.

00:01:47.110 --> 00:01:49.120
Actually, let me just
draw a diagram here.

00:01:49.120 --> 00:01:53.240
So let's say that this
block represents a whole,

00:01:53.240 --> 00:01:55.360
and let me shade in a 1/3 of it.

00:01:58.500 --> 00:02:00.230
So that's 1/3.

00:02:00.230 --> 00:02:02.330
We're going to multiply by 3.

00:02:02.330 --> 00:02:04.030
So we're going to
have 3 of these 1/3's.

00:02:04.030 --> 00:02:05.530
Or another way of
thinking about it,

00:02:05.530 --> 00:02:12.230
it's going to be 1/3 plus
another 1/3 plus another 1/3.

00:02:12.230 --> 00:02:15.180
That's our first 1/3, our
second 1/3, and our third 1/3.

00:02:15.180 --> 00:02:16.590
And we get the whole.

00:02:16.590 --> 00:02:19.100
This is 3/3, or 1.

00:02:19.100 --> 00:02:22.840
So this is going
to be equal to 1.

00:02:22.840 --> 00:02:24.600
So you use the exact same idea.

00:02:24.600 --> 00:02:27.470
If 1/3 times 3 is
equal to 1, then that

00:02:27.470 --> 00:02:35.060
means that 1 divided by
3 must be equal to 1/3.

00:02:38.410 --> 00:02:40.620
And this comes straight
out of how we first even

00:02:40.620 --> 00:02:42.060
thought about fractions.

00:02:42.060 --> 00:02:44.530
The first way that we ever
thought about fractions

00:02:44.530 --> 00:02:46.540
was, well, let's
start with a whole.

00:02:46.540 --> 00:02:48.710
And that whole would be our 1.

00:02:48.710 --> 00:02:51.520
And let's divide it into 3
equal sections, the same way

00:02:51.520 --> 00:02:54.790
that we divided this
8 into 4 equal groups.

00:02:54.790 --> 00:02:57.520
So if you divide this
into 3 equal sections,

00:02:57.520 --> 00:02:59.930
the size of each
of those sections

00:02:59.930 --> 00:03:04.755
is going to be exactly 1/3.

00:03:07.320 --> 00:03:09.480
Now, this leads to an
interesting question

00:03:09.480 --> 00:03:11.540
that might be popping
in your brain.

00:03:11.540 --> 00:03:13.380
Notice, we have 1
is the numerator,

00:03:13.380 --> 00:03:15.120
3 is the denominator,
and we just

00:03:15.120 --> 00:03:17.780
said that this is equal
to the numerator divided

00:03:17.780 --> 00:03:18.680
by the denominator.

00:03:18.680 --> 00:03:22.870
1 over 3 is the same
thing as 1 divided by 3.

00:03:22.870 --> 00:03:24.900
Is this always true
for a fraction?

00:03:24.900 --> 00:03:26.900
Well, let's just do the
same thought experiment,

00:03:26.900 --> 00:03:28.650
but let's do it with
a different fraction.

00:03:32.060 --> 00:03:37.150
Let's take 3/4 and
multiply it by 4.

00:03:37.150 --> 00:03:39.730
So multiply it by 4.

00:03:39.730 --> 00:03:43.260
So once again, let's see
if I could draw 1/4 here.

00:03:43.260 --> 00:03:45.100
Let me do this in a new color.

00:03:45.100 --> 00:03:48.070
So let's say that this block
right over here is a whole.

00:03:48.070 --> 00:03:50.250
We'll divide it into
four equal sections.

00:03:50.250 --> 00:03:52.820
So now I've divided
it into fourths.

00:03:52.820 --> 00:03:56.370
And let me copy and paste it
so I can use it multiple times.

00:03:56.370 --> 00:03:58.640
So copy.

00:03:58.640 --> 00:03:59.690
All right.

00:03:59.690 --> 00:04:03.064
Now, 3/4, that's
going to be-- we

00:04:03.064 --> 00:04:04.730
can assume-- I didn't
draw it perfectly.

00:04:04.730 --> 00:04:07.021
Actually, I could draw it a
little bit better than that

00:04:07.021 --> 00:04:10.340
just to make the four equal
sections actually look equal.

00:04:10.340 --> 00:04:13.040
So that looks like a
little bit better of a job.

00:04:13.040 --> 00:04:15.330
I'm trying to make them
four equal sections.

00:04:15.330 --> 00:04:17.589
Let me copy that one.

00:04:17.589 --> 00:04:21.200
So let me use it for later.

00:04:21.200 --> 00:04:22.450
Now, 3/4.

00:04:22.450 --> 00:04:26.260
This is four equal sections, and
3/4 represents three of them--

00:04:26.260 --> 00:04:29.120
one, two, three.

00:04:29.120 --> 00:04:32.170
But now we're going
to multiply it by 4.

00:04:32.170 --> 00:04:35.300
So we're going to
have 3/4 four times.

00:04:35.300 --> 00:04:37.180
So we're going to need
some more wholes here.

00:04:37.180 --> 00:04:38.513
So let's throw in another whole.

00:04:38.513 --> 00:04:40.270
So this is one 3/4.

00:04:40.270 --> 00:04:42.680
Now let me do the next
3/4 in another color.

00:04:42.680 --> 00:04:47.830
So that's a 1/4, that's a
second 1/4, that's a third 1/4.

00:04:47.830 --> 00:04:49.910
That's another 3/4.

00:04:49.910 --> 00:04:53.940
And now let's do-- so we've
done two 3/4 just now.

00:04:53.940 --> 00:04:55.310
Let me make it clear.

00:04:55.310 --> 00:04:59.380
This is the first 3/4,
and then this plus this

00:04:59.380 --> 00:05:01.540
is the second 3/4.

00:05:01.540 --> 00:05:03.070
Now let's do a third 3/4.

00:05:03.070 --> 00:05:08.080
And we're going to have to use
another whole right over here.

00:05:08.080 --> 00:05:10.250
And I will do that
in this color.

00:05:10.250 --> 00:05:14.090
So my third 3/4,
so here's a 1/4,

00:05:14.090 --> 00:05:16.680
here's my second 1/4,
here's a third 1/4.

00:05:16.680 --> 00:05:19.020
So in green, I have another 3/4.

00:05:19.020 --> 00:05:21.860
And now we need four 3/4.

00:05:21.860 --> 00:05:27.340
So let's do that in a color I
have not used yet, maybe white.

00:05:27.340 --> 00:05:32.490
So that's a 1/4, that's two
1/4, and that is three 1/4.

00:05:32.490 --> 00:05:44.460
So notice, now I have now I have
one 3/4, two 3/4, three 3/4,

00:05:44.460 --> 00:05:46.900
and four 3/4.

00:05:46.900 --> 00:05:49.220
And what did I do when
I got those four 3/4?

00:05:49.220 --> 00:05:50.220
Well, it's pretty clear.

00:05:50.220 --> 00:05:53.310
This is turned into 3 wholes.

00:05:53.310 --> 00:05:56.850
So this is equal to 3 wholes.

00:05:56.850 --> 00:06:00.480
Well, if 3/4 times
4 is equal to 3,

00:06:00.480 --> 00:06:09.603
that means that 3 divided
by 4 is equal to 3/4.

00:06:12.580 --> 00:06:14.410
So the same idea again.

00:06:14.410 --> 00:06:18.070
3 over 4 is the same
thing as 3 divided by 4.

00:06:18.070 --> 00:06:19.850
And in general, this is true.

00:06:19.850 --> 00:06:25.560
The fraction symbol here can
be interpreted as division.

00:06:25.560 --> 00:06:27.380
And looking at this
diagram right here,

00:06:27.380 --> 00:06:28.420
it made complete sense.

00:06:28.420 --> 00:06:30.880
If you started with
3 wholes, and you

00:06:30.880 --> 00:06:35.150
want to divide it into 4 equal
groups, one group, two groups,

00:06:35.150 --> 00:06:37.880
three groups, four
groups, each group

00:06:37.880 --> 00:06:43.150
is going to have 3/4 in it.


Multiplying two fractions: number line approach

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:05.080
In a previous video, we saw
that we could view 2/3 times 6

00:00:05.080 --> 00:00:08.460
as whatever number is
2/3 of the way to 6

00:00:08.460 --> 00:00:10.450
on the number line,
which we saw is 4.

00:00:10.450 --> 00:00:14.300
Or another way to think about
it is that 4 is 2/3 of 6.

00:00:14.300 --> 00:00:17.110
2/3 times 6 can be
viewed as-- well,

00:00:17.110 --> 00:00:20.530
how many do I have
if I take 2/3 of 6?

00:00:20.530 --> 00:00:23.110
Now, what we want to do now
is apply that same idea,

00:00:23.110 --> 00:00:25.800
but to multiply not a
fraction times a whole number,

00:00:25.800 --> 00:00:28.220
but a fraction times a fraction.

00:00:28.220 --> 00:00:35.060
So let's say that we wanted
to take 3/4 and multiply it

00:00:35.060 --> 00:00:38.290
by 1/2.

00:00:38.290 --> 00:00:41.000
And we know, of
course, the order

00:00:41.000 --> 00:00:42.460
that we multiply doesn't matter.

00:00:42.460 --> 00:00:46.330
This is the exact same
thing as 1/2 times 3/4.

00:00:49.940 --> 00:00:51.730
So to imagine
where this gets us,

00:00:51.730 --> 00:00:53.760
let's draw ourselves
a number line.

00:00:56.500 --> 00:00:58.250
And I'll do it pretty
large so that we

00:00:58.250 --> 00:01:00.680
have some space to work in.

00:01:00.680 --> 00:01:03.590
So that's 0.

00:01:03.590 --> 00:01:05.560
And then that is 1.

00:01:05.560 --> 00:01:08.940
And of course, our line
could keep on going.

00:01:08.940 --> 00:01:14.210
And let's first imagine 3/4
times 1/2 as 3/4 of the way

00:01:14.210 --> 00:01:14.850
to 1/2.

00:01:14.850 --> 00:01:17.840
So first let's plot
1/2 on our number line.

00:01:17.840 --> 00:01:21.380
Well, 1/2 is literally
halfway between 0 and 1.

00:01:21.380 --> 00:01:24.330
So that's 1/2 right over there.

00:01:24.330 --> 00:01:28.660
And how do we think about
3/4 of the way to 1/2?

00:01:28.660 --> 00:01:32.320
Well, what we could do is think
about well, what's 1/4 of 1/2?

00:01:32.320 --> 00:01:34.940
Well, we could divide this
part of the number line

00:01:34.940 --> 00:01:36.150
into 4 equal sections.

00:01:36.150 --> 00:01:38.240
So that's 2 equal sections.

00:01:38.240 --> 00:01:39.422
Now that's 4 equal sections.

00:01:39.422 --> 00:01:41.630
And while we're at it, let's
divide all of the halves

00:01:41.630 --> 00:01:43.690
into 4 equal sections.

00:01:43.690 --> 00:01:46.600
So let's divide all of the
halves into 4 equal sections.

00:01:46.600 --> 00:01:48.330
So that's 4 sections.

00:01:48.330 --> 00:01:49.810
And now let's do this one.

00:01:49.810 --> 00:01:52.960
I'm trying my best to
draw them equal sections.

00:01:52.960 --> 00:01:54.730
So I've taken each of
the halves and I've

00:01:54.730 --> 00:01:57.030
made them into 4 equal sections.

00:01:57.030 --> 00:02:04.504
So this point right
over here is 1/4 of 1/2.

00:02:04.504 --> 00:02:05.920
But that's not
what we care about.

00:02:05.920 --> 00:02:08.560
We want to get to 3/4 of 1/2.

00:02:08.560 --> 00:02:14.160
So we want to get
to 1, 2, 3/4 of 1/2.

00:02:14.160 --> 00:02:21.350
So this point right over here,
this is literally 3/4 times

00:02:21.350 --> 00:02:22.840
1/2.

00:02:22.840 --> 00:02:24.360
And this is, of
course, 1/2 here.

00:02:24.360 --> 00:02:25.359
But what number is this?

00:02:27.604 --> 00:02:29.020
And let me do this
in a new color.

00:02:29.020 --> 00:02:31.500
We can now visualize
it on the number line.

00:02:31.500 --> 00:02:34.040
But what number
is this actually?

00:02:34.040 --> 00:02:36.500
Well, a big clue is
that, well, before we

00:02:36.500 --> 00:02:39.290
had the section
between 0 and 1 divided

00:02:39.290 --> 00:02:42.550
into 2 equal sections when
we only had to plot 1/2.

00:02:42.550 --> 00:02:44.520
But then we took each of
those 2 equal sections

00:02:44.520 --> 00:02:46.360
and then split them
into 4 more sections.

00:02:49.630 --> 00:02:51.480
By doing that, we
now essentially

00:02:51.480 --> 00:02:54.890
have divided the section between
0 and 1 into 8 equal sections.

00:02:54.890 --> 00:02:57.280
So each of these
is actually 1/8.

00:02:57.280 --> 00:03:00.210
So this point right
over here is 1/8.

00:03:00.210 --> 00:03:02.210
This is 2/8.

00:03:02.210 --> 00:03:04.750
And then this is 3/8.

00:03:04.750 --> 00:03:06.660
And that's in line
with what we've

00:03:06.660 --> 00:03:09.130
seen about multiplying
fractions before.

00:03:09.130 --> 00:03:23.230
This should be equal to
3 times 1 over 4 times 2,

00:03:23.230 --> 00:03:25.010
which is equal to 3/8.

00:03:28.780 --> 00:03:31.790
And everything that
we're talking about,

00:03:31.790 --> 00:03:33.850
so we don't get
confused, this is all

00:03:33.850 --> 00:03:39.332
referring to this point right
over here on the number line.

00:03:39.332 --> 00:03:41.540
But what if we thought about
it the other way around?

00:03:41.540 --> 00:03:46.910
What if we thought about it
as 1/2 of the way to 3/4?

00:03:46.910 --> 00:03:51.650
So we could divide the space
between 0 and 1 into fourths.

00:03:51.650 --> 00:03:52.940
So let's do that.

00:03:52.940 --> 00:03:58.370
So that is 1/4, 2/4, 3/4.

00:03:58.370 --> 00:04:02.250
So this right over
here is the number 3/4.

00:04:02.250 --> 00:04:05.990
And we want to go half
of the way to 3/4.

00:04:05.990 --> 00:04:08.440
Well, what is half
of the way to 3/4?

00:04:08.440 --> 00:04:11.810
Well, we split this section
into 2 equal sections.

00:04:11.810 --> 00:04:14.050
So we could split
right over there.

00:04:16.760 --> 00:04:19.870
And we want to go exactly
one of those sections.

00:04:19.870 --> 00:04:24.520
1/2 of 3/4 gets us, once
again, right over here

00:04:24.520 --> 00:04:26.017
to this point-- 3/8.

00:04:26.017 --> 00:04:28.350
So either way you imagine it,
whether you're essentially

00:04:28.350 --> 00:04:32.530
taking 3/4 of 1/2, or saying
I'm going to go 3/4 of the way

00:04:32.530 --> 00:04:35.420
to 1/2, or you say I'm
going to go 1/2 of the way

00:04:35.420 --> 00:04:39.520
to 3/4, either way, hopefully
it now makes conceptual sense.

00:04:39.520 --> 00:04:41.770
You can visualize it, and
it makes numeric sense

00:04:41.770 --> 00:04:45.090
that this is going
to be equal to 3/8.


Multiplying fractions: visualizing

WEBVTT
Kind: captions
Language: en

00:00:00.302 --> 00:00:01.760
Let's think a little
bit about what

00:00:01.760 --> 00:00:03.530
it means to multiply fractions.

00:00:03.530 --> 00:00:08.920
Say I want to multiply
1/2 times 1/4.

00:00:08.920 --> 00:00:10.390
Well, one way to
think about this

00:00:10.390 --> 00:00:13.840
is we could view
this as 1/2 of a 1/4.

00:00:13.840 --> 00:00:15.240
And what do I mean there?

00:00:15.240 --> 00:00:19.100
Well let me take a whole,
let me take a whole here,

00:00:19.100 --> 00:00:21.770
and let me divide
it into fourths.

00:00:21.770 --> 00:00:24.300
So let me divide
it into fourths,

00:00:24.300 --> 00:00:28.040
so I'll divided into
4 equal sections.

00:00:28.040 --> 00:00:33.130
And so 1/4 would be 1 of
these 4 equal sections.

00:00:33.130 --> 00:00:35.510
But we want to take 1/2 of that.

00:00:35.510 --> 00:00:37.230
So how do we take half of that?

00:00:37.230 --> 00:00:39.530
Well, we could divide this
into 2 equal sections,

00:00:39.530 --> 00:00:41.340
and then just take 1 of them.

00:00:41.340 --> 00:00:43.890
So divide it into
2 equal sections,

00:00:43.890 --> 00:00:46.360
and then take 1 of them.

00:00:46.360 --> 00:00:53.570
So we're taking this pink area,
this whole pink area is 1/4,

00:00:53.570 --> 00:00:55.890
and now we're going
to take 1/2 of it.

00:00:55.890 --> 00:00:58.520
We're now going
to take 1/2 of it.

00:00:58.520 --> 00:01:01.220
So that's this yellow
square right over here.

00:01:01.220 --> 00:01:04.800
But what fraction of the whole
does this yellow represent?

00:01:04.800 --> 00:01:09.976
Well, it now represents 1
out of 1, 2, 3, 4, 5, 6, 7,

00:01:09.976 --> 00:01:11.800
8 equal sections.

00:01:11.800 --> 00:01:17.510
So this right over here, this
represents 1/8 of the whole.

00:01:17.510 --> 00:01:20.420
And so we see conceptually
that 1/2 times 1/4,

00:01:20.420 --> 00:01:26.080
it completely makes sense,
that 1/2 of 1/4 should be 1/8.

00:01:26.080 --> 00:01:28.670
And it hopefully makes
sense that you get this 8

00:01:28.670 --> 00:01:30.970
by multiplying
the 2 times the 4.

00:01:30.970 --> 00:01:32.850
You started with
4 equal sections,

00:01:32.850 --> 00:01:35.170
but then you divided each
of those 4 equal sections

00:01:35.170 --> 00:01:36.760
into 2 equal sections.

00:01:36.760 --> 00:01:40.310
So then you have 8
total equal sections

00:01:40.310 --> 00:01:42.305
that you split your whole into.

00:01:42.305 --> 00:01:43.930
Let's do another
example, but now let's

00:01:43.930 --> 00:01:46.010
multiply two
fractions that don't

00:01:46.010 --> 00:01:48.350
have 1's in the numerator.

00:01:48.350 --> 00:01:58.770
So let's multiply, let's
multiply 2/3 times 4/5.

00:01:58.770 --> 00:02:00.580
And I encourage you
now to pause the video

00:02:00.580 --> 00:02:03.740
and do something very
similar to what I just did.

00:02:03.740 --> 00:02:06.980
Try to represent 4/5
of a whole and then try

00:02:06.980 --> 00:02:11.200
to represent 2/3 of that
4/5 and see what fraction

00:02:11.200 --> 00:02:12.730
of the whole you actually have.

00:02:12.730 --> 00:02:14.927
So pause now.

00:02:14.927 --> 00:02:16.010
So let's think about this.

00:02:16.010 --> 00:02:18.320
Let's represent 4/5.

00:02:18.320 --> 00:02:21.320
So if I have a
whole like this, let

00:02:21.320 --> 00:02:24.610
me try to divide it
into 5 equal sections.

00:02:24.610 --> 00:02:28.570
5 equal sections, so let's say
that is 1 equal section, that

00:02:28.570 --> 00:02:33.157
is 2 equal sections, that is
3, 4, and 5-- I can do a better

00:02:33.157 --> 00:02:33.740
job than this.

00:02:33.740 --> 00:02:35.002
This is always the hard part.

00:02:35.002 --> 00:02:36.960
I'm trying my best to
make them look, at least,

00:02:36.960 --> 00:02:42.229
like equal sections--
2, 3, 4, and 5.

00:02:42.229 --> 00:02:43.520
I think you get the point here.

00:02:43.520 --> 00:02:45.840
I'm trying to make
them equal sections.

00:02:45.840 --> 00:02:47.770
And we want 4/5.

00:02:47.770 --> 00:02:50.630
So we want 4 of these
5 equal sections.

00:02:50.630 --> 00:02:56.170
So this would be 1 of the 5
equal sections, 2 of them,

00:02:56.170 --> 00:02:59.550
3 of them, and then 4 of them.

00:02:59.550 --> 00:03:02.010
So that right over there is 4/5.

00:03:02.010 --> 00:03:05.740
Now we can view this
as 2/3 of the 4/5.

00:03:05.740 --> 00:03:07.040
So how can we think about that?

00:03:07.040 --> 00:03:09.830
Well, we could take this section
and divide it into thirds.

00:03:09.830 --> 00:03:11.500
So let's do that.

00:03:11.500 --> 00:03:12.820
Divide it into thirds.

00:03:12.820 --> 00:03:15.270
So we're going divide it
into 3 equal sections.

00:03:15.270 --> 00:03:18.120
So that's 1/3, and then 2/3.

00:03:18.120 --> 00:03:19.870
So we took each of
the 5 equal sections,

00:03:19.870 --> 00:03:22.760
and we divided them
into 3 equal sections.

00:03:22.760 --> 00:03:26.280
Now what's going to
be 2/3 of the 4/5?

00:03:26.280 --> 00:03:30.020
Well, that's going to be
this part right over here.

00:03:30.020 --> 00:03:31.900
So let me make this clear.

00:03:31.900 --> 00:03:37.800
This is 1/3 of the 4/5.

00:03:37.800 --> 00:03:42.620
And then this would
be 2/3 of the 4/5.

00:03:42.620 --> 00:03:48.280
So this right over here, would
be 2/3 of the 4/5, or 2/3 times

00:03:48.280 --> 00:03:49.260
4/5.

00:03:49.260 --> 00:03:52.020
But what fraction of the
whole does that represent?

00:03:52.020 --> 00:03:55.770
Well, how many total, how many
total equal sections do we now

00:03:55.770 --> 00:03:56.750
have?

00:03:56.750 --> 00:04:04.696
Well, we have 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14,

00:04:04.696 --> 00:04:05.540
15.

00:04:05.540 --> 00:04:08.190
So we have 15 equal sections.

00:04:08.190 --> 00:04:09.400
I'm using a new color.

00:04:09.400 --> 00:04:11.470
We have 15 equal sections,
and that make sense.

00:04:11.470 --> 00:04:13.030
We started with
5 equal sections,

00:04:13.030 --> 00:04:16.529
but then we divided each of
those into 3 equal sections.

00:04:16.529 --> 00:04:20.370
So now we have 5 times
3 total equal sections.

00:04:20.370 --> 00:04:22.500
And then how many of
those are now colored in?

00:04:22.500 --> 00:04:24.263
Well, we see it's 2 times 4.

00:04:24.263 --> 00:04:29.370
1, 2, 3, 4, 5, 6, 7, 8.

00:04:29.370 --> 00:04:32.390
How many of them are in the
2/3 of the 4/5, I should say.

00:04:32.390 --> 00:04:36.380
And there's 8 of them, 8
of the 15 equals sections.

00:04:36.380 --> 00:04:37.740
And so there you have it.

00:04:37.740 --> 00:04:40.050
It should hopefully
now make visual sense,

00:04:40.050 --> 00:04:43.390
or it makes conceptual
sense, that 2/3 times 4/5--

00:04:43.390 --> 00:04:45.440
you can obviously compute
it by just multiplying

00:04:45.440 --> 00:04:47.320
the numerators, 2 times 4 is 8.

00:04:47.320 --> 00:04:50.660
And then multiplying the
denominators, 3 times 5 is 15--

00:04:50.660 --> 00:04:55.808
but hopefully this now makes
conceptual sense as 2/3 of 4/5.


Multiplying fractions and whole numbers: two approaches

WEBVTT
Kind: captions
Language: en

00:00:00.402 --> 00:00:01.860
Let's think a little
bit about what

00:00:01.860 --> 00:00:07.070
it means to multiply
2/3 times 6.

00:00:07.070 --> 00:00:11.370
One way to think about it
is to literally take six 2/3

00:00:11.370 --> 00:00:13.390
and add them together.

00:00:13.390 --> 00:00:15.930
This is six 2/3 right over here.

00:00:15.930 --> 00:00:18.230
And if we wanted to
actually compute this,

00:00:18.230 --> 00:00:20.400
this would be equal
to-- well, we're

00:00:20.400 --> 00:00:23.730
going to take these six
2's and add them together.

00:00:23.730 --> 00:00:30.840
So we could view it
as 2 times 6 over 3.

00:00:30.840 --> 00:00:34.740
2 times 6 over 3, which
is the same thing,

00:00:34.740 --> 00:00:41.010
of course, as 2, 4,
6, 8, 10, 12, 12/3.

00:00:43.790 --> 00:00:45.980
And what is 12/3 equal to?

00:00:45.980 --> 00:00:49.040
Well, we could rewrite
12 as-- so this is equal

00:00:49.040 --> 00:00:54.790
to-- we could rewrite 12
as 3 plus 3 plus 3 plus 3

00:00:54.790 --> 00:00:56.540
over the yellow 3.

00:00:59.930 --> 00:01:01.590
Let me do it like
this so I don't

00:01:01.590 --> 00:01:03.340
have to keep switching colors.

00:01:03.340 --> 00:01:07.450
This is going to be
the same thing as 3/3

00:01:07.450 --> 00:01:10.734
plus 3/3 plus 3/3 plus 3/3.

00:01:10.734 --> 00:01:12.400
And each of these are
obviously a whole.

00:01:12.400 --> 00:01:13.803
Each of these equal 1.

00:01:16.390 --> 00:01:21.330
That's 1 and that's 1, so this
is going to be equal to 4.

00:01:21.330 --> 00:01:25.260
So that's one way to
conceptualize 2/3 times 6.

00:01:25.260 --> 00:01:29.970
Another way to think
of it is as 2/3 of 6.

00:01:29.970 --> 00:01:31.530
So let's think about that.

00:01:31.530 --> 00:01:33.360
Let me draw a number line here.

00:01:33.360 --> 00:01:36.350
And I'm going to draw
the number line up to 6.

00:01:36.350 --> 00:01:39.330
So what I care about is the
section of the number line

00:01:39.330 --> 00:01:40.980
that goes to 6.

00:01:40.980 --> 00:01:42.600
So that looks pretty good.

00:01:42.600 --> 00:01:52.820
So this is 1, 2, 3, 4, 5, and 6.

00:01:52.820 --> 00:01:56.120
So if we want to
take 2/3 of 6, we

00:01:56.120 --> 00:01:58.650
can think of this whole
section of the number

00:01:58.650 --> 00:02:03.080
line between 0 and
6 as the whole.

00:02:03.080 --> 00:02:05.345
And then we want to
take 2/3 of that.

00:02:05.345 --> 00:02:06.220
So how do we do that?

00:02:06.220 --> 00:02:09.400
Well, we divide it into thirds,
to three equals sections.

00:02:09.400 --> 00:02:11.460
So that's one equal
section, two equal sections,

00:02:11.460 --> 00:02:12.880
and three equal sections.

00:02:12.880 --> 00:02:14.790
And we want two of those thirds.

00:02:14.790 --> 00:02:19.080
So we want 1/3 and 2/3.

00:02:19.080 --> 00:02:20.590
Now where does that get us?

00:02:20.590 --> 00:02:24.070
That gets us to 4.

00:02:24.070 --> 00:02:26.160
So we get, obviously,
to the same answer.

00:02:26.160 --> 00:02:28.437
We would be in a tough
situation if somehow we

00:02:28.437 --> 00:02:29.520
got two different answers.

00:02:29.520 --> 00:02:32.450
Either way, 2/3
times 6 or 6 times

00:02:32.450 --> 00:02:36.540
2/3, either way, that is
going to be equal to 4.

00:02:36.540 --> 00:02:38.710
But there are two different
ways of viewing this.

00:02:38.710 --> 00:02:43.470
This first way is literally
viewing it as 2/3 six times.

00:02:43.470 --> 00:02:47.700
And this way is we're taking
a fraction of the number 6.

00:02:47.700 --> 00:02:53.208
We're going 2/3 of the way to
6, which would get us to 4.


Subtracting decimals: example 2

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:04.816
Let's try to calculate
10.1 minus 3.93.

00:00:04.816 --> 00:00:06.440
And I encourage you
to pause this video

00:00:06.440 --> 00:00:08.350
and try it on your
own first, and then we

00:00:08.350 --> 00:00:11.590
can think about whether
we did it the same way.

00:00:11.590 --> 00:00:13.500
So let's just
rewrite it, aligning

00:00:13.500 --> 00:00:15.380
the decimal and the
different place values.

00:00:15.380 --> 00:00:19.910
So 10.1 minus-- the 3
is in the ones place,

00:00:19.910 --> 00:00:24.910
so I'll put it right
under the 0-- 3.93.

00:00:24.910 --> 00:00:27.030
Now, let's just try
to calculate this.

00:00:27.030 --> 00:00:30.400
Now, before we subtract, we
want all the numbers on top

00:00:30.400 --> 00:00:32.469
to be larger than the
numbers on the bottom.

00:00:32.469 --> 00:00:34.010
And we don't even
have a number here.

00:00:34.010 --> 00:00:35.910
We could stick a 0 here.

00:00:35.910 --> 00:00:37.850
Let me do that in a
different color here.

00:00:37.850 --> 00:00:41.450
We could stick a 0 here.

00:00:41.450 --> 00:00:45.400
10.1 is the same thing as 10.10,
but we still face an issue

00:00:45.400 --> 00:00:45.900
here.

00:00:45.900 --> 00:00:47.120
0 is less than 3.

00:00:47.120 --> 00:00:48.370
1 is less than 9.

00:00:48.370 --> 00:00:49.850
0 is less than 3.

00:00:49.850 --> 00:00:52.040
So we're going to do a
little bit of regrouping.

00:00:52.040 --> 00:00:54.280
So let's do that regrouping.

00:00:54.280 --> 00:00:58.720
So we could take a 10 away,
one 10 away, and then one 10

00:00:58.720 --> 00:01:00.490
is the same thing as 10 ones.

00:01:00.490 --> 00:01:02.790
So I could write a
10 in the ones place.

00:01:02.790 --> 00:01:04.430
And I could take
one of those ones

00:01:04.430 --> 00:01:06.380
away so I'm going
to have nine ones,

00:01:06.380 --> 00:01:08.550
and give that one
to the tenths place.

00:01:08.550 --> 00:01:10.722
Well, one is 10 tenths.

00:01:10.722 --> 00:01:15.029
10 tenths plus 1 tenth
is going to be 11 tenths.

00:01:15.029 --> 00:01:16.570
Now, I could take
one of those tenths

00:01:16.570 --> 00:01:20.730
away and give it to
the hundredths place.

00:01:20.730 --> 00:01:23.155
1 tenth is 10 hundredths.

00:01:25.700 --> 00:01:28.860
And now I have a higher
digit in the numerator,

00:01:28.860 --> 00:01:30.852
or at least as equal
in the numerator

00:01:30.852 --> 00:01:32.060
as I have in the denominator.

00:01:32.060 --> 00:01:34.912
So 10 minus 3 is 7.

00:01:34.912 --> 00:01:36.990
10 minus 9 is 1.

00:01:36.990 --> 00:01:38.730
I have the decimal.

00:01:38.730 --> 00:01:41.180
9 minus 3 is 6.

00:01:41.180 --> 00:01:43.610
And then I have
nothing over here.

00:01:43.610 --> 00:01:49.052
So 10.1 minus 3.93, 6.17.


Another example constructing box plot

WEBVTT
Kind: captions
Language: en

00:00:00.501 --> 00:00:01.941
- [Voiceover] Represent the following data

00:00:01.941 --> 00:00:05.041
using a box-and-whiskers plot.

00:00:05.041 --> 00:00:07.588
Once again, exclude the median
when computing the quartiles.

00:00:07.588 --> 00:00:10.041
And they gave us a bunch of data points,

00:00:10.041 --> 00:00:12.071
and it says, if it helps, you
might drag the numbers around,

00:00:12.071 --> 00:00:14.281
which I will do, because
that will be useful.

00:00:14.281 --> 00:00:15.324
And they say the order isn't checked,

00:00:15.324 --> 00:00:18.021
and that's because I'm doing
this on Khan Academy exercises.

00:00:18.021 --> 00:00:19.284
Up here in the top right,
where you can't see,

00:00:19.284 --> 00:00:20.905
there's actually a check answer.

00:00:20.905 --> 00:00:22.961
So I encourage you to use
the exercises yourself,

00:00:22.961 --> 00:00:24.941
but let's just use this as an example.

00:00:24.941 --> 00:00:27.921
So the first thing, if I'm
going to do a box-and-whiskers,

00:00:27.921 --> 00:00:29.301
I'm going to order these numbers.

00:00:29.301 --> 00:00:31.265
So let me order these numbers
from least to greatest.

00:00:31.265 --> 00:00:31.802
So let's see.

00:00:31.802 --> 00:00:36.682
There's a one here, and
we've got some twos.

00:00:36.682 --> 00:00:41.439
We've got some twos here and some threes,

00:00:42.243 --> 00:00:44.809
some threes, some four--

00:00:44.809 --> 00:00:47.060
I have one four and fives.

00:00:47.767 --> 00:00:48.515
I have a six.

00:00:48.515 --> 00:00:49.743
I have a seven.

00:00:49.743 --> 00:00:52.062
I have a couple of
eights, and I have a 10.

00:00:52.062 --> 00:00:52.842
So there you go.

00:00:52.842 --> 00:00:55.602
I have ordered these numbers
from least to greatest,

00:00:55.602 --> 00:00:58.872
and now, well just like that,
I can plot the whiskers,

00:00:58.872 --> 00:01:00.077
because I see the range.

00:01:00.077 --> 00:01:01.786
My lowest number is one.

00:01:01.786 --> 00:01:03.422
So my lowest number is one.

00:01:03.422 --> 00:01:04.811
My largest number is 10.

00:01:05.718 --> 00:01:08.162
So the whiskers help
me visualize the range.

00:01:08.162 --> 00:01:10.930
Now let me think about the
median of my data set is.

00:01:10.930 --> 00:01:15.462
So my median here is
going to be, let's see.

00:01:15.462 --> 00:01:18.282
I have one, two, three,
four, five, six, seven,

00:01:18.282 --> 00:01:22.742
eight, nine, 10, 11, 12, 13, 14 numbers.

00:01:22.742 --> 00:01:24.577
Since I have an even number of numbers,

00:01:24.577 --> 00:01:27.134
the middle two numbers are
going to help define my median,

00:01:27.134 --> 00:01:28.522
because there's no one middle number.

00:01:28.522 --> 00:01:31.902
I might say this number
right over here, this four,

00:01:31.902 --> 00:01:33.982
but notice, there's one, two, three, four,

00:01:33.982 --> 00:01:35.718
five, six, seven above it,

00:01:35.718 --> 00:01:39.702
and there's only one, two,
three, four, five, six below it.

00:01:39.702 --> 00:01:40.849
Same thing would have
been true for this five.

00:01:40.849 --> 00:01:42.080
So this four and five,

00:01:42.080 --> 00:01:44.582
the middle is actually
in between these two.

00:01:44.582 --> 00:01:46.502
So when you have an even
number of numbers like this,

00:01:46.502 --> 00:01:49.782
you take the middle two numbers,
this four and this five,

00:01:49.782 --> 00:01:51.702
and you take the mean of the two.

00:01:51.702 --> 00:01:55.002
So the mean of four and five
is going to be four-and-a-half.

00:01:55.002 --> 00:01:57.562
So that's going to be the
median of our entire data set,

00:01:57.562 --> 00:02:01.563
four-and-a-half, four-and-a-half.

00:02:01.563 --> 00:02:04.002
And now, I want to figure out the median

00:02:04.002 --> 00:02:07.202
of the bottom half of numbers
and the top half of numbers.

00:02:07.202 --> 00:02:08.782
And here they say exclude the median.

00:02:08.782 --> 00:02:10.002
Of course I'm going to exclude the median.

00:02:10.002 --> 00:02:12.357
It's not even included in
our data points right here,

00:02:12.357 --> 00:02:15.702
because our median is 4.5.

00:02:15.702 --> 00:02:17.868
So now let's take this
bottom half of numbers.

00:02:17.868 --> 00:02:20.742
Let's take this bottom half
of numbers right over here

00:02:20.742 --> 00:02:22.182
and find the middle.

00:02:22.182 --> 00:02:24.182
So this is the bottom seven numbers.

00:02:24.182 --> 00:02:26.142
And so the median of those is going to be

00:02:26.142 --> 00:02:28.098
the one which has three on either side,

00:02:28.098 --> 00:02:30.958
so it's going to be this
two right over here.

00:02:30.958 --> 00:02:33.822
So that right over there is kind of

00:02:33.822 --> 00:02:35.978
the left boundary of our box,

00:02:35.978 --> 00:02:37.543
and then for the right boundary,

00:02:37.543 --> 00:02:40.022
we need to figure out the middle
of our top half of numbers.

00:02:40.022 --> 00:02:42.482
Remember, four and five
were our middle two numbers.

00:02:42.482 --> 00:02:45.182
Our median is right in
between at four-and-a-half.

00:02:45.182 --> 00:02:47.322
So our top half of numbers
starts at this five

00:02:47.322 --> 00:02:48.382
and goes to this 10.

00:02:48.382 --> 00:02:49.242
Seven numbers.

00:02:49.242 --> 00:02:51.408
The middle one's going to
have three on both sides.

00:02:51.408 --> 00:02:53.762
The seven has three to the left,

00:02:53.762 --> 00:02:54.942
remember of the top half,

00:02:54.942 --> 00:02:56.922
and three to the right.

00:02:56.922 --> 00:02:58.268
And so the seven is,

00:02:58.268 --> 00:03:00.485
I guess you could say the
right side of our box.

00:03:00.485 --> 00:03:01.091
And we're done.

00:03:01.091 --> 00:03:03.442
We've constructed our
box-and-whiskers plot,

00:03:03.442 --> 00:03:06.488
which helps us visualize the entire range

00:03:06.488 --> 00:03:08.760
but also you could say the middle,

00:03:08.760 --> 00:03:11.942
roughly the middle half of our numbers.


Box and whisker plot exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.647 --> 00:00:02.170
- [Voiceover] Represent the following data

00:00:02.170 --> 00:00:05.545
using a box-and-whiskers plot.

00:00:05.545 --> 00:00:09.064
Exclude the median when
computing the quartiles.

00:00:09.064 --> 00:00:11.771
Alright, let's see if we can do this.

00:00:11.771 --> 00:00:13.187
So, we have a bunch of data, here,

00:00:13.187 --> 00:00:14.953
and they say if it helps,
you might drag the numbers

00:00:14.953 --> 00:00:16.078
to put them in a different order.

00:00:16.078 --> 00:00:18.076
So, we can drag these numbers
around, which is useful,

00:00:18.076 --> 00:00:20.249
because we will want to order them.

00:00:20.249 --> 00:00:22.057
The order isn't checked with your answer,

00:00:22.057 --> 00:00:24.518
and I'm doing this off of
the Khan Academy exercises,

00:00:24.518 --> 00:00:26.376
so I don't have my drawing tablet here.

00:00:26.376 --> 00:00:27.977
I just have my mouse, and I'm interacting

00:00:27.977 --> 00:00:30.067
with the exercise, which
I encourage you to do too,

00:00:30.067 --> 00:00:31.856
because the best way to
learn any of this stuff

00:00:31.856 --> 00:00:33.828
is to actually practice
it, and at Khan Academy,

00:00:33.828 --> 00:00:37.447
we have 150,000 exercises
for you to practice with.

00:00:37.447 --> 00:00:39.507
Anyway, so, let's do this.

00:00:39.507 --> 00:00:41.398
Let's order this thing,

00:00:41.398 --> 00:00:44.243
so we can figure out the range of numbers.

00:00:44.243 --> 00:00:45.427
What's the lowest and what's the highest?

00:00:45.427 --> 00:00:48.364
Let's see, there's a seven here.

00:00:48.364 --> 00:00:50.708
Then, let's see, we have some eights.

00:00:50.708 --> 00:00:53.321
We've got some eights goin' on.

00:00:53.321 --> 00:00:54.945
Then we have some nines,

00:00:54.945 --> 00:00:58.252
actually we have a bunch of
nines, we have four nines here.

00:00:58.252 --> 00:01:00.632
We have some nines, and then, let's see,

00:01:00.632 --> 00:01:02.166
13 is the largest number.

00:01:02.166 --> 00:01:03.963
There we go, we've ordered the numbers.

00:01:03.963 --> 00:01:06.241
So, our smallest number is seven, and this

00:01:06.241 --> 00:01:07.518
is what the whiskers are useful for.

00:01:07.518 --> 00:01:10.336
For helping us figure out
the entire range of numbers.

00:01:10.336 --> 00:01:12.058
Our smallest number is seven.

00:01:12.058 --> 00:01:16.155
Our largest number is
13, so we know the range.

00:01:16.155 --> 00:01:17.583
Now, let's plot the median,

00:01:17.583 --> 00:01:20.235
and this'll help us once getting

00:01:20.235 --> 00:01:21.873
this center line of our box,

00:01:21.873 --> 00:01:24.020
but then also we need to
do that, to figure out

00:01:24.020 --> 00:01:25.094
what these other lines are,

00:01:25.094 --> 00:01:26.891
that kind of define the box.

00:01:26.891 --> 00:01:29.576
To define the middle two
fourths of our number.

00:01:29.576 --> 00:01:31.676
Of our data, or the middle two quartiles.

00:01:31.676 --> 00:01:33.094
Roughly the middle two quartiles,

00:01:33.094 --> 00:01:35.718
it depends how some of
the numbers work out.

00:01:35.718 --> 00:01:37.980
But this middle line is going

00:01:37.980 --> 00:01:40.403
to be the median of our entire data set.

00:01:40.403 --> 00:01:42.217
Now, the median is just the middle number.

00:01:42.217 --> 00:01:45.268
If we sort them in order, median
is just the middle number.

00:01:45.268 --> 00:01:46.838
We have 11 numbers here.

00:01:46.838 --> 00:01:49.549
So, the middle one is gonna
have five on either side.

00:01:49.549 --> 00:01:51.606
It's just gonna be this nine.

00:01:51.606 --> 00:01:53.873
If we had 10 numbers here,

00:01:53.873 --> 00:01:56.113
if we had an even number of
numbers, you actually would

00:01:56.113 --> 00:01:58.761
of had two middle numbers,
and then to find the median,

00:01:58.761 --> 00:02:01.279
you'd found the mean of those two.

00:02:01.279 --> 00:02:02.730
If that last sentence was confusing,

00:02:02.730 --> 00:02:05.832
watch the videos on
Khan Academy, on median,

00:02:05.832 --> 00:02:08.468
and I go into much more detail on that.

00:02:08.468 --> 00:02:10.097
But here, I have 11 numbers,

00:02:10.097 --> 00:02:11.774
so my median is going
to be the middle one.

00:02:11.774 --> 00:02:13.794
It has five larger, five less,

00:02:13.794 --> 00:02:16.265
it's this nine, right over here.

00:02:16.265 --> 00:02:18.687
If I had my pen tablet, I would circle it.

00:02:18.687 --> 00:02:19.760
So, it's this nine.

00:02:19.760 --> 00:02:21.883
That is the median.

00:02:21.883 --> 00:02:26.208
Now, we need to figure out
what number is half way,

00:02:27.254 --> 00:02:28.718
what number is the median

00:02:28.718 --> 00:02:31.224
of the numbers in this bottom half?

00:02:31.224 --> 00:02:32.751
They told us to exclude the median

00:02:32.751 --> 00:02:34.516
when we compute the quartiles.

00:02:34.516 --> 00:02:35.456
So, this was the median.

00:02:35.456 --> 00:02:36.629
Let's ignore that.

00:02:36.629 --> 00:02:38.068
So, let's look at all of
the numbers below that.

00:02:38.068 --> 00:02:40.972
So, this nine, eight,
eight, eight, and seven.

00:02:40.972 --> 00:02:42.055
So, we have five numbers.

00:02:42.055 --> 00:02:44.615
What's the median of these five numbers?

00:02:44.615 --> 00:02:46.081
Well the median's the middle number.

00:02:46.081 --> 00:02:47.237
That is eight.

00:02:47.237 --> 00:02:50.693
So, the beginning of our second quartile

00:02:50.693 --> 00:02:52.526
is gonna be an eight, right over there.

00:02:52.526 --> 00:02:54.913
We do the same thing
for our third quartile.

00:02:54.913 --> 00:02:57.059
Remember, this was our median
of our entire data set.

00:02:57.059 --> 00:02:58.210
Let's exclude it.

00:02:58.210 --> 00:03:01.476
Let's look at the top half
of the numbers, so to speak.

00:03:01.476 --> 00:03:04.119
There's five numbers here, in order.

00:03:04.119 --> 00:03:06.894
So, the middle one, the
median of this is 10.

00:03:06.894 --> 00:03:09.622
So, that's gonna be the
top of our second quartile.

00:03:09.622 --> 00:03:10.887
Just like that, we're done.

00:03:10.887 --> 00:03:14.819
We have constructed our
box-and-whisker plot.


Quadrilateral on the coordinate plane

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:04.695
Use the line segments below
to create a quadrilateral, so

00:00:04.695 --> 00:00:07.880
a four-sided figure,
with vertices

00:00:07.880 --> 00:00:09.330
at the following points.

00:00:09.330 --> 00:00:11.470
So they give us four
coordinates right over here.

00:00:11.470 --> 00:00:14.140
So those have to
be where the sides

00:00:14.140 --> 00:00:16.660
of our quadrilateral intersect.

00:00:16.660 --> 00:00:19.940
So one is at the point 0, 9.

00:00:19.940 --> 00:00:22.020
So our x-coordinate is 0.

00:00:22.020 --> 00:00:22.930
So we're at 0.

00:00:22.930 --> 00:00:25.990
And then our y-coordinate, or
our vertical coordinate, is 9.

00:00:25.990 --> 00:00:31.690
So we need to have a
vertex right over here.

00:00:31.690 --> 00:00:35.005
Now, the next one
is at 0, negative 7.

00:00:38.354 --> 00:00:39.770
So let me just put
something here.

00:00:39.770 --> 00:00:43.300
We can later rearrange it to
see what actually makes sense.

00:00:43.300 --> 00:00:45.950
Then we have 8, negative 7.

00:00:45.950 --> 00:00:52.320
So 8, negative 7
Is right over here.

00:00:52.320 --> 00:00:54.610
It feels like this
might be the right edge.

00:00:54.610 --> 00:00:59.090
So I'll put its other point
right at 8, negative 7.

00:00:59.090 --> 00:01:01.375
And then finally, 8 comma 0.

00:01:04.810 --> 00:01:09.560
Well, 8 comma 0 Is
right over here.

00:01:09.560 --> 00:01:14.469
So it looks like I have a point
on every one of the vertices.

00:01:14.469 --> 00:01:16.260
So now I just have to
connect to these two,

00:01:16.260 --> 00:01:17.676
and I'll have a
four-sided figure.

00:01:17.676 --> 00:01:21.690
So let me take this there,
bring this one over.

00:01:21.690 --> 00:01:27.000
And let me move this one up,
and then bring this one down.

00:01:27.000 --> 00:01:28.010
And there you have it.

00:01:28.010 --> 00:01:30.860
I think I've got
my quadrilateral.


Parallelogram on the coordinate plane

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:05.980
You are graphing polygon
ABCD in the coordinate plane.

00:00:05.980 --> 00:00:09.800
The length of segment
AB must be the same

00:00:09.800 --> 00:00:13.530
as the length of segment
DC, and both segments

00:00:13.530 --> 00:00:15.820
are horizontal segments.

00:00:15.820 --> 00:00:19.710
The following are three of
the vertices of the polygon.

00:00:19.710 --> 00:00:22.730
Vertex A is at the point 1, 1.

00:00:25.320 --> 00:00:27.950
It puts us right over there.

00:00:27.950 --> 00:00:34.100
That is the vertex A. Vertex
C is at the point 4.5 comma

00:00:34.100 --> 00:00:39.990
4, so 4.5 along the
horizontal axis comma 4.

00:00:39.990 --> 00:00:46.920
So we go all the way up to 4.

00:00:46.920 --> 00:00:49.710
That right over
there is point C.

00:00:49.710 --> 00:00:55.430
Point D is at negative 1.5
comma 4, so negative 1.5

00:00:55.430 --> 00:00:58.750
along the horizontal or
the x-axis, we could say,

00:00:58.750 --> 00:01:04.604
negative 1.5 comma 4, so 4 along
the vertical or the y-axis.

00:01:04.604 --> 00:01:07.890
We go right over there.

00:01:07.890 --> 00:01:09.590
That's close enough.

00:01:09.590 --> 00:01:11.250
So that, of course,
is our y-axis.

00:01:11.250 --> 00:01:13.230
This is point D.

00:01:13.230 --> 00:01:16.990
And we need to figure out what
are the coordinates of point B

00:01:16.990 --> 00:01:20.860
if B must be in quadrant
I. And they tell us

00:01:20.860 --> 00:01:23.290
that the distance
from A to B must

00:01:23.290 --> 00:01:24.960
be the same as the
length of segment

00:01:24.960 --> 00:01:27.190
D to C in both our horizontals.

00:01:27.190 --> 00:01:30.200
So let's draw what
we know to draw.

00:01:30.200 --> 00:01:34.110
So DC, segment DC is this
segment right over here.

00:01:36.970 --> 00:01:38.500
And we see it's horizontal.

00:01:38.500 --> 00:01:41.880
Both of the vertical coordinates
are 4 at both vertex D

00:01:41.880 --> 00:01:47.389
and vertex C. So both of the
vertical coordinates are 4.

00:01:47.389 --> 00:01:48.680
Now what is the length of this?

00:01:48.680 --> 00:01:51.013
Because we're going to have
to construct another segment

00:01:51.013 --> 00:01:52.180
that has the same length.

00:01:52.180 --> 00:01:54.040
Well, along the
horizontal direction,

00:01:54.040 --> 00:01:58.790
we went from
negative 1.5 to 4.5.

00:01:58.790 --> 00:02:00.940
So how far did we go?

00:02:00.940 --> 00:02:06.630
Well, to go from negative
1.5 to 0, you go 1.5,

00:02:06.630 --> 00:02:09.410
and then you have
to go another 4.5.

00:02:09.410 --> 00:02:15.340
So this is going to be 4.5
plus 1.5, which is equal to 4

00:02:15.340 --> 00:02:21.470
plus 1 is 5, 0.5 plus 0.5
is 1, so 5 plus 1 is 6.

00:02:21.470 --> 00:02:26.865
So this distance right over
here is 6 of our units.

00:02:26.865 --> 00:02:28.740
Actually, let me put
the coordinates in here,

00:02:28.740 --> 00:02:32.080
just so it becomes a
little bit clearer.

00:02:32.080 --> 00:02:34.476
Let me do that in
something easier to see.

00:02:34.476 --> 00:02:35.850
This right over
here is the point

00:02:35.850 --> 00:02:46.750
4.5, 4, and this right over here
is the point negative 1.5, 4.

00:02:46.750 --> 00:02:48.690
Another way of thinking
about this distance

00:02:48.690 --> 00:02:50.277
is you could take
the end point--

00:02:50.277 --> 00:02:52.110
and we're really thinking
about the distance

00:02:52.110 --> 00:02:55.619
just along its horizontal line,
so the y-value does not change.

00:02:55.619 --> 00:02:57.660
It doesn't change in the
vertical direction, only

00:02:57.660 --> 00:02:59.020
the horizontal.

00:02:59.020 --> 00:03:02.600
So you really want to say,
if you start at negative 1.5

00:03:02.600 --> 00:03:05.530
and you get to 4.5,
how far have you gone?

00:03:05.530 --> 00:03:09.300
So you can just take your end
point, your end value, your end

00:03:09.300 --> 00:03:11.550
horizontal value or
your end x-value,

00:03:11.550 --> 00:03:14.790
and from that, you can
subtract your starting x-value.

00:03:14.790 --> 00:03:18.670
So you subtract negative 1.5.

00:03:18.670 --> 00:03:23.850
And this, of course, is equal
to 4.5 plus positive 1.5,

00:03:23.850 --> 00:03:28.700
which, once again,
is equal to 6.

00:03:28.700 --> 00:03:29.220
Fair enough.

00:03:29.220 --> 00:03:31.320
And let me draw some of
the rest of the polygon,

00:03:31.320 --> 00:03:34.624
just so that we see it
is indeed a polygon.

00:03:34.624 --> 00:03:36.040
We have this side
right over here.

00:03:36.040 --> 00:03:38.620
It looks like it's going
to be a parallelogram.

00:03:38.620 --> 00:03:41.300
We have this side
right over here.

00:03:41.300 --> 00:03:43.340
And we have to replace point B.

00:03:43.340 --> 00:03:46.550
Now, point B is going to
be someplace out here.

00:03:46.550 --> 00:03:49.460
It's going to have the
same vertical value

00:03:49.460 --> 00:03:51.950
or the same y-value as
point A. So its y-coordinate

00:03:51.950 --> 00:03:53.680
is going to be 1.

00:03:53.680 --> 00:03:57.320
So point B is going to
be out here someplace.

00:03:57.320 --> 00:03:58.735
Let me do this in a new color.

00:04:01.650 --> 00:04:03.790
I haven't used this orange yet.

00:04:03.790 --> 00:04:05.350
Actually, I have
used the orange yet.

00:04:05.350 --> 00:04:06.749
I haven't use the yellow.

00:04:06.749 --> 00:04:07.790
No, I've used the yellow.

00:04:07.790 --> 00:04:08.620
Let's see.

00:04:08.620 --> 00:04:09.820
I haven't used this green.

00:04:09.820 --> 00:04:12.630
Point B is going to
be someplace out here.

00:04:15.270 --> 00:04:17.538
We already know what
its y-coordinate is.

00:04:17.538 --> 00:04:19.079
It's a horizontal
line, so it's going

00:04:19.079 --> 00:04:23.370
to have to have the same exact
y-coordinate as point A. Point

00:04:23.370 --> 00:04:25.890
A's y-coordinate
was 1, so this is

00:04:25.890 --> 00:04:29.040
going to have to have
a y-coordinate of 1.

00:04:29.040 --> 00:04:35.691
Now the big question is, what
is its x-coordinate going to be?

00:04:35.691 --> 00:04:37.190
Let me do that in
a different color.

00:04:37.190 --> 00:04:40.370
It's going to have to be
whatever A's x-coordinate was.

00:04:43.590 --> 00:04:46.020
We see that A's
x-coordinate was 1.

00:04:46.020 --> 00:04:48.505
And it's going to have
to be that plus 6,

00:04:48.505 --> 00:04:50.380
because we're going to
move the same distance

00:04:50.380 --> 00:04:51.546
in the horizontal direction.

00:04:51.546 --> 00:04:53.240
This thing has to be 6.

00:04:53.240 --> 00:04:58.150
So if we start at 1,
we add 6, we get to 7.

00:04:58.150 --> 00:05:01.130
So what are the coordinates
of point B, especially

00:05:01.130 --> 00:05:03.050
if point B must
be in quadrant I?

00:05:03.050 --> 00:05:06.440
And notice we are definitely in
quadrant I. This is quadrant I,

00:05:06.440 --> 00:05:10.030
this is quadrant II,
this is quadrant III,

00:05:10.030 --> 00:05:11.990
and this is quadrant IV.

00:05:11.990 --> 00:05:15.662
The coordinate for
point B is 7 comma 1.


Volume of a rectangular prism: word problem

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:02.610
- [Instructor] Mario has a fish tank

00:00:02.610 --> 00:00:05.400
that is a right rectangular prism

00:00:05.400 --> 00:00:10.400
with base 15.6 centimeters
by 7.2 centimeters.

00:00:11.610 --> 00:00:12.990
So let's try to imagine that.

00:00:12.990 --> 00:00:15.120
So it's a right rectangular prism.

00:00:15.120 --> 00:00:15.953
This is a fish tank.

00:00:15.953 --> 00:00:17.520
Let me actually do it in blue.

00:00:17.520 --> 00:00:19.710
So it, one of the dimensions,

00:00:19.710 --> 00:00:21.630
that's not blue, that's orange.

00:00:21.630 --> 00:00:26.630
One of the dimensions is 15.6 centimeters.

00:00:27.420 --> 00:00:29.970
15.6 centimeters.

00:00:29.970 --> 00:00:34.457
And then the other dimension
of the base is 7.2 centimeters.

00:00:37.050 --> 00:00:38.280
7.2 centimeters.

00:00:38.280 --> 00:00:40.920
So this is the base right over here.

00:00:40.920 --> 00:00:42.570
So let me draw this,

00:00:42.570 --> 00:00:45.450
try to put some perspective in there.

00:00:45.450 --> 00:00:47.730
And, of course, it is a
right rectangular prism,

00:00:47.730 --> 00:00:50.310
this fish tank that Mario has.

00:00:50.310 --> 00:00:52.920
So it looks something like this.

00:00:52.920 --> 00:00:55.260
So this is his fish tank.

00:00:55.260 --> 00:00:58.200
Try to draw it as neatly as I can.

00:00:58.200 --> 00:01:03.200
So that's the top of the
fish tank, just like that.

00:01:04.260 --> 00:01:07.740
I think this does a
decent, respectable job

00:01:07.740 --> 00:01:09.660
of what this fish tank might look like.

00:01:09.660 --> 00:01:12.270
And let me erase this
thing right over here.

00:01:12.270 --> 00:01:13.290
And there we go.

00:01:13.290 --> 00:01:15.870
There is Mario's fish tank.

00:01:15.870 --> 00:01:16.800
There is his fish tank,

00:01:16.800 --> 00:01:18.240
and we can even make it look like glass.

00:01:18.240 --> 00:01:20.190
There you go, that looks nice.

00:01:20.190 --> 00:01:23.700
All right, the bottom of the
tank is filled with marbles.

00:01:23.700 --> 00:01:26.160
And the tank is then filled with water

00:01:26.160 --> 00:01:29.880
to a height of 6.4 centimeters.

00:01:29.880 --> 00:01:34.050
So the water is filled to a
height of 6.4 centimeters.

00:01:34.050 --> 00:01:36.960
So this is the water
when it's all filled up,

00:01:36.960 --> 00:01:39.300
6.4 centimeters.

00:01:39.300 --> 00:01:40.563
So let's draw that.

00:01:41.880 --> 00:01:43.770
And I'll make the water,

00:01:43.770 --> 00:01:45.810
well, maybe I should have made it

00:01:45.810 --> 00:01:47.040
a little more blue than this,

00:01:47.040 --> 00:01:49.050
but this gives you the picture.

00:01:49.050 --> 00:01:50.760
So the height of the
water right over here,

00:01:50.760 --> 00:01:53.010
actually, let me do that in a blue color.

00:01:53.010 --> 00:01:57.030
The height of the water
right over here is 6.4,

00:01:57.030 --> 00:01:59.400
6.4 centimeters.

00:01:59.400 --> 00:02:01.500
So that means that the distance

00:02:01.500 --> 00:02:05.130
from the bottom of the tank
to the top of, not the tank,

00:02:05.130 --> 00:02:09.240
but to the top of the
water, is 6.4 centimeters.

00:02:09.240 --> 00:02:12.390
6.4 centimeters.

00:02:12.390 --> 00:02:13.223
Fair enough.

00:02:13.223 --> 00:02:15.030
So that's the top of the water.

00:02:15.030 --> 00:02:16.170
When the marbles were removed,

00:02:16.170 --> 00:02:18.540
and it started off with
some marbles on the bottom.

00:02:18.540 --> 00:02:20.520
They don't tell us how many marbles.

00:02:20.520 --> 00:02:22.080
When the marbles are removed,

00:02:22.080 --> 00:02:26.940
the water level drops to a
height of 5.9 centimeters.

00:02:26.940 --> 00:02:28.290
So when they're removed,

00:02:28.290 --> 00:02:33.290
the water level drops by a
little bit, to 5.9 centimeters.

00:02:34.050 --> 00:02:35.490
5.9 centimeters.

00:02:35.490 --> 00:02:37.200
So it drops.

00:02:37.200 --> 00:02:38.607
It drops to 5.,

00:02:39.570 --> 00:02:41.340
it drops to 5.9,

00:02:41.340 --> 00:02:45.273
from 6.4 to 5.9 centimeters.

00:02:47.010 --> 00:02:51.420
What is the volume of the
water displaced by the marbles?

00:02:51.420 --> 00:02:52.920
So when you took the marbles out,

00:02:52.920 --> 00:02:55.110
the water dropped from 6.4.

00:02:55.110 --> 00:03:00.110
So it dropped from 6.4 centimeters
down to 5.9 centimeters.

00:03:02.760 --> 00:03:04.140
5.9 centimeters.

00:03:04.140 --> 00:03:05.610
So how much did it drop?

00:03:05.610 --> 00:03:08.580
Well, it dropped .5 centimeters,

00:03:08.580 --> 00:03:10.200
So it dropped,

00:03:10.200 --> 00:03:14.610
it dropped 0.5 centimeters.

00:03:14.610 --> 00:03:15.750
So what does that tell us about

00:03:15.750 --> 00:03:18.450
the volume of water
displaced by the marbles?

00:03:18.450 --> 00:03:20.610
Well, the volume of water
displaced by the marbles

00:03:20.610 --> 00:03:22.620
must be the amount of the,

00:03:22.620 --> 00:03:25.320
must be equivalent to this volume.

00:03:25.320 --> 00:03:28.260
This volume of this, I guess this,

00:03:28.260 --> 00:03:31.830
I guess this is another
rectangular prism that is the same,

00:03:31.830 --> 00:03:33.630
where the top area is the same

00:03:33.630 --> 00:03:35.940
as the base of this water tank.

00:03:35.940 --> 00:03:39.300
And then the height is the
height of the water drop.

00:03:39.300 --> 00:03:42.330
When you put the marbles
in, it takes up more volume,

00:03:42.330 --> 00:03:46.020
it pushes the water up by
that amount, by that volume.

00:03:46.020 --> 00:03:49.410
When you take it out, then
that volume gets replaced

00:03:49.410 --> 00:03:50.670
with the water down here,

00:03:50.670 --> 00:03:52.560
and then that volume goes back down,

00:03:52.560 --> 00:03:55.500
the water level goes
down to 5.9 centimeters.

00:03:55.500 --> 00:03:57.780
So we're essentially
trying to find the volume

00:03:57.780 --> 00:04:02.343
of a rectangular prism that is,

00:04:03.300 --> 00:04:06.210
that is equal to,

00:04:06.210 --> 00:04:10.710
so it's gonna be 15.6 by 7.2 by .5.

00:04:10.710 --> 00:04:12.210
And I haven't drawn it to scale yet,

00:04:12.210 --> 00:04:14.160
but I wanna see all of the measurements.

00:04:14.160 --> 00:04:19.160
So it's going to be 15.6
centimeters in this direction,

00:04:19.980 --> 00:04:23.940
it's going to be 7.2
centimeters in this direction,

00:04:23.940 --> 00:04:26.400
and it's going to be .5 centimeters high,

00:04:26.400 --> 00:04:29.160
0.5 centimeters high.

00:04:29.160 --> 00:04:30.510
So we know how to find volume,

00:04:30.510 --> 00:04:33.420
we just multiply the
length times the width

00:04:33.420 --> 00:04:35.010
times the height.

00:04:35.010 --> 00:04:37.620
So the volume in centimeter cubed,

00:04:37.620 --> 00:04:38.453
and we're (indistinct) scale,

00:04:38.453 --> 00:04:39.286
we're multiplying centimeters,

00:04:39.286 --> 00:04:40.470
times centimeters, times centimeters,

00:04:40.470 --> 00:04:42.630
so it's gonna be centimeters cubed,

00:04:42.630 --> 00:04:44.850
is going to be, so lemme write this down.

00:04:44.850 --> 00:04:49.850
The volume is going to be
15.6 times 7.2 times 0.5,

00:04:53.970 --> 00:04:56.400
and it's gonna be in centimeters cubed,

00:04:56.400 --> 00:04:59.070
or cubic centimeters, I
guess we could call them.

00:04:59.070 --> 00:05:02.310
Well, let's first multiply 7.2 times 0.5.

00:05:02.310 --> 00:05:04.110
We can do that in our head.

00:05:04.110 --> 00:05:06.510
This part right over
here is going to be 3.6,

00:05:06.510 --> 00:05:09.150
essentially just half of 7.2.

00:05:09.150 --> 00:05:13.490
So then this becomes 15.6 times 3.6.

00:05:14.340 --> 00:05:16.590
So let me just multiply that over here.

00:05:16.590 --> 00:05:21.570
So 15.6 times 3.6.

00:05:21.570 --> 00:05:23.970
So I'll ignore the decimals for a second.

00:05:23.970 --> 00:05:26.670
6 times 6 is 36,

00:05:26.670 --> 00:05:28.170
5 times 6 is 30,

00:05:28.170 --> 00:05:31.020
plus 3, is 33.

00:05:31.020 --> 00:05:32.370
1 times 6 is 6,

00:05:32.370 --> 00:05:34.113
plus 3, is 9.

00:05:34.113 --> 00:05:37.110
And then let's place a zero here.

00:05:37.110 --> 00:05:38.700
We're down in the 1s place,

00:05:38.700 --> 00:05:40.620
but I'm ignoring the decimals for now.

00:05:40.620 --> 00:05:43.983
3 times 6 is 18,

00:05:44.910 --> 00:05:46.397
3 times 5 is 15,

00:05:46.397 --> 00:05:48.225
plus one is 16,

00:05:48.225 --> 00:05:49.454
3 times 1 is 3,

00:05:49.454 --> 00:05:50.643
plus 1 is four.

00:05:51.540 --> 00:05:53.220
And then we get 6,

00:05:53.220 --> 00:05:58.050
3 plus 8 is 11, 16, 5.

00:05:58.050 --> 00:06:00.930
Now, if this was 156 times 36,

00:06:00.930 --> 00:06:03.750
this would be 5,616, but it's not.

00:06:03.750 --> 00:06:06.510
We have two numbers to the
right of the decimal point,

00:06:06.510 --> 00:06:10.410
one, two, so it's going to be 56.16.

00:06:10.410 --> 00:06:11.970
So the volume,

00:06:11.970 --> 00:06:14.280
the volume, and we
deserve a drum roll now,

00:06:14.280 --> 00:06:19.280
is 56.16 cubic, cubic centimeters.


Volume of a rectangular prism: fractional cubes

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:02.760
- [Instructor] So I have
this rectangular prism here,

00:00:02.760 --> 00:00:05.040
it's kind of the shape of
a brick or a fish tank,

00:00:05.040 --> 00:00:08.340
and it's made up of these unit cubes.

00:00:08.340 --> 00:00:10.560
And each of these unit cubes,

00:00:10.560 --> 00:00:14.100
we're saying is 1/4 of a foot

00:00:14.100 --> 00:00:17.430
by 1/4 of a foot by 1/4 of a foot.

00:00:17.430 --> 00:00:19.620
So you could almost imagine that this is,

00:00:19.620 --> 00:00:20.700
so let me write it this way,

00:00:20.700 --> 00:00:22.710
this is a fourth of a foot,

00:00:22.710 --> 00:00:26.940
by 1/4 of a foot, by 1/4 of a foot.

00:00:26.940 --> 00:00:31.940
Those are its length,
height, and width or depth,

00:00:32.130 --> 00:00:33.660
whatever you want to call it.

00:00:33.660 --> 00:00:35.970
So given that, what is the volume

00:00:35.970 --> 00:00:39.693
of this entire rectangular
prism going to be?

00:00:40.680 --> 00:00:42.210
So I'm assuming you've given a go at it.

00:00:42.210 --> 00:00:44.250
So there's a couple of
ways to think about it.

00:00:44.250 --> 00:00:47.070
You could first think about
the volume of each unit cube,

00:00:47.070 --> 00:00:48.930
and then think about how
many unit cubes there are.

00:00:48.930 --> 00:00:49.770
So let's do that.

00:00:49.770 --> 00:00:53.170
The unit cube, its volume is going to be

00:00:55.590 --> 00:01:00.120
1/4 of a foot times 1/4 of a foot

00:01:00.120 --> 00:01:03.390
times 1/4 of a foot.

00:01:03.390 --> 00:01:05.100
Or another way to think about it is

00:01:05.100 --> 00:01:10.100
it's going to be 1/4 times
1/4 times 1/4 cubic feet,

00:01:11.681 --> 00:01:13.980
which is often written as feet

00:01:13.980 --> 00:01:16.860
to the third power cubic feet.

00:01:16.860 --> 00:01:19.740
So 1/4 times 1/4 is 1/16

00:01:19.740 --> 00:01:22.560
times 1/4 is 1/64.

00:01:22.560 --> 00:01:26.960
So this is going to be 1/64 cubic feet,

00:01:29.040 --> 00:01:31.560
or 1/64 of a cubic foot.

00:01:31.560 --> 00:01:33.270
That's the volume of each of these.

00:01:33.270 --> 00:01:35.910
That's the volume of
each of these unit cubes.

00:01:35.910 --> 00:01:38.820
Now, how many of them are there?

00:01:38.820 --> 00:01:40.800
Well, you could view them
as kind of these two layers.

00:01:40.800 --> 00:01:44.730
The first layer has one, two, three, four,

00:01:44.730 --> 00:01:47.340
five, six, seven, eight.

00:01:47.340 --> 00:01:49.803
That's this first layer right over here.

00:01:50.880 --> 00:01:53.220
That's this first layer right over here.

00:01:53.220 --> 00:01:55.170
And then we have the
second layer down here,

00:01:55.170 --> 00:01:56.940
which would be another eight.

00:01:56.940 --> 00:01:59.100
So it's going to be 8 plus 8 or 16.

00:01:59.100 --> 00:02:00.780
So the total volume here,

00:02:00.780 --> 00:02:03.580
the total volume is going to be 16

00:02:05.856 --> 00:02:09.800
times 1/64 of a cubic foot,

00:02:10.800 --> 00:02:14.581
which is going to be equal to 16/64.

00:02:14.581 --> 00:02:19.581
16/64 cubic feet,

00:02:19.770 --> 00:02:21.698
cubic feet, which is the same thing,

00:02:21.698 --> 00:02:25.470
16/64 is the same thing as 1/4,

00:02:25.470 --> 00:02:28.200
divide the numerator and
the denominator by 16.

00:02:28.200 --> 00:02:32.400
This is the same thing
as 1/4 of a cubic foot,

00:02:32.400 --> 00:02:35.790
of a cubic foot, and that's our volume.

00:02:35.790 --> 00:02:38.100
Now, there's other ways that
you could have done this.

00:02:38.100 --> 00:02:40.140
You could have just thought
about the dimensions

00:02:40.140 --> 00:02:43.710
of the length, the width, and the height.

00:02:43.710 --> 00:02:46.050
The width right over here

00:02:46.050 --> 00:02:49.290
is going to be two times 1/4 feet,

00:02:49.290 --> 00:02:52.440
which is equal to one half of a foot.

00:02:52.440 --> 00:02:54.300
The height here is the same thing,

00:02:54.300 --> 00:02:57.060
it's two times, so it's gonna be two

00:02:57.060 --> 00:03:02.060
times 1/4 of a foot, which is equal to 2/4

00:03:02.274 --> 00:03:04.470
or 1/2 of a foot.

00:03:04.470 --> 00:03:06.300
And then the length,

00:03:06.300 --> 00:03:11.280
the length here is 4 times 1/4 of a foot.

00:03:11.280 --> 00:03:13.770
4 times 1/4 of a foot,

00:03:13.770 --> 00:03:16.770
well, that's equal to 4/4 of a foot,

00:03:16.770 --> 00:03:18.810
which is equal to one foot.

00:03:18.810 --> 00:03:20.714
So to figure out the volume,

00:03:20.714 --> 00:03:22.830
we could multiply.

00:03:22.830 --> 00:03:27.180
We could multiply the
length times the width,

00:03:27.180 --> 00:03:31.890
the length times the
width times the height.

00:03:31.890 --> 00:03:34.260
Times the height, and I
mean these little dots here,

00:03:34.260 --> 00:03:35.160
these aren't decimals,

00:03:35.160 --> 00:03:36.180
I've written them a little higher.

00:03:36.180 --> 00:03:37.050
These are another way,

00:03:37.050 --> 00:03:38.640
it's a shorthand for multiplication.

00:03:38.640 --> 00:03:41.250
Instead of writing a, this
kind of X looking thing,

00:03:41.250 --> 00:03:43.140
this cross looking thing.

00:03:43.140 --> 00:03:46.110
so the length is one,

00:03:46.110 --> 00:03:48.810
the width is half of a foot,

00:03:48.810 --> 00:03:50.970
so times one half.

00:03:50.970 --> 00:03:54.393
And then the height is another half.

00:03:55.320 --> 00:03:56.610
Let me do it this way.

00:03:56.610 --> 00:03:58.440
The height is another half.

00:03:58.440 --> 00:03:59.843
So what's 1 times 1/2 times 1/2?

00:04:00.691 --> 00:04:03.363
Well, that's going to be equal to 1/4.

00:04:04.320 --> 00:04:06.870
And this is a foot, this
is a foot, this is a foot.

00:04:06.870 --> 00:04:08.738
So foot, times foot, times foot,

00:04:08.738 --> 00:04:11.400
that's gonna be feet to the
third power or cubic feet,

00:04:11.400 --> 00:04:14.130
1/4 of a cubic foot.

00:04:14.130 --> 00:04:15.900
Either way, we got the same result,

00:04:15.900 --> 00:04:16.863
which is good.


Volume of a rectangular prism: fractional dimensions

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:03.300
- [Narrator] Let's see if
we can calculate the volume

00:00:03.300 --> 00:00:05.880
of this rectangular prism

00:00:05.880 --> 00:00:07.170
or, I guess, you'd think this thing,

00:00:07.170 --> 00:00:09.870
the shape of a brick or a
fish tank right over here.

00:00:09.870 --> 00:00:11.190
And what's interesting is now

00:00:11.190 --> 00:00:15.150
that the dimensions are actually
fractions, we have a width

00:00:15.150 --> 00:00:17.160
or, I guess, yeah, we
could call this the width.

00:00:17.160 --> 00:00:20.160
The width here is 3/5ths of a unit.

00:00:20.160 --> 00:00:22.950
The length here is 1 and 1/6th units,

00:00:22.950 --> 00:00:26.100
and the height here is 3/7ths of a unit.

00:00:26.100 --> 00:00:27.690
So I encourage you to pause this video

00:00:27.690 --> 00:00:31.200
and try to figure out the volume
of this figure on your own

00:00:31.200 --> 00:00:33.303
before we work through it together.

00:00:34.320 --> 00:00:36.600
So there's a couple of
ways to think about it.

00:00:36.600 --> 00:00:39.540
One way to think about it is you're trying

00:00:39.540 --> 00:00:41.970
to pack unit cubes in here.

00:00:41.970 --> 00:00:44.340
And one way to think
about how many unit cubes

00:00:44.340 --> 00:00:47.340
could fit in here is to
think about the area,

00:00:47.340 --> 00:00:51.690
is to think about the area
of this base right over here.

00:00:51.690 --> 00:00:54.560
So sometimes you'll see
volume is equal to the area

00:00:54.560 --> 00:00:58.590
of the base times the
height, times the height.

00:00:58.590 --> 00:01:02.100
This right over here is the height.

00:01:02.100 --> 00:01:05.370
And let me make it clear,
this is the area of the base.

00:01:05.370 --> 00:01:07.680
Area of the base times the height.

00:01:07.680 --> 00:01:09.810
Well, what's the area of the base?

00:01:09.810 --> 00:01:11.850
Well, the area of the
base is the same thing

00:01:11.850 --> 00:01:14.130
as the length times the width.

00:01:14.130 --> 00:01:15.840
So you might see it written like that.

00:01:15.840 --> 00:01:19.192
You might see it written as area of base

00:01:19.192 --> 00:01:24.192
is going to be your
length times your width,

00:01:24.570 --> 00:01:27.240
times your width.

00:01:27.240 --> 00:01:32.240
Width, length times
width is the same thing

00:01:32.400 --> 00:01:33.900
as your area of the base.

00:01:33.900 --> 00:01:35.460
So that's that right over there.

00:01:35.460 --> 00:01:39.150
And, of course, you still have
to multiply times the height.

00:01:39.150 --> 00:01:40.380
Or another way of thinking about it,

00:01:40.380 --> 00:01:43.020
you're gonna multiply your
length times your width

00:01:43.020 --> 00:01:44.160
times your height.

00:01:44.160 --> 00:01:46.620
You're gonna multiply the
three dimensions of this thing

00:01:46.620 --> 00:01:49.200
to figure out how many unit
cubes could fit into it

00:01:49.200 --> 00:01:51.960
to figure out the volume.

00:01:51.960 --> 00:01:53.310
So let's calculate it.

00:01:53.310 --> 00:01:56.610
The volume here is going
to be, what's our length?

00:01:56.610 --> 00:01:59.940
Our length is 1 and 1/6th units.

00:01:59.940 --> 00:02:03.090
Now, when I multiply
fractions as I'm about to do,

00:02:03.090 --> 00:02:04.740
I don't like to multiply mixed numbers.

00:02:04.740 --> 00:02:07.590
I like to write them
as improper fractions.

00:02:07.590 --> 00:02:11.670
So let me convert 1 and 1/6th
to an improper fraction.

00:02:11.670 --> 00:02:16.366
So 1 is the same thing as
6/6ths plus 1 is 7/6ths.

00:02:16.366 --> 00:02:18.630
So this is going to be 7/6.

00:02:18.630 --> 00:02:23.630
That's my length times
3/5ths, that's my width.

00:02:23.760 --> 00:02:28.260
So times 3/5ths, times the height,

00:02:28.260 --> 00:02:33.260
which is 3/7ths times the
height, which is 3/7ths.

00:02:34.410 --> 00:02:36.900
And we know when we multiply fractions,

00:02:36.900 --> 00:02:39.240
we can multiply the numerators.

00:02:39.240 --> 00:02:44.240
So it's gonna be 7
times 3 times 3 times 3.

00:02:47.130 --> 00:02:49.650
And the denominator, we can
just multiply the denominators.

00:02:49.650 --> 00:02:54.330
So it's going to be 6 times 5,

00:02:54.330 --> 00:02:59.330
6 times 5 times 7 times 7.

00:02:59.550 --> 00:03:01.380
Now, we could just multiply these out,

00:03:01.380 --> 00:03:03.570
but just to try to get an answer

00:03:03.570 --> 00:03:08.070
that is as simplified as I can make it.

00:03:08.070 --> 00:03:10.110
Let me, see, we have a 7 in the numerator

00:03:10.110 --> 00:03:12.120
and a 7 in the denominator.

00:03:12.120 --> 00:03:15.480
So let's divide the numerator
and the denominator by 7.

00:03:15.480 --> 00:03:19.650
So let's divide the numerator
and the denominator by 7.

00:03:19.650 --> 00:03:24.090
And what that does is that
becomes 1 and those become 1.

00:03:24.090 --> 00:03:27.206
And we also see both the
numerator denominator has 3.

00:03:27.206 --> 00:03:28.680
They're both divisible by 3.

00:03:28.680 --> 00:03:29.580
We see a 3 up here.

00:03:29.580 --> 00:03:31.080
We see a 3 over here.

00:03:31.080 --> 00:03:32.550
So let's divide both the numerator

00:03:32.550 --> 00:03:34.530
and the denominator by 3.

00:03:34.530 --> 00:03:37.650
So we divide by 3, divide by 3.

00:03:37.650 --> 00:03:39.960
3 divided by 3 is 1.

00:03:39.960 --> 00:03:44.960
6 divided by 3 is going to be equal to 2.

00:03:45.330 --> 00:03:47.760
So in our numerator,
what are we left with?

00:03:47.760 --> 00:03:49.080
This is going to be equal to,

00:03:49.080 --> 00:03:51.480
where we're just left with that green 3,

00:03:51.480 --> 00:03:56.480
it's going to be equal to 3/2 times 5.

00:03:57.030 --> 00:04:01.440
2 times 5 is 10, 2 times
5, right over here.

00:04:01.440 --> 00:04:06.300
So the volume over here
is 3/10ths units cubed.

00:04:06.300 --> 00:04:08.970
Or we could fit 3/10ths of a unit cube

00:04:08.970 --> 00:04:11.040
inside of this brick or this fish tank

00:04:11.040 --> 00:04:13.233
or whatever you want to call it.