Identifying a function's derivative example

WEBVTT
Kind: captions
Language: en

00:00:00.960 --> 00:00:03.700
I have a function
f of x here, and I

00:00:03.700 --> 00:00:05.940
want to think about
which of these curves

00:00:05.940 --> 00:00:08.980
could represent f prime
of x, could represent

00:00:08.980 --> 00:00:10.924
the derivative of f of x.

00:00:10.924 --> 00:00:12.340
Well, to think
about that, we just

00:00:12.340 --> 00:00:13.150
have to think
about, well, what is

00:00:13.150 --> 00:00:15.800
a slope of the tangent line
doing at each point of f of x

00:00:15.800 --> 00:00:18.480
and see if this
corresponds to that slope,

00:00:18.480 --> 00:00:22.060
if the value of these functions
correspond to that slope.

00:00:22.060 --> 00:00:24.130
So we can see when x
is equal to negative 4,

00:00:24.130 --> 00:00:27.350
the slope of the tangent
line is essentially vertical.

00:00:27.350 --> 00:00:29.490
So you could say it's
not really defined there.

00:00:29.490 --> 00:00:33.110
But as we go slightly to the
right of x equals negative 4,

00:00:33.110 --> 00:00:35.552
we just have a very,
very, very positive slope.

00:00:35.552 --> 00:00:37.260
So you could kind of
view it as our slope

00:00:37.260 --> 00:00:41.650
is going from infinity to very,
very positive to a little bit

00:00:41.650 --> 00:00:43.550
less positive to
a little bit less

00:00:43.550 --> 00:00:45.420
positive, to a little
bit less positive,

00:00:45.420 --> 00:00:47.380
to a little bit less positive.

00:00:47.380 --> 00:00:50.690
So which of these graphs
here have that property?

00:00:50.690 --> 00:00:53.140
Remember, this is trying
to graph the slope.

00:00:53.140 --> 00:00:56.790
So which of these functions down
here, which of these graphs,

00:00:56.790 --> 00:01:00.540
have a value that is essentially
kind of approaching infinity

00:01:00.540 --> 00:01:03.170
when x is equal to
negative 4, and then it

00:01:03.170 --> 00:01:07.117
gets less and less and less
positive as x goes to 0?

00:01:07.117 --> 00:01:09.700
So this one, it looks like it's
coming from negative infinity,

00:01:09.700 --> 00:01:11.741
and it's getting less and
less and less negative.

00:01:11.741 --> 00:01:14.600
So that doesn't seem to
meet our constraints.

00:01:14.600 --> 00:01:17.999
This one looks like it is
coming from positive infinity,

00:01:17.999 --> 00:01:20.040
and it's getting less and
less and less positive,

00:01:20.040 --> 00:01:21.900
so that seems to be OK.

00:01:21.900 --> 00:01:23.340
This has the same property.

00:01:23.340 --> 00:01:25.820
It's getting less and
less and less positive.

00:01:25.820 --> 00:01:28.230
This one right over here
starts very negative

00:01:28.230 --> 00:01:30.000
and gets less and less
and less negative.

00:01:30.000 --> 00:01:32.000
So we can rule that out.

00:01:32.000 --> 00:01:36.420
Now let's think about what
happens when x gets to 0.

00:01:36.420 --> 00:01:41.550
When x gets to 0, the
tangent line is horizontal.

00:01:41.550 --> 00:01:44.960
We're at a maximum point of
this curve right over here.

00:01:44.960 --> 00:01:49.480
The slope of a
horizontal line is 0.

00:01:49.480 --> 00:01:51.980
Remember, we're trying to look
for which one of these curves

00:01:51.980 --> 00:01:54.220
represent the value
of that slope.

00:01:54.220 --> 00:01:59.170
So which one of these curves
hit 0 when x is equal to 0?

00:01:59.170 --> 00:02:00.560
Well, this one doesn't.

00:02:00.560 --> 00:02:02.726
So the only candidate that
we have left is this one,

00:02:02.726 --> 00:02:05.380
and this one does hit
0 when x equals 0.

00:02:05.380 --> 00:02:07.310
And let's see if
it keeps satisfying

00:02:07.310 --> 00:02:09.780
what we need for f prime of x.

00:02:09.780 --> 00:02:12.130
So after that point,
it should start

00:02:12.130 --> 00:02:14.950
getting more and more negative.

00:02:14.950 --> 00:02:18.550
The slope should get more
and more and more negative,

00:02:18.550 --> 00:02:23.980
essentially approaching negative
infinity as x approaches 4.

00:02:23.980 --> 00:02:25.080
And we see that here.

00:02:25.080 --> 00:02:27.621
The value of this function is
getting more and more negative,

00:02:27.621 --> 00:02:31.530
and it's approaching negative
infinity as x approaches 4.

00:02:31.530 --> 00:02:33.190
So we'll go with this one.

00:02:33.190 --> 00:02:37.200
This looks like a pretty good
candidate for f prime of x.


Decimals: comparing place values

WEBVTT
Kind: captions
Language: en

00:00:01.830 --> 00:00:09.640
We have the number
7,346,521.032.

00:00:09.640 --> 00:00:12.000
And what I want
to think about is

00:00:12.000 --> 00:00:15.020
if I look at the same digit
in two different places,

00:00:15.020 --> 00:00:17.990
in particular, I'm going
to look at the digit 3 here

00:00:17.990 --> 00:00:22.680
and the digit 3 here, how much
more value does this left 3

00:00:22.680 --> 00:00:24.809
represent than this right 3?

00:00:24.809 --> 00:00:26.350
In order to think
about that, we have

00:00:26.350 --> 00:00:28.720
to think about place value.

00:00:28.720 --> 00:00:30.490
So let's write down
all the place values.

00:00:30.490 --> 00:00:32.940
So this right over here,
this is the ones place.

00:00:32.940 --> 00:00:34.430
Now, we could move to the right.

00:00:34.430 --> 00:00:36.790
And as we move to the
right in place values,

00:00:36.790 --> 00:00:40.060
each place represents 1/10
of the place before it.

00:00:40.060 --> 00:00:43.620
Or you could divide by 10 as
we're moving to the right.

00:00:43.620 --> 00:00:44.990
So this is the ones place.

00:00:44.990 --> 00:00:46.170
This is divide by 10.

00:00:46.170 --> 00:00:49.210
This is the 1/10 place,
or the tenths place.

00:00:49.210 --> 00:00:52.410
Divide by 10 again, this
is the hundredths place.

00:00:52.410 --> 00:00:56.230
Divide by 10 again, this
is the thousandths place.

00:00:56.230 --> 00:00:59.070
And that "s" I'm just saying
to be plural-- hundredths,

00:00:59.070 --> 00:01:00.570
thousandths.

00:01:00.570 --> 00:01:02.920
Now, if we go to the
left, now each place

00:01:02.920 --> 00:01:05.770
represents a factor of 10 more.

00:01:05.770 --> 00:01:09.420
So if this is ones, multiply
by 10, this is the tens place.

00:01:09.420 --> 00:01:11.220
This is the hundreds place.

00:01:11.220 --> 00:01:14.790
This is the thousands place.

00:01:14.790 --> 00:01:16.360
This is the ten thousands place.

00:01:16.360 --> 00:01:18.360
I'm going to have to write
a little bit smaller.

00:01:18.360 --> 00:01:22.060
This is the hundred
thousands place.

00:01:22.060 --> 00:01:24.140
And then the 7 is in
the millions place.

00:01:28.050 --> 00:01:30.960
So what does this number,
what does this 3 represent?

00:01:30.960 --> 00:01:32.800
Well, it's in the
hundred thousands place.

00:01:32.800 --> 00:01:39.430
It literally represents
3 hundred thousands,

00:01:39.430 --> 00:01:47.170
or you could say 300,000,
3 followed by five zeroes.

00:01:47.170 --> 00:01:49.990
Now, what does this 3 represent?

00:01:49.990 --> 00:01:52.110
It's in the hundredths place.

00:01:52.110 --> 00:01:54.340
It literally represents
3 hundredths.

00:01:54.340 --> 00:02:00.310
It represents 3
times 1/100, which

00:02:00.310 --> 00:02:05.900
is the same thing as 3, which
is equal to 3 over-- let

00:02:05.900 --> 00:02:08.240
me do the 3 in
that purple color.

00:02:08.240 --> 00:02:14.940
Which is the same
thing as 3/100,

00:02:14.940 --> 00:02:20.860
which is the same thing as 0.03.

00:02:20.860 --> 00:02:22.480
These are all
equivalent statements.

00:02:22.480 --> 00:02:24.490
Now let's try to answer
our original question.

00:02:24.490 --> 00:02:28.060
How much larger is this
3 than that 3 there?

00:02:28.060 --> 00:02:31.390
Well, one way to think about
it is how much would you

00:02:31.390 --> 00:02:35.270
have to multiply this 3 by
to get to this 3 over here?

00:02:35.270 --> 00:02:37.270
Well, one way to think
about is to look directly

00:02:37.270 --> 00:02:39.120
at place value.

00:02:39.120 --> 00:02:43.320
So we got to multiply by 10.

00:02:43.320 --> 00:02:44.820
Every time we
multiply by 10, that's

00:02:44.820 --> 00:02:47.880
equivalent to thinking about
shifting it to one place

00:02:47.880 --> 00:02:48.600
to the left.

00:02:48.600 --> 00:02:56.108
So we would have to multiply by
10 one, two, three, four, five,

00:02:56.108 --> 00:02:59.660
six, seven times.

00:02:59.660 --> 00:03:01.482
So multiplying by
10 seven times.

00:03:01.482 --> 00:03:02.440
Let me write this down.

00:03:02.440 --> 00:03:05.860
So this multiplied by 10 seven
times should be equal to this.

00:03:05.860 --> 00:03:07.280
Let me rewrite this.

00:03:07.280 --> 00:03:13.540
300,000 should be
equal to 3/100-- let

00:03:13.540 --> 00:03:15.270
me write it the same way.

00:03:15.270 --> 00:03:25.270
3/100 multiplied by 10 seven
times, so times 10 times

00:03:25.270 --> 00:03:29.600
10 times 10 times 10
times 10-- let's see,

00:03:29.600 --> 00:03:33.370
that's five times--
times 10 times 10.

00:03:33.370 --> 00:03:36.710
Now, multiplying
by 10 seven times

00:03:36.710 --> 00:03:40.090
is the same thing as multiplying
by 1 followed by seven zeroes.

00:03:40.090 --> 00:03:41.850
Every time you
multiply by 10, you're

00:03:41.850 --> 00:03:43.520
going to get another zero here.

00:03:43.520 --> 00:03:52.690
So this is the same
thing as 3/100 times 1

00:03:52.690 --> 00:03:58.150
followed by one, two, three,
four, five, six, seven zeroes.

00:03:58.150 --> 00:04:02.320
So this is literally
3/100 times 10 million.

00:04:02.320 --> 00:04:06.390
So let's see if this
actually is the case.

00:04:06.390 --> 00:04:13.180
Does this actually
equal 300,000?

00:04:13.180 --> 00:04:20.220
Well, if you divide 10 million
by 100, dividing 10 million

00:04:20.220 --> 00:04:22.956
by 100, or I guess you'd
say in the numerator,

00:04:22.956 --> 00:04:24.580
you have 10 million
and the denominator

00:04:24.580 --> 00:04:27.390
you have 100, if you were to
just multiply it like this,

00:04:27.390 --> 00:04:31.420
if you view this as 3 over
100 times 10 million over 1.

00:04:31.420 --> 00:04:33.632
Well, you divide the
numerator by 100,

00:04:33.632 --> 00:04:35.590
you're going to get rid
of two of these zeroes.

00:04:35.590 --> 00:04:37.131
Divide the denominator
by 100, you're

00:04:37.131 --> 00:04:38.570
going to get rid
of this 100 here.

00:04:38.570 --> 00:04:41.150
And so you're going to be
left with 3 times-- now

00:04:41.150 --> 00:04:42.650
we got to be careful
with the commas

00:04:42.650 --> 00:04:44.940
here, because since
I removed two zeroes,

00:04:44.940 --> 00:04:47.310
the commas are going
to be different.

00:04:47.310 --> 00:04:48.930
It's going to be
3 times-- we put

00:04:48.930 --> 00:04:52.790
our commas in the right
place, so just like that.

00:04:52.790 --> 00:04:58.075
So this simplified
to 3 times 100,000,

00:04:58.075 --> 00:05:00.420
which is indeed 300,000.

00:05:00.420 --> 00:05:02.040
So it did work out.

00:05:02.040 --> 00:05:05.856
Shifting the 3 one, two,
three, four, five, six,

00:05:05.856 --> 00:05:11.350
seven decimal places
makes that 3 worth 1

00:05:11.350 --> 00:05:14.630
followed by seven zeroes
more, or it essentially

00:05:14.630 --> 00:05:17.730
makes that 3 worth
10 million more.

00:05:17.730 --> 00:05:23.590
So this 3 represents 10 million
times the value of this 3.

00:05:23.590 --> 00:05:24.840
Let me write down the numbers.

00:05:24.840 --> 00:05:39.910
So this 3 is 10 million
times the value of that 3.


Volume word problem

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:05.480
A water tank is 12 feet high,
5 feet long, and 9 feet wide.

00:00:05.480 --> 00:00:09.190
A solid metal box which is
7 feet high, 4 feet long,

00:00:09.190 --> 00:00:12.530
and 8 feet wide is sitting
at the bottom of the tank.

00:00:12.530 --> 00:00:14.180
The tank is filled with water.

00:00:14.180 --> 00:00:16.457
What is the volume of
the water in the tank?

00:00:16.457 --> 00:00:17.540
So let's think about this.

00:00:17.540 --> 00:00:18.570
We have a water tank.

00:00:18.570 --> 00:00:19.620
It's 12 feet high.

00:00:19.620 --> 00:00:21.770
I'll try to draw this
as good as I can.

00:00:21.770 --> 00:00:23.260
So it's 12 feet high.

00:00:23.260 --> 00:00:24.525
It's 5 feet long.

00:00:24.525 --> 00:00:27.300
So this looks like
that's about 5 feet.

00:00:27.300 --> 00:00:28.860
And it's 9 feet wide.

00:00:28.860 --> 00:00:32.560
So this is my best rendition
of what a tank looks like.

00:00:32.560 --> 00:00:36.370
So the tank might look
something like this.

00:00:36.370 --> 00:00:38.910
That is my water tank.

00:00:38.910 --> 00:00:42.030
Let me draw it, draw
the whole thing.

00:00:42.030 --> 00:00:44.215
So there is my water tank.

00:00:44.215 --> 00:00:45.840
And I'm going to make
it transparent so

00:00:45.840 --> 00:00:49.350
that we can see what's
going on inside of the tank.

00:00:49.350 --> 00:00:52.132
So here we go.

00:00:52.132 --> 00:00:54.090
There's like a helicopter
outside or something,

00:00:54.090 --> 00:00:55.830
I don't know if
you all hear that.

00:00:55.830 --> 00:00:56.420
But let's see.

00:00:56.420 --> 00:01:04.340
So there is my water tank
12 feet high, 5 feet long,

00:01:04.340 --> 00:01:07.460
and 9 feet wide.

00:01:07.460 --> 00:01:09.220
And then they say
there's a solid metal

00:01:09.220 --> 00:01:13.300
box which is 7 feet high, 4
feet long, and 8 feet wide

00:01:13.300 --> 00:01:16.357
sitting at the bottom.

00:01:16.357 --> 00:01:17.690
So let's see if I can draw that.

00:01:17.690 --> 00:01:21.650
So let's say it's 4 feet wide.

00:01:21.650 --> 00:01:25.030
Or I guess they say 4
feet long, 7 feet high.

00:01:25.030 --> 00:01:27.750
So let's see, 4 feet might
look something like this.

00:01:27.750 --> 00:01:32.350
It's 7 feet high, which might
look something like that.

00:01:32.350 --> 00:01:34.530
7 feet high.

00:01:34.530 --> 00:01:36.860
Obviously, I'm not drawing
it perfectly to scale.

00:01:36.860 --> 00:01:38.690
7 feet high and 8 feet wide.

00:01:38.690 --> 00:01:42.430
So it might look
something like this

00:01:42.430 --> 00:01:45.690
as it's sitting in this tank.

00:01:45.690 --> 00:01:49.020
So this is that metal box.

00:01:49.020 --> 00:01:51.720
And they say it's
a solid metal box.

00:01:51.720 --> 00:01:53.590
It's not like any
water can fit in here.

00:01:53.590 --> 00:01:56.190
So let me make it as
a solid metal box.

00:01:56.190 --> 00:01:57.870
So this is a solid metal box.

00:01:57.870 --> 00:02:01.600
And then I'm going to fill
the whole thing with water.

00:02:01.600 --> 00:02:03.660
I'm going to pour
water into this thing.

00:02:03.660 --> 00:02:05.410
And the water's going
to start filling up.

00:02:05.410 --> 00:02:07.770
And it's going to fill up
all the volume of the tank

00:02:07.770 --> 00:02:10.101
except where the metal box is.

00:02:10.101 --> 00:02:12.100
It's not going to be able
to fill in that volume

00:02:12.100 --> 00:02:15.029
because the metal box is solid.

00:02:15.029 --> 00:02:16.070
So it's going to fill up.

00:02:16.070 --> 00:02:18.630
We're going to fill it,
slowly fill this thing up

00:02:18.630 --> 00:02:19.871
around the metal box.

00:02:19.871 --> 00:02:21.870
So what's the volume that
it's going to fill up?

00:02:21.870 --> 00:02:23.590
Well, it's going to
fill up the volume

00:02:23.590 --> 00:02:26.360
of the tank minus the
volume of the metal box.

00:02:26.360 --> 00:02:28.340
It couldn't fill in
the metal box volume.

00:02:28.340 --> 00:02:30.170
So let's figure
out what that is.

00:02:30.170 --> 00:02:32.760
The volume of the
tank is going to be

00:02:32.760 --> 00:02:39.000
9 foot by 5 foot times 12 feet.

00:02:39.000 --> 00:02:40.970
That's the volume of the tank.

00:02:40.970 --> 00:02:41.665
Tank volume.

00:02:44.320 --> 00:02:47.150
And from that, we want to
subtract the metal box volume.

00:02:47.150 --> 00:02:56.670
So minus 4 foot by
8 feet by 7 feet.

00:03:01.760 --> 00:03:03.430
This is 4 feet wide.

00:03:03.430 --> 00:03:09.110
It is 8 feet-- they
say it's 7 feet high.

00:03:09.110 --> 00:03:10.240
It's 4 feet long.

00:03:10.240 --> 00:03:12.980
And it's 8 feet wide.

00:03:12.980 --> 00:03:15.700
So this right over
here is the volume.

00:03:15.700 --> 00:03:18.270
I guess we could call
it the metal box volume.

00:03:21.730 --> 00:03:23.980
When you take the tank volume
and subtract out the box

00:03:23.980 --> 00:03:26.440
volume, that's how much the
water can actually fill in.

00:03:26.440 --> 00:03:28.490
So I only drew the
water partially filled.

00:03:28.490 --> 00:03:30.240
But once it's already
filled in, the water

00:03:30.240 --> 00:03:32.200
is going to go all the
way to the top here.

00:03:32.200 --> 00:03:33.600
And we'll fill in
everything except for where

00:03:33.600 --> 00:03:34.580
that blue box is.

00:03:34.580 --> 00:03:37.510
So let's figure out
what this value is.

00:03:37.510 --> 00:03:39.530
So 5 times 12 is 60.

00:03:39.530 --> 00:03:42.790
60 times 9 is 540.

00:03:42.790 --> 00:03:45.450
And then in blue
here, let's see.

00:03:45.450 --> 00:03:49.190
4 times 8 is 32.

00:03:49.190 --> 00:03:52.480
32 times 7 is 210.

00:03:52.480 --> 00:03:55.450
Plus 14, which is 224.

00:03:55.450 --> 00:03:57.970
So it's minus 224.

00:03:57.970 --> 00:03:59.017
Minus 224.

00:03:59.017 --> 00:03:59.850
Did I do that right?

00:03:59.850 --> 00:04:01.516
I don't want to make
a careless mistake?

00:04:01.516 --> 00:04:03.410
So 32 times 7.

00:04:03.410 --> 00:04:05.640
2 times 7 is 14.

00:04:05.640 --> 00:04:07.090
3, which is really a 30 here.

00:04:07.090 --> 00:04:11.770
30 times 7 is 210.

00:04:11.770 --> 00:04:14.160
Plus another 10 is 220.

00:04:14.160 --> 00:04:16.550
So 224.

00:04:16.550 --> 00:04:19.850
So this is going to be
equal to-- let's see.

00:04:19.850 --> 00:04:24.396
500 minus 200 is 300.

00:04:24.396 --> 00:04:29.670
And then 40 minus 24 is 16.

00:04:29.670 --> 00:04:31.380
316.

00:04:31.380 --> 00:04:34.616
And our units are in cubic feet.

00:04:34.616 --> 00:04:36.240
So the volume of the
water in the tank?

00:04:36.240 --> 00:04:39.360
316 cubic feet.


Volume of a rectangular prism or box examples

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:02.590
What is the volume of this box?

00:00:02.590 --> 00:00:04.352
Drag on the box to rotate it.

00:00:04.352 --> 00:00:05.310
So this is pretty neat.

00:00:05.310 --> 00:00:09.272
We can actually sit
and rotate this box.

00:00:09.272 --> 00:00:10.730
And here it looks
like everything's

00:00:10.730 --> 00:00:12.130
being measured in meters.

00:00:12.130 --> 00:00:16.420
So we want to measure our
volume in terms of cubic meters.

00:00:16.420 --> 00:00:18.150
That's going to be
our unit cube here.

00:00:18.150 --> 00:00:21.530
So when we want to think about
how many cubic meters could fit

00:00:21.530 --> 00:00:24.000
in this box, we've
already seen examples.

00:00:24.000 --> 00:00:27.600
You really just have to multiply
the three different dimensions

00:00:27.600 --> 00:00:28.749
of this box.

00:00:28.749 --> 00:00:30.540
So if you wanted the
number of cubic meters

00:00:30.540 --> 00:00:32.000
that could fit in
here, it's going

00:00:32.000 --> 00:00:39.960
to be six meters times 8
meters times 7 meters which

00:00:39.960 --> 00:00:42.700
is going to give you
something in cubic meters.

00:00:42.700 --> 00:00:44.340
So let's think
about what that is.

00:00:44.340 --> 00:00:48.580
6 times 8 is 48.

00:00:48.580 --> 00:00:50.980
Let me see if I can
do this in my head.

00:00:50.980 --> 00:00:55.750
48 times 7, that's
40 times 7, which

00:00:55.750 --> 00:01:03.010
is going to be 280 plus
8 times 7, which is 56,

00:01:03.010 --> 00:01:12.970
280 plus 56 is going to be 336.

00:01:12.970 --> 00:01:14.340
Let's check our answer.

00:01:14.340 --> 00:01:16.575
Let's do one more of these.

00:01:16.575 --> 00:01:17.950
So what's the
volume of this box?

00:01:17.950 --> 00:01:21.020
We'll once again, we have
its height at six feet.

00:01:21.020 --> 00:01:22.790
Now everything is
being measured in feet.

00:01:22.790 --> 00:01:27.520
We have it's width
being four feet.

00:01:27.520 --> 00:01:31.670
So we could multiple the height
times the width of four feet.

00:01:31.670 --> 00:01:34.850
And then we can multiply that
times its depth of two feet.

00:01:38.170 --> 00:01:42.420
So 6 times 4 is 24
times 2 is 48 feet.

00:01:42.420 --> 00:01:44.990
48, and I should say cubic feet.

00:01:44.990 --> 00:01:47.580
We're saying how many
cubic feet can fit in here?

00:01:47.580 --> 00:01:51.340
When we multiply the various
dimensions measured in feet,

00:01:51.340 --> 00:01:53.100
we're counting almost
how many of those

00:01:53.100 --> 00:01:57.230
cubic feet can
fit into this box.


Volume: measuring as area times length

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:01.890
I have this figure here.

00:00:01.890 --> 00:00:03.860
You could call it a
rectangular prism.

00:00:03.860 --> 00:00:05.560
And I want to
measure its volume.

00:00:05.560 --> 00:00:10.240
And I'm defining my unit cube
as being a 1 centimeter by 1

00:00:10.240 --> 00:00:12.130
centimeter by 1 centimeter cube.

00:00:12.130 --> 00:00:15.070
It has 1 centimeter width, 1
centimeter depth, 1 centimeter

00:00:15.070 --> 00:00:15.760
height.

00:00:15.760 --> 00:00:21.780
And I will call this, this is
equal to 1 cubic centimeter.

00:00:21.780 --> 00:00:24.640
So I want to measure this volume
in terms of cubic centimeters.

00:00:24.640 --> 00:00:26.340
We've already seen that we
can do that by saying, hey,

00:00:26.340 --> 00:00:28.040
how many of these
cubic centimeters

00:00:28.040 --> 00:00:32.060
can fit into this figure without
them overlapping in any way?

00:00:32.060 --> 00:00:34.170
So if we had this in our
hands, we could kind of

00:00:34.170 --> 00:00:35.872
try to go around it
and try to count it,

00:00:35.872 --> 00:00:37.580
but it's hard to see
here because there's

00:00:37.580 --> 00:00:40.210
some cubes that we can't
see behind the ones

00:00:40.210 --> 00:00:41.550
that we are seeing.

00:00:41.550 --> 00:00:43.880
So I'm going to try
different tactics at it.

00:00:43.880 --> 00:00:46.950
So first, let's just think
about what we can observe.

00:00:46.950 --> 00:00:48.780
So we see that this
one, if we measure

00:00:48.780 --> 00:00:51.300
its different
dimensions, its width,

00:00:51.300 --> 00:00:54.760
it's 2 of the unit length wide.

00:00:54.760 --> 00:00:57.030
So it's 2 centimeters wide.

00:00:57.030 --> 00:00:59.700
It's 4 of our unit length--
we're defining our unit length

00:00:59.700 --> 00:01:03.810
as a centimeter-- it's 4
of our unit length high.

00:01:03.810 --> 00:01:07.470
So this dimension right
over here is 4 centimeters.

00:01:07.470 --> 00:01:10.940
And it is 3 of our
unit length deep.

00:01:10.940 --> 00:01:15.280
So this dimension right
over here is 3 centimeters.

00:01:15.280 --> 00:01:17.170
So I want to explore
if we can somehow

00:01:17.170 --> 00:01:19.140
use these numbers
to figure out how

00:01:19.140 --> 00:01:23.330
many of these cubic centimeters
would fit into this figure.

00:01:23.330 --> 00:01:25.240
And the first way I'm
going to think about it

00:01:25.240 --> 00:01:27.160
is by looking at slices.

00:01:27.160 --> 00:01:32.190
So I'm going to take
this slice right

00:01:32.190 --> 00:01:36.070
over here of our
original figure.

00:01:36.070 --> 00:01:39.670
And let's think about
how using these numbers,

00:01:39.670 --> 00:01:43.130
we can figure out how many
unit cubes were in that slice.

00:01:43.130 --> 00:01:48.010
Well this is 2 centimeters wide
and it is 4 centimeters high.

00:01:51.676 --> 00:01:53.050
And you might be
saying, hey Sal,

00:01:53.050 --> 00:01:54.383
I could just count these things.

00:01:54.383 --> 00:01:57.344
I could get 8 squares here.

00:01:57.344 --> 00:01:58.760
But what if there
was a ton there?

00:01:58.760 --> 00:01:59.760
It would be a lot harder.

00:01:59.760 --> 00:02:01.384
And you might realize
well I could just

00:02:01.384 --> 00:02:03.740
multiply the width times the
height, that would give me

00:02:03.740 --> 00:02:06.630
the area of this
surface right over here.

00:02:06.630 --> 00:02:08.280
And it's only 1
deep so that also

00:02:08.280 --> 00:02:10.190
will give me the
number of cubes.

00:02:10.190 --> 00:02:10.930
So let's do that.

00:02:10.930 --> 00:02:12.530
Let's find the area here.

00:02:12.530 --> 00:02:14.416
Well that's going to be
2 centimeters times 4

00:02:14.416 --> 00:02:14.916
centimeters.

00:02:20.640 --> 00:02:23.950
That gives us the area of this.

00:02:23.950 --> 00:02:27.000
And then if we want to find
out the number of cubes,

00:02:27.000 --> 00:02:29.450
well that's also going to
be equivalent to the number

00:02:29.450 --> 00:02:30.090
of cubes.

00:02:30.090 --> 00:02:33.020
So we have 8 square
centimeters is this area,

00:02:33.020 --> 00:02:34.640
and the number of cubes is 8.

00:02:34.640 --> 00:02:36.890
And if we want the number
of cubes in the whole thing,

00:02:36.890 --> 00:02:39.420
we just have to multiply
by the number of slices.

00:02:39.420 --> 00:02:43.600
And we see that we need
one, two, three slices.

00:02:43.600 --> 00:02:46.110
This is 3 centimeters deep.

00:02:46.110 --> 00:02:49.500
So we're going to
multiply that times 3.

00:02:49.500 --> 00:02:51.790
So we took the area
of one surface.

00:02:51.790 --> 00:02:54.790
We took the area of this
surface right over here.

00:02:54.790 --> 00:02:57.310
And then we multiply by the
depth, that essentially gives

00:02:57.310 --> 00:02:59.660
us the number of cubes because
the area of this surface

00:02:59.660 --> 00:03:04.240
gives us the number of cubes in
an slice that is 1 cube deep.

00:03:04.240 --> 00:03:07.530
And then we would have to
have 3 slices like that.

00:03:07.530 --> 00:03:09.550
So we would have to
have this is 1 slice.

00:03:09.550 --> 00:03:13.710
We would have to have another
slice, another slice and then

00:03:13.710 --> 00:03:17.360
another slice in order to
construct the original figure.

00:03:17.360 --> 00:03:20.510
So 2 centimeters times 4
centimeters times 3 centimeters

00:03:20.510 --> 00:03:21.934
would give us our volume.

00:03:21.934 --> 00:03:23.100
Let's see if that works out.

00:03:23.100 --> 00:03:29.550
2 times 4 is 8 times 3 is 24.

00:03:29.550 --> 00:03:32.000
Let me do that in
that pink color.

00:03:32.000 --> 00:03:37.340
24 centimeters cubed, or I
could say cubic centimeters.

00:03:37.340 --> 00:03:39.410
So that's one way to
measure the volume.

00:03:39.410 --> 00:03:41.097
Now there's multiple
surfaces here.

00:03:41.097 --> 00:03:42.680
I happened to pick
this surface, but I

00:03:42.680 --> 00:03:43.950
could have picked another one.

00:03:43.950 --> 00:03:45.950
I could have picked this
surface right over here

00:03:45.950 --> 00:03:47.950
and done the exact same thing.

00:03:47.950 --> 00:03:51.070
So let's pick this surface
and do the exact same thing.

00:03:51.070 --> 00:03:56.996
This surface is 3
centimeters by 4 centimeters.

00:03:59.500 --> 00:04:01.890
Let me do that in
that blue color.

00:04:01.890 --> 00:04:05.530
Color changing is
always difficult.

00:04:05.530 --> 00:04:09.300
So its area is going to
be 12 square centimeters

00:04:09.300 --> 00:04:10.980
is the area of this surface.

00:04:10.980 --> 00:04:13.370
And 12 is also the
number of cubes

00:04:13.370 --> 00:04:14.853
that we have in that slice.

00:04:14.853 --> 00:04:17.019
And so how many slices do
we need like this in order

00:04:17.019 --> 00:04:18.769
to construct the
original figure?

00:04:18.769 --> 00:04:21.050
Well we need, it's
2 centimeters deep.

00:04:21.050 --> 00:04:25.530
This is only 1 centimeter
deep so we need two of them

00:04:25.530 --> 00:04:28.140
to construct the
original figure.

00:04:28.140 --> 00:04:31.410
So we can essentially find
the area of that first surface

00:04:31.410 --> 00:04:37.410
which was 3 times
4, and then multiply

00:04:37.410 --> 00:04:43.320
that times the width, times how
many of those slices you need,

00:04:43.320 --> 00:04:44.850
so times 2.

00:04:44.850 --> 00:04:46.730
And once again, this is
going to be 3 times 4

00:04:46.730 --> 00:04:49.227
is 12 times 2 is 24.

00:04:49.227 --> 00:04:50.935
I didn't write the
units this first time.

00:04:50.935 --> 00:04:52.640
But that's going to
give us the count

00:04:52.640 --> 00:04:54.960
of how many cubic
centimeters we have,

00:04:54.960 --> 00:04:56.470
how many unit cubes we can fit.

00:04:56.470 --> 00:04:59.846
So once again, this is
24 cubic centimeters.

00:04:59.846 --> 00:05:01.970
And you could imagine, you
could do the same thing,

00:05:01.970 --> 00:05:04.820
not with this surface,
not with this surface,

00:05:04.820 --> 00:05:06.480
but with the top surface.

00:05:06.480 --> 00:05:10.300
The top surface is
3 centimeters deep.

00:05:13.130 --> 00:05:14.610
And 2 centimeters wide.

00:05:17.950 --> 00:05:20.180
So you could view
its area or its area

00:05:20.180 --> 00:05:23.550
is going to be 3 centimeters
times 2 centimeters.

00:05:23.550 --> 00:05:27.096
So that area is-- let me do
it in the same colors-- 3

00:05:27.096 --> 00:05:34.670
centimeters times 2 centimeters
which is 6 square centimeters.

00:05:34.670 --> 00:05:36.170
And that also tells
you that there's

00:05:36.170 --> 00:05:39.820
going to be 6 cubes in
this one cube deep slice.

00:05:39.820 --> 00:05:42.140
But how many of these
slices do you need?

00:05:42.140 --> 00:05:46.202
Well you have this whole
thing is 4 centimeters tall,

00:05:46.202 --> 00:05:47.660
and this thing is
only 1 centimeter

00:05:47.660 --> 00:05:49.201
so you're going to
need four of them.

00:05:49.201 --> 00:06:00.620
So that's 2, 3, try to draw
it as neatly as I can, and 4.

00:06:00.620 --> 00:06:02.120
You're going to need 4 of these.

00:06:02.120 --> 00:06:03.200
So to figure out
the whole volume,

00:06:03.200 --> 00:06:04.991
you going to have to
take that and multiply

00:06:04.991 --> 00:06:07.050
that times 4 centimeters.

00:06:07.050 --> 00:06:10.865
So once again, 3 times 2 is
6 square centimeters times 4

00:06:10.865 --> 00:06:15.140
centimeters is 24
cubic centimeters.

00:06:15.140 --> 00:06:18.130
So it doesn't matter what
order you multiply these in.

00:06:18.130 --> 00:06:20.510
You could view this and
take the area of one side

00:06:20.510 --> 00:06:22.764
then multiply it
times the depth.

00:06:22.764 --> 00:06:24.930
Or you could take the surface
area of another height

00:06:24.930 --> 00:06:28.880
and multiply it times the height
or the width or the depth.

00:06:28.880 --> 00:06:30.450
And these are all the scenarios.

00:06:30.450 --> 00:06:33.500
But what it shows is that it
doesn't matter what order we

00:06:33.500 --> 00:06:36.820
multiply these
three dimensions in.

00:06:36.820 --> 00:06:38.780
You could take the 2
times the 4 first and then

00:06:38.780 --> 00:06:40.170
multiply it by the 3.

00:06:40.170 --> 00:06:42.595
Or you take the 3 times the
4 first and then multiply it

00:06:42.595 --> 00:06:43.500
by the 2.

00:06:43.500 --> 00:06:46.540
And or you could take
the 2 times the 3 first,

00:06:46.540 --> 00:06:48.182
and then multiply by the 4.

00:06:48.182 --> 00:06:49.640
When you're
multiplying, it doesn't

00:06:49.640 --> 00:06:52.450
matter what order
you're doing these in.

00:06:52.450 --> 00:06:55.800
And so if you have a
rectangular prism like this

00:06:55.800 --> 00:06:57.390
and you know it's
three dimensions,

00:06:57.390 --> 00:07:00.689
you know it's 2 centimeters
wide, 3 centimeters deep, and 4

00:07:00.689 --> 00:07:02.230
centimeters tall,
you could say, hey,

00:07:02.230 --> 00:07:05.270
the volume of this thing,
the number of unit cubes,

00:07:05.270 --> 00:07:07.680
the number of cubic
centimeters it can fit

00:07:07.680 --> 00:07:16.070
is going to be 2 centimeters
times 4 centimeters times 3

00:07:16.070 --> 00:07:18.780
centimeters, which
we've seen three times

00:07:18.780 --> 00:07:21.870
already is 24 cubic centimeters.


Volume: how to measure it

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:02.590
Human beings have
always realized

00:00:02.590 --> 00:00:04.700
that certain things are
longer than other things.

00:00:04.700 --> 00:00:06.630
For example, this
line segment looks

00:00:06.630 --> 00:00:08.820
longer than this line segment.

00:00:08.820 --> 00:00:11.470
But that's not so satisfying
just to make that comparison.

00:00:11.470 --> 00:00:12.490
You want to be
able to measure it.

00:00:12.490 --> 00:00:14.448
You want to be able to
quantify how much longer

00:00:14.448 --> 00:00:16.920
the second one is
than the first one.

00:00:16.920 --> 00:00:18.490
And how do we go
about doing that?

00:00:18.490 --> 00:00:21.770
Well, we define a unit length.

00:00:21.770 --> 00:00:24.410
So if we make this
our unit length,

00:00:24.410 --> 00:00:28.270
we say this is one unit, then
we could say how many of those

00:00:28.270 --> 00:00:30.430
the lengths are
each of these lines?

00:00:30.430 --> 00:00:32.729
So this first line
looks like it is--

00:00:32.729 --> 00:00:35.350
we could do one of those units
and then we could do it again,

00:00:35.350 --> 00:00:38.727
so it looks like
this is two units.

00:00:38.727 --> 00:00:40.810
While this third one looks
like we can get-- let's

00:00:40.810 --> 00:00:44.970
see that's 1, 2, 3 of the units.

00:00:44.970 --> 00:00:47.460
So this is three of the units.

00:00:47.460 --> 00:00:50.500
And right here, I'm
just saying units.

00:00:50.500 --> 00:00:53.490
Sometimes we've made conventions
to define a centimeter, where

00:00:53.490 --> 00:00:55.270
the unit might look
something like this.

00:00:55.270 --> 00:00:57.686
And it's going to look different
depending on your screen.

00:00:57.686 --> 00:01:01.330
Or we might have an inch that
looks something like this.

00:01:01.330 --> 00:01:02.890
Or we might have a
foot that I won't

00:01:02.890 --> 00:01:05.970
be able to fit on this screen
based on how big I've just

00:01:05.970 --> 00:01:07.600
drawn the inch or a meter.

00:01:07.600 --> 00:01:09.480
So there's different
units that you

00:01:09.480 --> 00:01:12.120
could use to
measure in terms of.

00:01:12.120 --> 00:01:14.580
But now let's think
about more dimensions.

00:01:14.580 --> 00:01:17.250
This is literally a
one-dimensional case.

00:01:17.250 --> 00:01:18.310
This is 1D.

00:01:18.310 --> 00:01:19.430
Why is it one dimension?

00:01:19.430 --> 00:01:23.970
Well, I can only measure length.

00:01:23.970 --> 00:01:26.350
But now let's go to a 2D case.

00:01:26.350 --> 00:01:28.720
Let's go to two
dimensions where objects

00:01:28.720 --> 00:01:33.730
could have a length and a
width or a width and a height.

00:01:33.730 --> 00:01:37.340
So let's imagine two figures
here that look like this.

00:01:37.340 --> 00:01:39.770
So let's say this
is one of them.

00:01:39.770 --> 00:01:43.260
This is one of them.

00:01:43.260 --> 00:01:47.692
And notice, it has a
width and it has a height.

00:01:47.692 --> 00:01:49.650
Or you could view it as
a width and the length,

00:01:49.650 --> 00:01:51.191
depending on how
you want to view it.

00:01:51.191 --> 00:01:53.970
So let's say this is one
figure right over here.

00:01:53.970 --> 00:01:55.805
And let's say this
is the other one.

00:01:55.805 --> 00:02:00.630
So this is the other
one right over here.

00:02:00.630 --> 00:02:04.580
Try to draw them
reasonably well.

00:02:04.580 --> 00:02:06.880
Now, once again, now
we're in two dimensions.

00:02:06.880 --> 00:02:10.250
And we want to say, well,
how much in two dimensions

00:02:10.250 --> 00:02:11.700
space is this taking up?

00:02:11.700 --> 00:02:14.147
Or how much area are each
of these two taking up?

00:02:14.147 --> 00:02:16.230
Well, once again, we could
just make a comparison.

00:02:16.230 --> 00:02:19.530
This second, if you viewed
them as carpets or rectangles,

00:02:19.530 --> 00:02:21.860
the second rectangle is
taking up more of my screen

00:02:21.860 --> 00:02:24.740
than this first one, but I
want to be able to measure it.

00:02:24.740 --> 00:02:26.270
So how would we measure it?

00:02:26.270 --> 00:02:28.889
Well, once again, we would
define a unit square.

00:02:28.889 --> 00:02:31.305
Instead of just a unit length,
we now have two dimensions.

00:02:31.305 --> 00:02:34.270
We have to define a unit square.

00:02:34.270 --> 00:02:37.840
And so we might make
our unit square.

00:02:37.840 --> 00:02:41.620
And the unit square we will
define as being a square,

00:02:41.620 --> 00:02:44.390
where its width
and its height are

00:02:44.390 --> 00:02:46.670
both equal to the unit length.

00:02:46.670 --> 00:02:52.310
So this is its width is one
unit and its height is one unit.

00:02:52.310 --> 00:02:56.250
And so we will often
call this 1 square unit.

00:02:56.250 --> 00:02:59.750
Oftentimes, you'll
say this is 1 unit.

00:02:59.750 --> 00:03:03.746
And you put this 2 up here, this
literally means 1 unit squared.

00:03:03.746 --> 00:03:05.120
And instead of
writing unit, this

00:03:05.120 --> 00:03:06.740
could've been a centimeter.

00:03:06.740 --> 00:03:09.420
So this would be 1
square centimeter.

00:03:09.420 --> 00:03:12.490
But now we can use this
to measure these areas.

00:03:12.490 --> 00:03:14.900
And just as we said
how many of this unit

00:03:14.900 --> 00:03:16.620
length could fit
on these lines, we

00:03:16.620 --> 00:03:19.797
could say, how many of these
unit squares can fit in here?

00:03:19.797 --> 00:03:21.880
And so here, we might take
one of our unit squares

00:03:21.880 --> 00:03:24.597
and say, OK, it fills
up that much space.

00:03:24.597 --> 00:03:26.180
Well, we need more
to cover all of it.

00:03:26.180 --> 00:03:28.740
Well, there, we'll put
another unit square there.

00:03:28.740 --> 00:03:31.370
We'll put another unit
square right over there.

00:03:31.370 --> 00:03:33.670
We'll put another unit
square right over there.

00:03:33.670 --> 00:03:36.530
Wow, 4 units squares
exactly cover this.

00:03:36.530 --> 00:03:38.170
So we would say that
this has an area

00:03:38.170 --> 00:03:45.950
of 4 square units
or 4 units squared.

00:03:45.950 --> 00:03:48.110
Now what about this
one right over here?

00:03:48.110 --> 00:03:58.910
Well, here, let's seem I could
fit 1, 2, 3, 4, 5, 6, 7, 8,

00:03:58.910 --> 00:03:59.940
and 9.

00:03:59.940 --> 00:04:06.710
So here I could fit 9
units, 9 units squared.

00:04:06.710 --> 00:04:07.710
Let's keep going.

00:04:07.710 --> 00:04:09.960
We live in a
three-dimensional world.

00:04:09.960 --> 00:04:12.320
Why restrict ourselves
to only one or two?

00:04:12.320 --> 00:04:14.890
So let's go to the 3D case.

00:04:14.890 --> 00:04:16.751
And once again,
when people say 3D,

00:04:16.751 --> 00:04:18.209
they're talking
about 3 dimensions.

00:04:18.209 --> 00:04:20.050
They're talking about
the different directions

00:04:20.050 --> 00:04:21.341
that you can measure things in.

00:04:21.341 --> 00:04:22.620
Here there's only length.

00:04:22.620 --> 00:04:26.340
Here there is length and
width or width and height.

00:04:26.340 --> 00:04:29.890
And here, there'll be
width and height and depth.

00:04:29.890 --> 00:04:37.910
So once again, if you have,
let's say, an object, and now

00:04:37.910 --> 00:04:39.910
we're in three dimensions,
we're in the world we

00:04:39.910 --> 00:04:43.120
live in that looks
like this, and then

00:04:43.120 --> 00:04:52.260
you have another object
that looks like this,

00:04:52.260 --> 00:04:57.730
it looks like this second
object takes up more space,

00:04:57.730 --> 00:05:01.120
more physical space than
this first object does.

00:05:01.120 --> 00:05:03.510
It looks like it
has a larger volume.

00:05:03.510 --> 00:05:05.050
But how do we
actually measure that?

00:05:05.050 --> 00:05:08.460
And remember, volume is just how
much space something takes up

00:05:08.460 --> 00:05:09.960
in three dimensions.

00:05:09.960 --> 00:05:14.340
Area is how much space something
takes up in two dimensions.

00:05:14.340 --> 00:05:16.660
Length is how much
space something takes up

00:05:16.660 --> 00:05:18.030
in one dimension.

00:05:18.030 --> 00:05:19.530
But when we think
about space, we're

00:05:19.530 --> 00:05:21.238
normally thinking
about three dimensions.

00:05:21.238 --> 00:05:24.250
So how much space would
you take up in the world

00:05:24.250 --> 00:05:25.370
that we live in?

00:05:25.370 --> 00:05:27.530
So just like we did before,
we can define, instead

00:05:27.530 --> 00:05:30.080
of a unit length
or unit area, we

00:05:30.080 --> 00:05:33.160
can define a unit
volume or unit cube.

00:05:33.160 --> 00:05:34.870
So let's do that.

00:05:34.870 --> 00:05:36.770
Let's define our unit cube.

00:05:36.770 --> 00:05:41.110
And here, it's a cube so its
length, width, and height

00:05:41.110 --> 00:05:43.170
are going to be the same value.

00:05:43.170 --> 00:05:45.400
So my best attempt
at drawing a cube.

00:05:45.400 --> 00:05:47.050
And they're all
going to be one unit.

00:05:47.050 --> 00:05:52.580
So it's going to be one
unit high, one unit deep,

00:05:52.580 --> 00:05:55.685
and one unit wide.

00:05:55.685 --> 00:05:57.560
And so to measure volume,
we could say, well,

00:05:57.560 --> 00:06:00.020
how many of these
unit cubes can fit

00:06:00.020 --> 00:06:02.200
into these different shapes?

00:06:02.200 --> 00:06:04.391
Well, this one right
over here, and you

00:06:04.391 --> 00:06:06.140
won't be able to
actually see all of them.

00:06:06.140 --> 00:06:08.300
I could essentially
break it down

00:06:08.300 --> 00:06:11.390
into-- so let me see
how well I can do this

00:06:11.390 --> 00:06:13.105
so that we can count them all.

00:06:13.105 --> 00:06:14.980
It's a little bit
harder to see them all

00:06:14.980 --> 00:06:17.850
because there's some
cubes that are behind us.

00:06:17.850 --> 00:06:20.180
But if you think of
it as two layers,

00:06:20.180 --> 00:06:22.980
so one layer would
look like this.

00:06:22.980 --> 00:06:25.060
One layer is going
to look like this.

00:06:25.060 --> 00:06:28.000
So imagine two things like this
stacked on top of each other.

00:06:28.000 --> 00:06:30.655
So this one's going to
have 1, 2, 3, 4 cubes.

00:06:30.655 --> 00:06:32.280
Now, this is going
to have two of these

00:06:32.280 --> 00:06:33.630
stacked on top of each other.

00:06:33.630 --> 00:06:37.370
So here you have 8 unit cubes.

00:06:37.370 --> 00:06:40.720
Or you could have 8
units cubed volume.

00:06:40.720 --> 00:06:42.140
What about here?

00:06:42.140 --> 00:06:46.150
If we try to fit
it all in-- let me

00:06:46.150 --> 00:06:47.525
see how well I could draw this.

00:06:50.735 --> 00:06:52.360
It's going to look
something like this.

00:06:55.570 --> 00:06:58.260
And obviously, this is
kind of a rough drawing.

00:06:58.260 --> 00:07:00.600
And so if we were to
try to take this apart,

00:07:00.600 --> 00:07:04.850
you would essentially have a
stack of three sections that

00:07:04.850 --> 00:07:07.840
would each look
something like this.

00:07:07.840 --> 00:07:10.030
My best attempt at drawing it.

00:07:10.030 --> 00:07:14.250
Three sections that
would look something

00:07:14.250 --> 00:07:19.220
like what I'm about to draw.

00:07:19.220 --> 00:07:22.130
So it would look like this.

00:07:22.130 --> 00:07:24.270
So if you took three of
these and stacked them

00:07:24.270 --> 00:07:26.830
on top of each other, you'd
get this right over here.

00:07:26.830 --> 00:07:32.050
And each of these have 1, 2, 3,
4, 5, 6, 7, 8, 9 cubes in it.

00:07:32.050 --> 00:07:38.280
9 times 3, you're going to
have 27 cubic units in this one

00:07:38.280 --> 00:07:38.966
right over here.

00:07:38.966 --> 00:07:40.840
So hopefully that helps
us think a little bit

00:07:40.840 --> 00:07:42.510
about how we measure
things especially

00:07:42.510 --> 00:07:45.960
how we measure things in
different number of dimensions,

00:07:45.960 --> 00:07:50.120
especially in three dimensions
when we call it volume.


Volume: measuring with unit cubes

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.190
We are asked, what is
the volume of this box?

00:00:04.190 --> 00:00:06.900
And they tell us that
each cube is a cubic foot.

00:00:06.900 --> 00:00:09.630
So just as a reminder,
volume is a measure

00:00:09.630 --> 00:00:12.560
of how much space does
something take up.

00:00:12.560 --> 00:00:15.690
And it's usually
measured in cubic units.

00:00:15.690 --> 00:00:17.730
And here we're talking
about cubic feet.

00:00:17.730 --> 00:00:20.270
And when they say that
each cube is a cubic foot,

00:00:20.270 --> 00:00:22.420
they're saying that each
of these little boxes,

00:00:22.420 --> 00:00:27.790
each of these cubes here, is
exactly a 1 foot by 1 foot

00:00:27.790 --> 00:00:30.110
by 1 foot cube.

00:00:30.110 --> 00:00:31.679
We're in three
dimensions right now.

00:00:31.679 --> 00:00:33.220
We're talking about
three dimensions.

00:00:33.220 --> 00:00:35.420
3D, you literally
need three dimensions.

00:00:35.420 --> 00:00:37.610
You need how tall you
are, how wide you are,

00:00:37.610 --> 00:00:38.820
and how deep you are.

00:00:38.820 --> 00:00:41.840
And for a cubic foot, each
of those dimensions is one.

00:00:41.840 --> 00:00:43.500
So each of these
are one cubic foot.

00:00:43.500 --> 00:00:47.470
The entire shape has one,
two, three, four, five,

00:00:47.470 --> 00:00:49.490
six cubic feet in it.

00:00:49.490 --> 00:00:54.250
So the volume of this
entire box is 6 cubic feet.

00:00:54.250 --> 00:00:56.080
And you see right here
the unit says feet,

00:00:56.080 --> 00:00:58.020
and it has a
superscript here of 3.

00:00:58.020 --> 00:00:59.367
This is feet to the third power.

00:00:59.367 --> 00:01:01.200
You could view this as
feet times feet times

00:01:01.200 --> 00:01:03.120
feet, or cubic feet.

00:01:03.120 --> 00:01:05.040
So this is 6 cubic feet.

00:01:05.040 --> 00:01:07.026
Let's do a few more of these.

00:01:07.026 --> 00:01:08.400
What is the volume
of this shape?

00:01:08.400 --> 00:01:13.020
So this one is quite
interesting right over here.

00:01:13.020 --> 00:01:15.620
You have-- let's see
if we can count these.

00:01:15.620 --> 00:01:20.316
So you have one, two, three,
four, five, six, seven,

00:01:20.316 --> 00:01:23.840
eight unit boxes,
or I guess these

00:01:23.840 --> 00:01:27.020
are cubic inches
for this example.

00:01:27.020 --> 00:01:30.020
So we have 8 cubic inches.

00:01:30.020 --> 00:01:32.620
This is a lot of fun, especially
when we get to rotate it.

00:01:32.620 --> 00:01:35.780
So here, each cube is
a cubic foot again.

00:01:35.780 --> 00:01:37.740
So we have one, two,
three cubic feet.

00:01:41.000 --> 00:01:42.190
Let's do one more of these.

00:01:42.190 --> 00:01:44.356
I like the ones that have
these kind of crazy shapes

00:01:44.356 --> 00:01:47.340
that we have to rotate to be
able to see all the boxes.

00:01:47.340 --> 00:01:49.610
So each of these
is a cubic meter,

00:01:49.610 --> 00:01:52.520
which means that in each
dimension, each of these boxes

00:01:52.520 --> 00:01:54.330
is a 1 by 1 by 1 meter.

00:01:54.330 --> 00:01:55.610
But how many boxes are there?

00:01:55.610 --> 00:01:56.110
Let's see.

00:01:56.110 --> 00:01:57.510
We have one.

00:01:57.510 --> 00:01:58.490
We have one.

00:01:58.490 --> 00:02:01.390
Let me see if I can get a view
where I can look at it better.

00:02:01.390 --> 00:02:07.030
We have one, two, three,
four, five, six boxes.

00:02:07.030 --> 00:02:11.290
So the total volume
here is 6 cubic meters.


Seeing multiplication as scaling with fractions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.110
We have three expressions here.

00:00:02.110 --> 00:00:04.130
This is 2/3 times 7/8.

00:00:04.130 --> 00:00:07.700
The second expression
is 8/7 times 2/3.

00:00:07.700 --> 00:00:13.090
This third expression is
5 times 2 over 3 times 5.

00:00:13.090 --> 00:00:15.720
And what I want you to do is
pause this video right now

00:00:15.720 --> 00:00:17.470
and think about which
of these expressions

00:00:17.470 --> 00:00:22.400
is the largest, which one is in
the middle in terms of value,

00:00:22.400 --> 00:00:24.699
and which one is the smallest.

00:00:24.699 --> 00:00:26.740
And I want you to think
about it without actually

00:00:26.740 --> 00:00:27.810
doing the calculation.

00:00:27.810 --> 00:00:29.210
If you could just look
at them and figure out

00:00:29.210 --> 00:00:31.793
which of these is the largest,
which of these is the smallest,

00:00:31.793 --> 00:00:33.360
and which of these
is in the middle.

00:00:33.360 --> 00:00:35.586
So pause the video now.

00:00:35.586 --> 00:00:37.210
Now, you might have
taken a shot at it.

00:00:37.210 --> 00:00:40.460
And I'll give you a little
bit of a hint in case

00:00:40.460 --> 00:00:41.590
you had trouble with it.

00:00:41.590 --> 00:00:44.850
All of these involve
multiplying something by 2/3.

00:00:44.850 --> 00:00:46.930
And you see a 2/3 here.

00:00:46.930 --> 00:00:48.495
You see a 2/3 here.

00:00:48.495 --> 00:00:52.150
And it might not be as obvious,
but you also see a 2/3 here.

00:00:52.150 --> 00:00:54.670
And let me rewrite that to
make it a little bit clearer.

00:00:54.670 --> 00:01:01.830
So this first expression could
be rewritten as 7/8 times 2/3.

00:01:01.830 --> 00:01:04.749
This second expression here
could be written as-- well,

00:01:04.749 --> 00:01:09.426
it's already written
as 8/7 times 2/3.

00:01:12.360 --> 00:01:14.180
And then, this
last expression, we

00:01:14.180 --> 00:01:20.760
could write it as, in
the numerator, 5 times 2.

00:01:20.760 --> 00:01:26.790
And then in the denominator,
it's over 5 times 3.

00:01:26.790 --> 00:01:34.670
5 times 3, which is of course
the same thing as 5/5 times

00:01:34.670 --> 00:01:37.610
2/3.

00:01:37.610 --> 00:01:39.620
So you see, all three
of these expressions

00:01:39.620 --> 00:01:41.936
involve something times 2/3.

00:01:41.936 --> 00:01:43.310
Now, looking at
it this way, does

00:01:43.310 --> 00:01:45.268
it become easier to pick
out which of these are

00:01:45.268 --> 00:01:47.237
the largest, which of
these are the smallest,

00:01:47.237 --> 00:01:49.070
and which of these are
someplace in between?

00:01:49.070 --> 00:01:50.444
I encourage you
to pause it again

00:01:50.444 --> 00:01:52.530
if you haven't
thought about it yet.

00:01:52.530 --> 00:01:54.820
So let's visualize each
of these expressions

00:01:54.820 --> 00:01:57.710
by first trying
to visualize 2/3.

00:01:57.710 --> 00:02:00.570
So let's say the height of
what I am drawing right now,

00:02:00.570 --> 00:02:04.510
let's say the height of this
bar right over here is 2/3.

00:02:04.510 --> 00:02:07.650
So this right over
here represents 2/3.

00:02:07.650 --> 00:02:11.075
The height here is 2/3.

00:02:14.480 --> 00:02:16.990
So first, let's think about
what this one on the right

00:02:16.990 --> 00:02:18.320
here represents.

00:02:18.320 --> 00:02:21.580
This is 5/5 times 2/3.

00:02:21.580 --> 00:02:22.750
Well, what's 5/5?

00:02:22.750 --> 00:02:24.880
5/5 is the same thing as 1.

00:02:24.880 --> 00:02:27.805
This is literally
just 1 times 2/3.

00:02:27.805 --> 00:02:29.430
This whole expression
is the same thing

00:02:29.430 --> 00:02:32.230
as 1 times 2/3, or
really, just 2/3.

00:02:32.230 --> 00:02:34.550
So this, the height
here, 2/3, this

00:02:34.550 --> 00:02:36.950
is the same thing as
this thing over here.

00:02:36.950 --> 00:02:38.610
This is going to
be equal to-- this

00:02:38.610 --> 00:02:42.590
could also be viewed as
5 times 2 over 3 times

00:02:42.590 --> 00:02:46.420
5, which was this first
expression right over here.

00:02:46.420 --> 00:02:50.720
Now, let's think about
what these would look like.

00:02:50.720 --> 00:02:53.630
So this is 7/8 times 2/3.

00:02:53.630 --> 00:02:56.340
So it's less than 8/8 times 2/3.

00:02:56.340 --> 00:02:58.570
It's less than 1 times 2/3.

00:02:58.570 --> 00:03:01.190
So we're going to
scale 2/3 down.

00:03:01.190 --> 00:03:03.470
This is going to
be less than 2/3.

00:03:03.470 --> 00:03:06.000
It's going to be 7/8 of 2/3.

00:03:06.000 --> 00:03:13.020
So this one right over here
would look something like this.

00:03:13.020 --> 00:03:14.295
Let me see if I can draw it.

00:03:14.295 --> 00:03:19.580
Yeah, it would look
something like this.

00:03:19.580 --> 00:03:23.720
If the yellow height is 2/3,
then this right over here,

00:03:23.720 --> 00:03:29.400
then this height right over
here-- let me make it clear.

00:03:29.400 --> 00:03:34.709
This height right over here
would be 7/8 times 2/3.

00:03:34.709 --> 00:03:36.750
Likewise, let's look at
this one right over here.

00:03:36.750 --> 00:03:41.210
Let's look at this one in
the middle, 8/7 times 2/3.

00:03:41.210 --> 00:03:45.260
Well, 8/7 is bigger than 7/7.

00:03:45.260 --> 00:03:46.510
It's more than 1.

00:03:46.510 --> 00:03:48.270
This is more than 2/3.

00:03:48.270 --> 00:03:50.610
This is 1 and 1/7 times 2/3.

00:03:50.610 --> 00:03:54.200
So it's going to be the same
height as 2/3 plus another 1/7.

00:03:54.200 --> 00:03:58.020
So it's going to look
something like this.

00:03:58.020 --> 00:04:00.200
It's going to look
something like this.

00:04:00.200 --> 00:04:05.520
So its height-- now we
scaled the 2/3 up because 8/7

00:04:05.520 --> 00:04:07.420
is greater than 1.

00:04:07.420 --> 00:04:10.140
So this right over
here, this height

00:04:10.140 --> 00:04:14.467
is going to be 8/7 times 2/3.

00:04:14.467 --> 00:04:16.050
So the way that you
could have spotted

00:04:16.050 --> 00:04:18.175
which of these is the
largest and which of these is

00:04:18.175 --> 00:04:21.727
the smallest is to say, well,
how are they scaling 2/3?

00:04:21.727 --> 00:04:23.560
This one right over
here, you're essentially

00:04:23.560 --> 00:04:25.330
multiplying 2/3 by 1.

00:04:25.330 --> 00:04:27.820
So you're just going to get 2/3.

00:04:27.820 --> 00:04:30.980
You're not scaling it up, or
you aren't scaling it down.

00:04:30.980 --> 00:04:33.990
This one right over here,
you're scaling 2/3 down.

00:04:33.990 --> 00:04:36.580
You are multiplying it
by something less than 1.

00:04:36.580 --> 00:04:39.404
If you multiply it by
something less than 1,

00:04:39.404 --> 00:04:41.070
then you're going to
be scaling it down.

00:04:41.070 --> 00:04:43.110
I should say, a positive
number or a number

00:04:43.110 --> 00:04:45.692
between 0 and 1--
less than 1-- then

00:04:45.692 --> 00:04:47.150
you're going to be
scaling it down.

00:04:47.150 --> 00:04:48.690
So this thing is scaled down.

00:04:48.690 --> 00:04:50.160
It's going to be the smallest.

00:04:50.160 --> 00:04:52.690
And here, you're multiplying
the 2/3 times a number

00:04:52.690 --> 00:04:55.200
bigger than 1, by 1 and 1/7.

00:04:55.200 --> 00:04:56.740
So you're going to scale it up.

00:04:56.740 --> 00:05:00.520
So this expression is the
largest, 8/7 times 2/3.

00:05:00.520 --> 00:05:03.050
The smallest is 2/3 times 7/8.

00:05:03.050 --> 00:05:06.240
And this one right over
here is in between.


Linear equation word problem

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.170
MacDonald had a farm with a
certain number of orange trees.

00:00:04.170 --> 00:00:07.620
He had to cut down 5 trees
to control the insects.

00:00:07.620 --> 00:00:10.690
Each of the remaining
trees produced 210 oranges,

00:00:10.690 --> 00:00:15.250
producing a total harvest
of 41,790 oranges.

00:00:15.250 --> 00:00:19.340
How many trees did MacDonald's
farm have initially?

00:00:19.340 --> 00:00:22.630
Let's let t equal what
they're asking is for.

00:00:22.630 --> 00:00:26.960
So this is the number
of trees initially.

00:00:29.670 --> 00:00:32.390
So he starts off with
t trees, but then they

00:00:32.390 --> 00:00:38.980
tell us that he has to cut down
5 trees to control the insects.

00:00:38.980 --> 00:00:41.090
So how many trees would
he have after that?

00:00:41.090 --> 00:00:43.650
Well, he started with t,
and he had to cut down 5,

00:00:43.650 --> 00:00:46.880
so he's going to have
t minus 5 trees now.

00:00:46.880 --> 00:00:52.660
Now, they tell us that each of
the remaining trees-- and we

00:00:52.660 --> 00:00:55.780
know there are t minus 5
remaining trees-- produced

00:00:55.780 --> 00:00:58.220
210 oranges.

00:00:58.220 --> 00:01:01.040
So each of these t
minus 5 trees are

00:01:01.040 --> 00:01:03.920
going to produce 210 oranges.

00:01:03.920 --> 00:01:07.350
So this is the number of
oranges that t minus 5 trees

00:01:07.350 --> 00:01:08.420
are going to produce.

00:01:08.420 --> 00:01:12.040
This is the number of trees
times the oranges per tree.

00:01:12.040 --> 00:01:14.170
So this is the total
number of oranges

00:01:14.170 --> 00:01:17.660
produced after
cutting the 5 trees.

00:01:17.660 --> 00:01:20.420
And then they tell
us that this ends up

00:01:20.420 --> 00:01:26.110
being a total harvest of 41,790.

00:01:26.110 --> 00:01:30.130
So this is equal to 41,790.

00:01:30.130 --> 00:01:31.570
So we've set up our equation.

00:01:31.570 --> 00:01:34.450
Now we just have to solve
for t, the number of trees

00:01:34.450 --> 00:01:36.964
that MacDonald initially had.

00:01:36.964 --> 00:01:38.380
So the first thing
I would do here

00:01:38.380 --> 00:01:41.905
is, well, I'm multiplying
this expression by 210.

00:01:41.905 --> 00:01:45.212
Well, why don't I just divide
both sides of this by 210?

00:01:45.212 --> 00:01:46.920
There's many ways that
you could do this.

00:01:46.920 --> 00:01:50.089
You could distribute the 210
and go in another direction.

00:01:50.089 --> 00:01:51.630
Actually, I will do
it both ways just

00:01:51.630 --> 00:01:53.940
to show that you
could do it both ways.

00:01:53.940 --> 00:02:00.280
So the first way, I'm going
to divide both sides by 210.

00:02:00.280 --> 00:02:04.215
The left-hand side
simplifies to t minus 5.

00:02:06.780 --> 00:02:08.900
The right-hand side--
let's see, what

00:02:08.900 --> 00:02:12.275
is 4,000-- I'm going to do
some long division here.

00:02:12.275 --> 00:02:21.390
I'll do it on the side,
so 41,790 divided by 210.

00:02:21.390 --> 00:02:23.010
Let's see, 210
does not go into 4.

00:02:23.010 --> 00:02:24.530
It does not going into 41.

00:02:24.530 --> 00:02:28.220
It goes into 417 one
time, because two times

00:02:28.220 --> 00:02:30.530
would be 420-- one time.

00:02:30.530 --> 00:02:33.340
1 times 210 is 210.

00:02:33.340 --> 00:02:34.640
You subtract.

00:02:34.640 --> 00:02:39.190
You get 207, and then
you bring down the 9.

00:02:39.190 --> 00:02:43.180
How many times does
210 go into 2,079?

00:02:43.180 --> 00:02:46.750
It looks like it would go
into it not quite 10 times.

00:02:46.750 --> 00:02:49.030
It looks like it would
go into it nine times.

00:02:49.030 --> 00:02:53.170
9 times 210 is going
to be-- let's see.

00:02:53.170 --> 00:02:54.510
9 times 0 is 0.

00:02:54.510 --> 00:02:56.100
9 times 1 is 9.

00:02:56.100 --> 00:02:58.940
9 times 2 is 18.

00:02:58.940 --> 00:03:01.420
And then we subtract again.

00:03:01.420 --> 00:03:03.710
9 minus 0 is 9.

00:03:03.710 --> 00:03:06.310
We have to regroup from
the thousands place,

00:03:06.310 --> 00:03:10.420
so let's take 1,000 from there.

00:03:10.420 --> 00:03:13.740
Let's give that 1,000
to the hundreds place,

00:03:13.740 --> 00:03:15.550
so it becomes 10 hundreds.

00:03:15.550 --> 00:03:17.910
But then we have to take
100 from the hundreds place,

00:03:17.910 --> 00:03:21.020
so this becomes 9, and
give to the tens place.

00:03:21.020 --> 00:03:24.370
So this becomes 17 tens, or 170.

00:03:24.370 --> 00:03:27.760
So 17 minus 9 is 8.

00:03:27.760 --> 00:03:30.450
9 minus 8 is 1.

00:03:30.450 --> 00:03:33.330
So we get 189.

00:03:33.330 --> 00:03:36.615
And now we can bring
down another 0.

00:03:39.585 --> 00:03:41.430
It's a little
off-center right now.

00:03:41.430 --> 00:03:46.820
And we already see that 210 goes
into 1,800 ninety-nine times.

00:03:46.820 --> 00:03:51.050
9 times 210 is 1,890.

00:03:51.050 --> 00:03:53.660
When we subtract, we
have no remainder.

00:03:53.660 --> 00:04:01.030
So what we get on the
right-hand side is 199.

00:04:01.030 --> 00:04:03.200
And now we just have
to add 5 to both sides.

00:04:03.200 --> 00:04:04.520
Remember, whatever we
have to do to one side,

00:04:04.520 --> 00:04:05.519
we have to do the other.

00:04:05.519 --> 00:04:07.679
Otherwise, the equality
wouldn't be equal anymore.

00:04:07.679 --> 00:04:10.220
They were equal before adding
5, so if you want them to still

00:04:10.220 --> 00:04:12.750
be equal, you have to do the
same thing to both sides.

00:04:12.750 --> 00:04:14.810
So the left-hand side becomes t.

00:04:14.810 --> 00:04:19.490
I'll do the t in
that purple color.

00:04:19.490 --> 00:04:22.500
And the right-hand
side becomes 204.

00:04:22.500 --> 00:04:25.810
So he started off
with 204 trees.

00:04:25.810 --> 00:04:28.100
Now, I told you there's
multiple ways to do this.

00:04:28.100 --> 00:04:30.550
Instead of dividing
both sides by 210,

00:04:30.550 --> 00:04:33.849
you could have decided
to distribute the 210.

00:04:33.849 --> 00:04:35.640
And then you would have
ended up with-- let

00:04:35.640 --> 00:04:37.800
me do another alternate
way of doing it.

00:04:37.800 --> 00:04:49.140
210 times t minus 5 times 210.

00:04:49.140 --> 00:04:51.820
Actually, let me just multiply
it out so we save some space.

00:04:51.820 --> 00:05:01.770
5 times 210 is 1,050-- minus
1,050 is equal to 41,790.

00:05:01.770 --> 00:05:05.720
And then you could add
1,050 to both sides.

00:05:05.720 --> 00:05:10.630
And so let me do that,
1,050 to both sides.

00:05:10.630 --> 00:05:16.150
1,050, not 150.

00:05:16.150 --> 00:05:21.870
The left-hand side, you're just
going to be left with 210t.

00:05:21.870 --> 00:05:24.460
While the right-hand
side, let's see,

00:05:24.460 --> 00:05:28.350
you're going to
be 0 plus 0 is 0.

00:05:28.350 --> 00:05:31.120
9 plus 5 is 14.

00:05:31.120 --> 00:05:36.910
1 plus 7 is 8-- 42,840.

00:05:36.910 --> 00:05:45.067
And now you can divide
both sides by 210.

00:05:45.067 --> 00:05:46.650
Now we know where
this is going to go.

00:05:46.650 --> 00:05:48.740
I could do the long
division again.

00:05:48.740 --> 00:05:52.370
t is going to be equal
to 42,840 divided

00:05:52.370 --> 00:05:57.170
210, which is equal to 204.


Constructing equations from proportions to solve problems

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:04.320
Rick just finished eating with
his family at a restaurant.

00:00:04.320 --> 00:00:07.990
The total bill for
his family was $52.

00:00:07.990 --> 00:00:10.140
And Rick was pretty
impressed with the service.

00:00:10.140 --> 00:00:13.830
So he's going to
leave a 25% tip.

00:00:13.830 --> 00:00:18.240
He wants to know
the total amount

00:00:18.240 --> 00:00:21.390
he should leave at the table.

00:00:21.390 --> 00:00:23.940
Select the equations Rick
can use and determine

00:00:23.940 --> 00:00:25.774
the total bill,
including the tip.

00:00:25.774 --> 00:00:28.190
So the unknown thing, the thing
that we should figure out,

00:00:28.190 --> 00:00:30.480
is the total amount he
should leave at the table.

00:00:30.480 --> 00:00:33.750
So let's just call that x.

00:00:33.750 --> 00:00:36.000
Now, there's a couple of
ways we could think about it.

00:00:36.000 --> 00:00:37.500
We could think about
it as the ratio

00:00:37.500 --> 00:00:39.820
between the total
amount he's going to pay

00:00:39.820 --> 00:00:41.405
and the actual bill.

00:00:45.620 --> 00:00:47.780
That should be the
same thing as the ratio

00:00:47.780 --> 00:00:49.850
between the respective ratios.

00:00:49.850 --> 00:00:53.810
So you could view the
actual amount of the bill

00:00:53.810 --> 00:00:55.850
as being 100%.

00:00:55.850 --> 00:00:57.870
That is 100% of the bill.

00:00:57.870 --> 00:00:59.510
But he doesn't want
to just pay 100%.

00:00:59.510 --> 00:01:04.445
He wants to pay
100% plus a 25% tip.

00:01:07.930 --> 00:01:10.310
So we could rewrite everything
that I just did here

00:01:10.310 --> 00:01:17.900
as x over 52 is equal
to 100% plus 25%--

00:01:17.900 --> 00:01:26.091
I could write that
as 125%-- over 100%.

00:01:26.091 --> 00:01:27.590
Now, if we look at
the choices here,

00:01:27.590 --> 00:01:29.340
this looks exactly like this.

00:01:29.340 --> 00:01:31.630
If we just swap the left
and right-hand side,

00:01:31.630 --> 00:01:36.770
we get 125% over 100%
is equal to x over 52.

00:01:36.770 --> 00:01:38.380
So we can select this one.

00:01:38.380 --> 00:01:39.840
And once again,
this makes sense.

00:01:39.840 --> 00:01:41.920
The ratio between
what he's going

00:01:41.920 --> 00:01:45.369
to pay with the tip to
100%, or the percentage

00:01:45.369 --> 00:01:46.910
that he's going to
pay if you include

00:01:46.910 --> 00:01:50.070
the tip relative
to 100% is going

00:01:50.070 --> 00:01:54.260
to be the actual amount he pays
relative to the actual bill.

00:01:54.260 --> 00:01:56.150
Now, let's see if any
of these other ones

00:01:56.150 --> 00:01:58.060
make sense over here.

00:01:58.060 --> 00:02:01.230
So here he says it's the
ratio of the actual bill

00:02:01.230 --> 00:02:06.406
at 100% is equal to the ratio
you're going to pay at 125%.

00:02:06.406 --> 00:02:07.530
Now, this makes sense, too.

00:02:07.530 --> 00:02:09.870
Let me just rewrite
it again for emphasis.

00:02:09.870 --> 00:02:11.890
So here they're
constructing a proportion

00:02:11.890 --> 00:02:14.610
where you have the
ratio of the total bill.

00:02:14.610 --> 00:02:17.310
And you're saying, hey, look,
that total bill is 100%.

00:02:21.380 --> 00:02:22.995
Actually, let me
write that in blue.

00:02:22.995 --> 00:02:26.400
The total bill is 100%.

00:02:26.400 --> 00:02:29.590
And we want to see
this ratio should

00:02:29.590 --> 00:02:32.450
be the same as the ratio
between the actual amount we're

00:02:32.450 --> 00:02:33.850
paying and 125%.

00:02:37.770 --> 00:02:41.200
So you increased this by 25%.

00:02:41.200 --> 00:02:45.680
Well, you're going to have
to increase this by 25%

00:02:45.680 --> 00:02:48.660
as well to get the actual
amount that you're paying.

00:02:48.660 --> 00:02:51.615
So this makes sense as well.

00:02:51.615 --> 00:02:52.490
Now, what about this?

00:02:52.490 --> 00:02:59.000
The ratio between 100% and 125%
should be equal to x over 52.

00:02:59.000 --> 00:03:02.630
Now, this does not make
sense, because this

00:03:02.630 --> 00:03:05.700
is the ratio between
the full price

00:03:05.700 --> 00:03:07.570
without paying the
tip and the tip.

00:03:07.570 --> 00:03:09.954
Well, here is the tip
and the full price.

00:03:09.954 --> 00:03:11.870
So if you swapped these
two, if you wrote this

00:03:11.870 --> 00:03:14.840
as 52 over x, then
you could check this.

00:03:14.840 --> 00:03:16.900
But here you're not
finding the ratio

00:03:16.900 --> 00:03:18.350
between the
corresponding things.

00:03:18.350 --> 00:03:20.460
So I would not check
this one there.

00:03:20.460 --> 00:03:22.070
But either way, our
last thing to do

00:03:22.070 --> 00:03:25.310
is actually to figure out how
much tip Rick should leave.

00:03:25.310 --> 00:03:27.930
So we need to
actually solve for x.

00:03:27.930 --> 00:03:30.640
And we could go back to
this original equation here.

00:03:30.640 --> 00:03:35.980
And if we multiply
both sides by 52,

00:03:35.980 --> 00:03:41.260
we get x is equal
to 125% divided

00:03:41.260 --> 00:03:50.850
by 100% is 1.25 times 52.

00:03:50.850 --> 00:03:52.870
And then I could
just multiply that.

00:03:52.870 --> 00:03:53.370
Let's see.

00:03:53.370 --> 00:03:54.640
52.

00:03:54.640 --> 00:03:58.200
4 goes into 52 13 times.

00:03:58.200 --> 00:04:02.677
So this is going to be-- well,
actually, let me just divide.

00:04:02.677 --> 00:04:03.760
Let me just multiply that.

00:04:03.760 --> 00:04:05.260
I don't want to
make a mistake here.

00:04:05.260 --> 00:04:11.470
So 1.25 times 52.

00:04:11.470 --> 00:04:13.940
2 times 5 is 10.

00:04:13.940 --> 00:04:17.880
2 times 2 is 4 plus 1 is 5.

00:04:17.880 --> 00:04:19.756
2 times 1 is 2.

00:04:19.756 --> 00:04:21.589
Let's throw a 0 down
here, because we're now

00:04:21.589 --> 00:04:23.722
multiplying by 50.

00:04:23.722 --> 00:04:26.840
5 times 5 is 25.

00:04:26.840 --> 00:04:30.920
5 times 2 is 10 plus 2 is 12.

00:04:30.920 --> 00:04:33.780
5 times 1 is 5 plus 1 is 6.

00:04:33.780 --> 00:04:36.260
So we get 0.

00:04:36.260 --> 00:04:37.440
Let's carry that 1.

00:04:37.440 --> 00:04:38.210
We get a 5.

00:04:38.210 --> 00:04:38.741
We get a 6.

00:04:38.741 --> 00:04:40.990
And then we have two numbers
behind the decimal point.

00:04:40.990 --> 00:04:42.882
It is 65.

00:04:42.882 --> 00:04:43.590
And that's right.

00:04:43.590 --> 00:04:47.780
That's 52 plus 1/4
of 52, which is 13.

00:04:47.780 --> 00:04:50.260
52 plus 13 is 65.

00:04:50.260 --> 00:04:57.380
So he should leave $65 if
we want to include the tip.


Multiplying and dividing in scientific notation (example)

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.960
We have 7 times 10 to
the fifth over 2 times 10

00:00:03.960 --> 00:00:08.360
to the negative 2 times
2.5 times 10 to the ninth.

00:00:08.360 --> 00:00:10.500
So let's try to simplify
this a little bit.

00:00:10.500 --> 00:00:13.230
And I'll start off by trying to
simplify this denominator here.

00:00:13.230 --> 00:00:16.764
So the numerator's just
7 times 10 to the fifth.

00:00:16.764 --> 00:00:18.430
And the denominator,
I just have a bunch

00:00:18.430 --> 00:00:20.680
of numbers that are being
multiplied times each other.

00:00:20.680 --> 00:00:22.229
So I can do it in any order.

00:00:22.229 --> 00:00:23.270
So let me swap the order.

00:00:23.270 --> 00:00:31.940
So I'm going to do over
2 times 2.5 times 10

00:00:31.940 --> 00:00:36.730
to the negative 2
times 10 to the ninth.

00:00:40.229 --> 00:00:42.770
And this is going to be equal
to-- so the numerator I haven't

00:00:42.770 --> 00:00:47.942
changed yet-- 7 times
10 to the fifth over--

00:00:47.942 --> 00:00:50.340
and here in the
denominator, 2 times--

00:00:50.340 --> 00:00:52.310
let me do this in
a new color now.

00:00:52.310 --> 00:00:55.520
2 times 2.5 is 5.

00:00:55.520 --> 00:00:58.420
And then 10 to the negative
2 times 10 to the ninth,

00:00:58.420 --> 00:01:00.441
when you multiply
two numbers that

00:01:00.441 --> 00:01:02.940
are being raised to exponents
and have the exact same base--

00:01:02.940 --> 00:01:05.580
so it's 10 to the negative 2
times 10 to the negative 9--

00:01:05.580 --> 00:01:07.160
we can add the exponents.

00:01:07.160 --> 00:01:09.500
So this is going to be
10 to the 9 minus 2,

00:01:09.500 --> 00:01:11.040
or 10 to the seventh.

00:01:11.040 --> 00:01:15.360
So times 10 to the seventh.

00:01:15.360 --> 00:01:17.310
And now we can
view this as being

00:01:17.310 --> 00:01:27.235
equal to 7 over 5 times
10 to the fifth over 10

00:01:27.235 --> 00:01:27.860
to the seventh.

00:01:36.392 --> 00:01:37.850
Let me do that in
that orange color

00:01:37.850 --> 00:01:39.860
to keep track of the colors.

00:01:39.860 --> 00:01:41.550
10 to the seventh.

00:01:41.550 --> 00:01:43.650
Now, what is 7 divided by 5?

00:01:43.650 --> 00:01:48.730
7 divided by 5 is equal to--
let's see, it's 1 and 2/5,

00:01:48.730 --> 00:01:50.840
or 1.4.

00:01:50.840 --> 00:01:52.400
So I'll just write it as 1.4.

00:01:52.400 --> 00:01:56.262
And then 10 to the fifth
divided by 10 to the seventh.

00:01:56.262 --> 00:01:57.970
So that's going to be
the same thing as--

00:01:57.970 --> 00:01:59.050
and there's two
ways to view this.

00:01:59.050 --> 00:02:01.008
You could view this as
10 to the fifth times 10

00:02:01.008 --> 00:02:01.830
to the negative 7.

00:02:01.830 --> 00:02:02.746
You add the exponents.

00:02:02.746 --> 00:02:03.980
You get 10 to the negative 2.

00:02:03.980 --> 00:02:06.920
Or you say, hey, look,
I'm dividing this by this.

00:02:06.920 --> 00:02:08.039
We have the same base.

00:02:08.039 --> 00:02:09.360
We can subtract exponents.

00:02:09.360 --> 00:02:11.680
So it's going to be
10 to the 5 minus 7,

00:02:11.680 --> 00:02:13.740
which is 10 to the negative 2.

00:02:13.740 --> 00:02:18.480
So this part right over here is
going to simplify to times 10

00:02:18.480 --> 00:02:20.860
to the negative 2.

00:02:20.860 --> 00:02:21.880
Now, are we done?

00:02:21.880 --> 00:02:25.560
Have we written what we have
here in scientific notation?

00:02:25.560 --> 00:02:27.640
It looks like we have.

00:02:27.640 --> 00:02:31.700
This value right over here is
greater than or equal to 1,

00:02:31.700 --> 00:02:36.910
but it is less
than or equal to 9.

00:02:36.910 --> 00:02:40.550
It's a digit between 1
and 9, including 1 and 9.

00:02:40.550 --> 00:02:45.450
And it's being multiplied
by 10 to some power.

00:02:45.450 --> 00:02:47.410
So it looks like we're done.

00:02:47.410 --> 00:02:52.420
This simplified to 1.4
times 10 to the negative 2.


Simplifying a complicated expression into scientific notation

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.140
Let's see if we can simplify
this expression right over

00:00:03.140 --> 00:00:05.979
here and write it in
scientific notation.

00:00:05.979 --> 00:00:07.520
So the first thing
I want to do, well

00:00:07.520 --> 00:00:09.353
I already have parts
of this expression that

00:00:09.353 --> 00:00:11.010
are written in
scientific notation.

00:00:11.010 --> 00:00:13.156
For my brain, to simplify
the multiplication,

00:00:13.156 --> 00:00:15.280
I like to write everything
into scientific notation

00:00:15.280 --> 00:00:17.190
and then do whatever
I need to do

00:00:17.190 --> 00:00:21.490
to get the final product
into scientific notation.

00:00:21.490 --> 00:00:23.730
So this part right
over here, 0.2,

00:00:23.730 --> 00:00:25.550
is not in scientific notation.

00:00:25.550 --> 00:00:27.300
In order for it to be
scientific notation,

00:00:27.300 --> 00:00:30.560
it would have to be some
number between 1 and 10,

00:00:30.560 --> 00:00:32.729
not including 10-- so
greater than or equal to 1,

00:00:32.729 --> 00:00:37.040
less than 10-- being
multiplied by some power of 10.

00:00:37.040 --> 00:00:39.279
And this is clearly less than 1.

00:00:39.279 --> 00:00:40.820
But we could just
view this as, look,

00:00:40.820 --> 00:00:42.810
this is in the tenths place.

00:00:42.810 --> 00:00:44.880
This is 2 times 1/10.

00:00:44.880 --> 00:00:46.730
1/10 is 10 to the negative 1.

00:00:46.730 --> 00:00:51.200
So this is the same
thing as 2 times 10

00:00:51.200 --> 00:00:54.864
to the negative 1-- same
thing as 2 times 1/10.

00:00:54.864 --> 00:00:56.280
Now if we look in
the denominator,

00:00:56.280 --> 00:00:57.580
in blue, we have this part.

00:00:57.580 --> 00:00:59.470
It is written in
scientific notation.

00:00:59.470 --> 00:01:01.940
But this green part is not.

00:01:01.940 --> 00:01:05.980
But we could easily write it
as-- this is five 10,000's.

00:01:05.980 --> 00:01:08.950
10,000 is 10 to the fourth.

00:01:08.950 --> 00:01:13.060
So this is the same thing as 5
times 10 to the fourth power.

00:01:13.060 --> 00:01:16.730
And you see that it has
one, two, three, four 0's.

00:01:16.730 --> 00:01:18.760
So now let's take the
product in the numerator

00:01:18.760 --> 00:01:19.892
and the denominator.

00:01:19.892 --> 00:01:21.350
So in the numerator,
I'm just going

00:01:21.350 --> 00:01:23.170
to swap the order in
which I'm multiplying.

00:01:23.170 --> 00:01:24.753
I'm just multiplying
a bunch of stuff.

00:01:24.753 --> 00:01:28.400
4.6 times 10 to the sixth times
2 times 10 to the negative 1.

00:01:28.400 --> 00:01:30.880
It doesn't matter what
order I multiply them in.

00:01:30.880 --> 00:01:36.470
So I could rewrite this
as 4.6 times 2 times

00:01:36.470 --> 00:01:44.000
2 times 10 to the sixth-- I'm
switching colors-- times 10

00:01:44.000 --> 00:01:44.870
to the negative 1.

00:01:47.750 --> 00:01:54.030
And then in the denominator,
let me just write the 5 times

00:01:54.030 --> 00:02:03.700
2.3 times 10 to the fourth
times 10 to the negative 2.

00:02:07.180 --> 00:02:11.210
And now let us attempt
to simplify this thing.

00:02:11.210 --> 00:02:15.040
So here, we have 4.6 times 2.

00:02:15.040 --> 00:02:15.900
Let me circle that.

00:02:15.900 --> 00:02:21.080
So 4.6 times 2 is 9.2.

00:02:21.080 --> 00:02:23.790
So that's 9.2.

00:02:23.790 --> 00:02:26.800
And then 10 to the sixth
times 10 to the negative 1--

00:02:26.800 --> 00:02:27.759
we have the same base.

00:02:27.759 --> 00:02:28.800
We're taking the product.

00:02:28.800 --> 00:02:30.560
We can add the
exponents-- is going

00:02:30.560 --> 00:02:33.340
to be 10 to the 6 minus 1
or 10 to the fifth power.

00:02:36.140 --> 00:02:38.230
So we've simplified
our numerator.

00:02:38.230 --> 00:02:42.980
And now in our
denominator, let's see.

00:02:42.980 --> 00:02:47.110
5 times 2.3-- 5 times 2 is 10.

00:02:47.110 --> 00:02:52.870
5 times 0.3 is 1.5, so
it's going to be 11.5.

00:02:52.870 --> 00:02:56.420
So this is going to be 11.5.

00:02:56.420 --> 00:03:00.080
And then if I multiply
10 to the fourth times 10

00:03:00.080 --> 00:03:03.290
to the negative 2, that's going
to be 10 to the 4 minus 2 or 10

00:03:03.290 --> 00:03:06.830
squared-- times 10
to the second power.

00:03:06.830 --> 00:03:09.150
And now I can divide
these two things.

00:03:09.150 --> 00:03:11.630
So this is going to
be equal to-- we'll

00:03:11.630 --> 00:03:15.100
have to think about
what 9.2 over 11.5 is.

00:03:15.100 --> 00:03:16.790
But actually let me
just do that right

00:03:16.790 --> 00:03:19.195
now, get a little practice
dividing decimals.

00:03:21.700 --> 00:03:25.150
Let me get some
real estate here.

00:03:25.150 --> 00:03:29.130
Let me do that in
the same color.

00:03:29.130 --> 00:03:36.220
9.2 divided by 11.5--
well if we multiply

00:03:36.220 --> 00:03:43.050
both of these times 10, that's
the exact same thing as 92

00:03:43.050 --> 00:03:44.602
divided by 115.

00:03:44.602 --> 00:03:46.060
We're essentially
moved the decimal

00:03:46.060 --> 00:03:48.220
to the right for both of them.

00:03:48.220 --> 00:03:50.630
And let me add some zeros
here because I suspect

00:03:50.630 --> 00:03:52.550
that I'm going to
get a decimal here.

00:03:52.550 --> 00:03:54.740
So let's think what
this is going to be.

00:03:54.740 --> 00:03:55.870
Let's think about this.

00:03:55.870 --> 00:03:57.710
Well 115 doesn't go into 9.

00:03:57.710 --> 00:03:59.180
It doesn't go into 92.

00:03:59.180 --> 00:04:01.820
It does go into 920.

00:04:01.820 --> 00:04:06.064
And I'm going to eyeball and
say that it will go eight times.

00:04:06.064 --> 00:04:07.230
Let's see if that works out.

00:04:07.230 --> 00:04:08.313
So I have my decimal here.

00:04:08.313 --> 00:04:09.550
That's a 0.

00:04:09.550 --> 00:04:12.830
8 times-- 8 times 5 is 40.

00:04:12.830 --> 00:04:16.130
8 times 11 is 88.

00:04:16.130 --> 00:04:19.196
And then 88 plus 4 is 92.

00:04:19.196 --> 00:04:21.700
Oh, it went in
exactly, very good.

00:04:21.700 --> 00:04:24.440
So 920, we have no remainder.

00:04:24.440 --> 00:04:32.460
So 9.2 divided by 11.5
simplified to 0.80.

00:04:32.460 --> 00:04:36.560
And then 10 to the fifth
divided by 10 to the second,

00:04:36.560 --> 00:04:38.450
we have the same base,
and we're dividing.

00:04:38.450 --> 00:04:39.910
So we can subtract
the exponents.

00:04:39.910 --> 00:04:44.022
That's going to be
10 to the 5 minus 2.

00:04:44.022 --> 00:04:45.730
So this right over
here is going to be 10

00:04:45.730 --> 00:04:50.130
to the third power-- so
times 10 to the third power.

00:04:50.130 --> 00:04:52.280
Now, are we done?

00:04:52.280 --> 00:04:56.270
Well in order to be done,
this number right over here

00:04:56.270 --> 00:04:59.840
needs to be greater than or
equal to 1 and less than 10.

00:04:59.840 --> 00:05:02.710
It is clearly not greater
than or equal to 1.

00:05:02.710 --> 00:05:05.120
So how can we rewrite this
as the product of something

00:05:05.120 --> 00:05:08.300
that is greater than or
equal to 1 and less than 10

00:05:08.300 --> 00:05:10.370
and some power of 10?

00:05:10.370 --> 00:05:13.290
Well this 8 right over here,
this is in the tenths place.

00:05:13.290 --> 00:05:17.500
It's 8/10, 8 times 1/10.

00:05:17.500 --> 00:05:20.560
So this is going to be the
same thing as 8 times 10

00:05:20.560 --> 00:05:22.610
to the negative 1 power.

00:05:22.610 --> 00:05:27.200
And then we have this 10 to
the third here-- so times 10

00:05:27.200 --> 00:05:29.240
to the third power.

00:05:29.240 --> 00:05:30.680
We'll do that in
that other color.

00:05:34.560 --> 00:05:36.160
And now we have the same base.

00:05:36.160 --> 00:05:37.470
Just add the exponents.

00:05:37.470 --> 00:05:42.130
So this is going to
be equal to 8 times 10

00:05:42.130 --> 00:05:47.565
to the 3 minus 1-- so
8 times 10 squared.

00:05:47.565 --> 00:05:48.190
And we're done.

00:05:48.190 --> 00:05:51.660
We've simplified our
original expression.


Red blood cells in human body (scientific notation word problem)

WEBVTT
Kind: captions
Language: en

00:00:01.040 --> 00:00:03.770
A human body has
5 liters of blood,

00:00:03.770 --> 00:00:06.700
40% of which is red blood cells.

00:00:06.700 --> 00:00:10.150
Each red blood cell has a volume
of approximately 90 times 10

00:00:10.150 --> 00:00:12.170
to the negative 15 liters.

00:00:12.170 --> 00:00:15.280
How many red blood cells
are there in a human body?

00:00:15.280 --> 00:00:17.530
Write your answer in
scientific notation,

00:00:17.530 --> 00:00:20.280
and round to two decimal places.

00:00:20.280 --> 00:00:25.030
So they tell us the total volume
of blood in the human body.

00:00:25.030 --> 00:00:27.585
We have 5 liters.

00:00:30.370 --> 00:00:35.020
And then they tell us that 40%
of that is red blood cells.

00:00:35.020 --> 00:00:39.620
So if we take the 5 liters
and we multiply by 40%,

00:00:39.620 --> 00:00:42.800
this expression right here
gives us the total volume

00:00:42.800 --> 00:00:47.090
of the red blood cells, 40%
of our total volume of blood.

00:00:47.090 --> 00:00:49.270
Now this is the total
volume of red blood cells.

00:00:49.270 --> 00:00:52.170
And we divide by the volume
of each red blood cell.

00:00:52.170 --> 00:00:54.870
Then we're going to get the
number of red blood cells.

00:00:54.870 --> 00:00:55.810
So let's do that.

00:00:55.810 --> 00:00:59.430
Let's divide by the volume
of each red blood cell.

00:00:59.430 --> 00:01:01.830
So the volume of
each red blood cell

00:01:01.830 --> 00:01:09.440
is 90 times 10 to the
negative 15 liters.

00:01:09.440 --> 00:01:11.380
So let's see if we can simplify.

00:01:11.380 --> 00:01:13.120
So one thing that we
can feel good about

00:01:13.120 --> 00:01:15.040
is that the units
actually do cancel out.

00:01:15.040 --> 00:01:17.722
We have liters in the numerator,
liters in the denominator,

00:01:17.722 --> 00:01:19.680
so we're going to get
just a pure number, which

00:01:19.680 --> 00:01:20.380
is what we want.

00:01:20.380 --> 00:01:23.520
We just want how many red blood
cells are actually in the body.

00:01:23.520 --> 00:01:27.720
So let's just focus
the numbers here.

00:01:27.720 --> 00:01:35.080
So 5 times 40%-- well 40%
is the same thing as 0.4.

00:01:35.080 --> 00:01:36.450
So let me write that down.

00:01:36.450 --> 00:01:40.446
This is the same thing as 0.4.

00:01:40.446 --> 00:01:44.260
5 times 0.4 is 2.

00:01:44.260 --> 00:01:47.420
So our numerator
simplifies to 2.

00:01:47.420 --> 00:01:52.420
And in the denominator, we have
90 times 10 to the negative 15,

00:01:52.420 --> 00:01:57.120
which definitely is not
in scientific notation.

00:01:57.120 --> 00:01:58.900
It looks like it, but
remember, in order

00:01:58.900 --> 00:02:03.230
to be in scientific
notation, this number

00:02:03.230 --> 00:02:06.010
has to be greater than or
equal to 1 and less than 10.

00:02:06.010 --> 00:02:07.720
It's clearly not less than 10.

00:02:07.720 --> 00:02:09.699
But we can convert this
to scientific notation

00:02:09.699 --> 00:02:10.919
very easily.

00:02:10.919 --> 00:02:14.120
90 is the same
thing as 9 times 10,

00:02:14.120 --> 00:02:16.820
or you could even say 9
times 10 to the first.

00:02:16.820 --> 00:02:21.500
And then you multiply that
times 10 to the negative 15.

00:02:21.500 --> 00:02:29.020
And then this
simplifies to 9 times--

00:02:29.020 --> 00:02:32.092
let's add these two exponents--
10 to the negative 14.

00:02:38.870 --> 00:02:40.884
And now we can actually divide.

00:02:40.884 --> 00:02:42.800
And let's simplify this
division a little bit.

00:02:42.800 --> 00:02:54.790
This is going to be the same
thing as 2/9 times 1 to t

00:02:54.790 --> 00:02:57.640
over 10 to the negative 14.

00:02:57.640 --> 00:03:00.490
Well what's 1 over 10
to the negative 14?

00:03:00.490 --> 00:03:02.460
Well that's just 10 to the 14.

00:03:02.460 --> 00:03:08.739
So this right over here is just
the same thing as 10 to the 14.

00:03:08.739 --> 00:03:11.280
Now you might say, OK, we just
have to figure out what 2/9 is

00:03:11.280 --> 00:03:11.905
and we're done.

00:03:11.905 --> 00:03:13.951
We've written this in
scientific notation.

00:03:13.951 --> 00:03:15.450
But you might have
already realized,

00:03:15.450 --> 00:03:19.180
look, 2/9 is not greater
than or equal to 1.

00:03:19.180 --> 00:03:23.110
How can we make this
greater than or equal to 1?

00:03:23.110 --> 00:03:26.820
Well we could multiply it by 10.

00:03:26.820 --> 00:03:28.390
If we multiply this
by 10, then we've

00:03:28.390 --> 00:03:30.790
got to divide this
by 10 to not change

00:03:30.790 --> 00:03:32.910
the value of this expression.

00:03:32.910 --> 00:03:36.320
But let's do that.

00:03:36.320 --> 00:03:38.610
So I'm going to
multiply this by 10,

00:03:38.610 --> 00:03:40.515
and I'm going to
divide this by 10.

00:03:40.515 --> 00:03:41.390
So I haven't changed.

00:03:41.390 --> 00:03:43.320
I've multiplied
and divided by 10.

00:03:43.320 --> 00:03:50.800
So this is equal to 20/9 times
10 to the 14th divided by 10

00:03:50.800 --> 00:03:54.440
is 10 to the 13th power.

00:03:54.440 --> 00:03:55.467
So what's 20/9?

00:03:55.467 --> 00:03:57.050
This is going to
give us a number that

00:03:57.050 --> 00:03:59.510
is greater than or equal
to 1 and less than 10.

00:03:59.510 --> 00:04:00.500
So let's figure it out.

00:04:00.500 --> 00:04:05.720
And I think they said round our
answer to two decimal places.

00:04:05.720 --> 00:04:06.960
So let's do that.

00:04:06.960 --> 00:04:12.130
So 20 divided by 9--
9 doesn't go into 2.

00:04:12.130 --> 00:04:13.770
It does go into 20 two times.

00:04:13.770 --> 00:04:15.830
2 times 9 is 18.

00:04:15.830 --> 00:04:16.880
Subtract.

00:04:16.880 --> 00:04:17.811
Get a remainder of 2.

00:04:17.811 --> 00:04:19.769
I think you see where
this show is going to go.

00:04:22.270 --> 00:04:24.230
9 goes into 20 two times.

00:04:24.230 --> 00:04:25.640
2 times 9 is 18.

00:04:25.640 --> 00:04:27.810
We're just going to
keep getting 2's.

00:04:27.810 --> 00:04:29.710
So we get another
2, bring down a 0.

00:04:29.710 --> 00:04:31.610
Nine goes into 20 two times.

00:04:31.610 --> 00:04:36.320
So this thing right over
here is really 2.2 repeating.

00:04:36.320 --> 00:04:38.640
But they said round
to two decimal places,

00:04:38.640 --> 00:04:43.680
so this is going to be
equal to 2.22 times 10

00:04:43.680 --> 00:04:46.100
to the 13th power.


Regrouping whole numbers

WEBVTT
Kind: captions
Language: en

00:00:00.284 --> 00:00:01.950
Let's think about
different ways that we

00:00:01.950 --> 00:00:10.080
can represent the number 675.

00:00:10.080 --> 00:00:11.930
So the most obvious
way is to just look

00:00:11.930 --> 00:00:13.980
at the different place values.

00:00:13.980 --> 00:00:16.079
So the 6 is in the
hundreds place.

00:00:16.079 --> 00:00:18.800
It literally represents 600.

00:00:18.800 --> 00:00:20.480
So that's 600.

00:00:20.480 --> 00:00:23.860
I'm going to do that
in the red color-- 600.

00:00:23.860 --> 00:00:25.210
The 7 is in the tens place.

00:00:25.210 --> 00:00:28.930
It represents 7 tens, or 70.

00:00:28.930 --> 00:00:30.930
And then the 5 in
the ones place.

00:00:30.930 --> 00:00:33.067
It represents 5.

00:00:33.067 --> 00:00:34.900
So let me copy and paste
this and then think

00:00:34.900 --> 00:00:38.170
about how we can regroup the
value in the different places

00:00:38.170 --> 00:00:40.423
to represent this
in different ways.

00:00:40.423 --> 00:00:42.720
So let me copy and
let me paste it,

00:00:42.720 --> 00:00:44.730
and maybe I'll do
it three times.

00:00:44.730 --> 00:00:48.950
So let me do it once, and
let me do it one more time.

00:00:48.950 --> 00:00:52.660
So one thing that we could do is
we could regroup from one place

00:00:52.660 --> 00:00:53.610
to the next.

00:00:53.610 --> 00:00:56.660
So, for example,
we could take if we

00:00:56.660 --> 00:01:01.940
wanted to-- we could take
1 from the hundreds place.

00:01:01.940 --> 00:01:05.870
Taking 1 from the hundreds
place will make this a 5.

00:01:05.870 --> 00:01:08.080
That's essentially
taking 100 away.

00:01:08.080 --> 00:01:11.130
So this is really
making this a 500.

00:01:11.130 --> 00:01:12.880
And we could give
that 100 to-- well,

00:01:12.880 --> 00:01:14.629
we could actually give
it to either place,

00:01:14.629 --> 00:01:16.590
but let's give it
to the tens place.

00:01:16.590 --> 00:01:22.867
So we're going to give
100 to the tens place.

00:01:22.867 --> 00:01:24.450
Now, if you give 100
to the tens place

00:01:24.450 --> 00:01:27.340
and you already had 70 there,
what's it going to be equal to?

00:01:27.340 --> 00:01:31.560
Well, it's going
to be equal to 170.

00:01:31.560 --> 00:01:33.430
Well, how would we
represent that is tens?

00:01:33.430 --> 00:01:37.010
Well, 170 is 17 tens.

00:01:37.010 --> 00:01:41.570
So we could just say
that 7 becomes 17.

00:01:41.570 --> 00:01:43.080
Now, we could keep doing that.

00:01:43.080 --> 00:01:45.990
We could regroup some of
this value in the tens place

00:01:45.990 --> 00:01:47.360
to the ones place.

00:01:47.360 --> 00:01:51.940
So, for example, we could
give 10 from the tens place

00:01:51.940 --> 00:01:53.190
and give it to the ones place.

00:01:53.190 --> 00:01:53.910
Let's do that.

00:01:53.910 --> 00:01:55.910
So let's take 10 away from here.

00:01:55.910 --> 00:01:58.510
So that becomes 160.

00:01:58.510 --> 00:02:00.750
This becomes 16.

00:02:00.750 --> 00:02:05.510
And let's give that
10 to the ones place.

00:02:05.510 --> 00:02:07.660
Well, what does the
ones place now become?

00:02:07.660 --> 00:02:10.979
Well, 10 plus 5 is 15.

00:02:10.979 --> 00:02:14.750
So this 5 is now a 15.

00:02:14.750 --> 00:02:17.220
Let's do another scenario.

00:02:17.220 --> 00:02:19.380
Let's do something nutty.

00:02:19.380 --> 00:02:23.170
Let's take 200 from
the hundreds place.

00:02:26.550 --> 00:02:32.390
So this is going to now 4, and
this is going to become 400.

00:02:32.390 --> 00:02:34.420
That's what this
4 now represents.

00:02:34.420 --> 00:02:39.660
And let's give 100
to the tens place.

00:02:39.660 --> 00:02:44.000
And let's give another
100 to the ones place.

00:02:44.000 --> 00:02:46.600
So in other words, I'm
just regrouping that 200.

00:02:46.600 --> 00:02:49.140
Those 200's, I've taken
from the hundreds place,

00:02:49.140 --> 00:02:51.680
and I'm going to give it
to these other places.

00:02:51.680 --> 00:02:55.370
So now the tens place
is going to be 170.

00:02:55.370 --> 00:02:58.340
We're going to have 170
here, which is 17 tens.

00:02:58.340 --> 00:03:01.690
So you could say that
the tens place is now 17.

00:03:01.690 --> 00:03:03.920
And now the ones
place, well, I had 5,

00:03:03.920 --> 00:03:05.760
and now I'm going
to add 100 to it.

00:03:05.760 --> 00:03:09.676
So it's going to be 105.

00:03:09.676 --> 00:03:13.080
And 105 ones is literally 105.

00:03:13.080 --> 00:03:18.260
And notice, this is 400 plus
17 tens, which is 170 plus 105.

00:03:18.260 --> 00:03:22.130
You add these together,
you're still going to get 675.

00:03:22.130 --> 00:03:24.850
Let's do that again.

00:03:24.850 --> 00:03:26.750
So let's take, I don't know.

00:03:26.750 --> 00:03:29.751
Let's take all of
the hundreds away.

00:03:29.751 --> 00:03:31.250
Let's take all of
the hundreds away.

00:03:31.250 --> 00:03:32.920
So that goes zero.

00:03:32.920 --> 00:03:34.090
That goes zero.

00:03:34.090 --> 00:03:38.070
And let's give 400
to the tens place.

00:03:38.070 --> 00:03:41.480
So let's give 400
to the tens place.

00:03:41.480 --> 00:03:44.630
Well, that'll make that 470.

00:03:44.630 --> 00:03:46.520
Let me do that in
that green color.

00:03:46.520 --> 00:03:52.710
That will make that 470, which
is the same thing as 47 tens.

00:03:52.710 --> 00:03:56.881
And then I still have 200
to give to the ones place.

00:03:56.881 --> 00:03:57.630
So let me do that.

00:03:57.630 --> 00:04:02.240
So I still have 200 to
give to the ones place.

00:04:02.240 --> 00:04:09.140
And so that's going
to go from 5 to 205.

00:04:09.140 --> 00:04:10.880
Once again, all
I've done is I've

00:04:10.880 --> 00:04:14.990
regrouped the value
in the number 675.

00:04:14.990 --> 00:04:17.334
Any of these still
represent 675.

00:04:17.334 --> 00:04:18.959
You can add these
numbers on the right.

00:04:18.959 --> 00:04:21.600
They all add up to 675.


Regrouping whole numbers example 2

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:02.400
Fill in the table
with whole numbers

00:00:02.400 --> 00:00:06.200
to write 430 in
five different ways.

00:00:06.200 --> 00:00:07.980
So let's think about
this a little bit.

00:00:07.980 --> 00:00:11.120
We could literally view
this 4 as 4 hundreds.

00:00:11.120 --> 00:00:12.530
So it's 4 hundreds.

00:00:12.530 --> 00:00:15.910
And the 3 is in the tens
place, so it's 3 tens.

00:00:15.910 --> 00:00:18.109
So this is 4
hundreds and 3 tens.

00:00:18.109 --> 00:00:19.900
But we could rearrange
things a little bit.

00:00:19.900 --> 00:00:22.540
For example, we might
only have 3 hundreds.

00:00:22.540 --> 00:00:25.140
So we took a hundred
from the hundreds place,

00:00:25.140 --> 00:00:27.050
and we could give it
to the tens place.

00:00:27.050 --> 00:00:30.570
So if you take that hundred
and give it to the tens place,

00:00:30.570 --> 00:00:33.040
a hundred is 10 tens.

00:00:33.040 --> 00:00:36.450
So you've got 10 tens, and
then you already had 3 tens.

00:00:36.450 --> 00:00:39.762
So you're going to have
13 tens right over here.

00:00:39.762 --> 00:00:41.220
Now, this seems a
little confusing.

00:00:41.220 --> 00:00:44.580
Remember, this is 430.

00:00:44.580 --> 00:00:47.910
This is 300 plus 130.

00:00:47.910 --> 00:00:49.890
That's still 430.

00:00:49.890 --> 00:00:55.150
Likewise, you could view
this as 200 plus 230.

00:00:55.150 --> 00:00:58.260
230 is the same
thing as 23 tens.

00:00:58.260 --> 00:01:02.764
Same way, you could view
this as 100 plus 330.

00:01:02.764 --> 00:01:06.090
33 tens is 330.

00:01:06.090 --> 00:01:12.460
And finally, you could view
it as 0 hundreds and 43 tens.

00:01:12.460 --> 00:01:16.350
43 tens is 430.


Regrouping whole numbers example 1

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.050
Fill in the blanks to
complete the equation.

00:00:03.050 --> 00:00:06.550
So they have 76,830 is
equal to 6 ten thousands

00:00:06.550 --> 00:00:11.640
plus blank thousands plus
8 hundreds plus 3 tens.

00:00:11.640 --> 00:00:15.130
So let's just
think about 76,830.

00:00:15.130 --> 00:00:28.020
So if we write 76,830-- I'm
going to try to color-code it.

00:00:28.020 --> 00:00:35.250
We could write this out as
being the same thing as 70,000

00:00:35.250 --> 00:00:38.810
plus 6,000-- the 7 is in
the ten thousands place.

00:00:38.810 --> 00:00:42.780
So it's 7 ten thousands,
which is 70,000, plus-- the 6

00:00:42.780 --> 00:00:47.862
is in the thousands place--
so it's 6,000, or 6 thousands,

00:00:47.862 --> 00:00:50.760
one could say, plus 800.

00:00:50.760 --> 00:00:52.481
Or you could say 8 hundreds.

00:00:52.481 --> 00:00:55.720
800 plus 3 tens, or 30.

00:00:59.780 --> 00:01:01.680
And then there's 0
ones, so we could just

00:01:01.680 --> 00:01:04.050
write that out as 0 if we like.

00:01:04.050 --> 00:01:05.519
Now, what we have
on the right here

00:01:05.519 --> 00:01:08.090
is almost what they have
on the right over here.

00:01:08.090 --> 00:01:08.712
Notice the 0.

00:01:08.712 --> 00:01:09.670
They didn't write that.

00:01:09.670 --> 00:01:11.160
So we can ignore that.

00:01:11.160 --> 00:01:13.380
The 0 really doesn't
affect the value.

00:01:13.380 --> 00:01:15.155
We have 3 tens.

00:01:15.155 --> 00:01:16.740
We have 3 tens.

00:01:19.340 --> 00:01:20.950
Let me do this in a
color you can see.

00:01:20.950 --> 00:01:22.280
You have 8 hundreds.

00:01:22.280 --> 00:01:25.260
You have 800.

00:01:25.260 --> 00:01:27.500
But then that's where
it starts to break down.

00:01:27.500 --> 00:01:28.700
They have blank thousands.

00:01:28.700 --> 00:01:30.080
Here we have 6 thousands.

00:01:30.080 --> 00:01:32.413
So we're going to have to
think about this a little bit.

00:01:32.413 --> 00:01:35.010
And then they have 6 ten
thousands, which is 60,000.

00:01:35.010 --> 00:01:39.555
Here we have 70 thousands.

00:01:39.555 --> 00:01:42.280
So we've got to work
through these two places

00:01:42.280 --> 00:01:43.290
right over here.

00:01:43.290 --> 00:01:45.040
So they have 6 ten
thousands, which

00:01:45.040 --> 00:01:47.090
is the same thing as 60,000.

00:01:47.090 --> 00:01:48.690
We have 70,000.

00:01:48.690 --> 00:01:50.860
So they essentially
took 10,000 from here

00:01:50.860 --> 00:01:52.314
and regrouped it someplace.

00:01:52.314 --> 00:01:53.980
And since these two
places are the same,

00:01:53.980 --> 00:01:56.188
they must have regrouped it
into the thousands place.

00:01:56.188 --> 00:01:57.430
So let's try to do that.

00:01:57.430 --> 00:02:01.710
So let's get rid of
10,000 out of the 70,000.

00:02:01.710 --> 00:02:04.480
It becomes 60,000.

00:02:04.480 --> 00:02:06.790
So this becomes a 6.

00:02:06.790 --> 00:02:10.190
And then we're regrouping
that into the thousands place.

00:02:10.190 --> 00:02:15.320
So we're going to add 10,000
to the thousands place.

00:02:15.320 --> 00:02:17.560
Now what's the thousands
place going to be then?

00:02:17.560 --> 00:02:22.280
Well, it's 10,000 plus 6,000.

00:02:22.280 --> 00:02:24.830
It's going to
become 16 thousands.

00:02:24.830 --> 00:02:27.980
This right over here is
going to become a 16.

00:02:27.980 --> 00:02:30.000
So now what we wrote
here looks just

00:02:30.000 --> 00:02:31.490
like what they wrote up here.

00:02:31.490 --> 00:02:36.780
This is 6 ten thousands
plus 16 thousands

00:02:36.780 --> 00:02:40.250
plus 8 hundreds plus 3 tens.

00:02:40.250 --> 00:02:42.110
And we're done.