Thinking about reasonable units to describe a rate

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.940
So I have a car here, and
let's say that in 3 hours,

00:00:03.940 --> 00:00:11.512
this car is able to
travel 150 kilometers.

00:00:11.512 --> 00:00:13.470
So what I want to think
about in this video is,

00:00:13.470 --> 00:00:17.130
what are some reasonable
ways to express

00:00:17.130 --> 00:00:21.055
the rate at which this car
traveled for those 3 hours?

00:00:21.055 --> 00:00:22.680
And I encourage you
to pause this video

00:00:22.680 --> 00:00:24.280
and think about it for yourself.

00:00:24.280 --> 00:00:25.970
One, you could
calculate the rate.

00:00:25.970 --> 00:00:27.670
But also think about
different units

00:00:27.670 --> 00:00:29.900
that you could use
to express that rate,

00:00:29.900 --> 00:00:32.012
and which ones would
be useful-- which ones

00:00:32.012 --> 00:00:33.470
what would be
reasonable, and which

00:00:33.470 --> 00:00:36.040
ones would be unreasonable.

00:00:36.040 --> 00:00:38.820
So let's just remind
ourselves what rate even is.

00:00:38.820 --> 00:00:43.410
So you could think
about distance

00:00:43.410 --> 00:00:49.270
as being equal to
rate times time.

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Or you could imagine if you
divide both sides by time,

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you could imagine that
distance divided by time

00:00:58.510 --> 00:00:59.955
is equal to rate.

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So they've given us a distance,
they've given us a time.

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So we could just divide the
distance divided by the time

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to figure out the rates.

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And I'm going to
keep the units here.

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Because it's really important
to recognize that the units,

00:01:15.670 --> 00:01:20.030
to some degree, they can also
be manipulated algebraically.

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Now they aren't variables, but
they follow the same rules,

00:01:24.500 --> 00:01:27.010
I guess I should say,
as a variable would.

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So for example, if
I said, look rate

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is distance divided by time.

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So I could say that my
rate in this situation

00:01:35.180 --> 00:01:44.025
is going to be 150 kilometers
divided by 3 hours.

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So if we just look at
the numeric part of this,

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what's 150 divided by 3?

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Well that's going to be 50.

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So this is going
to be equal to 50.

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But we can keep
the units the way

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that they are, right over here.

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This is 50 kilometers per hour.

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This is what I meant
by saying look,

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we're dividing this quantity,
expressed in kilometers,

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by this quantity,
expressed in hours.

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We can divide the numeric
part, 150 divided by 3.

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But then we could leave at
the units in that relationship

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that they were before.

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So you can kind of algebraically
keep them that way.

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And you'll see in a
second we're going

00:02:32.430 --> 00:02:34.660
to manipulate them
a lot more, using

00:02:34.660 --> 00:02:37.330
what's often called the
dimensional analysis.

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But anyway, this is a reasonable
way to express a rate.

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50 kilometers per hour.

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I can imagine this.

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I can imagine that
in 1 hour, you're

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going to go 50 kilometers.

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Let's think about other ways
that we could represent that.

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So 50 kilometers
per hour-- and this

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is where we're really going to
do some dimensional analysis

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with our units.

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So 50 kilometers per hour.

00:03:02.630 --> 00:03:04.980
Let's say we want to
express it in terms

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of kilometers per second.

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So how could we write
50 kilometers per hour,

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in terms of
kilometers per second?

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Well it's always good, actually,
as a first approximation,

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to just think about it.

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If you went this far in an hour,
then the number of kilometers

00:03:24.580 --> 00:03:28.100
you go in a second, is that
going to be less, or more?

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Well a second's a much,
much shorter period of time.

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There's 3,600
seconds in an hour.

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So you're going to go
1/3,600 of this distance.

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But let's think about
how we would actually

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work out with the units.

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Well, we want to get rid of
this hours in the denominator.

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And the plural,
obviously the grammar

00:03:47.460 --> 00:03:48.880
doesn't hold up
with the algebra,

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but this could be hour or hours.

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So we could think
about well, 1 hour--

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I'll write an hour in
the numerator that's

00:03:54.980 --> 00:03:57.730
going to cancel with this
hour in the denominator.

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But we want it in
terms of seconds.

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So 1 hour is equal
to how many seconds?

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Well 1 hour is equal
to 3,600 seconds.

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This is what I meant
by saying that using

00:04:12.610 --> 00:04:15.430
dimensional analysis, which
is what I'm doing right now,

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we can essentially
manipulate these units,

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as we would traditionally
do with a variable.

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So we have hours
divided by hours.

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And so when we do
the multiplication,

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we can multiply
the numeric parts.

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So we have 50 times
1, divided by 3,600.

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Let me write that.

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50 times 1 over 3,600.

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And then our units left
are kilometers per second.

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Or I could say seconds.

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So we can play around with
the plural and singular

00:04:52.180 --> 00:04:55.970
parts of it, but I'll just write
it as kilometers per second.

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And so this is 50/3,600.

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And this fits our intuition.

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In a second, you're going to go
1/3,600 as far as you would go

00:05:06.410 --> 00:05:07.670
in an hour.

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But let's actually think
about what this is equal to.

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50/3,600-- so this is
going to be the same thing,

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as-- Let me just
simplify it over here.

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So 50/3,600 is the
same thing as 5/360,

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which is the same thing as-- let
me write it this way-- 10/720.

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And I did that way because
that makes it clear

00:05:36.870 --> 00:05:38.328
that that's the
same thing as 1/72.

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So you could write
this as, you're going,

00:05:43.370 --> 00:05:55.500
this is equal to 1/72 of
a kilometer per second.

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Now I would claim
that this is not

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so reasonable of units for
this example right over here.

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1/72 of a kilometer
every second?

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That doesn't help me too much.

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I guess I'll know
that in 72 seconds

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I will have gone a kilometer.

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But this is something that's
kind of strange for me

00:06:12.410 --> 00:06:13.270
to conceptualize.

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If I wanted to get
my calculator out,

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1/72, 1 divided by
72-- if someone said,

00:06:21.500 --> 00:06:27.850
hey, I'm going 0.0139
kilometers in a second,

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that doesn't seem to make a
lot of conceptual sense to me.

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So I would say that
this, right over here,

00:06:33.020 --> 00:06:38.470
is a very reasonable
way to express our rate.

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This one seems like more
of an unreasonable way.

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But we could salvage this.

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Because we're going 1/72
of a kilometer per second.

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Now this is a small
number, but how could

00:06:48.870 --> 00:06:50.429
we make it much larger?

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Well, what if we
thought in terms of,

00:06:51.970 --> 00:06:54.010
not kilometers per
second, but if we

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thought in terms of
meters per second.

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A kilometer's 1,000 meters.

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So if we think about this in
terms of meters per second,

00:07:01.800 --> 00:07:05.410
we're going to get a
larger number here,

00:07:05.410 --> 00:07:08.460
in fact larger by
a factor of 1,000.

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So let's think about that.

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Let's try to convert this
kilometers to meters.

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So how would we do that?

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Well, once again, if
we have kilometers

00:07:15.520 --> 00:07:19.890
in the denominator
this kilometer

00:07:19.890 --> 00:07:21.379
will cancel with that kilometer.

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And we want a meter
in the numerator.

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So we want to think about
how many meters are there

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per kilometer.

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Well there's 1,000
meters per kilometer.

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Kilometers cancel out, and we
are left with 1,000 times 1--

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I'll just write that
as 1,000-- over 72.

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And now we're left
with in the numerator,

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we're left with the
unit meters per second.

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And I know I keep writing
second in different ways.

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Oftentimes actually you'll see
people write second like that.

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So actually let me
just go with that.

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So s is second, is seconds,
is sec, just like that.

00:08:01.150 --> 00:08:03.020
So is this fairly reasonable?

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Well actually, this
feels pretty good.

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Let's get our calculator out
and figure out what that is.

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So 1,000 divided by 72
gives us 13, if I round,

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that's about 13.9.

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So this is approximately equal
to 13.9 meters per second.

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Which I can visualize.

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I can imagine how
far 13.9 meters

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is, and of doing that
distance in one second.

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So this actually also
seems like a reasonable way

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to express the rate.

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So I could say hey, this thing's
going 50 kilometers per hour.

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I can imagine it going roughly
13.9 meters per second.

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So this is reasonable as well.

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But to say it's going 1/72
of a kilometer per second

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doesn't really
seem to make sense.

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And also, if I try to think
about it in terms of meters

00:08:58.880 --> 00:09:00.580
per hour, that also
would be strange.

00:09:00.580 --> 00:09:02.329
Actually, I encourage
you to calculate it.

00:09:02.329 --> 00:09:06.030
Try to convert this right
over here to meters per hour.

00:09:06.030 --> 00:09:09.510
Then we would say well, we
could use the same thing here.

00:09:09.510 --> 00:09:15.110
That's going to be 1,000
meters for every 1 kilometer.

00:09:15.110 --> 00:09:16.750
Kilometers cancel out.

00:09:16.750 --> 00:09:18.420
And I'm going to be
left with 50 times

00:09:18.420 --> 00:09:27.420
1,000 is 50,000 meters per hour.

00:09:27.420 --> 00:09:29.792
So I have trouble
imagining-- well,

00:09:29.792 --> 00:09:31.750
obviously if I convert
to kilometers in my head

00:09:31.750 --> 00:09:34.430
I could imagine it-- but this
is kind of a crazy large number.

00:09:34.430 --> 00:09:37.500
50,000 meters per hour.

00:09:37.500 --> 00:09:41.380
So at least in my eyes,
using kilometers per hour

00:09:41.380 --> 00:09:44.350
to describe this
rate seems useful.

00:09:44.350 --> 00:09:47.190
Describing this rate as meters
per second seems useful.

00:09:47.190 --> 00:09:51.280
But describing it as kilometers
per second, or meters per hour,

00:09:51.280 --> 00:09:53.999
seem a little bit unusual.


Figuring out units from formula

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.377
Now that we know that we can
treat units algebraically,

00:00:03.377 --> 00:00:04.960
let's see if we can
use that knowledge

00:00:04.960 --> 00:00:07.280
to figure out what the
units of a variable

00:00:07.280 --> 00:00:10.110
might be from some formula.

00:00:10.110 --> 00:00:14.520
So let's say that I am given
this formula, that capital K is

00:00:14.520 --> 00:00:21.380
equal to lowercase
b over s squared.

00:00:21.380 --> 00:00:26.530
And let's say that
we are also told

00:00:26.530 --> 00:00:30.360
that b is going to
be in centimeters,

00:00:30.360 --> 00:00:39.160
and s is going to be in
grams per centimeter.

00:00:39.160 --> 00:00:41.670
So that's the units of s
is grams per centimeter.

00:00:41.670 --> 00:00:44.870
B is in centimeters.

00:00:44.870 --> 00:00:48.140
What is the units
of K going to be?

00:00:48.140 --> 00:00:50.120
And I encourage you
to pause this video

00:00:50.120 --> 00:00:52.490
and think about this.

00:00:52.490 --> 00:00:56.200
Well, let's try it out with
just some value and some values

00:00:56.200 --> 00:00:58.770
for b and s, and
see what happens.

00:00:58.770 --> 00:01:04.420
Let's say that b is
equal to 1 centimeter,

00:01:04.420 --> 00:01:07.290
and I'm just using 1, because
it makes the math really easy.

00:01:07.290 --> 00:01:13.140
And let's say that s is
1 gram per centimeter.

00:01:13.140 --> 00:01:16.880
So, we know that
K is going to be

00:01:16.880 --> 00:01:23.550
equal to 1 centimeter-- that's
b-- over 1 gram per centimeter

00:01:23.550 --> 00:01:24.050
squared.

00:01:29.314 --> 00:01:30.980
Now, we've already
talked about the idea

00:01:30.980 --> 00:01:34.720
that we should treat our units
like algebraic quantities,

00:01:34.720 --> 00:01:41.660
so this should be equal to
1 centimeter over-- now,

00:01:41.660 --> 00:01:46.460
you could treat this
as 1 squared times--

00:01:46.460 --> 00:01:47.410
let me rewrite this.

00:01:47.410 --> 00:01:51.370
You could rewrite this
as 1 squared times grams

00:01:51.370 --> 00:01:55.310
per centimeter squared, and
this just comes out of the idea

00:01:55.310 --> 00:01:59.010
that if I have a
times b to some power,

00:01:59.010 --> 00:02:03.080
say, the second power, that's
equal to a squared b squared.

00:02:03.080 --> 00:02:07.220
And we also know that if I
have a/b to some power, say,

00:02:07.220 --> 00:02:12.100
squared, that's the same thing
as a squared over b squared.

00:02:12.100 --> 00:02:13.800
So I could rewrite
this as this is

00:02:13.800 --> 00:02:19.390
going to be equal to 1
centimeter over-- 1 squared is

00:02:19.390 --> 00:02:28.036
just 1-- 1 gram squared
per centimeter squared.

00:02:28.036 --> 00:02:29.660
Now, what is this
going to be equal to?

00:02:29.660 --> 00:02:31.620
Let me write it over here.

00:02:31.620 --> 00:02:35.730
Well, that's going to be 1
centimeter in our numerator.

00:02:35.730 --> 00:02:37.472
Let me do it in that
green color, just

00:02:37.472 --> 00:02:38.690
to keep track of things.

00:02:38.690 --> 00:02:42.060
1 centimeter from our numerator,
instead of dividing it

00:02:42.060 --> 00:02:44.930
by 1 gram squared
centimeter squared,

00:02:44.930 --> 00:02:48.400
I can multiply it times
the reciprocal of this.

00:02:48.400 --> 00:02:53.945
So times 1/1, which
is just going to be 1,

00:02:53.945 --> 00:02:55.410
but let me just
write it that way

00:02:55.410 --> 00:02:58.670
just so that you get that I'm
taking the reciprocal of 1.

00:02:58.670 --> 00:03:04.580
So 1/1 times the reciprocal of
grams squared per centimeter

00:03:04.580 --> 00:03:05.080
squared.

00:03:05.080 --> 00:03:06.788
Well, that's just
going to be centimeters

00:03:06.788 --> 00:03:09.720
squared per grams squared.

00:03:09.720 --> 00:03:11.380
So what's this going
to be equal to?

00:03:11.380 --> 00:03:17.990
Well, I have 1 times 1, so if I
take the numeric parts, 1 times

00:03:17.990 --> 00:03:20.320
1, it's just going to be 1.

00:03:20.320 --> 00:03:23.280
And now remember, we can
treat the units algebraically.

00:03:23.280 --> 00:03:26.980
I have a centimeters
times centimeters squared.

00:03:26.980 --> 00:03:31.290
Well, that's going to get me to
centimeters cubed, centimeters

00:03:31.290 --> 00:03:33.640
to the third power.

00:03:33.640 --> 00:03:36.190
And this isn't c times
n to the third power.

00:03:36.190 --> 00:03:37.897
This is centimeters
to the third power--

00:03:37.897 --> 00:03:39.480
or you could write
it like this if you

00:03:39.480 --> 00:03:43.140
wanted to be clear about it--
per-- and over here we still

00:03:43.140 --> 00:03:47.680
have grams squared--
per grams squared.

00:03:47.680 --> 00:03:50.680
So the units, I just picked
the numbers 1 arbitrarily,

00:03:50.680 --> 00:03:52.390
so the 1 isn't
what matters here.

00:03:52.390 --> 00:03:54.144
I just picked those numbers.

00:03:54.144 --> 00:03:55.810
What matters here is
what are the units.

00:03:55.810 --> 00:03:57.810
What are the units
for this formula when

00:03:57.810 --> 00:04:01.200
b is in centimeters and s
is in grams per centimeters?

00:04:01.200 --> 00:04:05.180
We see that K is
going to be expressed

00:04:05.180 --> 00:04:13.820
in terms of cubic
centimeters per gram squared.

00:04:13.820 --> 00:04:15.830
And I can't tell
you offhand kind

00:04:15.830 --> 00:04:18.730
of an intuitive physical
representation of that,

00:04:18.730 --> 00:04:20.760
but you at least know the units.

00:04:20.760 --> 00:04:23.160
You have cubic
centimeters, which

00:04:23.160 --> 00:04:27.030
you can view as a volume--
divided by, I guess,

00:04:27.030 --> 00:04:28.170
a mass squared.

00:04:28.170 --> 00:04:31.560
And a mass squared is a little
bit less intuitive for me,

00:04:31.560 --> 00:04:33.450
but at least we know
what the units of K

00:04:33.450 --> 00:04:37.940
would be in this formula
given the units of b and s.


Unit conversion word problem: roadtrip

WEBVTT
Kind: captions
Language: en

00:00:00.940 --> 00:00:06.660
Your car gets 25
miles per gallon,

00:00:06.660 --> 00:00:12.270
and you want to go on
a 400-mile road trip.

00:00:12.270 --> 00:00:17.550
Right now, gas
costs $3 per gallon.

00:00:17.550 --> 00:00:22.040
How much will the gas
for your road trip cost?

00:00:22.040 --> 00:00:22.600
So let's see.

00:00:22.600 --> 00:00:27.000
They tell us they're going
on a 400-mile road trip.

00:00:27.000 --> 00:00:28.960
400 miles.

00:00:28.960 --> 00:00:31.660
So the first thing that I'd
want to think about is, well,

00:00:31.660 --> 00:00:33.080
how many gallons am I using?

00:00:33.080 --> 00:00:35.080
And then once I know how
many gallons I'm using,

00:00:35.080 --> 00:00:36.720
I know it's $3 per
gallon, so I can

00:00:36.720 --> 00:00:39.680
multiply the number
of gallons by $3.

00:00:39.680 --> 00:00:41.960
So to figure out the
number of gallons,

00:00:41.960 --> 00:00:43.990
would I want to
multiply 400 miles--

00:00:43.990 --> 00:00:47.770
would I want to multiply that
times the miles per gallon,

00:00:47.770 --> 00:00:51.320
which is 25, or would I want
to multiply by the gallons

00:00:51.320 --> 00:00:53.260
per mile?

00:00:53.260 --> 00:00:56.009
Well, if I multiply by
the gallons per mile,

00:00:56.009 --> 00:00:57.550
and I multiply that
times 400, then I

00:00:57.550 --> 00:00:58.840
would get the number of gallons.

00:00:58.840 --> 00:01:00.131
So let's just think about that.

00:01:00.131 --> 00:01:05.300
I want to multiply
that times-- and I'll

00:01:05.300 --> 00:01:13.950
write the units first--
the gallons per mile.

00:01:13.950 --> 00:01:16.510
And what are the
gallons per mile?

00:01:16.510 --> 00:01:19.340
Well, we have 25
miles per gallon.

00:01:19.340 --> 00:01:24.940
We have 25 miles
for every 1 gallon.

00:01:24.940 --> 00:01:28.370
Or you could say we have 1
gallon for every 25 miles.

00:01:28.370 --> 00:01:30.300
So I really just
took the reciprocal

00:01:30.300 --> 00:01:35.092
of 25 miles per gallon and
made it 1/25 gallons per mile.

00:01:35.092 --> 00:01:37.300
Now what do we get when we
multiply these two things?

00:01:37.300 --> 00:01:38.758
The whole purpose
was to figure out

00:01:38.758 --> 00:01:40.350
how many gallons
we're going to use.

00:01:40.350 --> 00:01:42.540
Well, we see that
our miles cancel out.

00:01:42.540 --> 00:01:44.760
Miles cancel out with miles.

00:01:44.760 --> 00:01:48.360
And then I have 400
times 1/25 gallons,

00:01:48.360 --> 00:01:51.900
which is the same thing as
400 divided by 25 gallons.

00:01:51.900 --> 00:01:57.360
So this is equal to
400/25 gallons, which

00:01:57.360 --> 00:02:03.970
is the same thing as 400 divided
by 25 is equal to 16 gallons.

00:02:03.970 --> 00:02:06.550
Now it's always important
to do a reality check here,

00:02:06.550 --> 00:02:09.320
not just to try to
blindly cancel out units.

00:02:09.320 --> 00:02:12.700
Does this actually
makes sense that 16

00:02:12.700 --> 00:02:15.780
is a much lower number than 400?

00:02:15.780 --> 00:02:16.750
Well, sure it does.

00:02:16.750 --> 00:02:18.380
And actually, if you have any
experience filling up a car,

00:02:18.380 --> 00:02:20.630
you would sense that, OK,
well, that's about the size.

00:02:20.630 --> 00:02:23.050
On around 16
gallons, a car tends

00:02:23.050 --> 00:02:26.542
to go 300 or 400 miles if it
gets pretty good fuel mileage.

00:02:26.542 --> 00:02:28.250
So that just makes
sense from experience.

00:02:28.250 --> 00:02:30.590
And it also make sense
based on how it's stated.

00:02:30.590 --> 00:02:33.480
You get 25 miles per gallon.

00:02:33.480 --> 00:02:35.590
So you're going to
need fewer gallons

00:02:35.590 --> 00:02:37.140
than you're going to need miles.

00:02:37.140 --> 00:02:39.710
So this all seems to
make sense so far.

00:02:39.710 --> 00:02:41.490
But we haven't answered
their question.

00:02:41.490 --> 00:02:43.800
They want to know, how
much will my trip cost?

00:02:43.800 --> 00:02:45.300
Right now, we've just
figured out how much fuel

00:02:45.300 --> 00:02:46.150
we're going to use.

00:02:46.150 --> 00:02:50.004
So then we could
take our 16 gallons.

00:02:50.004 --> 00:02:51.420
And to figure out
the dollar cost,

00:02:51.420 --> 00:02:53.461
are we going to multiply
it by dollars per gallon

00:02:53.461 --> 00:02:55.030
or gallons per dollar?

00:02:55.030 --> 00:02:57.680
Well, if we're thinking
just about unit conversion,

00:02:57.680 --> 00:03:02.330
we want to multiply times
the dollars per gallon.

00:03:02.330 --> 00:03:04.660
So I could write it like this.

00:03:04.660 --> 00:03:06.860
I could write it like
dollars per gallon.

00:03:06.860 --> 00:03:09.420
Actually, let me just write
out the word "dollar."

00:03:09.420 --> 00:03:11.700
Dollars per gallon.

00:03:11.700 --> 00:03:13.990
The units will cancel out.

00:03:13.990 --> 00:03:15.570
And it also makes sense.

00:03:15.570 --> 00:03:17.210
Whatever number of
dollars per gallon,

00:03:17.210 --> 00:03:18.580
I multiply it times
the number of gallons,

00:03:18.580 --> 00:03:20.440
and that's going to tell me
how much it's going to cost.

00:03:20.440 --> 00:03:22.106
This happens at the
fuel pump every day.

00:03:22.106 --> 00:03:23.300
Hey, it's $3 per gallon.

00:03:23.300 --> 00:03:24.633
I'm going to fill up 16 gallons.

00:03:24.633 --> 00:03:26.400
Hey, 3 times 16.

00:03:26.400 --> 00:03:27.400
So let's do that.

00:03:27.400 --> 00:03:30.290
So it's $3 per gallon.

00:03:30.290 --> 00:03:32.330
We see the gallons cancel out.

00:03:32.330 --> 00:03:40.170
And we are left
with 16 times $3,

00:03:40.170 --> 00:03:44.070
which is the same thing as $48.

00:03:44.070 --> 00:03:46.125
And we are done.


Multi-step unit conversion word problem

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:06.310
A squirrel is running across
the road at 12 feet per second.

00:00:06.310 --> 00:00:10.920
It needs to run 9 feet
to get across the road.

00:00:10.920 --> 00:00:15.230
How long will it take the
squirrel to run 9 feet?

00:00:15.230 --> 00:00:17.120
Round to the nearest
hundredth of a second.

00:00:17.120 --> 00:00:18.040
Fair enough.

00:00:18.040 --> 00:00:21.150
A car is 50 feet away
from the squirrel-- OK,

00:00:21.150 --> 00:00:24.480
this is a high-stakes word
problem-- driving toward it

00:00:24.480 --> 00:00:27.730
at a speed of 100
feet per second.

00:00:27.730 --> 00:00:30.880
How long will it take
the car to drive 50 feet?

00:00:30.880 --> 00:00:33.370
Round to the nearest
hundredth of a second.

00:00:33.370 --> 00:00:36.600
Will the squirrel make
it 9 feet across the road

00:00:36.600 --> 00:00:38.660
before the car gets there?

00:00:38.660 --> 00:00:40.480
So this definitely is
high stakes, at least

00:00:40.480 --> 00:00:42.012
for the squirrel.

00:00:42.012 --> 00:00:43.470
So let's answer
the first question.

00:00:43.470 --> 00:00:45.140
Let's figure out
how long will it

00:00:45.140 --> 00:00:48.730
take the squirrel to run 9 feet.

00:00:48.730 --> 00:00:49.750
So let's think about it.

00:00:56.460 --> 00:01:00.082
So the squirrel's
got to go 9 feet,

00:01:00.082 --> 00:01:02.040
and we want to figure
out how many seconds it's

00:01:02.040 --> 00:01:03.030
going to take.

00:01:03.030 --> 00:01:07.950
So would we divide or
multiply this by 12?

00:01:07.950 --> 00:01:10.780
Well, to think about that, you
could think about the units

00:01:10.780 --> 00:01:13.060
where we want to get an
answer in terms of seconds.

00:01:13.060 --> 00:01:14.800
We want to figure
out time, so it'd

00:01:14.800 --> 00:01:22.810
be great if we could multiply
this times seconds per foot.

00:01:22.810 --> 00:01:26.080
Then the feet will cancel out,
and I'll be left with seconds.

00:01:26.080 --> 00:01:28.740
Now, right over here, we're
told that the squirrel can

00:01:28.740 --> 00:01:33.490
run at 12 feet per second,
but we want seconds per foot.

00:01:33.490 --> 00:01:37.290
So the squirrel, every second,
so they go 12 feet per second,

00:01:37.290 --> 00:01:41.426
then we could also say 1
second per every 12 feet.

00:01:41.426 --> 00:01:42.550
So let's write it that way.

00:01:42.550 --> 00:01:44.299
So it's essentially
the reciprocal of this

00:01:44.299 --> 00:01:46.600
because the units are
the reciprocal of this.

00:01:46.600 --> 00:01:51.170
So, it's 1 second
for every 12 feet.

00:01:51.170 --> 00:01:54.030
Notice, all I did is I took this
information right over here, 12

00:01:54.030 --> 00:01:57.110
feet per second, and I wrote
it as second per foot--

00:01:57.110 --> 00:02:01.550
12 feet for every 1 second,
1 second for every 12 feet.

00:02:01.550 --> 00:02:05.010
What's useful about this is
this will now give me the time

00:02:05.010 --> 00:02:07.230
it takes for the
squirrel in seconds.

00:02:07.230 --> 00:02:10.400
So the feet cancel
out with the feet,

00:02:10.400 --> 00:02:17.250
and I am left with 9 times
1/12, which is 9/12 seconds.

00:02:17.250 --> 00:02:19.320
And 9/12 seconds
is the same thing

00:02:19.320 --> 00:02:24.130
as 3/4 seconds, which is
the same thing as 0.75

00:02:24.130 --> 00:02:28.490
seconds for the squirrel
to cross the street.

00:02:28.490 --> 00:02:30.810
Now let's think about the car.

00:02:30.810 --> 00:02:32.670
So now let's think
about the car.

00:02:32.670 --> 00:02:34.060
And it's the exact same logic.

00:02:34.060 --> 00:02:37.110
They tell us that the
car is 50 feet away.

00:02:37.110 --> 00:02:41.650
So the squirrel is trying
to cross the road like that,

00:02:41.650 --> 00:02:44.810
and the car is 50 feet
away coming in like that,

00:02:44.810 --> 00:02:48.450
and we want to figure out if
the squirrel will survive.

00:02:48.450 --> 00:02:51.090
So the car is 50 feet away.

00:02:51.090 --> 00:02:52.990
So it's 50 feet away.

00:02:52.990 --> 00:02:55.830
We want to figure out the
time it'll take to travel that

00:02:55.830 --> 00:02:57.010
50 feet.

00:02:57.010 --> 00:02:58.770
Once again, we would
want it in seconds.

00:02:58.770 --> 00:03:01.240
So we would want
seconds per feet.

00:03:01.240 --> 00:03:05.750
So we would want to multiply
by seconds per foot.

00:03:08.340 --> 00:03:11.750
They give us the speed
in feet per second, 100

00:03:11.750 --> 00:03:13.414
feet per second.

00:03:13.414 --> 00:03:15.080
And so we just have
to realize that this

00:03:15.080 --> 00:03:22.440
is 100 feet for every 1 second,
or 1/100 seconds per feet.

00:03:22.440 --> 00:03:26.215
This is once again
just this information,

00:03:26.215 --> 00:03:27.590
but we took the
reciprocal of it,

00:03:27.590 --> 00:03:29.173
because we don't
want feet per second,

00:03:29.173 --> 00:03:30.890
we want seconds per feet.

00:03:30.890 --> 00:03:33.680
And if we do that,
that cancels with that,

00:03:33.680 --> 00:03:36.910
and we're left with
50/100 seconds.

00:03:36.910 --> 00:03:43.030
So this is 50/100
is 0.50 seconds.

00:03:43.030 --> 00:03:45.710
And so now let's answer the
question, this life and death

00:03:45.710 --> 00:03:47.825
situation for the squirrel.

00:03:47.825 --> 00:03:51.360
Will the squirrel make
it 9 feet across the road

00:03:51.360 --> 00:03:53.440
before the car gets there?

00:03:53.440 --> 00:03:57.320
Well, it's going to take the
squirrel 0.75 seconds to cross,

00:03:57.320 --> 00:04:00.480
and it's going to take the
car only half a second.

00:04:00.480 --> 00:04:03.490
So the car is going to get to
where the squirrel is crossing

00:04:03.490 --> 00:04:05.320
before the squirrel
has a chance to get

00:04:05.320 --> 00:04:07.240
all the way across the road.

00:04:07.240 --> 00:04:12.500
So unfortunately for the
squirrel, the answer is no.


Dependent and independent variables exercise: express the graph as an equation

WEBVTT
Kind: captions
Language: en

00:00:00.651 --> 00:00:01.150
Let's see.

00:00:01.150 --> 00:00:03.500
We have this graph
over here with t

00:00:03.500 --> 00:00:07.220
is the independent variable
on the horizontal axis

00:00:07.220 --> 00:00:09.967
and d is the dependent
variable on the vertical axis.

00:00:09.967 --> 00:00:11.300
And then they have a table here.

00:00:11.300 --> 00:00:13.740
Looks like this table
corresponds to this graph.

00:00:13.740 --> 00:00:19.090
When t equals 1, d is 40,
when t is equal to 2, d is 80.

00:00:19.090 --> 00:00:22.760
So these points correspond
to points on this line.

00:00:22.760 --> 00:00:26.560
And then they explain to us,
you are buying a gym membership.

00:00:26.560 --> 00:00:29.600
The membership
costs $40 per month.

00:00:29.600 --> 00:00:33.710
In the graph and table above, d
is the total number of dollars

00:00:33.710 --> 00:00:35.360
that you pay for
your gym membership,

00:00:35.360 --> 00:00:38.630
so that's d right over there,
and t is the time in months

00:00:38.630 --> 00:00:40.240
that you keep the membership.

00:00:40.240 --> 00:00:42.340
Write an equation for
the amount of money

00:00:42.340 --> 00:00:44.920
d that you pay for
your gym membership,

00:00:44.920 --> 00:00:47.490
if you keep the
membership for t months.

00:00:47.490 --> 00:00:48.440
And you see that here.

00:00:48.440 --> 00:00:49.870
If you're one
month, you pay $40,

00:00:49.870 --> 00:00:52.420
two months you pay
another $40, you pay $80.

00:00:52.420 --> 00:00:54.960
You see that in the
graph right over here.

00:00:54.960 --> 00:00:56.810
If you have 0 months,
you pay nothing.

00:00:56.810 --> 00:01:00.710
Then one month, $40, the
next month another $40

00:01:00.710 --> 00:01:02.060
getting you to $80.

00:01:02.060 --> 00:01:05.040
So if I were to write
this as an equation,

00:01:05.040 --> 00:01:07.660
the dependent variable
here is the amount

00:01:07.660 --> 00:01:09.070
that I pay in dollars.

00:01:09.070 --> 00:01:13.110
So that is going to
depend on the time.

00:01:13.110 --> 00:01:15.290
And how is it going to
depend on the time in months?

00:01:15.290 --> 00:01:19.660
Well I'm going to pay
$40 per month times

00:01:19.660 --> 00:01:20.590
the number of months.

00:01:20.590 --> 00:01:22.910
So I can either write it as
40 with a little asterisk

00:01:22.910 --> 00:01:27.050
sign, which is Shift 8--
and then I put t there--

00:01:27.050 --> 00:01:30.000
or I could literally put
just a t right over there.

00:01:30.000 --> 00:01:33.910
And I think that's right,
because if my time is 0

00:01:33.910 --> 00:01:36.580
according to this table and
according to this graph,

00:01:36.580 --> 00:01:37.251
I pay nothing.

00:01:37.251 --> 00:01:39.250
If there was some initial
membership charge then

00:01:39.250 --> 00:01:41.310
maybe we would add
that membership charge

00:01:41.310 --> 00:01:43.310
and you would pay the
monthly fee after that.

00:01:43.310 --> 00:01:45.840
But here this looks about
right, that the dollars

00:01:45.840 --> 00:01:51.210
paid is equal to 40
times the time in months.

00:01:51.210 --> 00:01:53.870
The time in months is
the independent variable.

00:01:53.870 --> 00:01:55.740
It drives the
dependent variable,

00:01:55.740 --> 00:01:58.640
the number of dollars you pay.


Dependent and independent variables exercise: the basics

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.210
On your math quiz, you earn
5 points for each question

00:00:04.210 --> 00:00:05.900
that you answer correctly.

00:00:05.900 --> 00:00:09.850
In the table above, q represents
the number of questions

00:00:09.850 --> 00:00:12.380
that you answer correctly
on your math quiz,

00:00:12.380 --> 00:00:15.290
and p represents the
total number of points

00:00:15.290 --> 00:00:16.960
that you score on your quiz.

00:00:16.960 --> 00:00:18.900
The relationship between
these two variables

00:00:18.900 --> 00:00:21.430
can be expressed by the
following equation--

00:00:21.430 --> 00:00:25.320
p is equal to 5q, where
p is the points you get

00:00:25.320 --> 00:00:27.760
and q is the number of questions
you answered correctly.

00:00:27.760 --> 00:00:29.300
And you could see
that in the table.

00:00:29.300 --> 00:00:31.980
If q is 0, if you got
no questions right,

00:00:31.980 --> 00:00:33.290
you get 0 points.

00:00:33.290 --> 00:00:36.445
If you got no questions
right, well, 5 times 0

00:00:36.445 --> 00:00:37.910
is going to be 0.

00:00:37.910 --> 00:00:41.880
If you get one question
right, well, 1 times 5 is 5.

00:00:41.880 --> 00:00:43.930
You get 5 points per question.

00:00:43.930 --> 00:00:46.705
Two questions right,
well, 2 times 5 is 10.

00:00:49.270 --> 00:00:50.970
3 times 5 is 15.

00:00:50.970 --> 00:00:52.600
So this all makes sense.

00:00:52.600 --> 00:00:55.010
So then they ask us, which
of the following statements

00:00:55.010 --> 00:00:55.510
are true?

00:00:55.510 --> 00:00:57.427
Check all that apply.

00:00:57.427 --> 00:00:58.510
So let's think about this.

00:00:58.510 --> 00:01:00.490
They say the
dependent variable is

00:01:00.490 --> 00:01:03.450
the number of points you score.

00:01:03.450 --> 00:01:05.390
So when you think about
what's happening here,

00:01:05.390 --> 00:01:07.250
is your number of
points you score

00:01:07.250 --> 00:01:11.420
is being driven by how many
questions you get right.

00:01:11.420 --> 00:01:14.610
It's not like somehow the
teacher says you got 15 points

00:01:14.610 --> 00:01:16.860
and now you have to get
exactly three questions right.

00:01:16.860 --> 00:01:17.943
It's the other way around.

00:01:17.943 --> 00:01:19.730
The number of
questions you get right

00:01:19.730 --> 00:01:21.970
is the independent
variable, and that's

00:01:21.970 --> 00:01:24.176
driving the number
of points you score.

00:01:24.176 --> 00:01:25.550
So the number of
points you score

00:01:25.550 --> 00:01:29.460
is the dependent variable.

00:01:29.460 --> 00:01:32.630
And typically, the convention is
to have the dependent variable

00:01:32.630 --> 00:01:36.020
be equal to some
expression involving

00:01:36.020 --> 00:01:37.080
the independent variable.

00:01:37.080 --> 00:01:38.670
And you see that
right over here.

00:01:38.670 --> 00:01:40.869
p is dependent on
what happens to q.

00:01:40.869 --> 00:01:42.410
Depending on the
number of questions,

00:01:42.410 --> 00:01:44.260
you multiply it by
5, and you get p.

00:01:44.260 --> 00:01:47.520
So the dependent variable is
the number of points you score.

00:01:47.520 --> 00:01:49.080
The dependent
variable is the number

00:01:49.080 --> 00:01:50.410
of questions you
answer correctly.

00:01:50.410 --> 00:01:51.909
No, we've already
talked about that.

00:01:51.909 --> 00:01:53.700
That's the independent variable.

00:01:53.700 --> 00:01:58.020
The independent variable is
the number of points you score.

00:01:58.020 --> 00:01:58.910
No, that's not right.

00:01:58.910 --> 00:01:59.845
That's the dependent variable.

00:01:59.845 --> 00:02:01.510
The independent
variable is a number

00:02:01.510 --> 00:02:03.090
of questions you
answer correctly.

00:02:03.090 --> 00:02:03.881
Well, that's right.

00:02:03.881 --> 00:02:06.360
That's what's driving
the dependent variable.

00:02:06.360 --> 00:02:08.660
And we can check our answer.


Writing algebraic expressions example 3

WEBVTT
Kind: captions
Language: en

00:00:00.688 --> 00:00:03.270
PROBLEM: "Write a mathematical expression which

00:00:03.270 --> 00:00:10.686
corresponds to x time y minus a times b times c."

00:00:10.686 --> 00:00:13.398
So let's just think this through: "x times y."

00:00:13.398 --> 00:00:14.545
Well I could write that –

00:00:14.545 --> 00:00:16.844
I could either write that as x × y,

00:00:16.844 --> 00:00:19.314
or I could just write that as xy.

00:00:19.314 --> 00:00:22.227
And from that, I'm going to subtract a times b times c.

00:00:22.227 --> 00:00:24.996
So: minus a times ...

00:00:24.996 --> 00:00:27.827
And I could write it a [asterisk] b [asterisk] c.

00:00:27.827 --> 00:00:28.876
I could write it like that.

00:00:28.876 --> 00:00:32.493
Or I could just write -abc.

00:00:32.493 --> 00:00:34.252
And you can see how it's being interpreted

00:00:34.252 --> 00:00:35.602
by the computer right over here.

00:00:35.602 --> 00:00:36.435
It understands that I'm saying

00:00:36.435 --> 00:00:40.264
x times y minus a times b times c (or xy - abc).

00:00:40.264 --> 00:00:42.408
And I'm done.


Writing algebraic expressions word problem example 2

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.640
The volume of a cube
is equal to its length

00:00:03.640 --> 00:00:05.830
times its width
times its height.

00:00:05.830 --> 00:00:10.230
A cube with side x has length
x, width x, and height x.

00:00:10.230 --> 00:00:12.360
So its volume is equal
to x times x times

00:00:12.360 --> 00:00:14.670
x, or x to the third.

00:00:14.670 --> 00:00:20.660
Shane takes a big cube of
butter with side length 5

00:00:20.660 --> 00:00:21.880
from the fridge.

00:00:21.880 --> 00:00:24.213
So I'm assuming-- well, they're
not giving us the units.

00:00:24.213 --> 00:00:25.580
Maybe it's 5 centimeters.

00:00:25.580 --> 00:00:28.646
With side length
5 from the fridge.

00:00:28.646 --> 00:00:30.020
Maybe I'll go with
5 centimeters.

00:00:30.020 --> 00:00:31.120
That seems reasonable.

00:00:31.120 --> 00:00:34.370
Then he cuts out a little
cube of side length 2

00:00:34.370 --> 00:00:37.170
from one of the
corners of the cube.

00:00:37.170 --> 00:00:38.780
Write an expression
that describes

00:00:38.780 --> 00:00:40.800
the volume of butter left.

00:00:40.800 --> 00:00:41.800
So let's think about it.

00:00:41.800 --> 00:00:44.870
The original volume of
butter that was-- they're

00:00:44.870 --> 00:00:46.270
saying side length 5.

00:00:46.270 --> 00:00:48.400
I'm visualizing it
as 5 centimeters.

00:00:48.400 --> 00:00:51.480
Its volume is going
to be 5 times 5 times

00:00:51.480 --> 00:00:56.740
5, which we could just write
as 5 to the third power.

00:00:56.740 --> 00:01:00.320
And then he takes out a
little chunk from that.

00:01:00.320 --> 00:01:02.810
And the volume that
he takes out is

00:01:02.810 --> 00:01:05.099
the volume of that little cube.

00:01:05.099 --> 00:01:07.530
And that little cube
has side length 2.

00:01:07.530 --> 00:01:10.510
So its volume is going
to be 2 times 2 times 2.

00:01:10.510 --> 00:01:14.510
Or another way of writing
that is 2 to the third power.

00:01:14.510 --> 00:01:18.700
So it's going to be 5 to the
third minus 2 to the third.


Writing algebraic expressions word problem

WEBVTT
Kind: captions
Language: en

00:00:03.400 --> 00:00:05.460
Mary Jeanne is the
system administrator

00:00:05.460 --> 00:00:06.800
for a big website.

00:00:06.800 --> 00:00:09.780
The website requires multiple
web servers and database

00:00:09.780 --> 00:00:11.040
servers to run.

00:00:11.040 --> 00:00:13.970
Each web server
costs $40 per month,

00:00:13.970 --> 00:00:17.187
and each database
costs $60 per month.

00:00:17.187 --> 00:00:19.520
To get an idea of the monthly
bill her company will have

00:00:19.520 --> 00:00:23.730
to pay, Mary Jeanne
makes a little table.

00:00:23.730 --> 00:00:27.200
So this is the number of
potential web servers,

00:00:27.200 --> 00:00:29.220
and this is the number
of the database servers.

00:00:29.220 --> 00:00:31.510
So these are
different scenarios.

00:00:31.510 --> 00:00:33.240
What is the expression
that describes

00:00:33.240 --> 00:00:39.090
a monthly hosting bill for n web
servers and m database servers?

00:00:39.090 --> 00:00:41.090
So let's think about
what's going on right here.

00:00:41.090 --> 00:00:44.460
When she has one web server
and one database server,

00:00:44.460 --> 00:00:47.800
well, she has to pay $40 per
month for that one web server

00:00:47.800 --> 00:00:51.340
and $60 a month for that
one database server.

00:00:51.340 --> 00:00:53.810
Now, when you add that
together, you get $100.

00:00:53.810 --> 00:00:56.030
Now, here, in the
second scenario,

00:00:56.030 --> 00:00:58.260
she has two web servers
and one database server.

00:00:58.260 --> 00:01:00.270
So here, she's
multiplying 40 times

00:01:00.270 --> 00:01:02.330
2, because that's how
many web servers she has.

00:01:02.330 --> 00:01:05.330
So she's going to have to pay
$80 a month for the web servers

00:01:05.330 --> 00:01:08.710
and still $60 per month for
the one database server.

00:01:08.710 --> 00:01:10.460
Let's skip to one
that's more interesting.

00:01:10.460 --> 00:01:13.790
So here she has five web servers
and two database servers.

00:01:13.790 --> 00:01:17.690
So she pays $48 per month for
each of the five web servers.

00:01:17.690 --> 00:01:21.180
And she pays $60 a month
for each of the two database

00:01:21.180 --> 00:01:21.990
servers.

00:01:21.990 --> 00:01:24.240
So if you wanted to write
an expression that describes

00:01:24.240 --> 00:01:27.760
the monthly hosting bill for
n web servers and m database

00:01:27.760 --> 00:01:30.170
servers, well, you see
here, you see this pattern.

00:01:30.170 --> 00:01:34.450
She pays $40 per month for
every web server that she has.

00:01:34.450 --> 00:01:36.600
So if she has n
web servers, she's

00:01:36.600 --> 00:01:41.830
going to pay $40 per month
per server times n servers.

00:01:41.830 --> 00:01:46.260
And then she's paying $60 a
month per database server.

00:01:46.260 --> 00:01:48.420
So if she has m
database servers,

00:01:48.420 --> 00:01:51.660
she's going to pay $60
per month per server times

00:01:51.660 --> 00:01:53.580
the number of servers.

00:01:53.580 --> 00:01:55.940
So this right over
here is the expression

00:01:55.940 --> 00:01:57.610
for the monthly
hosting billing cost.

00:01:57.610 --> 00:02:00.490
And you could see
that right over here.

00:02:00.490 --> 00:02:03.420
If you view this is as n,
the number of web servers,

00:02:03.420 --> 00:02:10.669
and this as m, you see that
when n is 20 and m is 10,

00:02:10.669 --> 00:02:15.890
then you get 40 times 20
plus 60 times 10-- 40 times n

00:02:15.890 --> 00:02:18.550
plus 60 times m.

00:02:18.550 --> 00:02:21.870
And then you would
get your actual value.


Equivalent forms of expressions

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.660
Which expressions are equivalent
to 2 times the quantity

00:00:04.660 --> 00:00:07.810
4f plus 2g?

00:00:07.810 --> 00:00:09.430
Mark all that apply.

00:00:09.430 --> 00:00:12.190
So the first one
here is 8f plus 4g.

00:00:12.190 --> 00:00:14.100
So could I manipulate
this somehow,

00:00:14.100 --> 00:00:17.550
in a valid way, so
I get 8f plus 4g?

00:00:17.550 --> 00:00:20.170
Well, the most obvious thing
is I just distribute the 2.

00:00:20.170 --> 00:00:22.050
So 2 times this
whole quantity is

00:00:22.050 --> 00:00:29.190
going to be 2 times 4f
plus 2 times 2g, which

00:00:29.190 --> 00:00:33.660
is equal to 8f plus 4g.

00:00:33.660 --> 00:00:38.100
So this expression is indeed
equivalent to that expression

00:00:38.100 --> 00:00:39.520
right over there.

00:00:39.520 --> 00:00:41.020
Now let's see what
they tried to do.

00:00:41.020 --> 00:00:44.210
2f times this thing
right over here.

00:00:44.210 --> 00:00:46.760
Is that equivalent
to that over there?

00:00:46.760 --> 00:00:48.620
Well, it doesn't look
like it's going to be.

00:00:48.620 --> 00:00:50.460
And you could even try to
distribute this right over

00:00:50.460 --> 00:00:50.960
here.

00:00:50.960 --> 00:00:53.240
You're going to get--
if we distribute the 2f,

00:00:53.240 --> 00:00:57.720
you'll get 2f times 4 is 8f.

00:00:57.720 --> 00:01:02.410
And 2f times 2g
is plus 4fg, which

00:01:02.410 --> 00:01:05.930
is very different
than 8f plus 4g.

00:01:05.930 --> 00:01:08.450
This is 8f plus 4fg.

00:01:08.450 --> 00:01:13.380
So this one is not an
equivalent expression.

00:01:13.380 --> 00:01:16.040
And here that you
have 8f plus 2g.

00:01:16.040 --> 00:01:17.740
Well, 8f plus 2g,
we already know,

00:01:17.740 --> 00:01:20.554
is different than 8f
plus 4g, And 8f plus 4g

00:01:20.554 --> 00:01:21.720
is an equivalent expression.

00:01:21.720 --> 00:01:23.344
And these two things
aren't equivalent,

00:01:23.344 --> 00:01:25.670
so we can cross that out.

00:01:25.670 --> 00:01:29.210
Now they have a 4
times 2f plus g.

00:01:29.210 --> 00:01:31.870
Well, what happens if we
were to factor a 4 out

00:01:31.870 --> 00:01:33.770
of 8f plus 4g, which
we already know

00:01:33.770 --> 00:01:36.120
is equivalent to our
original expression?

00:01:36.120 --> 00:01:38.290
So if you try to factor
out a 4 right over here,

00:01:38.290 --> 00:01:41.450
so you divide 8
by 4, you get 2f.

00:01:41.450 --> 00:01:44.400
And you divide 4g
by 4, you get g.

00:01:44.400 --> 00:01:46.100
And you can't just
only divide by 4.

00:01:46.100 --> 00:01:47.960
Then you then have to
multiply by 4 in order

00:01:47.960 --> 00:01:50.043
to not change the actual
value of the expressions.

00:01:50.043 --> 00:01:52.250
So all we did is we divided
by 4 and then multiplied

00:01:52.250 --> 00:01:54.540
by 4, which doesn't
change the actual value.

00:01:54.540 --> 00:01:57.590
Or you could think of it
as we undistributed a 4.

00:01:57.590 --> 00:01:59.240
We factored out a 4.

00:01:59.240 --> 00:02:02.470
So 4 times 2f plus 4g
is indeed the same thing

00:02:02.470 --> 00:02:04.620
as 8f plus 4g, which
is, we already know,

00:02:04.620 --> 00:02:07.050
is the same thing as
our original expression.

00:02:07.050 --> 00:02:09.080
And you can distribute
this to verify that.

00:02:09.080 --> 00:02:13.840
So this is also a
valid expression.

00:02:13.840 --> 00:02:16.280
Let's do a couple more of these.

00:02:16.280 --> 00:02:18.560
Fill in the blank to
produce an expression

00:02:18.560 --> 00:02:22.140
equivalent to mu
plus mu plus mu.

00:02:22.140 --> 00:02:25.050
Well, I have 3 mus
right over here.

00:02:25.050 --> 00:02:28.770
So this is literally just
going to be 3 mu, 3 times

00:02:28.770 --> 00:02:29.650
the variable mu.

00:02:29.650 --> 00:02:31.949
That's all that's going
on right over there.

00:02:31.949 --> 00:02:33.240
Let's do a couple more of this.

00:02:33.240 --> 00:02:34.760
That was pretty fast.

00:02:34.760 --> 00:02:38.740
Which expressions are equivalent
to 6l plus 5m minus 3n?

00:02:38.740 --> 00:02:40.860
Mark all that apply.

00:02:40.860 --> 00:02:42.360
So let's look at this first one.

00:02:42.360 --> 00:02:49.310
If I were to distribute the 3,
I would get 6l minus 3n plus 5.

00:02:49.310 --> 00:02:51.470
And that and that
are equivalent.

00:02:51.470 --> 00:02:54.370
If you just change the
order-- oh, a plus 5m.

00:02:54.370 --> 00:02:55.800
Let me be very careful here.

00:02:55.800 --> 00:02:57.174
So if I were to
distribute the 3,

00:02:57.174 --> 00:03:02.507
6l minus 3n plus 5m-- and this
expression and this expression

00:03:02.507 --> 00:03:04.090
are going to be
equivalent if you just

00:03:04.090 --> 00:03:08.450
swap the negative 3m and the 5m,
which you could completely do.

00:03:08.450 --> 00:03:09.450
Addition is commutative.

00:03:09.450 --> 00:03:12.760
It doesn't matter which
order you actually add in.

00:03:12.760 --> 00:03:15.157
So this is legitimate.

00:03:15.157 --> 00:03:15.740
Now let's see.

00:03:15.740 --> 00:03:18.890
3n plus 6l.

00:03:18.890 --> 00:03:21.400
So already, something
goes shady here.

00:03:21.400 --> 00:03:24.879
Here you have a minus 3n, or you
could view it as a negative 3n.

00:03:24.879 --> 00:03:26.420
Here you have a
positive 3n, and they

00:03:26.420 --> 00:03:27.670
don't fix it anywhere else.

00:03:27.670 --> 00:03:31.770
So this does not seem like
a legitimate expression.

00:03:31.770 --> 00:03:33.270
And they also have
a negative 5m,

00:03:33.270 --> 00:03:35.190
while it was a positive
5m right over there.

00:03:35.190 --> 00:03:37.290
So that's definitely
not the case.

00:03:37.290 --> 00:03:39.410
So here you have
5m, a positive 5m.

00:03:39.410 --> 00:03:42.160
Well, you have a positive
5m right over there.

00:03:42.160 --> 00:03:45.380
Then you have plus 6l minus 3n.

00:03:45.380 --> 00:03:48.390
Putting these parentheses,
these are essentially

00:03:48.390 --> 00:03:51.110
reassociating what
operation I would do first.

00:03:51.110 --> 00:03:53.360
But you could actually
remove the parentheses here,

00:03:53.360 --> 00:03:56.424
and it won't change the value.

00:03:56.424 --> 00:03:58.840
You could think about it as
distributing the positive sign

00:03:58.840 --> 00:04:00.910
or distributing the
positive 1 here.

00:04:00.910 --> 00:04:04.490
It would just become 5m plus
6l minus 3n, which is just

00:04:04.490 --> 00:04:06.190
a reordering of this
right over here.

00:04:06.190 --> 00:04:07.870
But it's completely legitimate.

00:04:07.870 --> 00:04:10.570
So let me write that
right over there.

00:04:10.570 --> 00:04:12.656
And then they have 5l.

00:04:12.656 --> 00:04:14.530
And if you distribute
this, this would be 3m.

00:04:14.530 --> 00:04:17.209
So they mixed everything up.

00:04:17.209 --> 00:04:18.089
This is 6l.

00:04:18.089 --> 00:04:18.720
This is a 5l.

00:04:18.720 --> 00:04:24.070
So this one right
over here is no good.

00:04:24.070 --> 00:04:24.960
This is a lot of fun.

00:04:24.960 --> 00:04:26.470
Let's keep going.

00:04:26.470 --> 00:04:29.690
If we take the
expression 2 times

00:04:29.690 --> 00:04:33.420
a plus 2b-- in parentheses--
and ignore the parentheses,

00:04:33.420 --> 00:04:36.100
we can write another
expression, 2a plus 2b,

00:04:36.100 --> 00:04:37.420
if you were to do that.

00:04:37.420 --> 00:04:42.445
Is 2a plus 2b equivalent to 2
times a plus 2b in parentheses?

00:04:42.445 --> 00:04:44.570
So when they're saying all
of this thing over here,

00:04:44.570 --> 00:04:45.790
they're saying,
hey, look, I'm just

00:04:45.790 --> 00:04:47.165
this irresponsible
mathematician.

00:04:47.165 --> 00:04:49.250
And I like to just
ignore parentheses

00:04:49.250 --> 00:04:51.480
without thinking about it fully.

00:04:51.480 --> 00:04:54.700
And if I did, I would
just get 2a plus 2b.

00:04:54.700 --> 00:04:56.210
Can I do this?

00:04:56.210 --> 00:04:58.280
And you might
already be imagining,

00:04:58.280 --> 00:05:00.820
based on the tone of my
voice, what the answer is.

00:05:00.820 --> 00:05:02.740
And to think about it, you just
have to realize, well, look.

00:05:02.740 --> 00:05:04.535
You're multiplying 2
not just times the a.

00:05:04.535 --> 00:05:07.350
You're multiplying 2 times
the entire quantity a plus 2b.

00:05:07.350 --> 00:05:09.460
You have to distribute that 2.

00:05:09.460 --> 00:05:15.800
This is going to be equal to 2
times a plus 2 times 2b, which

00:05:15.800 --> 00:05:18.830
is equal to 2a plus 4b.

00:05:18.830 --> 00:05:20.550
So you can't just
ignore that there.

00:05:20.550 --> 00:05:23.560
2a plus 4b is very
different than 2a plus 2b.

00:05:23.560 --> 00:05:25.380
So are these two
things equivalent?

00:05:25.380 --> 00:05:26.840
No.

00:05:26.840 --> 00:05:30.790
No, this is seriously
irresponsible mathematics.


Evaluating an algebraic expression in a word problem

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.730
The surface area of a
cube is equal to the sum

00:00:03.730 --> 00:00:06.937
of the areas of its six sides.

00:00:06.937 --> 00:00:08.020
Let's just visualize that.

00:00:08.020 --> 00:00:09.830
I like to visualize things.

00:00:09.830 --> 00:00:14.290
So if that's the cube,
we can see three sides.

00:00:14.290 --> 00:00:15.854
Three sides are facing us.

00:00:15.854 --> 00:00:17.270
But then if it was
transparent, we

00:00:17.270 --> 00:00:20.160
see that there are actually
six sides of a cube.

00:00:20.160 --> 00:00:22.960
So there's this one--
one, two, three in front--

00:00:22.960 --> 00:00:25.390
and then one--
this is the bottom.

00:00:25.390 --> 00:00:27.540
This is in the back, and
this is also in the back.

00:00:27.540 --> 00:00:29.040
So you have three
sides of the cube.

00:00:29.040 --> 00:00:30.900
So I believe what
they're saying.

00:00:30.900 --> 00:00:33.470
The surface area of a
cube with side length

00:00:33.470 --> 00:00:36.415
x-- so if this is
x, if this is x,

00:00:36.415 --> 00:00:40.610
if this is x-- is given by
the expression 6x squared.

00:00:40.610 --> 00:00:41.570
That also makes sense.

00:00:41.570 --> 00:00:45.380
The area of each side is going
to be x times x is x squared,

00:00:45.380 --> 00:00:46.380
and there's six of them.

00:00:46.380 --> 00:00:48.680
So it's going to be 6x squared.

00:00:48.680 --> 00:00:51.470
Jolene has two
cube-shaped containers

00:00:51.470 --> 00:00:52.910
that she wants to paint.

00:00:52.910 --> 00:00:55.740
One cube has side length 2.

00:00:55.740 --> 00:00:57.870
So this is one cube
right over here.

00:00:57.870 --> 00:01:00.240
I'll do my best to draw it.

00:01:00.240 --> 00:01:03.500
So this right over
here has side length 2,

00:01:03.500 --> 00:01:05.379
so that's its dimensions.

00:01:05.379 --> 00:01:08.760
The other cube has
side length 1.5.

00:01:08.760 --> 00:01:11.370
So the other cube is going
to be a little bit smaller.

00:01:11.370 --> 00:01:14.390
It has side length 1.5.

00:01:14.390 --> 00:01:20.090
So it's 1.5 by 1.5 by 1.5.

00:01:20.090 --> 00:01:23.470
What is the total surface
area that she has to paint?

00:01:23.470 --> 00:01:25.750
Well, we know that the
surface area of each cube

00:01:25.750 --> 00:01:27.990
is going to be 6x
squared, where x

00:01:27.990 --> 00:01:30.000
is the dimensions of that cube.

00:01:30.000 --> 00:01:33.180
So the surface area of
this cube right over here

00:01:33.180 --> 00:01:35.630
is going to be 6.

00:01:35.630 --> 00:01:38.235
And now-- let me do it in
that color of that cube--

00:01:38.235 --> 00:01:40.990
it's going to be
6 times x, where

00:01:40.990 --> 00:01:44.230
x is the dimension of the cube.

00:01:44.230 --> 00:01:46.270
And then the cube all
has the same dimensions,

00:01:46.270 --> 00:01:48.750
so its length, width, and
depth is all the same.

00:01:48.750 --> 00:01:50.350
So for this cube,
the surface area

00:01:50.350 --> 00:01:52.845
is going to be 6
times 2 squared.

00:01:57.177 --> 00:01:58.760
And then the surface
area of this cube

00:01:58.760 --> 00:02:04.230
is going to be 6
times 1.5 squared.

00:02:06.799 --> 00:02:09.090
And if we want the total
surface area she has to paint,

00:02:09.090 --> 00:02:11.400
it's going to be the
sum of the two cubes.

00:02:11.400 --> 00:02:14.030
So we're just going to
add these two things.

00:02:14.030 --> 00:02:16.530
And so if we were to compute
this first one right over here,

00:02:16.530 --> 00:02:18.870
this is going to be 6 times 4.

00:02:18.870 --> 00:02:20.690
This is 24.

00:02:20.690 --> 00:02:22.481
And this one right
over here, this

00:02:22.481 --> 00:02:23.980
is going to be a
little bit hairier.

00:02:23.980 --> 00:02:24.800
Let's see.

00:02:24.800 --> 00:02:27.360
15 times 15 is 225.

00:02:27.360 --> 00:02:33.430
So 1.5 times 1.5 is 2.25.

00:02:33.430 --> 00:02:37.540
So 1.5 squared is 2.25.

00:02:37.540 --> 00:02:42.610
And 2.25 times 6-- so let
me just multiply that out.

00:02:42.610 --> 00:02:46.790
2.25 times 6.

00:02:46.790 --> 00:02:47.300
Let's see.

00:02:47.300 --> 00:02:49.775
We're going to have
6 times 5 is 30.

00:02:49.775 --> 00:02:54.000
6 times 2 is 12, plus 3 is 15.

00:02:54.000 --> 00:02:56.830
6 times 2 is 12, plus 1 is 13.

00:02:56.830 --> 00:02:59.980
I have two numbers behind
the decimal-- 13.5.

00:02:59.980 --> 00:03:03.540
So it's going to be 13.5.

00:03:03.540 --> 00:03:05.940
And if I add these
two together, this

00:03:05.940 --> 00:03:08.380
is going to be equal to
the total surface area

00:03:08.380 --> 00:03:15.944
that she has got to paint, is
going to be 37.5 square-- well,

00:03:15.944 --> 00:03:17.610
I guess they're not
giving us the units.

00:03:17.610 --> 00:03:21.130
Well, 37.5 is going to be the
total area of square units

00:03:21.130 --> 00:03:24.100
of whatever the
units happen to be.


Khan Academy Talent Search 2015 Winners

WEBVTT
Kind: captions
Language: en

00:00:00.361 --> 00:00:03.347
- Hi, I'm Dan Quinn, originally from McLean, Virginia,

00:00:03.347 --> 00:00:07.252
and I created the video about the tears of wine

00:00:07.252 --> 00:00:09.644
that form around the edge of wine glasses.

00:00:09.644 --> 00:00:12.287
Most people who drink wine have seen the weird drops

00:00:12.287 --> 00:00:14.906
that form around the side of the glass.

00:00:14.906 --> 00:00:18.911
Traditionally, they're called "tears" or "legs" of wine.

00:00:18.911 --> 00:00:21.977
But, what most people don't realize is that,

00:00:21.977 --> 00:00:23.948
if you watch patiently, these drops are

00:00:23.948 --> 00:00:25.879
actually moving around.

00:00:25.879 --> 00:00:28.276
My favorite part is when I'll be at a party or something,

00:00:28.276 --> 00:00:30.327
and someone has a glass of wine

00:00:30.327 --> 00:00:32.724
and they're twirling it around and they see the legs of wine

00:00:32.724 --> 00:00:34.192
or the tears of wine and they say,

00:00:34.192 --> 00:00:36.647
"Oh, Dan. Come explain this to us."

00:00:36.647 --> 00:00:39.002
- Students, before lab, I can see them

00:00:39.002 --> 00:00:41.441
in the hallways watching my videos.

00:00:41.441 --> 00:00:43.961
And then also, during lab, sometimes they'll, like,

00:00:43.961 --> 00:00:45.789
have it on their phone and they'll be like,

00:00:45.789 --> 00:00:47.375
"Well, what did he do in the video?

00:00:47.375 --> 00:00:49.141
Just bring it up on your phone."

00:00:49.141 --> 00:00:51.315
- My students think it's super cool.

00:00:51.315 --> 00:00:52.898
(laughs)

00:00:52.898 --> 00:00:57.091
- The second they heard that I had a video on YouTube,

00:00:57.091 --> 00:00:59.510
they were, like, really excited.

00:00:59.510 --> 00:01:01.541
The mushy food lands in your stomach,

00:01:01.541 --> 00:01:03.795
where gastric acid and chemicals

00:01:03.795 --> 00:01:06.193
break down the food even more.

00:01:07.534 --> 00:01:08.917
- To be a really good teacher,

00:01:08.917 --> 00:01:10.968
you just have to be a really good communicator

00:01:10.968 --> 00:01:14.199
and you need to also be able to relate to your students.

00:01:14.199 --> 00:01:18.099
- I think that a good teacher is engaged and supportive

00:01:18.099 --> 00:01:21.301
and responsible and should treat

00:01:21.301 --> 00:01:24.169
his or her students with respect.

00:01:24.169 --> 00:01:25.881
- If you put yourself in an ivory tower,

00:01:25.881 --> 00:01:27.592
or spew and force wisdom onto the masses,

00:01:27.592 --> 00:01:29.477
like, that's not gunna help anybody.

00:01:29.477 --> 00:01:32.077
The truth is your students are just as smart as you are,

00:01:32.077 --> 00:01:33.398
you just happen to know

00:01:33.398 --> 00:01:35.023
a little bit more than them for the moment.

00:01:35.023 --> 00:01:36.405
- You know, when you're teaching it,

00:01:36.405 --> 00:01:39.188
you know it all very well but when you're teaching it

00:01:39.188 --> 00:01:42.703
to someone else, you have to think of what they're thinking

00:01:42.703 --> 00:01:45.567
and, when I make the videos, I, sort of, keep in mind

00:01:45.567 --> 00:01:48.311
what questions pop into their head first.

00:01:48.311 --> 00:01:51.846
If you're telling me that we have a seeing threshold

00:01:51.846 --> 00:01:53.999
just like how we having a hearing threshold,

00:01:53.999 --> 00:01:57.249
wouldn't that mean that there's light waves?

00:01:57.249 --> 00:02:00.521
I think what's really important is that they love it

00:02:00.521 --> 00:02:02.167
because, when a teacher loves something

00:02:02.167 --> 00:02:05.560
and they teach it to you, you, sort of, love it too.

00:02:05.560 --> 00:02:06.669
- Usually, when I create a new video,

00:02:06.669 --> 00:02:08.526
it's because I found something that's really exciting

00:02:08.526 --> 00:02:09.886
that I wanna talk about to people.

00:02:09.886 --> 00:02:12.060
And so, I, kind of, can't not talk about it

00:02:12.060 --> 00:02:13.362
and I wanna get it all into one thing

00:02:13.362 --> 00:02:15.128
that can be my packaged, you know,

00:02:15.128 --> 00:02:17.120
delivery of this idea to the world.

00:02:17.120 --> 00:02:19.293
- The thing that excites me the most

00:02:19.293 --> 00:02:23.906
about video making is actually YouTube comments.

00:02:23.906 --> 00:02:29.086
Because of the nature of the content that I produced,

00:02:29.086 --> 00:02:32.541
it didn't really attract a whole lot of trolls.

00:02:32.541 --> 00:02:35.974
So, all of the comments were just uniformly constructive

00:02:35.974 --> 00:02:38.575
or helpful or they would say, "David, you screwed up here."

00:02:38.575 --> 00:02:40.993
But they would say it, like, real politely.

00:02:40.993 --> 00:02:44.629
- Other fantastic comments include, "That was tolerable."

00:02:44.629 --> 00:02:46.783
and "I did not want to blow my brains out

00:02:46.783 --> 00:02:48.815
after watching your video."

00:02:48.815 --> 00:02:50.236
Those are from harsher critics.

00:02:50.236 --> 00:02:52.005
- Ok, so one of my biggest worries

00:02:52.005 --> 00:02:54.057
about putting things on YouTube,

00:02:54.057 --> 00:02:57.043
you know, YouTube's notorious for critical comments.

00:02:57.043 --> 00:02:58.973
And I don't care about that.

00:02:58.973 --> 00:03:01.046
That's much better than no comments, in my mind.

00:03:01.046 --> 00:03:03.870
But, I was worried that my mom might, you know,

00:03:03.870 --> 00:03:05.230
she's naturally gunna watch videos.

00:03:05.230 --> 00:03:07.119
She cares about what her son's doing.

00:03:07.119 --> 00:03:08.556
And I don't want her to see people

00:03:08.556 --> 00:03:09.923
saying bad things about her son.

00:03:09.923 --> 00:03:11.387
No mother wants that.

00:03:11.387 --> 00:03:13.682
So, I was a little bit stressed about that

00:03:13.682 --> 00:03:15.104
to start 'cuz I don't care.

00:03:15.104 --> 00:03:17.826
That's just critical feedback, especially with education.

00:03:17.826 --> 00:03:19.981
So, it was a bit of a relief, when, at least, you know,

00:03:19.981 --> 00:03:23.415
so far, people are, by and large, very, very nice about it.

00:03:23.415 --> 00:03:26.441
- The initial feedback I got from my students

00:03:26.441 --> 00:03:30.221
about the videos and providing that has just been amazing.

00:03:30.221 --> 00:03:34.244
And I never thought that it would be as well received.

00:03:34.244 --> 00:03:36.275
And, of course, I also never thought that

00:03:36.275 --> 00:03:38.917
that would lead to being here at Khan Academy.

00:03:38.917 --> 00:03:41.476
- And so, I would encourage anyone

00:03:41.476 --> 00:03:43.751
who wants to do this, who thinks they have something to say

00:03:43.751 --> 00:03:47.521
about history, science, mathematics,

00:03:47.521 --> 00:03:51.066
to just go out there and give it to the world.

00:03:51.066 --> 00:03:54.398
- I feel like teaching is the best way to learn sometimes.

00:03:54.398 --> 00:03:55.961
- Each professor, each teacher,

00:03:55.961 --> 00:03:59.173
each tutor has their own style and, I think,

00:03:59.173 --> 00:04:01.143
any person can bring out their own style,

00:04:01.143 --> 00:04:02.606
from their personality, from the way

00:04:02.606 --> 00:04:03.966
they think about things.

00:04:03.966 --> 00:04:06.019
That's something that only they can offer

00:04:06.019 --> 00:04:07.421
and they can offer it the best.

00:04:07.421 --> 00:04:09.189
- I hope my videos show that, you know,

00:04:09.189 --> 00:04:12.236
if you break it down into one step after the other,

00:04:12.236 --> 00:04:16.366
you can really understand even the most complex phenomenon.


Comparing dot plots, histograms, and box plots

WEBVTT
Kind: captions
Language: en

00:00:00.147 --> 00:00:01.935
- [Voiceover] What I wanna do with this video is look

00:00:01.935 --> 00:00:04.907
at some examples of data represented in different ways,

00:00:04.907 --> 00:00:07.670
and think about which representation is the best,

00:00:07.670 --> 00:00:10.921
or can help us answer different questions?

00:00:10.921 --> 00:00:12.779
So we see this first example.

00:00:12.779 --> 00:00:14.984
A statistician recorded the length of each

00:00:14.984 --> 00:00:17.623
of Pixar's first 14 films.

00:00:17.623 --> 00:00:21.757
The statistician made a dot plot, each dot is a film,

00:00:21.757 --> 00:00:23.961
a histogram, and a box plot

00:00:23.961 --> 00:00:26.423
to display the running time data.

00:00:26.423 --> 00:00:30.254
Which display could be used to find the median?

00:00:30.254 --> 00:00:31.531
To find the median.

00:00:31.531 --> 00:00:34.572
All right, so let's look at these displays.

00:00:34.572 --> 00:00:38.171
So over here we see, this is the dot plot.

00:00:38.171 --> 00:00:40.400
We have a dot for each of the 14 films.

00:00:40.400 --> 00:00:43.996
So one film had a running time of 81 minutes.

00:00:43.996 --> 00:00:44.882
We see that there.

00:00:44.882 --> 00:00:47.064
One film had a running time of 92.

00:00:47.064 --> 00:00:50.083
One had a running time of 93.

00:00:50.083 --> 00:00:52.544
We see one had a running time of 95.

00:00:52.544 --> 00:00:55.841
We see two had running times of 96 minutes,

00:00:55.841 --> 00:00:57.931
and so on and so forth.

00:00:57.931 --> 00:01:01.042
So I claim that I could use this to figure out the median,

00:01:01.042 --> 00:01:03.875
because I could make a list of all of the running times

00:01:03.875 --> 00:01:05.941
of the films, I could order them,

00:01:05.941 --> 00:01:07.503
and then I could find the middle value.

00:01:07.503 --> 00:01:08.565
I could literally make a list.

00:01:08.565 --> 00:01:12.234
I could write down 81, and then write down 92,

00:01:12.234 --> 00:01:14.927
then write down 93, then write down 95,

00:01:14.927 --> 00:01:16.947
then I could write down 96 twice,

00:01:16.947 --> 00:01:19.302
and then I could write down 98,

00:01:19.302 --> 00:01:20.430
then I could write down 100.

00:01:20.430 --> 00:01:22.682
I think you see where this is going.

00:01:22.682 --> 00:01:24.540
I could write out the entire list,

00:01:24.540 --> 00:01:26.537
and then I could find the middle values.

00:01:26.537 --> 00:01:31.134
So the dot plot, I could definitely use to find the median.

00:01:31.134 --> 00:01:32.574
Now, what about the histogram?

00:01:32.574 --> 00:01:35.313
This is the histogram right over here.

00:01:35.313 --> 00:01:38.146
And the key here is, for a median, to figure out a median,

00:01:38.146 --> 00:01:40.415
I just need to figure out a list of numbers.

00:01:40.415 --> 00:01:41.722
I need to figure out a list of numbers.

00:01:41.722 --> 00:01:45.089
So here, I don't know, they say I have one film

00:01:45.089 --> 00:01:47.410
that's between 80 and 85,

00:01:47.410 --> 00:01:49.339
but I don't know its exact running time.

00:01:49.339 --> 00:01:52.101
Its running time might have been 81 minutes,

00:01:52.101 --> 00:01:54.608
its running time might have been 84 minutes.

00:01:54.608 --> 00:01:58.299
So I don't know here, and so I can't really make a list

00:01:58.299 --> 00:02:00.111
of the running times of the films

00:02:00.111 --> 00:02:01.504
and find the middle values,

00:02:01.504 --> 00:02:02.683
so I don't think I'm gonna be able

00:02:02.683 --> 00:02:04.609
to do it using the histogram.

00:02:04.609 --> 00:02:08.586
Now, with the box plot right over here,

00:02:08.586 --> 00:02:10.165
so I'm not gonna click histogram.

00:02:10.165 --> 00:02:11.721
With the box plot over here,

00:02:11.721 --> 00:02:14.358
I might not be able to make a list of all the values,

00:02:14.358 --> 00:02:17.781
but the box plot explicitly tells us what the median is.

00:02:17.781 --> 00:02:20.428
This middle line in the middle of the box,

00:02:20.428 --> 00:02:23.168
that tells us the median is, what is this,

00:02:23.168 --> 00:02:26.744
this median is, if this is 100, this is 99.

00:02:26.744 --> 00:02:29.832
So this is 95, 96, 97, 98, 99.

00:02:29.832 --> 00:02:32.061
It explicitly tells us the median is 99.

00:02:32.061 --> 00:02:34.894
This is actually the easiest for calculating the median.

00:02:34.894 --> 00:02:36.287
So I'll go with the box plot.

00:02:36.287 --> 00:02:38.353
So the histogram is of no use to me

00:02:38.353 --> 00:02:40.001
if I wanna calculate the median.

00:02:40.001 --> 00:02:41.929
Let's do a couple more of these.

00:02:41.929 --> 00:02:44.808
Nam owns a used car lot.

00:02:44.808 --> 00:02:46.596
He checked the odometers of the cars

00:02:46.596 --> 00:02:49.127
and recorded how far they had driven.

00:02:49.127 --> 00:02:51.936
He then created both a histogram and a box plot

00:02:51.936 --> 00:02:55.268
to display the same data, both diagrams are shown below.

00:02:55.998 --> 00:02:58.194
Which display can be used

00:02:58.194 --> 00:03:00.889
to find how many vehicles had driven

00:03:00.889 --> 00:03:04.139
more than 200,000 kilometers?

00:03:04.139 --> 00:03:06.158
So how many vehicles had driven

00:03:06.158 --> 00:03:09.525
more than 200,000 kilometers?

00:03:09.525 --> 00:03:13.264
So it looks like here in this histogram,

00:03:13.264 --> 00:03:16.700
I have three vehicles that were between 200 and 250,

00:03:16.700 --> 00:03:20.415
and then I have two vehicles that are between 250 and 300.

00:03:20.415 --> 00:03:22.158
So it looks pretty clear that I have five vehicles,

00:03:22.158 --> 00:03:26.105
three that had a mileage between 200,000 and 250,000,

00:03:26.105 --> 00:03:27.938
and then I had two that had mileage

00:03:27.938 --> 00:03:30.115
between 250,000 and 300,000.

00:03:30.115 --> 00:03:31.507
So I may be able to answer the question.

00:03:31.507 --> 00:03:36.368
Five vehicles had a mileage more than 200,000,

00:03:36.368 --> 00:03:40.036
and so I would say that the histogram is pretty useful.

00:03:40.036 --> 00:03:42.915
But let's verify that the box plot isn't so useful.

00:03:42.915 --> 00:03:45.029
So I wanna know how many vehicles had a mileage

00:03:45.029 --> 00:03:47.141
more than 200,000.

00:03:47.141 --> 00:03:50.880
Well, I know that if I have a mileage more than 200,000,

00:03:50.880 --> 00:03:54.943
I'm going to be in the fourth quartile,

00:03:54.943 --> 00:03:58.194
but I don't know how many values I have sitting there

00:03:58.194 --> 00:04:01.583
in the fourth quartile just looking at this data over here,

00:04:01.583 --> 00:04:04.671
so that's not gonna be useful for answering that question.

00:04:04.671 --> 00:04:06.227
Let's look at the second question.

00:04:06.227 --> 00:04:07.690
Which display can be used

00:04:07.690 --> 00:04:09.518
to find that the median distance,

00:04:09.518 --> 00:04:11.478
which display can be used to find

00:04:11.478 --> 00:04:12.478
that the median distance

00:04:12.478 --> 00:04:15.420
was approximately 140,000 kilometers?

00:04:15.420 --> 00:04:16.744
Well, to calculate the median,

00:04:16.744 --> 00:04:18.717
you essentially wanna be able to list all of the numbers

00:04:18.717 --> 00:04:20.251
and then find the middle number.

00:04:20.251 --> 00:04:23.315
And over here, I can't list all of the numbers.

00:04:23.315 --> 00:04:25.405
I know that there's three values that are

00:04:25.405 --> 00:04:27.541
between zero and 50,000 kilometers,

00:04:27.541 --> 00:04:28.585
but I don't know what they are.

00:04:28.585 --> 00:04:30.792
Could be 10,000, 10,000, 10,000.

00:04:30.792 --> 00:04:34.483
It could be 10,000, 15,000, and 40,000.

00:04:34.483 --> 00:04:37.020
I don't know what they are, and so if I can't list all

00:04:37.020 --> 00:04:39.142
of these things and put them in order,

00:04:39.146 --> 00:04:42.228
I really am going to have trouble finding the middle value.

00:04:42.228 --> 00:04:45.392
The middle value, it's going to be

00:04:45.392 --> 00:04:47.783
in this range right around here,

00:04:47.783 --> 00:04:49.687
but I don't know exactly what it's going to be.

00:04:49.687 --> 00:04:50.964
The histogram is not useful,

00:04:50.964 --> 00:04:53.918
because throwing all the values into these buckets.

00:04:53.918 --> 00:04:56.258
While on the box plot, it explicitly,

00:04:56.258 --> 00:04:58.116
it directly tells me the median value.

00:04:58.116 --> 00:05:00.508
This line right over here, the middle of the box,

00:05:00.508 --> 00:05:02.737
this tells us the median value,

00:05:02.737 --> 00:05:05.034
and we see that the median value here,

00:05:05.034 --> 00:05:08.425
this is 140,000 kilometers.

00:05:08.425 --> 00:05:11.341
Right, this is 100, 110, 120, 130,

00:05:11.341 --> 00:05:16.011
140,000 kilometers is the median mileage for the cars.

00:05:16.011 --> 00:05:18.808
And so the box plot clearly...

00:05:20.835 --> 00:05:23.242
clearly gives us that data.


Examples analyzing clusters, gaps, peaks and outliers for distributions

WEBVTT
Kind: captions
Language: en

00:00:00.135 --> 00:00:01.772
- [Voiceover] In this video, I wanna do some examples

00:00:01.772 --> 00:00:04.615
looking at distributions, in particular, different features

00:00:04.615 --> 00:00:08.489
in distributions like clusters, gaps, and peaks.

00:00:08.489 --> 00:00:10.549
So over here, I wanna do some examples.

00:00:10.549 --> 00:00:12.699
Which of the following are accurate descriptions

00:00:12.699 --> 00:00:14.550
of the distribution below?

00:00:14.550 --> 00:00:16.787
Select all that apply.

00:00:16.787 --> 00:00:20.060
So the first statement is the distribution has an outlier.

00:00:20.060 --> 00:00:23.163
So an outlier is a data point that's way off

00:00:23.163 --> 00:00:24.930
of where the other data points are,

00:00:24.930 --> 00:00:27.077
it's way larger or way smaller

00:00:27.077 --> 00:00:28.619
than where all of the other data points

00:00:28.619 --> 00:00:31.109
seem to be clustered and if we look over here,

00:00:31.109 --> 00:00:34.448
we have a lot of data points between zero and six.

00:00:34.448 --> 00:00:35.842
And let's just think about what they're measuring:

00:00:35.842 --> 00:00:38.411
this is shelf time for each apple

00:00:38.411 --> 00:00:41.921
at Gorg's Grocier.

00:00:41.921 --> 00:00:45.183
So, for example, we see there's one, two, three, four,

00:00:45.183 --> 00:00:49.667
five, six, seven apples that have a shelf life

00:00:49.667 --> 00:00:52.877
of zero days, so (laughs), they're about to go bad.

00:00:52.877 --> 00:00:56.709
You see you have one, two, three, four, five, six, seven,

00:00:56.709 --> 00:00:59.299
eight apples that are gonna be good for another day.

00:00:59.299 --> 00:01:00.987
You have two apples that are gonna be good

00:01:00.987 --> 00:01:03.968
for another six days, and you have one apple

00:01:03.968 --> 00:01:06.020
that's gonna be good for 10 days, and this is unusual.

00:01:06.020 --> 00:01:10.127
This is an outlier here, it has a way larger shelf life

00:01:10.127 --> 00:01:12.339
than all of the other data, so I would say

00:01:12.339 --> 00:01:14.076
this definitely does have an outlier.

00:01:14.076 --> 00:01:15.445
We just have this one data point

00:01:15.445 --> 00:01:18.147
sitting all the way to the right, way larger,

00:01:18.147 --> 00:01:20.633
way more shelf life than everything else, so it definitely

00:01:20.633 --> 00:01:23.874
has an outlier, and this one would be the outlier.

00:01:23.874 --> 00:01:26.772
The distribution has a cluster from four to six days.

00:01:26.772 --> 00:01:29.767
And we indeed do see a cluster from four to six days.

00:01:29.767 --> 00:01:33.392
A cluster, you can imagine, it's a grouping of data

00:01:33.392 --> 00:01:35.780
that's sitting there, or you have a grouping of apples

00:01:35.780 --> 00:01:37.857
that have a shelf life between four and six days,

00:01:37.857 --> 00:01:40.478
and you definitely do see that cluster there.

00:01:40.478 --> 00:01:42.040
And since I already selected two things,

00:01:42.040 --> 00:01:44.671
I'm definitely not gonna select none of the above.

00:01:44.671 --> 00:01:45.963
Let me check my answer.

00:01:45.963 --> 00:01:50.413
Let me do a few more of these.

00:01:50.413 --> 00:01:52.805
Which of the following are accurate descriptions

00:01:52.805 --> 00:01:54.336
of the distribution below?

00:01:54.336 --> 00:01:57.458
And once again we're going to select all that apply.

00:01:57.458 --> 00:02:00.163
So the distribution has an outlier.

00:02:00.163 --> 00:02:02.103
So let's see this distribution.

00:02:02.103 --> 00:02:04.281
I do have a data point here that is at the high end

00:02:04.281 --> 00:02:06.608
and I have another data point here that's at the low end,

00:02:06.608 --> 00:02:08.697
but I don't have any data points that are sitting

00:02:08.697 --> 00:02:12.254
far above or far below the bulk of the data.

00:02:12.254 --> 00:02:14.258
If I had a data point that was out here, then yeah,

00:02:14.258 --> 00:02:16.417
I would say that was an outlier to the right,

00:02:16.417 --> 00:02:19.355
or a positive outlier, if I had a data point way to the left

00:02:19.355 --> 00:02:21.507
off the screen over here, maybe that would be an outlier,

00:02:21.507 --> 00:02:23.459
but I don't really see any obvious outliers.

00:02:23.459 --> 00:02:27.450
All of the data, it's pretty clustered together.

00:02:27.450 --> 00:02:30.676
So I would not say that the distribution has an outlier.

00:02:30.676 --> 00:02:34.728
The distribution has a peak at 22 degrees.

00:02:34.728 --> 00:02:36.745
Yeah, it does indeed look like we have,

00:02:36.745 --> 00:02:38.199
and let's just look at what we're actually measuring:

00:02:38.199 --> 00:02:42.619
high temperature each day in Edgeton, Iowa in July.

00:02:42.619 --> 00:02:44.996
So it does indeed look like we have the most number

00:02:44.996 --> 00:02:49.060
of days that had a high temperature at 22,

00:02:49.060 --> 00:02:51.925
most number of days in July had a high tempurature

00:02:51.925 --> 00:02:55.853
at 22 degrees Celsius, so that is a peak.

00:02:55.853 --> 00:02:57.710
You can see it, if you imagine this as kind of a mountain

00:02:57.710 --> 00:02:59.488
this is a peak right here, this is a high point.

00:02:59.488 --> 00:03:03.856
You have, at least locally, the most number of days

00:03:03.856 --> 00:03:06.748
at 22 degrees Celsius.

00:03:06.748 --> 00:03:08.862
So I would say it definitely has a peak there.

00:03:08.862 --> 00:03:10.746
Since I selected something, I'm not gonna select

00:03:10.746 --> 00:03:12.066
none of the above.

00:03:12.066 --> 00:03:14.486
Let's do a couple more of these.

00:03:14.486 --> 00:03:16.308
Which of the following are accurate descrptions

00:03:16.308 --> 00:03:17.827
of the distribution below?

00:03:17.827 --> 00:03:20.629
So the first one, the distribution has an outlier.

00:03:20.629 --> 00:03:21.244
So...

00:03:21.934 --> 00:03:26.004
number of guests by day at Seth's Sandwich Shop.

00:03:26.004 --> 00:03:28.594
So, let's see, the lowest...

00:03:29.514 --> 00:03:31.884
They have no days...

00:03:31.884 --> 00:03:36.074
No days where he had between zero and 19 guests,

00:03:36.074 --> 00:03:38.888
no days where he had between 20 and 39 guests,

00:03:38.888 --> 00:03:40.343
looks like there's about nine days

00:03:40.343 --> 00:03:42.417
where he had between 40 and 59 guests,

00:03:42.417 --> 00:03:45.625
looks like 20 days where he had between 60 and 79 guests,

00:03:45.625 --> 00:03:48.011
all the way where it looks like maybe 8 days

00:03:48.011 --> 00:03:51.441
that he had between 180 and 199 guests.

00:03:51.441 --> 00:03:54.402
But the question of outliers, there doesn't seem to be

00:03:54.402 --> 00:03:57.833
any day where he had an unusual number of guests.

00:03:57.833 --> 00:03:59.774
There's not a day that's way out here,

00:03:59.774 --> 00:04:02.061
where he had, like, 500 guests.

00:04:02.061 --> 00:04:05.713
So I would say this distribution does not have an outlier.

00:04:05.713 --> 00:04:09.272
The distribution has a cluster from zero to 39 guests.

00:04:09.272 --> 00:04:13.557
So zero to 39 guests is right over here, zero to 39 guests.

00:04:13.557 --> 00:04:16.958
And there is no days where he had between zero and 39 guests

00:04:16.958 --> 00:04:19.730
neither zero to 19, or 20 to 39.

00:04:19.730 --> 00:04:21.063
So there's definitely not a cluster there.

00:04:21.063 --> 00:04:23.743
I would say that the cluster would be between days

00:04:23.743 --> 00:04:27.851
that had between 40 and 199 guests.

00:04:27.851 --> 00:04:30.029
Definitely not zero and 39, there was no days

00:04:30.029 --> 00:04:32.469
that were between zero and 39 guests.

00:04:32.469 --> 00:04:35.903
So I would say none of the above very confidently.

00:04:35.903 --> 00:04:37.875
Let's do one more of these.

00:04:37.875 --> 00:04:39.550
Which of the following are accurate descriptions

00:04:39.550 --> 00:04:41.053
of the distribution below?

00:04:41.053 --> 00:04:42.224
(laughs) Alright.

00:04:42.224 --> 00:04:45.905
The distribution has a peak from 12 to 13 points.

00:04:45.905 --> 00:04:50.650
Let me see what this is measuring, what this data is about.

00:04:50.650 --> 00:04:54.950
Test scores by student in Mrs. Frine's class.

00:04:54.950 --> 00:04:58.484
So you had one student who got between a zero and a one

00:04:58.484 --> 00:05:01.049
on a 20-point scale, so got between,

00:05:01.049 --> 00:05:04.926
I guess out of 20 questions, got between zero and one point.

00:05:04.926 --> 00:05:07.278
And then you see that there's no students got

00:05:07.278 --> 00:05:09.932
between two and three, or four and five, or six and seven.

00:05:09.932 --> 00:05:12.142
Then we have another student who got between eight and nine,

00:05:12.142 --> 00:05:14.560
looks like three students got between 10 and 11,

00:05:14.560 --> 00:05:16.435
and then we keep increasing, this looks like about

00:05:16.435 --> 00:05:20.600
12 students got either a 16 or a 17,

00:05:20.600 --> 00:05:21.748
or something in between maybe,

00:05:21.748 --> 00:05:25.303
if you could get decimal points on that test.

00:05:25.303 --> 00:05:29.316
And then it looks like 10 students got from 18 to 19.

00:05:29.316 --> 00:05:31.877
Alright, so this says the distribution has a peak

00:05:31.877 --> 00:05:35.986
from 12 to 13 points, 12 to 13 points,

00:05:35.986 --> 00:05:37.952
there were five students, but this isn't a peak.

00:05:37.952 --> 00:05:40.591
If you just go to 14 to 15 points, you have more students.

00:05:40.591 --> 00:05:41.611
So this is definitely not a peak.

00:05:41.611 --> 00:05:43.833
If you were looking at this as a mountain of some kind,

00:05:43.833 --> 00:05:46.078
you definitely wouldn't describe this point as a peak.

00:05:46.078 --> 00:05:48.165
You would say this distribution has a peak,

00:05:48.165 --> 00:05:49.462
it has the most number of students

00:05:49.462 --> 00:05:51.327
who got between 16 and 17 points,

00:05:51.327 --> 00:05:54.414
so that's the peak right there, not 12 to 13 points.

00:05:54.414 --> 00:05:57.137
So I would not select that first choice.

00:05:57.137 --> 00:05:59.614
The distribution has an outlier.

00:05:59.614 --> 00:06:01.480
Well, yeah, look at this: you have this outlier.

00:06:01.480 --> 00:06:05.411
Most of the students scored between eight and 19 points,

00:06:05.411 --> 00:06:07.104
and then you have this one student

00:06:07.104 --> 00:06:09.338
who got between zero and one, it's really an outlier.

00:06:09.338 --> 00:06:11.293
You even see this when you look at it visually,

00:06:11.293 --> 00:06:13.669
it's not even connected to the rest of the distribution.

00:06:13.669 --> 00:06:15.130
It's way to the left.

00:06:15.130 --> 00:06:17.464
If something is way to the left or way to the right,

00:06:17.464 --> 00:06:21.655
that's an outlier if it's unusually low or unusually high.

00:06:21.655 --> 00:06:23.733
So I would say this distribution definitely does

00:06:23.733 --> 00:06:26.207
have an outlier, and I'm not gonna pick none of the above

00:06:26.207 --> 00:06:27.905
since I found a choice.

00:06:29.135 --> 00:06:30.761
And I think we're all done.


Impact on median and mean when removing lowest value example

WEBVTT
Kind: captions
Language: en

00:00:00.363 --> 00:00:02.348
- "Ana played five rounds of golf

00:00:02.348 --> 00:00:04.559
"and her lowest score was an 80.

00:00:04.559 --> 00:00:08.425
" The scores of the first four rounds and the lowest round

00:00:08.425 --> 00:00:10.589
"are shown in the following dot plot."

00:00:10.589 --> 00:00:12.720
And we see it right over here.

00:00:12.720 --> 00:00:14.328
The lowest round she scores an 80,

00:00:14.328 --> 00:00:16.789
she also scores a 90 once, a 92 once,

00:00:16.789 --> 00:00:19.426
a 94 once, and a 96 once.

00:00:19.426 --> 00:00:22.286
"It was discovered that Ana broke some rules when she scored

00:00:22.286 --> 00:00:25.385
"80, so that score", so I guess cheating didn't help her,

00:00:25.385 --> 00:00:27.684
"so that score will be removed from the data set."

00:00:27.684 --> 00:00:29.631
So they removed that 80 right over there.

00:00:29.631 --> 00:00:32.810
We're just left with the scores from the other four rounds.

00:00:32.810 --> 00:00:35.286
"How will the removal of the lowest round

00:00:35.286 --> 00:00:38.025
"affect the mean and the median?"

00:00:38.025 --> 00:00:40.564
So let's actually think about the median first.

00:00:40.564 --> 00:00:42.528
So the median is the middle number.

00:00:42.528 --> 00:00:44.741
So over here when you had five data points

00:00:44.741 --> 00:00:46.947
the middle data point is gonna be the one

00:00:46.947 --> 00:00:49.130
that has two to the left and two to the right.

00:00:49.130 --> 00:00:52.857
So the median up here is going to be 92.

00:00:52.857 --> 00:00:57.479
The median up there is 92.

00:00:57.479 --> 00:00:59.372
And what's the median once you remove this?

00:00:59.372 --> 00:01:01.100
Now you only have four data points.

00:01:01.100 --> 00:01:02.463
When you're trying to find the median

00:01:02.463 --> 00:01:03.926
of an even number of numbers

00:01:03.926 --> 00:01:05.817
you look at the middle two numbers.

00:01:05.817 --> 00:01:08.811
So that's a 92 and a 94.

00:01:08.811 --> 00:01:10.529
And then you take the average of them.

00:01:10.529 --> 00:01:12.934
You go halfway between them to figure out the median.

00:01:12.934 --> 00:01:15.981
So the median here is going to be,

00:01:15.981 --> 00:01:17.572
let me do that a little bit clearer.

00:01:17.572 --> 00:01:18.754
The median over here is going to be

00:01:18.754 --> 00:01:22.088
halfway between 92 and 94 which is 93.

00:01:22.088 --> 00:01:26.865
So the median, the median is 93.

00:01:27.630 --> 00:01:29.980
Median is 93.

00:01:29.980 --> 00:01:32.180
So removing the lowest data point

00:01:32.180 --> 00:01:35.272
in this case increased the median.

00:01:35.272 --> 00:01:37.910
So the median, let me write it down here.

00:01:37.910 --> 00:01:40.263
So the median increased by a little bit.

00:01:40.854 --> 00:01:42.693
The median increases.

00:01:42.693 --> 00:01:45.638
Now what's going to happen to the mean?

00:01:45.638 --> 00:01:48.677
What's going to happen to the mean?

00:01:48.677 --> 00:01:50.535
Well one way to think about it without having to do

00:01:50.535 --> 00:01:53.965
any calculations is if you remove a number

00:01:53.965 --> 00:01:57.275
that is lower than the mean, lower than the existing mean,

00:01:57.275 --> 00:01:59.098
and I haven't calculated what the existing mean is,

00:01:59.098 --> 00:02:02.268
but if you remove that the mean is going to go up.

00:02:02.268 --> 00:02:04.182
The mean is going to go up.

00:02:04.182 --> 00:02:05.691
So hopefully that gives you some intuition.

00:02:05.691 --> 00:02:07.814
If you removed a number that's larger than the mean

00:02:07.814 --> 00:02:11.959
your mean is, your mean is going to go down

00:02:11.959 --> 00:02:14.010
cause you don't have that large number anymore.

00:02:14.010 --> 00:02:16.420
If you remove a number that's lower than the mean,

00:02:16.420 --> 00:02:19.283
well you take that out, you don't have that small number

00:02:20.031 --> 00:02:22.590
bringing the average down and so the mean will go up.

00:02:22.590 --> 00:02:24.638
But let's verify it mathematically.

00:02:24.638 --> 00:02:26.986
So let's calculate the mean over here.

00:02:26.986 --> 00:02:30.550
So we're gonna add 80, plus 90,

00:02:30.550 --> 00:02:37.247
plus 92, plus 94, plus 96.

00:02:37.247 --> 00:02:38.859
Those are our data points.

00:02:38.859 --> 00:02:43.581
And that gets us: two plus four is six, plus six is 12.

00:02:43.581 --> 00:02:47.801
And then we have one plus eight is nine, and this is,

00:02:47.801 --> 00:02:49.798
so these are nine and then you have another nine,

00:02:49.798 --> 00:02:51.237
another nine, another nine, another nine.

00:02:51.237 --> 00:02:54.385
You essentially have, this is five nines right over here.

00:02:54.385 --> 00:02:59.644
So this is going to be 452.

00:02:59.644 --> 00:03:03.292
So that's the sum of the scores of these five rounds,

00:03:03.292 --> 00:03:05.730
and then you divide it by the number of rounds you have.

00:03:05.730 --> 00:03:08.588
So it would be 452 divided by five.

00:03:08.588 --> 00:03:13.839
So 452 divided by five

00:03:13.839 --> 00:03:15.875
is going to give us, five goes into,

00:03:15.875 --> 00:03:18.564
it doesn't go into four, it goes into 45 nine times.

00:03:18.564 --> 00:03:23.127
Nine times five is 45, you subtract, get zero,

00:03:23.127 --> 00:03:24.459
bring down the two.

00:03:24.459 --> 00:03:27.628
Five goes into two zero times, zero times five is,

00:03:27.628 --> 00:03:29.845
zero times five is zero, subtract.

00:03:29.845 --> 00:03:34.043
You have two left over, so you can say that the mean here,

00:03:34.043 --> 00:03:38.837
the mean here is 90 and 2/5.

00:03:39.645 --> 00:03:43.057
Not nine and 2/5, 90 and 2/5.

00:03:43.057 --> 00:03:46.230
So the mean is right around here.

00:03:46.230 --> 00:03:49.943
So that's the mean of these data points right over there.

00:03:49.943 --> 00:03:53.295
And if you remove it what is the mean going to be?

00:03:53.295 --> 00:03:57.996
So here we're just going to take our 90, plus our 92,

00:03:57.996 --> 00:04:03.075
plus our 94, plus our 96, add 'em together.

00:04:03.075 --> 00:04:06.937
So let's see, two plus four plus six is 12.

00:04:06.937 --> 00:04:10.087
And then you add these together you're gonna get 37.

00:04:10.087 --> 00:04:13.045
372 divided by four, cause I have

00:04:13.045 --> 00:04:15.007
four data points now, not five.

00:04:15.007 --> 00:04:16.853
Four goes into, let me do this

00:04:16.853 --> 00:04:18.370
in a place where you can see it.

00:04:18.370 --> 00:04:23.203
So four goes into 372,

00:04:23.203 --> 00:04:25.141
goes into 37 nine times.

00:04:25.141 --> 00:04:28.368
Nine times four is 36, subtract, you get a one.

00:04:28.368 --> 00:04:31.139
Bring down the two, it goes exactly three times.

00:04:31.139 --> 00:04:33.256
Three times four is 12.

00:04:33.256 --> 00:04:34.544
You have no remainder.

00:04:34.544 --> 00:04:36.379
So the median and the mean here

00:04:36.379 --> 00:04:38.997
are both, so this is also the mean.

00:04:38.997 --> 00:04:41.529
The mean here is also 93.

00:04:41.529 --> 00:04:43.820
So you see that the median,

00:04:43.820 --> 00:04:46.676
the median went from 92 to 93, it increased.

00:04:46.676 --> 00:04:50.321
The mean went from 90 and 2/5 to 93.

00:04:50.321 --> 00:04:53.589
So the mean increased by more than the median.

00:04:53.589 --> 00:04:56.433
They both increased but the mean increased by more.

00:04:56.433 --> 00:04:58.515
And it makes sense cause this number was way,

00:04:58.515 --> 00:05:00.828
way below all of these over here.

00:05:00.828 --> 00:05:02.349
So you could imagine if you take this out

00:05:02.349 --> 00:05:04.235
the mean should increase by a good amount.

00:05:04.235 --> 00:05:05.891
But let's see which of these choices

00:05:05.891 --> 00:05:07.955
are what we just described.

00:05:08.590 --> 00:05:11.541
"Both the mean and the median will decrease", nope.

00:05:11.541 --> 00:05:14.734
"Both the mean and the median will decrease", nope.

00:05:14.734 --> 00:05:16.932
"Both the mean and the median will increase,

00:05:16.932 --> 00:05:19.927
"but the mean will increase by more than the median."

00:05:19.927 --> 00:05:23.155
That's exactly, that's exactly, what happened.

00:05:23.155 --> 00:05:26.957
The mean went from 90 and 2/5 or 90.4,

00:05:26.957 --> 00:05:30.776
went from 90.4 or 90 and 2/5 to 93.

00:05:30.776 --> 00:05:33.411
And then the median only increased by one.

00:05:33.411 --> 00:05:35.732
So this is the right answer.