Explicit & recursive formulas for geometric sequences

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:02.000
- [Voiceover] So, this table here

00:00:02.000 --> 00:00:03.200
where you're given a bunch of Ns,

00:00:03.200 --> 00:00:04.620
N equals one, two, three, four,

00:00:04.620 --> 00:00:06.930
and we get the corresponding G of N.

00:00:06.930 --> 00:00:08.800
And one way to think about
it is that this function,

00:00:08.800 --> 00:00:13.290
G, defines a sequence where N
is the term of the sequence.

00:00:13.290 --> 00:00:15.200
So for example, we could
say this is the same thing

00:00:15.200 --> 00:00:18.910
as the sequence where
the first term is 168,

00:00:18.910 --> 00:00:23.910
second term is 84, third term is 42,

00:00:24.079 --> 00:00:29.079
and fourth term is 21,
and we keep going on,

00:00:29.330 --> 00:00:30.830
and on, and on.

00:00:30.830 --> 00:00:32.930
Now, let's think about what
type of a sequence this is.

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If we think of it as starting at 168,

00:00:36.160 --> 00:00:38.890
and how do we go from 168 to 84?

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Well, one way, you could
say we subtract at 84,

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but another way to think about it is you

00:00:42.790 --> 00:00:44.960
multiply it by one half.

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So, times one half.

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And then to go from 84 to 42,

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you multiply by one half again.

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Times one half.

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And to go from 42 to 21, you
multiply by one half again.

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So, this right over here
is a geometric series.

00:00:59.560 --> 00:01:02.160
We're starting at a term
and every successive term

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is the previous term
times, it's often called

00:01:06.350 --> 00:01:09.580
the common ratio, times one half.

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So, how can we write G
of N, how can we define

00:01:12.320 --> 00:01:14.860
this explicitly in terms of N?

00:01:14.860 --> 00:01:16.050
And I encourage you to pause the video

00:01:16.050 --> 00:01:17.420
and think about how to do that.

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So, construct a, so,
if I say G of N equals,

00:01:21.181 --> 00:01:23.190
think of a function
definition that describes

00:01:23.190 --> 00:01:25.490
what we've just seen here starting at 168,

00:01:25.490 --> 00:01:28.060
and then multiplying
by one half every time

00:01:28.060 --> 00:01:29.920
you add a new term.

00:01:29.920 --> 00:01:34.180
Well, one way to think
about it is we start at 168,

00:01:34.790 --> 00:01:38.330
and then we're gonna multiply by one half,

00:01:38.330 --> 00:01:42.230
we're gonna multiply by one
half a certain number of times.

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So, we could view the exponent
as the number of times

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we multiply by one half.

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And how many times are we
gonna multiply by one half?

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The first term, we multiply
by one half zero times.

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The second term, we multiply
by one half one time.

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Third term, we multiply
by one half two times.

00:01:58.820 --> 00:02:01.878
Fourth term, we multiply
by one half three times.

00:02:01.878 --> 00:02:04.300
So, the figure, it seems
like whatever term we're on,

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we're multiplying by one half,
that term minus one times.

00:02:08.630 --> 00:02:10.120
And you can see that this works.

00:02:10.120 --> 00:02:12.430
If N is equal to one, you're going to have

00:02:12.430 --> 00:02:14.660
one minus one, that's just gonna be zero.

00:02:14.660 --> 00:02:16.300
One half to the zero's just one.

00:02:16.300 --> 00:02:18.260
So, you're just gonna get a 168.

00:02:18.260 --> 00:02:20.320
If N is two, well, two minus one,

00:02:20.320 --> 00:02:22.940
you're gonna multiply
by one half one time,

00:02:22.940 --> 00:02:24.690
which you see right over here, N is three,

00:02:24.690 --> 00:02:27.130
you're gonna multiply by one half twice.

00:02:27.130 --> 00:02:31.750
Three minus two is, or,
three minus one is two.

00:02:31.750 --> 00:02:33.230
You're gonna multiply by one half twice,

00:02:33.230 --> 00:02:35.490
and you see that right over there.

00:02:35.490 --> 00:02:39.790
So, this feels like a really
nice explicit definition

00:02:39.790 --> 00:02:42.050
for this geometric series.

00:02:42.050 --> 00:02:43.400
And you can think of it in other ways,

00:02:43.400 --> 00:02:47.740
you could write this
as G of N is equal to,

00:02:48.120 --> 00:02:50.220
let's see, one way you could write it, as,

00:02:50.220 --> 00:02:52.920
you could write it as 168,
and I'm just algebraically

00:02:52.920 --> 00:02:56.350
manipulating it over
two to the N minus one.

00:02:56.350 --> 00:02:57.675
Another way you could think about it is,

00:02:57.675 --> 00:03:01.090
well, let's use our exponent
properties a little bit,

00:03:01.090 --> 00:03:05.030
we could say G of N is
equal to, let's see,

00:03:05.030 --> 00:03:07.290
one half to the N minus
one, that's the same thing

00:03:07.290 --> 00:03:09.494
as one half, let me write this.

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It's equal to 168.

00:03:12.520 --> 00:03:14.221
Lemme do this in a different color.

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So, this part right over
here is the same thing

00:03:16.760 --> 00:03:18.470
as one half to the N.

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So, times one half to
the N, times one half

00:03:22.760 --> 00:03:24.300
to the negative one.

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One half to the negative one.

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Well, one half to the negative one

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is just two, is just two, so,

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this is times two.

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So, we could rewrite this whole thing as

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168 times two is what?

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336?

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336, did I do that right?

00:03:43.200 --> 00:03:47.500
160 times two would be 320, plus 16,

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two times eight, so yeah, 336.

00:03:50.030 --> 00:03:52.660
And then times one half to the N.

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Times one half to the N.

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So, these are equivalent statements.

00:03:57.970 --> 00:03:59.750
This one makes a little
bit more intuitive sense,

00:03:59.750 --> 00:04:02.400
it kinda jumps out at you,
you're starting at 168

00:04:02.400 --> 00:04:04.190
and you're multiplying by one half.

00:04:04.190 --> 00:04:06.581
Whatever term you are minus one times.

00:04:06.581 --> 00:04:09.220
But this is algebraically
equivalent to this,

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to our original one.

00:04:10.930 --> 00:04:13.850
But, can we also define
G of N recursively?

00:04:13.850 --> 00:04:17.050
And I encourage you to pause
the video and try to do that.

00:04:17.050 --> 00:04:19.010
In a lot of ways, the recursive definition

00:04:19.010 --> 00:04:20.690
is a little bit more straight
forward, so let's do that.

00:04:20.690 --> 00:04:23.890
G, well, I'll make the
recursive function a different,

00:04:23.890 --> 00:04:26.230
well, I got, I'll stick
with G of N since it's

00:04:26.230 --> 00:04:28.080
on this table right over here.

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G of N is equal to, and so, let's see,

00:04:33.030 --> 00:04:36.879
if we're going to, when N equals one,

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if N is equal to one,
we're starting at 168.

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168, and if N is greater than one

00:04:47.470 --> 00:04:49.790
and a whole number, so, if N, so,

00:04:49.790 --> 00:04:52.420
we're, this is gonna be defined
over all positive integers,

00:04:52.420 --> 00:04:56.080
and whole number, what are we gonna do?

00:04:56.479 --> 00:04:58.430
Well, we're gonna take
one half and multiply it

00:04:58.430 --> 00:05:00.080
times the previous term.

00:05:00.080 --> 00:05:05.080
So, it's gonna be one half
times G of N minus one.

00:05:06.172 --> 00:05:07.460
And you can verify that this works.

00:05:07.460 --> 00:05:09.690
If N is equal to one, we
just go right over here,

00:05:09.690 --> 00:05:11.300
it's gonna be 168.

00:05:11.300 --> 00:05:14.650
G of two is gonna be
one half times G of one,

00:05:14.650 --> 00:05:17.230
which is, of course, 168.

00:05:17.230 --> 00:05:19.180
so, 168 times one half is 84.

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G of three is gonna be
one half times G of two,

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which it is, G of three is
one half times G of two.

00:05:26.930 --> 00:05:28.730
So, this is how we would define,

00:05:28.730 --> 00:05:32.290
this is the explicit
definition of this sequence,

00:05:32.290 --> 00:05:36.450
this is a recursive function
to define this sequence.


Enzyme cofactors and coenzymes

WEBVTT
Kind: captions
Language: en

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- [Voiceover] We've already
spent a couple of videos

00:00:01.884 --> 00:00:03.858
talking about enzymes,
and what I want to do

00:00:03.858 --> 00:00:06.736
in this video is dig a
little bit deeper and focus

00:00:06.736 --> 00:00:09.949
on some actors that actually help enzymes.

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And just as a reminder, enzymes

00:00:11.567 --> 00:00:14.493
are around to help reactions to proceed,

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to lower their activation
energies, to make the reactions

00:00:18.486 --> 00:00:21.574
happen more frequently
or to happen faster.

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Now, we've already seen
examples of enzymes,

00:00:24.198 --> 00:00:26.776
and just to frame things
in our brain properly,

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sometimes in a textbook you'll
see an enzyme like this,

00:00:29.283 --> 00:00:31.070
you'll see a drawing like this.

00:00:31.070 --> 00:00:34.461
And people will call this the enzyme,

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they'll call this the enzyme,

00:00:36.441 --> 00:00:39.759
and then they'll call this right,

00:00:39.759 --> 00:00:41.812
they'll say okay, and it's acting on

00:00:41.812 --> 00:00:43.879
some kind of a substrate right over here,

00:00:43.879 --> 00:00:45.435
it's going to do something to that.

00:00:45.435 --> 00:00:48.986
And this is nice for a very
abstract, textbook idea

00:00:48.986 --> 00:00:52.331
of a substrate locking
into an enzyme like this,

00:00:52.331 --> 00:00:53.863
but this isn't actually what it

00:00:53.863 --> 00:00:55.605
looks like in a biological system.

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We have to remind ourselves,
when people talk about enzymes

00:00:58.485 --> 00:01:00.180
they're talking about proteins.

00:01:00.180 --> 00:01:03.918
Now there are these kind of
RNA enzymes called ribozymes

00:01:03.918 --> 00:01:06.588
but the great majority, when
we're talking about enzymes,

00:01:06.588 --> 00:01:08.864
we tend to be talking about proteins.

00:01:08.864 --> 00:01:11.440
And we spent a lot of time
talking about how proteins

00:01:11.440 --> 00:01:13.808
are these structures,
there's polypeptides,

00:01:13.808 --> 00:01:16.503
and all the side chains
of the various amino acids

00:01:16.503 --> 00:01:19.400
fold the proteins in all
sorts of different ways.

00:01:19.400 --> 00:01:22.377
So a better drawing
for something like this

00:01:22.377 --> 00:01:26.325
would be this protein that's
all folded in different ways,

00:01:26.325 --> 00:01:28.623
maybe has some alpha helices here,

00:01:28.623 --> 00:01:30.782
maybe it has some beta
sheets right over here.

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It's all this kinda crazy
stuff right over there.

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And then the substrate might be some type

00:01:35.891 --> 00:01:40.837
of a molecule, that is it
gets embedded in the protein.

00:01:40.837 --> 00:01:43.784
And you see some examples right over here.

00:01:43.784 --> 00:01:46.781
This is actually a
hexokinase model and you see,

00:01:46.781 --> 00:01:49.613
at least you can see a
little bit of the ATP

00:01:49.613 --> 00:01:51.912
right over there, and it's
a little harder to see

00:01:51.912 --> 00:01:54.722
the glucose that's going
to be phosphorylated.

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And this reaction is being
facilitated by this big

00:01:57.276 --> 00:01:59.784
protein structure, the hexokinase.

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Now, what we're going to
focus on in this video

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is that, when we talk about
an enzyme, and we're talking

00:02:04.362 --> 00:02:07.023
about proteins, we're talking
about a chain of amino acids,

00:02:07.023 --> 00:02:09.717
but there's often other
parts of the enzyme

00:02:09.717 --> 00:02:11.668
that aren't officially proteins.

00:02:11.668 --> 00:02:14.338
And we even saw that when
we talked about hexokinases,

00:02:14.338 --> 00:02:16.962
when we talked about the
phosphorylation of glucose,

00:02:16.962 --> 00:02:19.887
we said hey, the way that it
lowers the activation energy

00:02:19.887 --> 00:02:23.629
is you have these positive magnesium ions,

00:02:23.629 --> 00:02:26.187
these positive magnesium ions,

00:02:26.187 --> 00:02:28.825
that can keep the electrons
in the phosphate groups

00:02:28.825 --> 00:02:32.192
a little bit busy, draw them away, so that

00:02:32.192 --> 00:02:35.930
this hydroxyl group
right over here can bond

00:02:35.930 --> 00:02:37.625
with this phosphate and not be

00:02:37.625 --> 00:02:38.995
interfered with these electrons.

00:02:38.995 --> 00:02:42.153
Well these magnesium ions right over here,

00:02:42.153 --> 00:02:45.288
they aren't officially
part of the protein.

00:02:45.288 --> 00:02:48.190
These are what we call cofactors.

00:02:48.190 --> 00:02:51.836
So you might have a
cofactor right over there

00:02:51.836 --> 00:02:55.969
that latches onto the
broader protein to become

00:02:55.969 --> 00:02:58.244
part of the enzyme, and
you actually need that

00:02:58.244 --> 00:03:00.775
for the reaction to proceed,
it plays a crucial role here.

00:03:00.775 --> 00:03:03.978
So another drawing in
the textbook, you'll see

00:03:03.978 --> 00:03:06.673
something like this, or even,
they'll draw, they'll say

00:03:06.673 --> 00:03:09.204
okay, in order for this
reaction to proceed, yes,

00:03:09.204 --> 00:03:11.943
you need the substrate, but
you also need the cofactor.

00:03:11.943 --> 00:03:14.080
The cofactor.

00:03:14.080 --> 00:03:17.121
And once again, it
sounds like a fancy word,

00:03:17.121 --> 00:03:20.790
but all it means is a
non-protein part of an enzyme.

00:03:20.790 --> 00:03:25.248
It's another molecule or
ion or atom that is involved

00:03:25.248 --> 00:03:27.779
in letting the enzyme perform its function

00:03:27.779 --> 00:03:31.123
that it's not formally
a part of an amino acid

00:03:31.123 --> 00:03:34.373
or part of a side chain
or part of the protein,

00:03:34.373 --> 00:03:35.604
but it's another thing that needs

00:03:35.604 --> 00:03:37.925
to be there to help catalyze the reaction.

00:03:37.925 --> 00:03:39.835
We saw that with hexokinase, you had

00:03:39.835 --> 00:03:44.656
magnesium ions that the complex picks up.

00:03:44.656 --> 00:03:47.607
And this is why, when people
talk about your vitamins

00:03:47.607 --> 00:03:49.766
and minerals, a lot of
the vitamins and minerals

00:03:49.766 --> 00:03:54.526
that you need, they actually
act as cofactors for enzymes.

00:03:54.526 --> 00:03:56.987
And so you could even see it
in this drawing over here,

00:03:56.987 --> 00:04:00.469
at least based on what I read
these are the magnesium ions

00:04:00.469 --> 00:04:04.720
in green right over here,
and these are cofactors.

00:04:04.720 --> 00:04:07.459
These are cofactors.

00:04:07.459 --> 00:04:12.057
So cofactor, non-protein
part of your actual enzyme.

00:04:12.057 --> 00:04:14.890
Now, we can subdivide cofactors even more.

00:04:14.890 --> 00:04:17.653
We can divide them into organic cofactors

00:04:17.653 --> 00:04:20.142
and inorganic cofactors.

00:04:20.142 --> 00:04:25.035
So if you have cofactors, we've
seen an inorganic cofactor,

00:04:25.035 --> 00:04:27.822
a lot of these ions,
you'll see magnesium ions,

00:04:27.822 --> 00:04:29.913
you'll see sodium ions,
you'll see calcium ions,

00:04:29.913 --> 00:04:32.164
you'll see all sorts of
things acting as cofactors,

00:04:32.164 --> 00:04:35.717
often times to distract
electrons, or to keep them busy

00:04:35.717 --> 00:04:37.668
so that electrons can proceed.

00:04:37.668 --> 00:04:39.107
But you can also have organic ones,

00:04:39.107 --> 00:04:41.986
you can also have organic molecules.

00:04:41.986 --> 00:04:44.054
Remember, organic
molecules, these are just,

00:04:44.054 --> 00:04:45.540
they'll involve carbon, they have

00:04:45.540 --> 00:04:47.328
chains of carbons and other things.

00:04:47.328 --> 00:04:50.741
And cofactors that are organic molecules,

00:04:50.741 --> 00:04:53.054
we call them coenzymes.

00:04:53.054 --> 00:04:54.771
Coenzymes.

00:04:54.771 --> 00:04:58.134
And there's a bunch of
examples of coenzymes.

00:04:58.134 --> 00:05:00.787
This right over here is

00:05:00.797 --> 00:05:03.967
the enzyme lactate dehydrogenase

00:05:04.517 --> 00:05:06.206
and it has a coenzyme, and this coenzyme

00:05:06.206 --> 00:05:09.294
you are going to see a lot
in your biological careers,

00:05:09.294 --> 00:05:12.359
NAD, right over here.

00:05:12.359 --> 00:05:15.841
Notice, this isn't just an
ion, it is an entire molecule.

00:05:15.841 --> 00:05:18.348
It has carbon in it, that's
why we call it organic.

00:05:18.348 --> 00:05:22.550
And it is not formally
protein, it's not part

00:05:22.550 --> 00:05:25.341
of the amino acids that
make up the protein,

00:05:25.341 --> 00:05:27.015
so that's what makes it a cofactor,

00:05:27.015 --> 00:05:29.661
and since it's an entire organic molecule,

00:05:29.661 --> 00:05:31.495
we call this a coenzyme.

00:05:31.495 --> 00:05:34.956
Coenzyme.

00:05:34.956 --> 00:05:37.905
But like any cofactor, it plays a role

00:05:37.905 --> 00:05:39.762
in actually allowing the enzyme

00:05:39.762 --> 00:05:41.991
to do its function, to
facilitate a reaction.

00:05:41.991 --> 00:05:45.103
And this particular coenzyme, NAD,

00:05:45.103 --> 00:05:46.518
which you're going to see a lot,

00:05:46.518 --> 00:05:49.885
it helps facilitate the
transfer of hydride ions.

00:05:49.885 --> 00:05:53.044
Hydride ions never, or very
seldom, exist by themselves,

00:05:53.044 --> 00:05:55.203
but it's a hydrogen with an extra

00:05:55.203 --> 00:05:58.918
electron, so it has a negative charge.

00:05:58.918 --> 00:06:02.603
So it allows the transfer of this group

00:06:03.443 --> 00:06:06.435
from a substrate or to a substrate,

00:06:06.435 --> 00:06:09.361
and that's because NAD can accept

00:06:09.361 --> 00:06:14.121
a hydride anion right
over here and become NADH.

00:06:14.121 --> 00:06:16.170
And if you want to see
its broader structure,

00:06:16.170 --> 00:06:17.656
it's actually quite fascinating.

00:06:17.656 --> 00:06:19.304
I'll probably do a whole video on NAD

00:06:19.304 --> 00:06:21.162
because in so many textbooks growing up

00:06:21.162 --> 00:06:24.157
I just saw NAD and NADH and
I'm like what is this thing?

00:06:24.157 --> 00:06:25.876
And it's a fascinating molecule.

00:06:25.876 --> 00:06:27.431
So what it can do is
it can actually pick up

00:06:27.431 --> 00:06:29.938
the hydride anion right over here

00:06:29.938 --> 00:06:32.006
at this carbon, you can
actually form another bond

00:06:32.006 --> 00:06:34.931
with the hydrogen, and I'll
do that in a future video,

00:06:34.931 --> 00:06:36.695
I'll show the mechanism for it.

00:06:36.695 --> 00:06:38.228
But it's a pretty cool molecule

00:06:38.228 --> 00:06:39.459
and I like to actually
look at this molecule

00:06:39.459 --> 00:06:41.363
and remember, the whole
focus of this is coenzymes,

00:06:41.363 --> 00:06:43.824
but we see these patterns
throughout biology

00:06:43.824 --> 00:06:48.467
because the name, nicotinamide
adenine dinucleotide

00:06:48.467 --> 00:06:50.976
exactly describes what it is.

00:06:50.976 --> 00:06:55.976
Nicotinamide, right down here, that is

00:06:56.027 --> 00:07:01.027
this piece of the molecule,
and this is the part

00:07:01.076 --> 00:07:03.987
that can accept a hydride
or let go of a hydride,

00:07:03.987 --> 00:07:07.028
so you could say this is the
active part of the molecule.

00:07:07.028 --> 00:07:11.487
Adenine, our good old friend,
we've seen adenine in DNA,

00:07:11.487 --> 00:07:15.898
in RNA, in ATP, so this is our

00:07:15.898 --> 00:07:18.871
good old friend adenine, right over here.

00:07:18.871 --> 00:07:21.593
And it says dinucleotide,
cause we actually have

00:07:21.593 --> 00:07:23.635
two nucleotides paired together,

00:07:23.635 --> 00:07:25.494
their phosphate groups are tied together.

00:07:25.494 --> 00:07:27.188
And there's a couple cool
ways to think about this.

00:07:27.188 --> 00:07:28.721
You have an adenine right over here,

00:07:28.721 --> 00:07:30.811
you have a ribose, you
have a phosphate group.

00:07:30.811 --> 00:07:35.527
If you just looked at this piece,

00:07:36.420 --> 00:07:39.307
right over here, if you looked
at this right over here,

00:07:39.307 --> 00:07:41.815
this is your building block,

00:07:41.815 --> 00:07:46.621
or this could be a building block, of RNA,

00:07:46.621 --> 00:07:48.643
if you have an adenine right over there.

00:07:48.643 --> 00:07:53.078
And if you include, let me undo this,

00:07:53.078 --> 00:07:56.770
if you include all of
this, this right over here,

00:07:56.770 --> 00:07:59.951
this is ADP, well the reason
why it's called dinucleotide

00:07:59.951 --> 00:08:01.762
is you can also divide it the other way.

00:08:01.762 --> 00:08:04.479
You can say, alright
you have one nucleotide

00:08:04.479 --> 00:08:08.635
that has nicotinamide right over here,

00:08:08.635 --> 00:08:11.722
so that's one of the nucleotides,

00:08:11.722 --> 00:08:14.533
and then the other nucleotide
is right over here,

00:08:14.533 --> 00:08:17.250
the one that involves adenine,

00:08:17.250 --> 00:08:19.479
that's why it's called dinucleotide.

00:08:19.479 --> 00:08:22.659
So hopefully this makes NAD
less of a mysterious molecule,

00:08:22.659 --> 00:08:25.330
we'll see it in the future,
but I like to look at it

00:08:25.330 --> 00:08:27.350
because it's got all these patterns,

00:08:27.350 --> 00:08:28.998
it's got all these components that you see

00:08:28.998 --> 00:08:30.460
over and over again,
and you see it in ATP,

00:08:30.460 --> 00:08:32.667
you see it in RNA, over and over again.

00:08:32.667 --> 00:08:36.916
But this isn't the only
cofactor or coenzyme.

00:08:37.896 --> 00:08:39.834
There are many many others,
in fact when people say

00:08:39.834 --> 00:08:41.096
take your vitamins and your minerals,

00:08:41.096 --> 00:08:43.162
that tends to be because
they are cofactors.

00:08:43.162 --> 00:08:45.530
Vitamin C is a very important cofactor

00:08:45.530 --> 00:08:47.528
to be involved in enzymes that,

00:08:47.528 --> 00:08:48.619
well I won't go into all of the

00:08:48.619 --> 00:08:49.919
different things that it can do.

00:08:49.919 --> 00:08:51.614
These are two different
views of vitamin C,

00:08:51.614 --> 00:08:53.797
a space-filling model and
this is a ball-and-stick

00:08:53.797 --> 00:08:55.491
model right over here of vitamin C.

00:08:55.491 --> 00:08:58.278
Folic acid, once again,
two different views,

00:08:58.278 --> 00:09:02.532
but these are all
coenzymes, they all work,

00:09:02.532 --> 00:09:05.455
you know if you have a
protein right over here

00:09:05.455 --> 00:09:09.006
that you know it's all this
really complex structure,

00:09:09.006 --> 00:09:12.444
maybe you have some substrates,
but to help facilitate,

00:09:12.444 --> 00:09:14.905
let me do the substrates
in a different color,

00:09:16.205 --> 00:09:18.198
so maybe you have some substrates,

00:09:18.198 --> 00:09:20.080
so these are the things that the enzyme

00:09:20.080 --> 00:09:21.682
is trying to catalyze the reactions for.

00:09:21.682 --> 00:09:23.980
But then you could have some ions,

00:09:23.980 --> 00:09:26.210
which would, you know, you
could kind of view these

00:09:26.210 --> 00:09:29.438
as you would view these,
you would view the ions

00:09:29.438 --> 00:09:34.070
as cofactors, and you could
have organic cofactors,

00:09:35.130 --> 00:09:37.503
like the vitamin C, or other
things that we talked about

00:09:37.503 --> 00:09:41.287
that are also involved and help
facilitating the mechanism,

00:09:41.287 --> 00:09:43.516
or help facilitate the reaction.

00:09:43.516 --> 00:09:45.908
And once again, sometimes
it might be to help

00:09:45.908 --> 00:09:48.834
stabilize some charge,
sometimes it might be

00:09:48.834 --> 00:09:51.318
to be an electron acceptor or donor,

00:09:51.318 --> 00:09:52.897
or a whole series of different things.

00:09:52.897 --> 00:09:56.473
They can actually act as part
of the reaction mechanism.


Video games and violence

WEBVTT
Kind: captions
Language: en

00:00:01.190 --> 00:00:03.900
Lucio wants to test whether
playing violent video

00:00:03.900 --> 00:00:06.110
games makes people more violent.

00:00:06.110 --> 00:00:09.790
He asks his friends whether
they play violent video games

00:00:09.790 --> 00:00:12.660
and whether they have been
in a fight in the last month.

00:00:12.660 --> 00:00:15.890
He recorded the results
in the table shown below.

00:00:15.890 --> 00:00:18.280
Fill in the table to show
the fraction of each group

00:00:18.280 --> 00:00:20.680
of students who have
been in a fight.

00:00:20.680 --> 00:00:22.780
Then decide whether
there is an association

00:00:22.780 --> 00:00:25.520
between violent video games
and getting in a fight

00:00:25.520 --> 00:00:26.819
amongst Lucio's friends.

00:00:26.819 --> 00:00:28.360
So let's see what
they're doing here.

00:00:28.360 --> 00:00:30.920
So students who
play video games--

00:00:30.920 --> 00:00:33.645
fractions who have been in a
fight, fraction who haven't.

00:00:33.645 --> 00:00:35.520
Students who don't play
violent video games--

00:00:35.520 --> 00:00:38.460
fraction who have been in a
fight, fraction who haven't.

00:00:38.460 --> 00:00:40.090
So let's answer the
first part of this.

00:00:40.090 --> 00:00:42.200
Students who play
violent video games.

00:00:42.200 --> 00:00:43.990
So let's look at those students.

00:00:43.990 --> 00:00:46.300
So the students who play
violent video games--

00:00:46.300 --> 00:00:50.000
it looks like Ellen plays
violent video games.

00:00:50.000 --> 00:00:54.690
Actually, let me just focus on
the data that we care about.

00:00:54.690 --> 00:00:55.560
So Ellen.

00:00:55.560 --> 00:00:59.500
So let's look at all the people
who play violent video games.

00:00:59.500 --> 00:01:00.422
So let's see.

00:01:00.422 --> 00:01:01.880
This column is
violent video games.

00:01:01.880 --> 00:01:03.290
So we have a yes here.

00:01:03.290 --> 00:01:06.550
So actually, we have both
of these right over here.

00:01:06.550 --> 00:01:11.170
And then we have down here.

00:01:11.170 --> 00:01:12.460
And then that's all of them.

00:01:12.460 --> 00:01:17.470
There's 1, 2, 3, 4, 5 people
who play violent video games.

00:01:17.470 --> 00:01:20.495
Now what fraction of them
have been in a fight?

00:01:20.495 --> 00:01:24.760
Well, it looks like 1 out of
the 5 have been in a fight.

00:01:24.760 --> 00:01:26.910
The rest of them have
not been in a fight.

00:01:26.910 --> 00:01:29.490
So we could say 1/5--
let's just write that down.

00:01:29.490 --> 00:01:31.040
So 1/5 have been in a fight.

00:01:33.720 --> 00:01:36.640
Fraction who haven't-- 4/5.

00:01:36.640 --> 00:01:37.930
So that's all these other nos.

00:01:37.930 --> 00:01:39.340
They play violent
video games, but they

00:01:39.340 --> 00:01:40.340
haven't been in a fight.

00:01:40.340 --> 00:01:42.740
1, 2, 3, 4-- 4/5.

00:01:42.740 --> 00:01:44.720
So students who don't
play violent video games.

00:01:44.720 --> 00:01:46.340
Well, that's everyone else.

00:01:46.340 --> 00:01:48.352
And let's see how many
data points that is.

00:01:48.352 --> 00:01:56.720
That is 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15.

00:01:56.720 --> 00:01:59.040
So there's a total
of 15 students.

00:01:59.040 --> 00:02:01.620
And how many of them
have been in a fight?

00:02:01.620 --> 00:02:02.260
So let's see.

00:02:02.260 --> 00:02:04.810
We have 1.

00:02:04.810 --> 00:02:10.350
And then let's see,
1, and 2, and 3.

00:02:10.350 --> 00:02:13.100
So 3 out of the 15
have been in a fight.

00:02:13.100 --> 00:02:16.360
So 3 out of 15 is the
same thing as 1 out of 5.

00:02:16.360 --> 00:02:18.720
Those are equivalent fractions.

00:02:18.720 --> 00:02:20.400
And then the
fraction who haven't?

00:02:20.400 --> 00:02:22.890
Well, that's just going
to be everyone else.

00:02:22.890 --> 00:02:28.800
That's going to be 12
out of 15 or 4 out of 5.

00:02:28.800 --> 00:02:34.670
So based on Lucio's data-- and
this wasn't a huge sample size,

00:02:34.670 --> 00:02:35.360
obviously.

00:02:35.360 --> 00:02:37.830
He only found 5 kids who were
playing violent video games.

00:02:37.830 --> 00:02:39.413
And 1 of them had
gotten into a fight.

00:02:39.413 --> 00:02:43.350
So this isn't a
super rigorous study.

00:02:43.350 --> 00:02:45.510
But at least based
on his data, if we're

00:02:45.510 --> 00:02:47.710
trying to decide whether
there's an association

00:02:47.710 --> 00:02:50.050
between violent games
and getting into a fight

00:02:50.050 --> 00:02:53.290
amongst Lucio's friends, it
doesn't seem like there is.

00:02:53.290 --> 00:02:56.310
It seems like relatively,
whether or not

00:02:56.310 --> 00:02:59.490
they play violent video
games or not, 1/5 of them

00:02:59.490 --> 00:03:01.200
have been in a fight
in the last month.

00:03:01.200 --> 00:03:03.220
So it really doesn't
seem any difference.

00:03:03.220 --> 00:03:06.990
If this number was, I
don't know, 4/5, or 5/5,

00:03:06.990 --> 00:03:11.030
or all of them, then I'd say,
hey, even with Lucio's fairly

00:03:11.030 --> 00:03:12.940
small sample, I
would say, hey, maybe

00:03:12.940 --> 00:03:15.099
there is some type of
a strong association

00:03:15.099 --> 00:03:17.140
between playing violent
video games and fighting.

00:03:17.140 --> 00:03:19.933
But here you really
don't see any difference.


Interpreting a trend line

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.290
Shira's math test
included a survey question

00:00:03.290 --> 00:00:05.150
asking how many
hours students had

00:00:05.150 --> 00:00:07.000
spent studying for the test.

00:00:07.000 --> 00:00:09.090
The graph below shows
the relationship

00:00:09.090 --> 00:00:11.520
between how many hours
students spent studying

00:00:11.520 --> 00:00:13.360
and their score on the test.

00:00:13.360 --> 00:00:16.900
Shira drew the line below to
show the trend in the data.

00:00:16.900 --> 00:00:20.940
Assuming the line is correct,
what does the line slope of 15

00:00:20.940 --> 00:00:22.060
mean?

00:00:22.060 --> 00:00:23.190
So let's see.

00:00:23.190 --> 00:00:26.400
The horizontal axis is
time studying in hours.

00:00:26.400 --> 00:00:29.080
The vertical axis is
scores on the test.

00:00:29.080 --> 00:00:32.409
And each of these blue
dots represent the time

00:00:32.409 --> 00:00:34.040
and the score for
a given student.

00:00:34.040 --> 00:00:36.670
So this student right over
here spent-- I don't know,

00:00:36.670 --> 00:00:39.270
it looks like they spend
about 0.6 hours studying.

00:00:39.270 --> 00:00:41.260
And they didn't do
too well on the exam.

00:00:41.260 --> 00:00:43.920
They look like they got
below a 45, looks like a 43

00:00:43.920 --> 00:00:46.010
or a 44 on the exam.

00:00:46.010 --> 00:00:50.180
This student over here spent
almost 4 and 1/2 hours studying

00:00:50.180 --> 00:00:55.180
and got, looks like, a 94,
close to a 95 on the exam.

00:00:55.180 --> 00:00:56.940
And what Shira
did is try to draw

00:00:56.940 --> 00:00:59.660
a line that tries
to fit this data.

00:00:59.660 --> 00:01:01.620
And it seems like it
does a pretty good job

00:01:01.620 --> 00:01:05.440
of at least showing
the trend in the data.

00:01:05.440 --> 00:01:09.170
Now, slope of 15 means
that if I'm on the line--

00:01:09.170 --> 00:01:11.040
so let's say I'm here--
and if I increase

00:01:11.040 --> 00:01:13.515
in the horizontal
direction by 1-- so there,

00:01:13.515 --> 00:01:16.180
I increase the horizontal
direction by 1--

00:01:16.180 --> 00:01:19.272
I should be increasing in
the vertical direction by 15.

00:01:19.272 --> 00:01:19.980
And you see that.

00:01:19.980 --> 00:01:25.170
If we increase by one hour here,
we increase by 15% on the test.

00:01:25.170 --> 00:01:28.470
Now, what that means is
that the trend it shows

00:01:28.470 --> 00:01:31.960
is that, in general,
along this trend,

00:01:31.960 --> 00:01:34.675
if someone studies
an extra hour,

00:01:34.675 --> 00:01:36.175
then if we're going
with that trend,

00:01:36.175 --> 00:01:38.130
then, hey, it seems
reasonable that they

00:01:38.130 --> 00:01:42.630
might expect to see a
15% gain on their test.

00:01:42.630 --> 00:01:44.850
Now, let's see which of
these are consistent.

00:01:44.850 --> 00:01:47.120
In general, students
who didn't study at all

00:01:47.120 --> 00:01:50.064
got scores of about
15 on the test.

00:01:50.064 --> 00:01:50.730
Well, let's see.

00:01:50.730 --> 00:01:54.434
This is neither true--
these are the people who

00:01:54.434 --> 00:01:56.850
didn't study at all, and they
didn't get a 15 on the test.

00:01:56.850 --> 00:01:59.800
And that's definitely
not what this 15 implies.

00:01:59.800 --> 00:02:02.680
This doesn't say what the people
who didn't study at all get.

00:02:02.680 --> 00:02:04.900
So this one is not true.

00:02:04.900 --> 00:02:06.410
That one is not true.

00:02:06.410 --> 00:02:07.600
Let's try this one.

00:02:07.600 --> 00:02:11.210
If one student studied for one
hour more than another student,

00:02:11.210 --> 00:02:14.130
the student who studied
more got exactly

00:02:14.130 --> 00:02:16.140
15 more points on the test.

00:02:16.140 --> 00:02:18.330
Well, this is getting
closer to the spirit

00:02:18.330 --> 00:02:19.780
of what the slope means.

00:02:19.780 --> 00:02:23.090
But this word "exactly" is
what, at least in my mind,

00:02:23.090 --> 00:02:24.512
messes this choice up.

00:02:24.512 --> 00:02:26.970
Because this isn't saying that
it's a guarantee that if you

00:02:26.970 --> 00:02:30.440
study an hour extra that you'll
get 15% more on the test.

00:02:30.440 --> 00:02:32.650
This is just saying that
this is the general trend

00:02:32.650 --> 00:02:33.940
that this line is seeing.

00:02:33.940 --> 00:02:35.727
So it's not guaranteed.

00:02:35.727 --> 00:02:37.560
For example, we could
find this student here

00:02:37.560 --> 00:02:40.380
who studied exactly two hours.

00:02:40.380 --> 00:02:43.660
And if we look at the students
who studied for three hours,

00:02:43.660 --> 00:02:45.900
well, there's no one
exactly at three hours.

00:02:45.900 --> 00:02:48.020
But some of them-- so
this was, let's see,

00:02:48.020 --> 00:02:50.180
the student who
was at two hours.

00:02:50.180 --> 00:02:53.390
You go to three hours,
there's no one exactly there.

00:02:53.390 --> 00:02:56.094
But there's going to be
students who got better

00:02:56.094 --> 00:02:57.760
than what would be
expected and students

00:02:57.760 --> 00:02:58.960
who might get a
little bit worse.

00:02:58.960 --> 00:03:00.793
Notice, there's points
above the trend line,

00:03:00.793 --> 00:03:02.540
and there's points
below the trend line.

00:03:02.540 --> 00:03:04.260
So this "exactly,"
you can't say it's

00:03:04.260 --> 00:03:07.570
guaranteed an hour
more turns into 15%.

00:03:07.570 --> 00:03:08.780
Let's try this choice.

00:03:08.780 --> 00:03:10.780
In general, studying
for one extra hour

00:03:10.780 --> 00:03:14.590
was associated with a 15-point
improvement in test score.

00:03:14.590 --> 00:03:16.670
That feels about right.

00:03:16.670 --> 00:03:19.167
In general, studying
for 15 extra hours

00:03:19.167 --> 00:03:21.500
was associated with a 1-point
improvement in test score.

00:03:21.500 --> 00:03:23.890
Well, no, that would get the
slope the other way around.

00:03:23.890 --> 00:03:25.570
So that's definitely
not the case.

00:03:25.570 --> 00:03:26.653
So let's check our answer.

00:03:26.653 --> 00:03:28.480
And we got it right.


Smoking in 1945

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:03.910
The graph below shows the
percentage of American adults

00:00:03.910 --> 00:00:05.580
who smoke over time.

00:00:05.580 --> 00:00:07.170
Assuming the trend
shown in the data

00:00:07.170 --> 00:00:11.440
has been consistent
since 1945, use the graph

00:00:11.440 --> 00:00:13.570
to estimate the percentage
of American adults

00:00:13.570 --> 00:00:15.980
who smoked in 1945.

00:00:15.980 --> 00:00:17.640
So let's see what's
going on here.

00:00:17.640 --> 00:00:21.946
The horizontal axis here,
they say years since 1965.

00:00:21.946 --> 00:00:23.320
So at this point
right over here,

00:00:23.320 --> 00:00:25.680
this is 0 years since 1965.

00:00:25.680 --> 00:00:28.330
So this really represents 1965.

00:00:28.330 --> 00:00:30.030
And we see it
looks like around--

00:00:30.030 --> 00:00:31.750
let's see, if I
were to eyeball it,

00:00:31.750 --> 00:00:35.567
it looks like it's around 42%
of Americans, just looking

00:00:35.567 --> 00:00:36.150
at this graph.

00:00:36.150 --> 00:00:38.170
I know that's not
an exact number.

00:00:38.170 --> 00:00:42.910
Roughly 41% or 42% of
Americans smoked in 1965

00:00:42.910 --> 00:00:45.250
based on this graph.

00:00:45.250 --> 00:00:48.910
And then five years
later, this would be 1970.

00:00:48.910 --> 00:00:53.010
10 years later,
that would be 1975.

00:00:53.010 --> 00:00:55.150
And they don't sample
the data, or we

00:00:55.150 --> 00:00:57.120
don't have data from
every given year.

00:00:57.120 --> 00:00:59.662
This is just from some of the
years that we happen to have.

00:00:59.662 --> 00:01:01.120
But what is clear,
it looks like we

00:01:01.120 --> 00:01:04.400
have a negative linear
relationship right over here,

00:01:04.400 --> 00:01:06.950
that it would not be
difficult to fit a line.

00:01:06.950 --> 00:01:08.630
So let me try to do that.

00:01:08.630 --> 00:01:10.450
So I'm just going to
eyeball it and try

00:01:10.450 --> 00:01:13.510
to fit a line to this data.

00:01:13.510 --> 00:01:16.350
So our line might look
something like that.

00:01:16.350 --> 00:01:19.065
So it looks like a pretty strong
negative linear relationship.

00:01:19.065 --> 00:01:21.560
When I say it's a negative
linear relationship,

00:01:21.560 --> 00:01:24.160
we see that as time
increases, the percentage

00:01:24.160 --> 00:01:26.870
of smokers in the
US is decreasing.

00:01:26.870 --> 00:01:30.980
So that's what makes it
a negative relationship.

00:01:30.980 --> 00:01:32.290
Now, what are they asking?

00:01:32.290 --> 00:01:34.640
They want to estimate the
percentage of American adults

00:01:34.640 --> 00:01:37.100
who smoked in 1945.

00:01:37.100 --> 00:01:40.800
Well, 1945 would be
to the left of 0.

00:01:40.800 --> 00:01:43.830
So we could even
think of it as if 1945

00:01:43.830 --> 00:01:46.620
is 20 years before 1965.

00:01:46.620 --> 00:01:48.660
So let me see if
I can draw that.

00:01:48.660 --> 00:01:53.940
So 20 years before 1965.

00:01:53.940 --> 00:01:58.540
Let's see, this would be 5
years before 1965, 10 years,

00:01:58.540 --> 00:02:03.010
15 years, 20 years before 1965.

00:02:03.010 --> 00:02:05.450
So I could even put that as
negative 20 right over here.

00:02:05.450 --> 00:02:09.370
Negative 20 years since 1965
you could view as 20 years

00:02:09.370 --> 00:02:10.979
before 1965.

00:02:10.979 --> 00:02:14.464
So that would represent
1945 right over there.

00:02:14.464 --> 00:02:16.380
And one thing that we
could do is very roughly

00:02:16.380 --> 00:02:20.220
just try to extend this negative
linear relationship backwards.

00:02:20.220 --> 00:02:22.290
And they allow us
to do that by saying

00:02:22.290 --> 00:02:24.770
assuming the trend shown on
the data has been consistent.

00:02:24.770 --> 00:02:26.610
So the trend has
been consistent.

00:02:26.610 --> 00:02:28.390
This line represents the trend.

00:02:28.390 --> 00:02:30.680
So let's just keep
going backwards,

00:02:30.680 --> 00:02:36.100
keep going backwards at the same
rate, so something like that.

00:02:36.100 --> 00:02:40.000
I want to make
sure that it looks

00:02:40.000 --> 00:02:43.349
like it's the same
rate right over here.

00:02:43.349 --> 00:02:44.890
And you could just
try to eyeball it.

00:02:44.890 --> 00:02:48.410
You could say, well, let's
see, 20 years ago, 1945.

00:02:48.410 --> 00:02:51.880
If I were to extend
that line backwards,

00:02:51.880 --> 00:02:55.570
it looks like there
were about 52%

00:02:55.570 --> 00:02:58.200
of the population was smoking.

00:02:58.200 --> 00:03:01.212
It seems like we're about
52% right over here.

00:03:01.212 --> 00:03:02.670
Another way to
think about it would

00:03:02.670 --> 00:03:06.720
be to actually try to
calculate the rate of decline.

00:03:06.720 --> 00:03:08.590
And let's say we do it
over every 20 years,

00:03:08.590 --> 00:03:10.298
because that will be
useful because we're

00:03:10.298 --> 00:03:11.740
going 20 years back.

00:03:11.740 --> 00:03:14.640
So if we go 20 years from
this point, so this is 1965,

00:03:14.640 --> 00:03:17.010
you go 20 years in the future.

00:03:17.010 --> 00:03:22.820
So that is 10 years, and
then that is 20 years.

00:03:22.820 --> 00:03:25.974
So my change in the
horizontal is 20 years.

00:03:25.974 --> 00:03:27.390
What's the change
in the vertical?

00:03:27.390 --> 00:03:33.270
Well, it looks like we have a
decrease of a little bit more

00:03:33.270 --> 00:03:33.950
than 10%.

00:03:33.950 --> 00:03:37.450
It looks like it's
11% or 12% decrease.

00:03:37.450 --> 00:03:40.652
So I'll just say
minus 11% right there.

00:03:40.652 --> 00:03:42.110
And let's see if
that's consistent.

00:03:42.110 --> 00:03:44.030
If we were to go
another 20 years.

00:03:44.030 --> 00:03:48.320
So if we go another 20 years,
it looks like once again we've

00:03:48.320 --> 00:03:50.390
gone down by about 10%.

00:03:50.390 --> 00:03:52.442
So that looks like roughly 10%.

00:03:52.442 --> 00:03:54.400
If we're following the
line, it should actually

00:03:54.400 --> 00:03:55.780
be the same number.

00:03:55.780 --> 00:03:58.330
So let me write it this way.

00:03:58.330 --> 00:04:02.670
It's approximately down 10%.

00:04:02.670 --> 00:04:04.330
So that little
squiggly line, I'm

00:04:04.330 --> 00:04:08.860
just saying approximately
negative 10% every 20 years.

00:04:13.810 --> 00:04:21.870
So if you go back 20 years, you
should increase your percentage

00:04:21.870 --> 00:04:23.600
by 20%.

00:04:23.600 --> 00:04:27.100
So this should go up by-- or
you should increase your percent

00:04:27.100 --> 00:04:28.840
by 10%, I should say.

00:04:28.840 --> 00:04:31.022
So if we started at
41% or 42%, once again,

00:04:31.022 --> 00:04:32.980
this was what we saw when
we just eyeballed it,

00:04:32.980 --> 00:04:35.530
you should get to 51% or 52%.

00:04:35.530 --> 00:04:39.040
So my estimate of the
percentage of American adults

00:04:39.040 --> 00:04:44.285
who smoked in 1945
would be 51% or 52%.


Algebraic midpoint of a segment exercise

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:05.090
We're told that K is the
midpoint of segment, JL.

00:00:05.090 --> 00:00:08.960
So that tells us
that segment JK is

00:00:08.960 --> 00:00:11.550
going to be congruent
to segment KL,

00:00:11.550 --> 00:00:14.390
that they're going to be
the exact same length.

00:00:14.390 --> 00:00:18.120
And they tell us that segment
JK is equal to 8x minus 8.

00:00:18.120 --> 00:00:23.250
So this distance right over
here is equal to 8x minus 8.

00:00:23.250 --> 00:00:26.410
And then they tell
us that segment KL

00:00:26.410 --> 00:00:31.430
is equal to 7x minus 6, that its
length is equal to 7x minus 6.

00:00:31.430 --> 00:00:35.310
So this length right
over here is 7x minus 6.

00:00:35.310 --> 00:00:37.350
Because K is the
midpoint, we know

00:00:37.350 --> 00:00:40.460
that this length must
be equal to this length.

00:00:40.460 --> 00:00:44.390
So to find JL, we just need
to find the whole length.

00:00:44.390 --> 00:00:46.027
We defined what x is.

00:00:46.027 --> 00:00:47.610
If we know what x
is, then we're going

00:00:47.610 --> 00:00:49.776
to know what this length
is and what this length is.

00:00:49.776 --> 00:00:51.410
And we could either
double one of them

00:00:51.410 --> 00:00:54.570
or add them together to find the
length of the entire segment.

00:00:54.570 --> 00:00:56.612
So first, let's figure out x.

00:00:56.612 --> 00:00:58.820
And the best way to figure
out x is based on the fact

00:00:58.820 --> 00:01:09.540
that we know that 8x minus
8 is equal to 7x minus 6.

00:01:09.540 --> 00:01:12.040
And I just want to reemphasize,
how do we know that?

00:01:12.040 --> 00:01:15.470
Well, they told us that
K is the midpoint of JL.

00:01:15.470 --> 00:01:16.970
This is the midpoint,
which tells us

00:01:16.970 --> 00:01:21.130
that this distance is equal to
this distance, or 8x minus 8

00:01:21.130 --> 00:01:24.280
is equal to 7x minus 6.

00:01:24.280 --> 00:01:26.440
Now, to figure out
x, we just have

00:01:26.440 --> 00:01:28.250
to do a little bit of algebra.

00:01:28.250 --> 00:01:30.610
So let's see what we can
do to simplify things.

00:01:30.610 --> 00:01:32.694
So if we want to get all
the x terms on one side--

00:01:32.694 --> 00:01:35.151
let's say we're going to get
it on the left side-- we could

00:01:35.151 --> 00:01:36.875
subtract 7x from both sides.

00:01:36.875 --> 00:01:38.310
So let's do that.

00:01:38.310 --> 00:01:41.700
Let's subtract 7x
from both sides.

00:01:41.700 --> 00:01:45.120
And if we want to get all of
the constant terms on one side,

00:01:45.120 --> 00:01:47.160
well, let's add 8
to both sides so

00:01:47.160 --> 00:01:49.690
that we don't have this
negative 8 right over here.

00:01:49.690 --> 00:01:53.800
So let's add 8 to
both sides, and let's

00:01:53.800 --> 00:01:55.000
see what we are left with.

00:01:55.000 --> 00:01:57.620
On the left-hand side, you
don't have an 8 anymore,

00:01:57.620 --> 00:02:01.050
and 8x minus 7x is just an x.

00:02:01.050 --> 00:02:04.280
And that's going to be equal
to-- on the right-hand side,

00:02:04.280 --> 00:02:06.760
you don't have any 7x's anymore.

00:02:06.760 --> 00:02:10.810
And negative 6 plus 8 is just 2.

00:02:10.810 --> 00:02:13.130
So we get x is equal to 2.

00:02:13.130 --> 00:02:14.390
But we're not done yet.

00:02:14.390 --> 00:02:15.680
They didn't say solve for x.

00:02:15.680 --> 00:02:18.490
They said find JL.

00:02:18.490 --> 00:02:21.270
So JL is just going to
be the sum of JK and KL,

00:02:21.270 --> 00:02:24.520
or since K is the midpoint,
it would just be double JK

00:02:24.520 --> 00:02:26.440
or double KL.

00:02:26.440 --> 00:02:28.170
So let's figure out either way.

00:02:28.170 --> 00:02:34.060
So now we can figure out that
JK, the length of segment JK,

00:02:34.060 --> 00:02:37.370
is equal to 8 times 2 minus 8.

00:02:39.960 --> 00:02:41.460
We know that x is 2 now.

00:02:41.460 --> 00:02:45.960
So this is equal to--
well, this is 16 minus 8.

00:02:45.960 --> 00:02:49.264
This right over here
is just equal to 8.

00:02:49.264 --> 00:02:50.680
And if we wanted
to figure out JL,

00:02:50.680 --> 00:02:54.570
we know that this is halfway,
so that this must be 8 as well.

00:02:54.570 --> 00:02:57.600
And the length of the entire
thing, the length of JL

00:02:57.600 --> 00:02:59.187
must be 16.

00:02:59.187 --> 00:03:01.270
And if you wanted to spend
extra time to make sure

00:03:01.270 --> 00:03:03.300
that all of the
math is consistent,

00:03:03.300 --> 00:03:05.790
you could put 2 into
this right over here.

00:03:05.790 --> 00:03:09.570
And 7 times 2 is
14 minus 6 is 8.

00:03:09.570 --> 00:03:12.130
So, once again, you can
verify from another direction

00:03:12.130 --> 00:03:15.390
that this length of
segment KL is also 8.

00:03:15.390 --> 00:03:17.727
8 plus 8 is 16.


Linear equation using segment

WEBVTT
Kind: captions
Language: en

00:00:01.260 --> 00:00:04.290
So we have this line
segment JL here,

00:00:04.290 --> 00:00:07.150
and we have point K that
also sits on that segment.

00:00:07.150 --> 00:00:10.790
And they tell us that JK
is equal to 7x plus 9.

00:00:10.790 --> 00:00:16.900
So this distance right over
here is equal to 7x plus 9.

00:00:16.900 --> 00:00:19.750
Then they tell us that
JL is equal to 114.

00:00:19.750 --> 00:00:22.090
So JL is the entire
length of the segment.

00:00:22.090 --> 00:00:26.750
So this entire thing
is equal to 114.

00:00:26.750 --> 00:00:29.770
Then they tells us that
KL is equal to 9x plus 9.

00:00:29.770 --> 00:00:34.090
So this right over here
is equal to 9x plus 9.

00:00:34.090 --> 00:00:37.030
And they say find KL.

00:00:37.030 --> 00:00:40.220
So they essentially want us to
figure out what does 9x plus 9

00:00:40.220 --> 00:00:41.060
equal?

00:00:41.060 --> 00:00:44.080
And to figure that out, we have
to figure out what x equals.

00:00:44.080 --> 00:00:47.490
And lucky for us, they've given
us all the information we need.

00:00:47.490 --> 00:00:52.825
They tell us that the
entire segment is 114 long.

00:00:52.825 --> 00:00:54.200
And they don't
give us any units.

00:00:54.200 --> 00:00:55.190
It's just 114.

00:00:55.190 --> 00:00:57.600
It has a length of 114.

00:00:57.600 --> 00:01:02.079
And we know that this segment,
JK plus KL, added together,

00:01:02.079 --> 00:01:04.370
is going to be equal to the
length of the entire thing.

00:01:04.370 --> 00:01:10.999
So we could say that
7x plus 9-- actually

00:01:10.999 --> 00:01:12.040
let me write it this way.

00:01:12.040 --> 00:01:17.410
We could write that JK, so
the length of segment JK

00:01:17.410 --> 00:01:28.870
plus the length of segment KL
is going to be equal to 114.

00:01:28.870 --> 00:01:34.500
And we know that the length
of segment JFK is 7x plus 9.

00:01:34.500 --> 00:01:38.075
We know that the length of
segment KL is 9x plus 9.

00:01:41.950 --> 00:01:45.056
And this is going
to be equal to 114.

00:01:45.056 --> 00:01:46.430
And now we just
have to break out

00:01:46.430 --> 00:01:48.527
a little bit of our
algebraic skills.

00:01:48.527 --> 00:01:50.110
So the first thing
we might want to do

00:01:50.110 --> 00:01:54.530
is, let's see, we have two
terms that have x's in them.

00:01:54.530 --> 00:02:01.260
If I have 7 x's, and I were to
add that to another 9 x's, that

00:02:01.260 --> 00:02:05.190
means I'm going to have 16 x's.

00:02:05.190 --> 00:02:08.169
And then if I have-- so
this is just plain old 9,

00:02:08.169 --> 00:02:09.970
and I add it to another 9.

00:02:09.970 --> 00:02:12.030
That's going to give me 18.

00:02:12.030 --> 00:02:17.480
And that's going
to be equal to 114.

00:02:17.480 --> 00:02:20.660
Now I just subtract
18 from both sides.

00:02:20.660 --> 00:02:22.580
Let me do that explicitly.

00:02:22.580 --> 00:02:25.420
So I subtract 18
from both sides.

00:02:25.420 --> 00:02:30.260
On the left-hand side I have 16
x, and on the right-hand side,

00:02:30.260 --> 00:02:33.481
I'm going to have 114 minus 18.

00:02:33.481 --> 00:02:35.855
Well, if I were to subtract
14, that would get me to 100.

00:02:35.855 --> 00:02:37.700
And I'm going to subtract
4 more than that,

00:02:37.700 --> 00:02:40.630
so that's going to get me to 96.

00:02:40.630 --> 00:02:42.890
And now we just divide
both sides by 16.

00:02:48.210 --> 00:02:51.396
And let's see, this looks
like 96 divided by 16--

00:02:51.396 --> 00:02:52.770
we could do it
explicitly, but it

00:02:52.770 --> 00:02:54.530
looks like it's going to be 6.

00:02:54.530 --> 00:02:56.022
6 times 10 is 60.

00:02:56.022 --> 00:02:57.970
6 times 6 is 36.

00:02:57.970 --> 00:03:00.620
60 plus 36 is 96.

00:03:00.620 --> 00:03:04.340
So we get x is equal to 6.

00:03:04.340 --> 00:03:05.900
Now we're not done yet.

00:03:05.900 --> 00:03:07.260
We're not looking just for x.

00:03:07.260 --> 00:03:09.140
We're looking for
the length of KL.

00:03:09.140 --> 00:03:11.230
KL is 9x plus 9.

00:03:11.230 --> 00:03:12.980
Let me write that down.

00:03:12.980 --> 00:03:16.500
So KL is equal to 9x plus 9.

00:03:16.500 --> 00:03:20.920
They told us that right over
there, is equal to 9x plus 9.

00:03:20.920 --> 00:03:22.930
We just figured out
that x is equal to 6.

00:03:22.930 --> 00:03:27.200
So this is equal to
9 times 6 plus 9.

00:03:27.200 --> 00:03:35.180
And this is equal to-- 9 times
6 is 54, plus 9, is equal to 63.

00:03:35.180 --> 00:03:39.330
So KL is equal to 63.


Representing and comparing rates

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:04.360
Maria and Nadia drive from
Philadelphia to Toronto

00:00:04.360 --> 00:00:05.850
to visit their friend.

00:00:05.850 --> 00:00:08.440
They take two days for
the trip, stopping along

00:00:08.440 --> 00:00:10.120
the way for sightseeing.

00:00:10.120 --> 00:00:13.070
To conserve on gas
mileage, they drive

00:00:13.070 --> 00:00:16.350
at a constant speed
for the entire trip.

00:00:16.350 --> 00:00:18.910
All right, which of the
following equations, where

00:00:18.910 --> 00:00:22.290
x represents hours,
and y represents miles,

00:00:22.290 --> 00:00:25.420
represents a speed
that is greater

00:00:25.420 --> 00:00:27.590
than Maria and Nadia's speed?

00:00:27.590 --> 00:00:29.440
Select all that apply.

00:00:29.440 --> 00:00:31.900
So let's think about
what's happening here.

00:00:31.900 --> 00:00:37.880
So on day one, they travel for
4 hours and they go 240 miles.

00:00:37.880 --> 00:00:39.910
So if we wanted to
figure out their speed,

00:00:39.910 --> 00:00:42.280
you would figure out
distance divided by time.

00:00:42.280 --> 00:00:45.060
So they travel 240
miles in 4 hours.

00:00:50.230 --> 00:00:54.780
240 miles divided by 4
hours is 60 miles per hour.

00:00:54.780 --> 00:00:56.322
So they were going
60 miles per hour.

00:00:56.322 --> 00:00:58.738
And assuming they went on a
constant speed the whole time,

00:00:58.738 --> 00:01:01.090
they should have been going
60 miles per hour on day two

00:01:01.090 --> 00:01:01.589
as well.

00:01:01.589 --> 00:01:02.790
And we can verify that.

00:01:02.790 --> 00:01:07.020
60 miles per hour times 5
hours is indeed 300 miles,

00:01:07.020 --> 00:01:09.350
or 300 divided by 5 is 60.

00:01:09.350 --> 00:01:11.440
So they went 60 miles per hour.

00:01:11.440 --> 00:01:15.280
So which of these equations
represent a relation

00:01:15.280 --> 00:01:19.000
between time elapsed
in hours and distance

00:01:19.000 --> 00:01:22.890
that represents faster
than 60 miles per hour.

00:01:22.890 --> 00:01:25.651
Well, here, you're
taking the time times 60

00:01:25.651 --> 00:01:27.650
to get distance, which
is what we did over here.

00:01:27.650 --> 00:01:31.220
Time times 60 to get distance,
which is exactly their speed.

00:01:31.220 --> 00:01:33.490
But we care about a
speed that is greater.

00:01:33.490 --> 00:01:36.440
And so pretty much all of these
other coefficients are faster.

00:01:36.440 --> 00:01:39.950
So here you're taking your
time times 65 miles per hour

00:01:39.950 --> 00:01:42.170
to get-- So you're going
to go more distance, more

00:01:42.170 --> 00:01:44.350
than 60 miles per hour,
in this situation.

00:01:44.350 --> 00:01:46.350
Here, you're going
70 miles per hour.

00:01:46.350 --> 00:01:48.847
Here you're going
80 miles per hour.

00:01:48.847 --> 00:01:49.680
And we got it right.

00:01:49.680 --> 00:01:51.640
Let's do one more.

00:01:51.640 --> 00:01:53.860
Some vinyl records,
let's call them

00:01:53.860 --> 00:01:59.250
oldies, rotate at the rate
of 78 revolutions per minute.

00:01:59.250 --> 00:02:00.840
The chart below
shows revolutions

00:02:00.840 --> 00:02:02.720
per minute for three
different tracks

00:02:02.720 --> 00:02:05.720
on another type of vinyl
record called goodies.

00:02:05.720 --> 00:02:10.419
Which has a greater rate of
revolution, oldies or goodies?

00:02:10.419 --> 00:02:12.234
So oldies are at 78
revolutions per minute.

00:02:12.234 --> 00:02:14.400
Let's think about these
revolutions per minute right

00:02:14.400 --> 00:02:15.290
over here.

00:02:15.290 --> 00:02:19.960
So in three minutes, this
one makes 135 revolutions.

00:02:19.960 --> 00:02:21.140
So how many per minute?

00:02:21.140 --> 00:02:24.650
So if we divide revolutions by
minutes, you're going to get,

00:02:24.650 --> 00:02:28.330
let's see, 3 goes into 135--
it looks like it would go into,

00:02:28.330 --> 00:02:31.240
let's see, 3 goes
into 120 40 times.

00:02:31.240 --> 00:02:33.860
So it looks like
it goes 45 times.

00:02:33.860 --> 00:02:37.570
So it's 45 revolutions
per minute for track one.

00:02:37.570 --> 00:02:39.520
Track two is also 45.

00:02:39.520 --> 00:02:43.400
4 times 45, if I have 45
revolutions per minute times 4

00:02:43.400 --> 00:02:45.440
minutes, that's 160 plus 20.

00:02:45.440 --> 00:02:46.360
Yep, that's 180.

00:02:46.360 --> 00:02:49.000
So these are all 45
revolutions per minute

00:02:49.000 --> 00:02:53.530
you can multiply 5 minutes
times 45 revolutions per minute.

00:02:53.530 --> 00:02:55.320
You're going to get 225.

00:02:55.320 --> 00:02:58.290
So the oldies go at 78 RPM.

00:02:58.290 --> 00:03:00.150
So they do 78
revolutions in a minute,

00:03:00.150 --> 00:03:02.540
while the goodies go 45.

00:03:02.540 --> 00:03:06.390
So the oldies do more
revolutions in a minute.

00:03:06.390 --> 00:03:12.380
So oldies have a greater rate
of revolutions per minute,

00:03:12.380 --> 00:03:16.540
or they just have a greater
rate of revolutions per minute.


Comparing rates

WEBVTT
Kind: captions
Language: en

00:00:00.940 --> 00:00:05.240
Which is less-- the unit rate
of the equation y equals 6.5x

00:00:05.240 --> 00:00:07.642
or the unit rate of
the graph shown below?

00:00:07.642 --> 00:00:09.350
So when they're talking
about unit rate--

00:00:09.350 --> 00:00:11.391
and they're actually a
little bit ambiguous here.

00:00:11.391 --> 00:00:13.500
They should have been
clearer in this question.

00:00:13.500 --> 00:00:16.640
I'm assuming they're
asking us about the unit

00:00:16.640 --> 00:00:19.400
rate at which y changes
with respect to x.

00:00:19.400 --> 00:00:22.680
Or how much does y
change for a change of 1

00:00:22.680 --> 00:00:24.640
in x, the unit rate.

00:00:24.640 --> 00:00:27.300
And over here, you
see when x changes 1,

00:00:27.300 --> 00:00:29.940
y is going to change by 6.5.

00:00:29.940 --> 00:00:34.620
Every time x increases by 1,
y is going to increase by 6.5.

00:00:34.620 --> 00:00:38.050
Or you could say the unit rate
of change of y with respect

00:00:38.050 --> 00:00:42.440
to x is 6.5 for
every 1 change in x.

00:00:42.440 --> 00:00:45.750
In this graph right over
here, as x changes 1,

00:00:45.750 --> 00:00:50.340
as x increases 1, y increases it
looks like by about 3 and 1/2.

00:00:50.340 --> 00:00:53.590
x increases by 1, y
increases by 3 and 1/2.

00:00:53.590 --> 00:00:55.300
So the unit rate
of change here of y

00:00:55.300 --> 00:01:04.400
with respect to x is 3 and 1/2
for every unit increase in x.

00:01:04.400 --> 00:01:08.950
So this line is increasing at a
slower rate than this equation.

00:01:08.950 --> 00:01:13.260
Or y in this line is increasing
at a slower rate with respect

00:01:13.260 --> 00:01:16.750
to x than y is increasing with
respect to x in this equation

00:01:16.750 --> 00:01:17.910
right over here.

00:01:17.910 --> 00:01:20.390
So the unit rate of
the graph is less

00:01:20.390 --> 00:01:23.580
than the unit rate
of the equation.


Approximating with powers of 10

WEBVTT
Kind: captions
Language: en

00:00:00.503 --> 00:00:03.080
- [Voiceover] In 2014,
the population of the

00:00:03.080 --> 00:00:08.080
United States was about
318,900,000 people.

00:00:09.267 --> 00:00:11.067
Choose the best approximation of the

00:00:11.067 --> 00:00:14.898
United States population in 2014.

00:00:14.898 --> 00:00:16.941
So they say three times
10 to the eighth people

00:00:16.941 --> 00:00:18.767
or three times 10 to the ninth people.

00:00:18.767 --> 00:00:19.766
This is a big difference.

00:00:19.766 --> 00:00:20.775
This is a factor of 10.

00:00:20.775 --> 00:00:24.073
This is 10 times this right over here.

00:00:24.073 --> 00:00:25.651
So which one's closer?

00:00:25.651 --> 00:00:29.135
So if I were to round this to the nearest

00:00:29.135 --> 00:00:34.135
hundred million, this
would go to 300,000,000.

00:00:38.416 --> 00:00:42.173
300,000,000, this is
three times 100,000,000.

00:00:42.173 --> 00:00:44.912
This is the same thing as three times,

00:00:44.912 --> 00:00:49.637
three times 100,000,000.

00:00:49.637 --> 00:00:52.204
And 100,000,000 we see has eight zeros.

00:00:52.204 --> 00:00:55.593
So this is the same thing as
three times 10 to the eighth.

00:00:55.593 --> 00:00:58.115
Three times 10 to the eighth power,

00:00:58.115 --> 00:01:00.087
which they have right over here.

00:01:00.087 --> 00:01:01.000
Three times 10 to the ninth,

00:01:01.000 --> 00:01:03.147
that would be 3,000,000,000.

00:01:03.147 --> 00:01:04.646
That would be almost 10 times

00:01:04.646 --> 00:01:05.945
as large as this.

00:01:05.945 --> 00:01:07.779
So we would definitely
wanna go with this one.

00:01:07.779 --> 00:01:08.987
Three times 10 to the eight people

00:01:08.987 --> 00:01:10.206
is a pretty good approximation.

00:01:10.206 --> 00:01:12.331
This is 300,000,000 people.

00:01:12.331 --> 00:01:15.454
So we're kind of rounding
to the nearest 100,000,000.

00:01:15.454 --> 00:01:18.008
In 2014, the population of the world

00:01:18.008 --> 00:01:23.008
was about 7,125,000,000 people.

00:01:23.355 --> 00:01:24.669
Choose the best approximation

00:01:24.669 --> 00:01:27.871
of the world population in 2014.

00:01:27.871 --> 00:01:28.951
Well here we're dealing with,

00:01:28.951 --> 00:01:30.275
this is billions here,

00:01:30.275 --> 00:01:32.504
so let me just round
to the nearest billion.

00:01:32.504 --> 00:01:36.025
So we could say this is
roughly 7,000,000,000.

00:01:37.751 --> 00:01:39.445
So approximately 7,000,000,000

00:01:39.445 --> 00:01:41.904
if we're rounding to the nearest billion.

00:01:42.984 --> 00:01:44.812
A billion is 10 to the ninth.

00:01:44.812 --> 00:01:47.262
We see we have nine zeros here.

00:01:47.262 --> 00:01:48.202
So this is going to be equal

00:01:48.202 --> 00:01:52.753
to seven times 10 to the ninth power.

00:01:52.753 --> 00:01:54.215
Which is this right over here.

00:01:54.215 --> 00:01:55.841
This would be 700,000,000.

00:01:55.841 --> 00:01:57.662
This is less than a 10th

00:01:57.662 --> 00:01:59.712
of the actual world population.

00:01:59.712 --> 00:02:01.291
This is 7,000,000,000,

00:02:01.291 --> 00:02:02.950
which is, if you're rounding
to the nearest billion,

00:02:02.950 --> 00:02:06.024
how many people there were in 2014.

00:02:06.376 --> 00:02:08.062
Approximately how many times larger

00:02:08.062 --> 00:02:10.012
was the world population,

00:02:10.012 --> 00:02:12.487
how many times larger
was the world population

00:02:12.487 --> 00:02:16.388
than the United States population in 2014?

00:02:16.388 --> 00:02:18.925
Well what was the world population?

00:02:18.925 --> 00:02:21.003
Once again, they're taking approximately.

00:02:21.003 --> 00:02:23.533
They're saying two, 20, 200.

00:02:23.533 --> 00:02:24.950
So if we're saying approximately,

00:02:24.950 --> 00:02:26.227
see the world population,

00:02:26.227 --> 00:02:27.912
if we're using the approximation,

00:02:27.912 --> 00:02:31.673
was seven times 10 to the ninth power,

00:02:31.673 --> 00:02:33.008
and the United States population,

00:02:33.008 --> 00:02:35.922
we said it was roughly, in 2014,

00:02:35.922 --> 00:02:37.755
three times 10 to the eighth.

00:02:37.755 --> 00:02:42.755
So we could divide the world population,

00:02:42.934 --> 00:02:45.039
or our approximation of
the world population,

00:02:45.039 --> 00:02:47.993
by our approximation of the
United States population.

00:02:47.993 --> 00:02:50.652
So our approximation
of the world population

00:02:50.652 --> 00:02:52.277
divided by the approximation

00:02:52.277 --> 00:02:53.960
of the United States population,

00:02:53.960 --> 00:02:56.758
three times 10 to the eight power,

00:02:56.758 --> 00:02:58.500
well what's that going to be?

00:02:58.500 --> 00:03:01.019
This is going to be approximately,

00:03:01.019 --> 00:03:04.653
well, this is actually
going to be exactly 7/3.

00:03:04.653 --> 00:03:08.214
So we could say seven divided by three,

00:03:08.214 --> 00:03:11.174
that's 7/3, times,

00:03:11.174 --> 00:03:12.950
what's 10 to the ninth

00:03:12.950 --> 00:03:15.678
divided by 10 to the eighth power?

00:03:15.678 --> 00:03:17.353
Well that's just going to be 10.

00:03:17.353 --> 00:03:20.580
10 to the ninth is 10
to the eighth times 10,

00:03:20.580 --> 00:03:21.973
and if we divide that by 10 to the eighth,

00:03:21.973 --> 00:03:23.761
we're just gonna be left with 10.

00:03:23.761 --> 00:03:25.292
Or, if we think about exponent properties,

00:03:25.292 --> 00:03:27.318
we have the same base, 10 to the ninth,

00:03:27.318 --> 00:03:29.037
divided by 10 to the eighth,

00:03:29.037 --> 00:03:31.033
you can subtract this, the exponent

00:03:31.033 --> 00:03:31.613
in the denominator,

00:03:31.613 --> 00:03:33.494
from the exponent in the numerator.

00:03:33.494 --> 00:03:36.419
So it's going to be 10 to
the nine minus eight power,

00:03:36.419 --> 00:03:37.476
10 to the first.

00:03:37.476 --> 00:03:39.255
This is going to be times 10.

00:03:39.255 --> 00:03:40.417
Well that's not exactly what they have

00:03:40.417 --> 00:03:42.820
over here, but we see 7/3,

00:03:42.820 --> 00:03:45.919
this is 2 1/3 times 10.

00:03:45.919 --> 00:03:50.919
So we could say this is
equal to 2 1/3 times 10,

00:03:52.956 --> 00:03:55.059
which is approximately equal to,

00:03:55.059 --> 00:03:58.635
which is approximately
equal to two times 10

00:03:58.635 --> 00:03:59.934
if we're just rounding,

00:03:59.934 --> 00:04:02.258
once again, this is all
for the sake of roughness,

00:04:02.258 --> 00:04:04.336
which is 20,

00:04:04.336 --> 00:04:05.540
which is 20,

00:04:05.540 --> 00:04:08.449
which is that right over there.


Multiplying multiples of powers of 10

WEBVTT
Kind: captions
Language: en

00:00:00.399 --> 00:00:03.365
- [Voiceover] I would like
to multiply nine times 10

00:00:03.365 --> 00:00:07.151
to the ninth power times negative two,

00:00:07.151 --> 00:00:10.122
times ten to the negative third power.

00:00:10.122 --> 00:00:12.770
And so I encourage you, pause the video,

00:00:12.770 --> 00:00:15.009
see if you can work through this.

00:00:15.804 --> 00:00:16.593
Alright.

00:00:16.593 --> 00:00:18.683
So the first thing I would want to do is

00:00:18.683 --> 00:00:20.401
let's just change the
order of multiplication.

00:00:20.401 --> 00:00:22.652
Let's multiply the nine
and the negative two first

00:00:22.652 --> 00:00:24.395
and then we can multiply
the 10 to the ninth

00:00:24.395 --> 00:00:26.554
and 10 to the negative third power.

00:00:26.554 --> 00:00:28.156
So if I were to change the order,

00:00:28.156 --> 00:00:31.012
we could write this as nine times,

00:00:31.012 --> 00:00:33.124
nine times,

00:00:33.124 --> 00:00:35.679
nine times negative two.

00:00:35.679 --> 00:00:38.325
Nine times negative two

00:00:38.325 --> 00:00:40.857
times 10 to the ninth power.

00:00:40.857 --> 00:00:42.905
Times ten, let me do it in that,

00:00:43.770 --> 00:00:44.930
in this color.

00:00:44.930 --> 00:00:46.858
Times ten to the ninth power.

00:00:46.858 --> 00:00:50.434
Times 10 to the negative third power.

00:00:50.434 --> 00:00:53.558
Times 10 to the negative third power.

00:00:54.469 --> 00:00:57.440
Now what's nine times negative two?

00:00:57.440 --> 00:00:59.043
Well, if nine times two is 18,

00:00:59.043 --> 00:01:01.528
nine times negative two
is gonna be negative 18.

00:01:01.528 --> 00:01:03.222
So that's negative 18.

00:01:03.222 --> 00:01:08.214
And what is 10 to the ninth
times 10 to the negative three?

00:01:08.214 --> 00:01:10.955
Well if I have a number
raised to some exponent

00:01:10.955 --> 00:01:12.789
times that same number, I have 10,

00:01:12.789 --> 00:01:15.156
I have the same base for both of these,

00:01:15.156 --> 00:01:17.752
time that same number
raised to another exponent,

00:01:17.752 --> 00:01:20.144
this is going to be the
same thing as that number,

00:01:20.144 --> 00:01:23.418
as that base, raised to
the sum of these exponents.

00:01:23.418 --> 00:01:26.297
This comes straight out of
our exponents properties.

00:01:26.297 --> 00:01:29.246
So this is going to be 10 to the,

00:01:29.246 --> 00:01:30.593
10 to the

00:01:30.593 --> 00:01:34.470
nine plus negative three power.

00:01:34.470 --> 00:01:36.463
Nine plus, actually, let me
use those colors so you see

00:01:36.463 --> 00:01:38.970
where that nine and that
negative three came from.

00:01:38.970 --> 00:01:39.888
Nine

00:01:40.619 --> 00:01:42.994
plus negative three power.

00:01:43.834 --> 00:01:45.879
Nine plus,

00:01:45.879 --> 00:01:47.597
this color, I'm having
trouble changing colors.

00:01:47.597 --> 00:01:49.942
Nine plus negative three power.

00:01:49.942 --> 00:01:51.289
Now what's this gonna be?

00:01:51.289 --> 00:01:54.238
Well this is going to
be equal to negative 18

00:01:54.238 --> 00:01:55.152
times

00:01:56.110 --> 00:01:58.083
times 10 to the,

00:01:58.083 --> 00:01:59.152
nine plus negative three,

00:01:59.152 --> 00:02:00.985
same thing as nine minus three,

00:02:00.985 --> 00:02:02.216
which is six.

00:02:02.216 --> 00:02:04.817
Negative 18 times 10 to the sixth power.

00:02:04.817 --> 00:02:05.741
And if you wanted to think about

00:02:05.741 --> 00:02:06.924
well what number is this?

00:02:06.924 --> 00:02:09.269
10 to the sixth, that's
one with six zeros,

00:02:09.269 --> 00:02:10.478
this is a million.

00:02:10.478 --> 00:02:12.196
Negative 18 times a million,

00:02:12.196 --> 00:02:14.402
it's gonna be negative 18 million.

00:02:14.402 --> 00:02:17.908
Or we could say negative
18 times 10 the the sixth.

00:02:17.908 --> 00:02:18.651
Either way.

00:02:18.651 --> 00:02:20.439
But this is another way we
could have written this.

00:02:20.439 --> 00:02:23.596
We could have written this as negative 18,

00:02:23.596 --> 00:02:25.211
negative 18,

00:02:26.564 --> 00:02:29.560
let me write the zeros with
the green, just for fun.

00:02:29.560 --> 00:02:31.152
Negative 18.

00:02:32.249 --> 00:02:35.314
Negative 18 million.

00:02:35.314 --> 00:02:39.191
Either way is a legitimate
way to represent this number.


Comparing values with calculator

WEBVTT
Kind: captions
Language: en

00:00:00.411 --> 00:00:02.186
- [Voiceover] My question to you is,

00:00:02.186 --> 00:00:04.567
"Which of these two values is greater?"

00:00:04.567 --> 00:00:06.557
22.9%?

00:00:06.557 --> 00:00:10.631
Or, the square root of 0.45?

00:00:10.631 --> 00:00:12.347
And I encourage you to first try to see

00:00:12.347 --> 00:00:14.553
if you can think about
this without a calculator.

00:00:14.553 --> 00:00:18.433
And then use a calculator
to see which one is larger.

00:00:19.173 --> 00:00:21.365
Let's first try to do
it without a calculator.

00:00:21.365 --> 00:00:23.453
So one thing that you
could try to say is--

00:00:23.453 --> 00:00:24.566
Well look--

00:00:24.806 --> 00:00:29.806
You could say that 22.9%

00:00:29.896 --> 00:00:30.837
that's going to be less than--

00:00:30.837 --> 00:00:34.274
I'm just gonna pick an
arbitrary number here.

00:00:34.274 --> 00:00:36.701
Let's say that's less than 30%.

00:00:36.701 --> 00:00:37.896
So that's going to be less than--

00:00:37.896 --> 00:00:39.533
and I picked 30% because it's

00:00:39.533 --> 00:00:43.677
easy to calculate 30%,
or the square of 30%.

00:00:43.677 --> 00:00:46.659
So that's less than 30%.

00:00:46.659 --> 00:00:50.664
Or, it's another way of
saying that's less than 0.3.

00:00:50.664 --> 00:00:52.162
0.3

00:00:52.162 --> 00:00:56.457
And then, we can try to
compare 0.3 to this thing here.

00:00:56.457 --> 00:00:59.580
And if 0.3 is less than this,

00:00:59.580 --> 00:01:02.203
well, then 22.9% is going
to be less than this

00:01:02.203 --> 00:01:04.535
'cause it's less than 0.3.

00:01:04.535 --> 00:01:06.693
So why don't I change the problem to that?

00:01:06.693 --> 00:01:08.467
Let's compare 0.3--

00:01:09.007 --> 00:01:10.325
0.3

00:01:10.325 --> 00:01:14.471
to square root-- to the square root of--

00:01:14.471 --> 00:01:16.512
let me do that thing pink color--

00:01:16.980 --> 00:01:20.900
to the square root of 0.45.

00:01:21.367 --> 00:01:23.840
And now I can use the squaring technique.

00:01:23.840 --> 00:01:26.158
What happens if I square
each of these quantities?

00:01:26.158 --> 00:01:30.558
If I square this, 0.3 times 0.3,

00:01:30.558 --> 00:01:32.520
let's see, three times three is nine.

00:01:32.520 --> 00:01:33.855
But you're multiplying two things

00:01:33.855 --> 00:01:35.213
that each have one digit to the right,

00:01:35.213 --> 00:01:36.838
so you're gonna have
two digits to the right.

00:01:36.838 --> 00:01:39.044
So it's going to be .09.

00:01:39.044 --> 00:01:41.935
And if you were to square this over here--

00:01:41.935 --> 00:01:43.640
I'll just keep using the white--

00:01:43.640 --> 00:01:44.917
if you were to square
this right over here,

00:01:44.917 --> 00:01:46.253
what's that going to be?

00:01:46.253 --> 00:01:49.515
Well that's just going to be 0.45.

00:01:49.515 --> 00:01:53.299
So .09 is clearly less than 0.45.

00:01:53.299 --> 00:01:57.453
Or, 0.3 squared is clearly
less than square root of--

00:01:59.084 --> 00:02:04.084
0.3 squared is less than the
square root of 0.45 squared.

00:02:04.807 --> 00:02:08.325
And so we know that 0.3
is going to be less than,

00:02:08.325 --> 00:02:13.288
is going to be less than
the square root of 0.45.

00:02:13.595 --> 00:02:16.764
And so now we can say that 22.9%,

00:02:16.764 --> 00:02:18.517
if it's less than this,
and this is less than this,

00:02:18.517 --> 00:02:22.390
well 22.9% must be less than that.

00:02:22.390 --> 00:02:23.378
Now another way you can do it,

00:02:23.378 --> 00:02:25.373
you could take out a calculator--

00:02:25.373 --> 00:02:26.674
you could take out a calculator.

00:02:26.674 --> 00:02:28.856
I'll do that. Just for kicks.

00:02:28.856 --> 00:02:30.819
So let me get the calculator out.

00:02:30.819 --> 00:02:34.813
So, 22.9%, that's the same thing as--

00:02:34.813 --> 00:02:39.560
that's the same thing as 0.229.

00:02:39.560 --> 00:02:41.452
So we really just want
to compare this quantity

00:02:41.452 --> 00:02:43.577
to the square root of 0.45.

00:02:43.577 --> 00:02:44.970
And there are two ways we can do it.

00:02:44.970 --> 00:02:46.293
We could do it the way I started.

00:02:46.293 --> 00:02:50.107
You could square this and see
if it's greater than 0.45.

00:02:50.716 --> 00:02:51.657
So let's do that.

00:02:51.657 --> 00:02:52.770
You could just square it.

00:02:52.770 --> 00:02:53.514
And you see "no."

00:02:53.514 --> 00:02:57.099
This is 0.05.

00:02:57.099 --> 00:02:59.229
Which is clearly less than 0.45.

00:02:59.229 --> 00:03:01.371
And so that would validate this.

00:03:01.371 --> 00:03:03.054
Or, you could do it the other way around.

00:03:03.054 --> 00:03:04.157
You could just use your calculator

00:03:04.157 --> 00:03:06.212
to calculate the square root of this.

00:03:06.212 --> 00:03:08.157
So you could say

00:03:08.157 --> 00:03:10.897
0.45

00:03:10.897 --> 00:03:12.574
and then take the square root.

00:03:12.574 --> 00:03:14.759
The square root of this
value right over here

00:03:14.759 --> 00:03:16.582
is approximately--

00:03:16.582 --> 00:03:19.830
is approximately 0.67.

00:03:20.962 --> 00:03:25.962
So this thing right over
here is approximately 0.67.

00:03:26.244 --> 00:03:28.566
Which is approximately 67%,

00:03:28.566 --> 00:03:31.247
which is clearly greater than 22.9%.

00:03:31.247 --> 00:03:33.349
So a bunch of ways that
you could approach it.

00:03:33.349 --> 00:03:35.845
It is nice to be able to estimate things.

00:03:35.845 --> 00:03:36.530
Think in your head.

00:03:36.530 --> 00:03:38.746
So that if you didn't have
access to a calculator

00:03:38.746 --> 00:03:40.627
you could get a general sense of,

00:03:40.627 --> 00:03:44.343
"Hey, would you rather have a 22.9%

00:03:44.343 --> 00:03:45.665
off the price of something?"

00:03:45.665 --> 00:03:48.847
Or, maybe some type of
new store could say,

00:03:48.847 --> 00:03:52.341
"Hey, sales! Square root
of 0.45 off of all goods."

00:03:52.341 --> 00:03:53.398
I don't know.
(chuckles)

00:03:53.398 --> 00:03:54.803
That can be an interesting thing.

00:03:54.803 --> 00:03:56.938
Could be very confusing for customers.

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But, anyway. Hopefully this was helpful.


Sal Khan has conversation with MIT Dean of Admissions

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:04.312
- So I'm here with Stu Schmill,
Dean of Admissions at MIT.

00:00:04.312 --> 00:00:07.121
Thanks for joining me.

00:00:07.121 --> 00:00:08.862
- Sure, happy to, Sal.

00:00:08.862 --> 00:00:10.817
- So what I wanted to ask you about is,

00:00:10.817 --> 00:00:13.424
it's just the general
notion of, well, of course,

00:00:13.424 --> 00:00:15.686
college admissions, but in particular,

00:00:15.686 --> 00:00:17.288
there tends to be some stereotypes

00:00:17.288 --> 00:00:20.594
in terms of what makes
a competitive applicant,

00:00:20.594 --> 00:00:23.401
and I'll just throw out,
at least my stereotype of,

00:00:23.401 --> 00:00:26.874
you take, i don't know,
five, six AP courses

00:00:26.874 --> 00:00:27.996
and you do really well in them,

00:00:27.996 --> 00:00:30.434
you have a very high SAT score,

00:00:30.434 --> 00:00:32.509
your GPA is near perfect,

00:00:32.509 --> 00:00:34.739
you're close to the top of your class,

00:00:34.739 --> 00:00:37.142
you're vice-president or president

00:00:37.142 --> 00:00:39.976
of three or four clubs,

00:00:41.196 --> 00:00:43.975
you play first viola in
the symphony or whatever,

00:00:43.975 --> 00:00:45.352
whatever it might be--

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- Or all of those, yes.

00:00:46.497 --> 00:00:48.026
- All of those, so how much truth

00:00:48.026 --> 00:00:51.713
is there to that and or not?

00:00:52.573 --> 00:00:55.205
- Well, I think the things
that you've just described

00:00:55.205 --> 00:00:58.352
are components

00:00:58.352 --> 00:01:01.470
that a certain student might have

00:01:01.470 --> 00:01:05.716
as part of their
application, and it is not

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the first way that we think about

00:01:09.451 --> 00:01:11.947
evaluating students.

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So, we think about
trying to enroll students

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that are going to succeed

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academically on our campuses,

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who are going to be
well-matched to our institution,

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so that the kinds of things that motivates

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those students, that get them excited,

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are the things and the
culture on our campus,

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and then we think about
what kind of contributions

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that those students might be
able to make to our campuses,

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which might mean any particular kinds of

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talent that they have, but also

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how engaged they're going to be

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in the life of the university.

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So that's really the lens
that we're looking through

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and students can manifest those things

00:01:58.972 --> 00:02:03.337
and demonstrate those
things in different ways.

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So, you talked about a number of

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AP classes and things like that

00:02:08.740 --> 00:02:11.534
that students might have, and for us,

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we're looking at the choices
that students have made

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around their academics but
also their non-academics,

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so academically, we want to see

00:02:21.534 --> 00:02:24.296
that students are challenging themselves

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in the things that really interest them,

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and students can do that in
any number of different ways.

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So AP classes may be one,
but there are other ways

00:02:36.110 --> 00:02:37.286
of doing that as well,

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and not all students have
even access to AP classes.

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And it's not as though
there's a particular tally

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or score sheet that we have

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where we're looking at a student's record

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and saying they have
to have certain markers

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because, really, there are all
different kinds of markers,

00:03:00.315 --> 00:03:02.905
all different ways that
students can demonstrate

00:03:02.905 --> 00:03:04.577
these different talents.

00:03:04.577 --> 00:03:09.209
So if certainly, academically,
as I said earlier,

00:03:09.209 --> 00:03:12.544
not all students have
access to taking AP classes,

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so there may be other ways

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that students can demonstrate to us

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that academically
they're going to be ready

00:03:17.745 --> 00:03:20.620
to succeed on our campus.

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- So there are these two realities.

00:03:22.647 --> 00:03:24.309
On one extreme, you have,

00:03:24.309 --> 00:03:26.278
and this is maybe the
more traditional reality,

00:03:26.278 --> 00:03:28.370
that students taking a lot
of very rigorous courses,

00:03:28.370 --> 00:03:31.180
doing a lot of homework, staying
until 2:00 in the morning,

00:03:31.180 --> 00:03:32.423
they're very stressed out,

00:03:32.423 --> 00:03:35.780
they, frankly, don't have
time to pursue their passions.

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On the other side, you could have students

00:03:37.452 --> 00:03:40.845
who aren't in a traditional system maybe,

00:03:40.845 --> 00:03:42.914
and maybe they don't even
have traditional grades,

00:03:42.914 --> 00:03:43.944
but they're able to show evidence

00:03:43.944 --> 00:03:47.322
of their academic knowledge
through a standardized test,

00:03:47.322 --> 00:03:49.361
through, maybe they take things like

00:03:49.361 --> 00:03:52.575
the AP tests themselves, maybe
independent study for it,

00:03:52.575 --> 00:03:54.584
maybe they take some college courses,

00:03:54.584 --> 00:03:56.601
but they have more time
to pursue their passions

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and build portfolios,

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I mean, on some levels,
basically what I'm hearing,

00:03:59.458 --> 00:04:01.010
it sounds like that second
string might even be

00:04:01.010 --> 00:04:02.949
at an advantage.

00:04:02.949 --> 00:04:07.019
- I think students that
are overloading themselves,

00:04:07.019 --> 00:04:08.605
their coursework,

00:04:08.605 --> 00:04:11.404
where they don't leave any
time for anything else,

00:04:11.404 --> 00:04:14.121
are definitely doing
themselves a disservice.

00:04:14.121 --> 00:04:16.391
I think students, again,
challenge themselves

00:04:16.391 --> 00:04:18.411
in the areas that interest them,

00:04:18.411 --> 00:04:20.650
but I think students also really need to

00:04:20.650 --> 00:04:25.236
leave themselves time and capacity

00:04:25.236 --> 00:04:28.559
to pursue more independent projects.

00:04:28.559 --> 00:04:31.086
Independent, I don't mean by themselves,

00:04:31.086 --> 00:04:32.360
independent projects could mean

00:04:32.360 --> 00:04:34.078
with other people, certainly,

00:04:34.078 --> 00:04:37.306
but these other projects
that allow students

00:04:37.306 --> 00:04:40.604
to use some creativity

00:04:40.604 --> 00:04:43.473
and independent thought

00:04:43.473 --> 00:04:45.859
in coming up with some
new and interesting things

00:04:45.859 --> 00:04:49.794
in whatever topics interest them,

00:04:49.794 --> 00:04:50.899
those are the kinds of things

00:04:50.899 --> 00:04:53.894
that we also really like to see

00:04:53.894 --> 00:04:56.482
students submit as part
of their application,

00:04:56.482 --> 00:04:57.627
we really like to see it,

00:04:57.627 --> 00:05:01.819
and those kinds of projects
can really help students,

00:05:01.819 --> 00:05:04.758
because it can show them, it can show us,

00:05:04.758 --> 00:05:07.029
kinds of things that
students might want to pursue

00:05:07.029 --> 00:05:09.211
once they come to campus,

00:05:09.211 --> 00:05:11.480
and, so, on the academic side,

00:05:11.480 --> 00:05:13.322
really, we're just looking for evidence

00:05:13.322 --> 00:05:15.418
that students are talented and capable,

00:05:15.418 --> 00:05:19.238
and that they're gonna
do well, and beyond that,

00:05:19.238 --> 00:05:22.339
we're looking at the rest of it, and those

00:05:23.259 --> 00:05:26.096
portfolios that students can submit

00:05:26.096 --> 00:05:28.456
showing the independent work they've done

00:05:28.456 --> 00:05:30.995
can really enhance their applications.

00:05:30.995 --> 00:05:33.022
- So a student, just to make it concrete,

00:05:33.022 --> 00:05:35.073
a student with strong SAT scores,

00:05:35.073 --> 00:05:36.664
ones that we show are ACT scores,

00:05:36.664 --> 00:05:39.427
that show evidence that they'll
do well at a place like MIT,

00:05:39.427 --> 00:05:41.278
strong, maybe, AP scores,

00:05:41.278 --> 00:05:44.981
and the rigorous ones that
they have an interest in,

00:05:44.981 --> 00:05:47.871
plus a portfolio of things that they've,

00:05:47.871 --> 00:05:49.101
substantive things that they've done,

00:05:49.101 --> 00:05:50.571
not just, kind of, cookbook projects,

00:05:50.571 --> 00:05:54.201
but things that will
really show their passions,

00:05:54.201 --> 00:05:57.937
they would have a strong application.

00:05:57.937 --> 00:06:01.225
- Those students would have
a very strong application.

00:06:01.225 --> 00:06:03.127
- So that brings up another
interesting dimension,

00:06:03.127 --> 00:06:05.364
I mean, at the other extreme end of this,

00:06:05.364 --> 00:06:07.695
you do have some home schooled students,

00:06:07.695 --> 00:06:09.872
you have this unschooling movement,

00:06:09.872 --> 00:06:11.084
but how do you evaluate those students,

00:06:11.084 --> 00:06:13.153
and actually even compare
them relative to the students

00:06:13.153 --> 00:06:14.531
who are taking the five AP classes

00:06:14.531 --> 00:06:16.883
in a traditional college prep school?

00:06:16.883 --> 00:06:17.551
- Right.

00:06:17.551 --> 00:06:19.204
So, I think thinking about
home schooled students

00:06:19.204 --> 00:06:22.672
is a great way

00:06:22.672 --> 00:06:24.964
to demonstrate the fact that

00:06:24.964 --> 00:06:27.575
it's not just about the AP classes

00:06:27.575 --> 00:06:30.056
because we have home schooled students

00:06:30.056 --> 00:06:32.305
who were very successful

00:06:32.305 --> 00:06:34.674
and who do successful in
our process, certainly,

00:06:34.674 --> 00:06:38.481
but also become very
successful once at MIT.

00:06:38.481 --> 00:06:41.932
We do need evidence, though,
that they're going to do well,

00:06:41.932 --> 00:06:43.448
and there are a number of ways

00:06:43.448 --> 00:06:47.248
that students can demonstrate that to us.

00:06:47.248 --> 00:06:50.471
It is one of the reasons why
we require standardized tests,

00:06:50.471 --> 00:06:53.114
I mean, that's one piece of information.

00:06:53.114 --> 00:06:56.640
It's not all of it, but it's one.

00:06:56.640 --> 00:06:57.652
Some students, right,

00:06:57.652 --> 00:07:01.489
who are not in traditional
schooling pathways

00:07:01.489 --> 00:07:05.408
will still take classes

00:07:05.408 --> 00:07:06.485
in less traditional ways,

00:07:06.485 --> 00:07:10.071
so some will take classes at
local colleges, for example,

00:07:10.071 --> 00:07:14.065
and be able to demonstrate
their ability that way.

00:07:14.065 --> 00:07:17.457
And then there are other various kinds of,

00:07:17.457 --> 00:07:19.091
of ways students can do it,

00:07:19.091 --> 00:07:22.377
for example, the American Math Competition

00:07:22.377 --> 00:07:25.821
offers these competitions
where students who have a

00:07:25.821 --> 00:07:29.439
high ability in math can
do well on these exams

00:07:29.439 --> 00:07:33.846
and progress towards and
through the Olympiad program,

00:07:33.846 --> 00:07:35.903
and it's a way for students
to demonstrate, again,

00:07:35.903 --> 00:07:38.789
their talents, at least in
math, so that's one example,

00:07:38.789 --> 00:07:40.487
and there are other ways like that.

00:07:40.487 --> 00:07:44.831
So we're open to evaluating
all kinds of evidence

00:07:45.661 --> 00:07:48.267
that students can present to us

00:07:48.267 --> 00:07:50.097
to demonstrate

00:07:50.097 --> 00:07:54.664
their ability to succeed
once they come here.

00:07:55.144 --> 00:07:58.072
- Well, thank you, I think
that this is really exciting.

00:07:58.072 --> 00:07:59.330
I think I have some young
people I need to call

00:07:59.330 --> 00:08:00.829
and tell them about it.
(Stu laughs)

00:08:00.829 --> 00:08:02.834
- That's great, send them our way.

00:08:02.834 --> 00:08:04.126
(Sal laughs softly)