Enzymes

WEBVTT
Kind: captions
Language: en

00:00:00.305 --> 00:00:03.788
- There are all sorts of
reactions in biological systems

00:00:03.788 --> 00:00:06.442
that are energetically favorable,

00:00:06.442 --> 00:00:08.563
but they're still not
going to happen quickly

00:00:08.563 --> 00:00:10.596
or even happen on their own,

00:00:10.596 --> 00:00:13.679
and the phosphorylation of
glucose is an example of that.

00:00:13.679 --> 00:00:15.671
We go into some detail into that

00:00:15.671 --> 00:00:18.210
on the video on coupled reactions,

00:00:18.210 --> 00:00:19.366
and I think we actually called that

00:00:19.366 --> 00:00:21.989
The Phosphorylation of
Glucose 6-Phosphate,

00:00:21.989 --> 00:00:23.564
but it's super important because

00:00:23.564 --> 00:00:25.568
by putting the phosphate
group on a glucose,

00:00:25.568 --> 00:00:27.037
it's ready to be the input

00:00:27.037 --> 00:00:30.679
to a whole series of
biological mechanisms,

00:00:30.679 --> 00:00:32.485
it allows the glucose to be tagged

00:00:32.485 --> 00:00:35.184
so it's going to be hard for
it to escape the cell again,

00:00:35.184 --> 00:00:38.530
and it's fairly straightforward mechanism,

00:00:38.530 --> 00:00:42.090
where you have a lone pair of electrons

00:00:42.090 --> 00:00:45.839
on this hydroxyl group right over here,

00:00:45.839 --> 00:00:48.859
and then it attempts to, if
it's in the right configuration,

00:00:48.859 --> 00:00:51.555
it could form a bond with the phosphorus

00:00:51.555 --> 00:00:52.898
in the phosphate group.

00:00:52.898 --> 00:00:54.748
Now, the reason why it
doesn't happen on its own,

00:00:54.748 --> 00:00:56.311
even though it's energetically favorable,

00:00:56.311 --> 00:00:58.234
once you form the bond, you have,

00:00:58.234 --> 00:01:01.519
electrons are gonna be able
to go to a lower energy state.

00:01:01.519 --> 00:01:04.056
So it has a negative delta G.

00:01:04.056 --> 00:01:07.993
If this is the molecules
before the reaction,

00:01:07.993 --> 00:01:10.507
this is how much free energy
they have before the reaction,

00:01:10.507 --> 00:01:12.718
after the reaction, they
have less free energy,

00:01:12.718 --> 00:01:15.499
they have been able to release energy,

00:01:15.499 --> 00:01:17.028
so this is something
that we would consider

00:01:17.028 --> 00:01:20.588
to be spontaneous, but for
the reaction to happen,

00:01:20.588 --> 00:01:23.629
you need a little bit of energy
to be put into the system.

00:01:23.629 --> 00:01:25.579
We call this our activation energy.

00:01:25.579 --> 00:01:27.553
You might say, "Well, why is that?"

00:01:27.553 --> 00:01:29.622
Well, we have electrons
that want to form a bond

00:01:29.622 --> 00:01:30.813
with this phosphorus,

00:01:30.813 --> 00:01:33.437
but this phosphorus is
surrounded by negative charges.

00:01:33.437 --> 00:01:36.432
This oxygen right over
here has a negative charge.

00:01:36.432 --> 00:01:39.903
This oxygen right over
here has a negative charge,

00:01:39.903 --> 00:01:41.760
and as you can imagine,
electrons don't like

00:01:41.760 --> 00:01:45.296
being around other electrons,
like charges repel each other,

00:01:45.296 --> 00:01:47.932
so in order for this reaction to occur,

00:01:47.932 --> 00:01:50.555
or for it to occur more frequently,

00:01:50.555 --> 00:01:52.553
it has to be catalyzed.

00:01:52.553 --> 00:01:55.841
A catalyst is anything that
makes a reaction happen faster,

00:01:55.841 --> 00:01:58.940
or even allows the
reaction to happen at all,

00:01:58.940 --> 00:02:01.936
and when we talk about
catalysts in biological systems,

00:02:01.936 --> 00:02:03.835
we're typically talking about,

00:02:03.835 --> 00:02:06.832
we're typically talking about Enzymes.

00:02:06.832 --> 00:02:08.271
Enzymes.

00:02:08.271 --> 00:02:11.882
And the way that an Enzyme
might catalyze this reaction,

00:02:11.882 --> 00:02:13.147
we actually talk about it,

00:02:13.147 --> 00:02:16.021
and when we talk about coupled reactions,

00:02:16.021 --> 00:02:18.790
it'll maybe can provide
some positive charges.

00:02:18.790 --> 00:02:21.027
It could provide some positive charges

00:02:21.027 --> 00:02:25.426
around these negative charges
to pull them further away

00:02:25.426 --> 00:02:27.975
to create space

00:02:27.975 --> 00:02:30.935
so that you can actually
have the reaction proceed,

00:02:30.935 --> 00:02:32.498
and so what an Enzyme would do,

00:02:32.498 --> 00:02:34.896
it would make this curve, instead
of having this hump on it,

00:02:34.896 --> 00:02:37.182
the curve would more like this,

00:02:37.182 --> 00:02:41.258
so that the reaction can just proceed.

00:02:41.258 --> 00:02:42.940
But what are these Enzymes?

00:02:42.940 --> 00:02:45.332
These things that can maybe,

00:02:45.332 --> 00:02:47.388
it could place some interesting charge

00:02:47.388 --> 00:02:50.235
that can allow the reaction
to happen a certain way,

00:02:50.235 --> 00:02:52.915
it might bend the
molecules in a certain way

00:02:52.915 --> 00:02:55.830
to expose some bonds,

00:02:55.830 --> 00:02:59.370
it might have a more
acidic or basic environment

00:02:59.370 --> 00:03:02.435
that might be more
favorable for the reaction.

00:03:02.435 --> 00:03:05.546
What are these seemingly magical things?

00:03:05.546 --> 00:03:08.008
Well, at a very high
level, they tend to be

00:03:08.008 --> 00:03:11.018
these protein complexes, plus
or minus a few other things,

00:03:11.018 --> 00:03:14.545
so you can view them as proteins and

00:03:14.545 --> 00:03:18.549
maybe sometimes, they'll be
multiple polypeptide chains

00:03:18.549 --> 00:03:20.316
put together, they might have some other

00:03:20.316 --> 00:03:22.917
ions associated with them,
but for the most part,

00:03:22.917 --> 00:03:25.816
they are proteins, and the molecules

00:03:25.816 --> 00:03:27.254
that are going to react,

00:03:27.254 --> 00:03:29.729
that are going to bind to the proteins,

00:03:29.729 --> 00:03:32.127
we call these the Substrates.

00:03:32.127 --> 00:03:35.134
So these, and this
reaction, (mumbles) glucose

00:03:35.134 --> 00:03:39.635
and the ATP, these are
going to be the Substrates.

00:03:39.855 --> 00:03:42.260
So you can imagine

00:03:42.260 --> 00:03:43.392
the Enzyme that does this,

00:03:43.392 --> 00:03:44.997
and the general term for the Enzyme

00:03:44.997 --> 00:03:49.462
that helps phosphorylate a
sugar molecule like this,

00:03:49.462 --> 00:03:51.294
we call it hexokinase.

00:03:51.294 --> 00:03:53.589
So it might be this crazy-looking,

00:03:53.589 --> 00:03:55.620
this crazy-looking protein, we're gonna

00:03:55.620 --> 00:03:58.598
take better looks at
this in a few moments,

00:03:58.598 --> 00:04:02.030
but the ATP might bind

00:04:02.030 --> 00:04:03.181
to it right over there.

00:04:03.181 --> 00:04:05.596
ATP is one of the Substrates,

00:04:05.596 --> 00:04:08.772
and then the glucose might bind

00:04:08.772 --> 00:04:10.351
to it right over there,

00:04:10.351 --> 00:04:12.947
and so these two Substrates bind,

00:04:12.947 --> 00:04:15.071
and the area where all
of this is going on,

00:04:15.071 --> 00:04:17.191
we call that the Active Site.

00:04:17.191 --> 00:04:20.854
So the Active Site, because
that's where all the action is,

00:04:20.854 --> 00:04:22.933
the Active Site.

00:04:22.933 --> 00:04:25.829
And often, when you have
the Substrates bind,

00:04:25.829 --> 00:04:28.314
they're able to interact with the protein

00:04:28.314 --> 00:04:30.591
to make the fit even stronger,

00:04:30.591 --> 00:04:32.192
to make it even more,

00:04:34.142 --> 00:04:37.226
more suitable for the
reaction to take place,

00:04:37.226 --> 00:04:40.233
and so the whole protein
might bend a little bit

00:04:40.233 --> 00:04:42.632
to kind of lock these two
in place a little bit more,

00:04:42.632 --> 00:04:44.892
and we call that Induced fit.

00:04:44.892 --> 00:04:48.537
Induced fit.

00:04:48.537 --> 00:04:51.165
And so, where would these
positive charges come from?

00:04:51.165 --> 00:04:53.661
Well these would be things
that are the side chains

00:04:53.661 --> 00:04:57.094
of the different amino
acids on the actual,

00:04:57.094 --> 00:05:00.101
on the polypeptide chain on the protein,

00:05:00.101 --> 00:05:02.180
and it could even be other
ions that get involved,

00:05:02.180 --> 00:05:03.859
in fact, in particular,

00:05:03.859 --> 00:05:06.078
to facilitate the
phosphorylation of glucose,

00:05:06.078 --> 00:05:08.688
a magnesium ion might be involved

00:05:08.688 --> 00:05:10.463
to help draw some positive charge away,

00:05:10.463 --> 00:05:12.940
but there's other
positively charged groups

00:05:12.940 --> 00:05:15.533
that help draw charge
away so that the reaction

00:05:15.533 --> 00:05:18.424
is more likely to occur.

00:05:18.424 --> 00:05:21.122
So that's what enzymes
are, and they tend to be

00:05:21.122 --> 00:05:23.691
optimally working in
certain pH environments

00:05:23.691 --> 00:05:25.096
or certain temperatures.

00:05:25.096 --> 00:05:28.165
In general, the higher temperatures
allow more interactions,

00:05:28.165 --> 00:05:29.307
things are bumping around more,

00:05:29.307 --> 00:05:31.559
but if temperatures get
a little bit too high,

00:05:31.559 --> 00:05:33.688
the protein or the Enzyme
might stop working,

00:05:33.688 --> 00:05:34.679
it might denature,

00:05:34.679 --> 00:05:39.118
it might lose its actual structure.

00:05:39.118 --> 00:05:41.188
And what I want now give
you an appreciation for

00:05:41.188 --> 00:05:44.485
is how beautiful and complex
these structures are.

00:05:44.485 --> 00:05:45.922
You should appreciate
what I'm showing you.

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These are in your cells!

00:05:47.134 --> 00:05:49.570
These are in your,

00:05:49.570 --> 00:05:51.413
look at your hand, look at

00:05:51.413 --> 00:05:53.925
everything around you,
there's a lot of this stuff

00:05:53.925 --> 00:05:56.386
going on inside of you,
so hopefully it gives

00:05:56.386 --> 00:05:58.341
an appreciation for the complexity

00:05:58.341 --> 00:06:00.558
of you as a biological
system, but frankly,

00:06:00.558 --> 00:06:02.024
all biological systems.

00:06:02.024 --> 00:06:04.474
So this right over here,
this is a visualization

00:06:04.474 --> 00:06:07.469
of a hexokinase, one variety of it,

00:06:07.469 --> 00:06:09.164
and just to get a sense of scale,

00:06:09.164 --> 00:06:11.485
this is a glucose molecule,

00:06:11.485 --> 00:06:14.125
and this right over here is an ATP,

00:06:14.125 --> 00:06:16.927
and so they will bind, these
are the two Substrates,

00:06:16.927 --> 00:06:19.226
they will bind at the Active Site.

00:06:19.226 --> 00:06:22.349
You might have the Induced
fit, where this fits around it.

00:06:22.349 --> 00:06:23.619
It draws some charge away,

00:06:23.619 --> 00:06:25.999
it might bend the
molecules in a certain way

00:06:25.999 --> 00:06:28.865
so that they're more likely to interact,

00:06:28.865 --> 00:06:30.526
bring these things close together,

00:06:30.526 --> 00:06:31.994
and so you're gonna
have the reaction occur

00:06:31.994 --> 00:06:33.596
and then once the reaction occurs,

00:06:33.596 --> 00:06:36.523
they're not gonna want to bind
to the Substrates anymore.

00:06:36.523 --> 00:06:38.652
I guess you could say the
products, at that point,

00:06:38.652 --> 00:06:40.134
and then they're gonna let go of them,

00:06:40.134 --> 00:06:41.625
and then the Enzyme has a change,

00:06:41.625 --> 00:06:43.161
and that's an important
property of an Enzyme.

00:06:43.161 --> 00:06:45.495
It's not like it just has
one use and it goes away,

00:06:45.495 --> 00:06:48.389
it can keep doing this over
and over and over again.

00:06:48.389 --> 00:06:52.411
One Enzyme will do this many,
many, many, many, many times

00:06:52.411 --> 00:06:54.082
in its actual life.

00:06:54.082 --> 00:06:55.599
And so now what I want
to show you is a little

00:06:55.599 --> 00:06:58.502
three-dimensional visualization
that I got from a website,

00:06:58.502 --> 00:07:00.468
so let me go get that.

00:07:00.468 --> 00:07:01.610
Go ahead and pause my recording

00:07:01.610 --> 00:07:05.485
so I could get to this little
simulation or this model,

00:07:05.485 --> 00:07:07.962
and this is actually a hexokinase as well,

00:07:07.962 --> 00:07:09.372
and hexokinase is come into,

00:07:09.372 --> 00:07:11.670
in a bunch of different varieties,

00:07:11.670 --> 00:07:13.091
but this is a pretty neat thing to look at

00:07:13.091 --> 00:07:15.157
and this has been visualized differently,

00:07:15.157 --> 00:07:17.249
and when you look up
protein images on the web,

00:07:17.249 --> 00:07:18.901
or anywhere, you'll see them sometimes

00:07:18.901 --> 00:07:20.368
with these ball and stick models,

00:07:20.368 --> 00:07:22.880
sometimes you'll see them in
these space-filling model,

00:07:22.880 --> 00:07:24.603
sometimes you'll see
them with this kind of,

00:07:24.603 --> 00:07:26.025
where you the very structures,

00:07:26.025 --> 00:07:28.892
and you notice the alpha helices here

00:07:28.892 --> 00:07:31.306
that we studied when we talked
about protein structures,

00:07:31.306 --> 00:07:33.175
and you can also see some beta sheets,

00:07:33.175 --> 00:07:34.928
but this gives you an appreciation of

00:07:34.928 --> 00:07:37.691
the binding sites and how
these things might interact.

00:07:37.691 --> 00:07:38.681
This right over here,

00:07:38.681 --> 00:07:40.996
that is a molecule of ATP,

00:07:40.996 --> 00:07:43.067
and then right next to it, I believe,

00:07:43.067 --> 00:07:45.806
if I'm looking at that right,
that is a molecule of glucose,

00:07:45.806 --> 00:07:48.352
and notice they have bound,

00:07:48.352 --> 00:07:49.632
they are the two Substrates,

00:07:49.632 --> 00:07:52.023
they have bound at the Active Site,

00:07:52.023 --> 00:07:54.370
and now, they can
interact with each other,

00:07:54.370 --> 00:07:57.441
the Enzyme, the hexokinase in this case,

00:07:57.441 --> 00:08:00.065
can help facilitate the
reaction that we care about,

00:08:00.065 --> 00:08:02.298
the phosphorylation of glucose.

00:08:02.298 --> 00:08:05.236
So hopefully, images
like this, and like this,

00:08:05.236 --> 00:08:07.330
give you an appreciation for how complex

00:08:07.330 --> 00:08:11.099
and how beautiful these
things actually are.


Multiplication, division word problems: pedaling how fast?

WEBVTT
Kind: captions
Language: en

00:00:01.100 --> 00:00:05.050
Mark pedals his bike
70 times in a minute.

00:00:05.050 --> 00:00:06.920
Which of these are
reasonable estimates

00:00:06.920 --> 00:00:08.280
about Mark's peddling?

00:00:08.280 --> 00:00:09.330
Mark all that applies.

00:00:09.330 --> 00:00:11.230
So let's just think
about this a little bit.

00:00:11.230 --> 00:00:13.610
70 times in a minute.

00:00:13.610 --> 00:00:16.440
So there are 60
seconds in a minute.

00:00:16.440 --> 00:00:18.100
So if he's doing 70
times in a minute,

00:00:18.100 --> 00:00:19.810
if he was doing 60
times in a minute,

00:00:19.810 --> 00:00:22.830
then he would be
peddling once per second.

00:00:22.830 --> 00:00:24.800
Since he's doing 70
times in a minute,

00:00:24.800 --> 00:00:28.030
he's doing a little bit
more than once per second.

00:00:28.030 --> 00:00:30.099
So maybe that'll help
us right over here.

00:00:30.099 --> 00:00:31.890
So here it says Mark
peddles his bike what?

00:00:31.890 --> 00:00:33.081
4,000 times every second.

00:00:33.081 --> 00:00:33.830
Well, that's nuts.

00:00:33.830 --> 00:00:36.260
If he peddled his bike
4,000 times every second,

00:00:36.260 --> 00:00:38.270
you'd have a way huger
number right over here.

00:00:38.270 --> 00:00:41.010
You'd have 4,000
times per second times

00:00:41.010 --> 00:00:42.410
60 seconds per minute.

00:00:42.410 --> 00:00:44.490
You would have a huge
number right over here.

00:00:44.490 --> 00:00:46.230
If it's 70 times in
a minute, we just

00:00:46.230 --> 00:00:47.730
reasoned that he
peddles a little

00:00:47.730 --> 00:00:48.980
over one time in a second.

00:00:48.980 --> 00:00:52.260
So this is definitely not right.

00:00:52.260 --> 00:00:55.200
Mark pedals his bike
about once every hour.

00:00:55.200 --> 00:00:57.230
Well, an hour is 60 minutes.

00:00:57.230 --> 00:00:58.720
So if he does 70
times in a minute,

00:00:58.720 --> 00:01:01.166
he's going to do 60
times 70 in an hour.

00:01:01.166 --> 00:01:03.540
So this is also crazy, that
he only does once in an hour.

00:01:03.540 --> 00:01:05.710
This is unbelievably slow.

00:01:05.710 --> 00:01:08.610
Mark pedals his bike
about once every second.

00:01:08.610 --> 00:01:10.920
Well, we've already
reasoned that there's

00:01:10.920 --> 00:01:15.950
60 seconds in a minute, and
he does 70 times in a minute.

00:01:15.950 --> 00:01:19.550
So he does a little over--
if you divide 70 by 60,

00:01:19.550 --> 00:01:26.530
so you could divide
70 by 60, and you're

00:01:26.530 --> 00:01:29.180
going to get a number
that's a little bit over 1

00:01:29.180 --> 00:01:30.782
time per second.

00:01:30.782 --> 00:01:31.490
So this is right.

00:01:31.490 --> 00:01:32.300
We're estimating.

00:01:32.300 --> 00:01:33.810
We don't get exact numbers.

00:01:33.810 --> 00:01:35.720
It's about once every second.

00:01:35.720 --> 00:01:37.520
So that seems legitimate.

00:01:37.520 --> 00:01:40.667
Mark pedals his bike about
4,000 times every hour.

00:01:40.667 --> 00:01:41.750
So let's think about that.

00:01:41.750 --> 00:01:44.540
If he pedals 70
times in a minute,

00:01:44.540 --> 00:01:47.130
and then there are 60
minutes in an hour,

00:01:47.130 --> 00:01:51.680
this is going to be
equal to 70 times 60.

00:01:51.680 --> 00:01:55.140
Well, we can multiply
the 7 times the 6,

00:01:55.140 --> 00:01:59.000
and we get 7 times 6 is 42.

00:01:59.000 --> 00:02:00.830
And we have two zeroes here.

00:02:00.830 --> 00:02:04.200
So we have exactly
4,200 times every hour.

00:02:04.200 --> 00:02:06.160
So as an estimate,
this is pretty close.

00:02:06.160 --> 00:02:09.650
So I would go with that as well.


Multiplication, division word problems: how many pieces of pizzas?

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:03.950
Beatrice is ordering
pizzas for a party.

00:00:03.950 --> 00:00:07.830
Each pizza is cut
into 12 pieces.

00:00:07.830 --> 00:00:09.770
Beatrice wants to
have enough pizza

00:00:09.770 --> 00:00:12.533
so that each person
can have 4 pieces.

00:00:15.600 --> 00:00:18.140
And 16 people will
be at the party.

00:00:18.140 --> 00:00:20.530
A lot of information
in this problem.

00:00:20.530 --> 00:00:28.180
If Beatrice buys 7 pizzas, how
many extra pieces will there

00:00:28.180 --> 00:00:31.440
be after each person
eats 4 pieces?

00:00:31.440 --> 00:00:34.380
So to do this, let's think
about how many total pieces

00:00:34.380 --> 00:00:35.632
she will have.

00:00:35.632 --> 00:00:37.840
And then let's think about
how many total pieces will

00:00:37.840 --> 00:00:40.130
be eaten by the 16 people.

00:00:40.130 --> 00:00:43.250
And then we'll figure how
much that she had extra.

00:00:43.250 --> 00:00:46.065
So first, let's think about
how many total pieces.

00:00:46.065 --> 00:00:48.690
And I encourage you to pause the
video and try to figure it out

00:00:48.690 --> 00:00:49.910
yourself.

00:00:49.910 --> 00:00:51.430
She ordered 7 pizzas.

00:00:51.430 --> 00:00:54.100
How many total pieces
are there going to be?

00:00:54.100 --> 00:00:56.050
Well, you have 7 pizzas.

00:00:56.050 --> 00:00:58.155
7-- that's the number of pizzas.

00:01:02.430 --> 00:01:08.820
And if you have 12
pieces per pizza,

00:01:08.820 --> 00:01:12.800
you multiply by 12 to get
the total number of pieces.

00:01:12.800 --> 00:01:20.090
So this is number
of pieces per pizza.

00:01:20.090 --> 00:01:26.040
And 7 times 12 is equal
to 84 total pieces.

00:01:30.010 --> 00:01:31.780
So that's how many
pieces she's got.

00:01:31.780 --> 00:01:35.350
Now let's think about how many
get eaten by the 16 people.

00:01:35.350 --> 00:01:37.720
So you have 16 people.

00:01:37.720 --> 00:01:39.840
16-- that's the
number of people.

00:01:43.250 --> 00:01:45.720
And this little symbol,
that's shorthand for number.

00:01:45.720 --> 00:01:49.170
16 people, and they
each eat 4 pieces.

00:01:49.170 --> 00:01:53.220
So 16 people times
4 pieces per person.

00:01:53.220 --> 00:01:56.215
So number of pieces per person.

00:02:01.460 --> 00:02:05.110
16 times 4 is 64.

00:02:05.110 --> 00:02:10.840
So that's 64--
not 84-- 64 pieces

00:02:10.840 --> 00:02:14.750
are eaten by the 16 people.

00:02:14.750 --> 00:02:16.760
So how much does
she have left over?

00:02:16.760 --> 00:02:19.740
Well, if you start with 84
pieces and 64 get eaten,

00:02:19.740 --> 00:02:21.945
84 minus 64 is 20.

00:02:24.870 --> 00:02:26.660
We deserve a drum roll now.

00:02:26.660 --> 00:02:36.674
You have 20 extra pieces after
everyone eats their share.


Multiplication, division word problems: how many field goals?

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:06.150
Blair scored 144 points
this football season.

00:00:06.150 --> 00:00:09.084
He scored all his points
with 50-yard field goal--

00:00:09.084 --> 00:00:10.500
that's actually
pretty impressive.

00:00:10.500 --> 00:00:11.000
50.

00:00:11.000 --> 00:00:13.210
That's close to record breaking.

00:00:13.210 --> 00:00:15.140
Even in the National
Football League,

00:00:15.140 --> 00:00:17.440
I think the record is like
60 yards or something.

00:00:17.440 --> 00:00:20.410
So Blair seems like a fairly
impressive individual.

00:00:20.410 --> 00:00:23.800
He scored all his points with
50-yard field goal kicks.

00:00:26.547 --> 00:00:28.130
I'm just underlining
that, because I'm

00:00:28.130 --> 00:00:30.400
impressed by how far
those field goals are,

00:00:30.400 --> 00:00:32.850
which are each worth 3 points.

00:00:32.850 --> 00:00:35.486
So a field goal in
American football,

00:00:35.486 --> 00:00:36.860
I should clarify
for those of you

00:00:36.860 --> 00:00:39.276
from other countries who think
we're talking about soccer,

00:00:39.276 --> 00:00:41.720
a field goal is
worth 3 points each.

00:00:41.720 --> 00:00:48.240
He played in 16
games this season.

00:00:48.240 --> 00:00:52.080
How many field goals did
Blair make in each game?

00:00:52.080 --> 00:00:54.940
Assume he made the same
number in each game.

00:00:54.940 --> 00:00:59.070
So this guy is an incredibly
consistent field goal kicker.

00:00:59.070 --> 00:01:00.430
He makes 50-yard field goals.

00:01:00.430 --> 00:01:02.490
He makes the same in every game.

00:01:02.490 --> 00:01:04.480
So let's think about
how many field goals did

00:01:04.480 --> 00:01:06.087
he make in each game.

00:01:06.087 --> 00:01:07.920
So there's a couple of
ways you can do this.

00:01:07.920 --> 00:01:09.730
We could first figure
out, well, how many

00:01:09.730 --> 00:01:11.800
field goals did he
make in the season?

00:01:11.800 --> 00:01:17.300
So he scored 144 points total.

00:01:17.300 --> 00:01:21.620
And if we divide that by 3,
we divide that by the fact

00:01:21.620 --> 00:01:25.110
that a field goal gets you
3 points per field goal.

00:01:25.110 --> 00:01:27.300
So the total number
of points divided

00:01:27.300 --> 00:01:29.170
by the points per
field goal, this

00:01:29.170 --> 00:01:32.030
should give us the total
number of field goals he had.

00:01:32.030 --> 00:01:36.590
So this right over here is
going to be equal to-- so 3

00:01:36.590 --> 00:01:38.500
goes into 144.

00:01:38.500 --> 00:01:40.350
Let me do this on the
side right over here.

00:01:40.350 --> 00:01:41.620
3 goes into 144.

00:01:41.620 --> 00:01:43.920
3 doesn't go into 1.

00:01:43.920 --> 00:01:46.120
It goes into 14 four times.

00:01:46.120 --> 00:01:48.080
4 times 3 is 12.

00:01:48.080 --> 00:01:52.720
Subtract 14 minus 12
is 2, bring down a 4.

00:01:52.720 --> 00:01:54.125
3 goes into 24 perfectly.

00:01:54.125 --> 00:01:56.030
It goes exactly 8 times.

00:01:56.030 --> 00:01:57.650
8 times 3 is 24.

00:01:57.650 --> 00:01:59.760
We have no remainder.

00:01:59.760 --> 00:02:04.980
So 144 divided by
3 is equal to 48.

00:02:04.980 --> 00:02:09.564
So all season, he has
kicked 48 field goals.

00:02:13.477 --> 00:02:15.560
Now, we still haven't
answered the other question.

00:02:15.560 --> 00:02:17.032
How many did he
make in each game?

00:02:17.032 --> 00:02:19.240
And we're assuming it's the
same number in each game.

00:02:19.240 --> 00:02:22.330
So this is the total
number across 16 games.

00:02:22.330 --> 00:02:25.350
To find out the average
he makes in each game,

00:02:25.350 --> 00:02:27.640
we divide this by 16.

00:02:27.640 --> 00:02:35.840
So let's take 48 total field
goals divided by 16 games

00:02:35.840 --> 00:02:40.290
is going to be-- well,
48 divided by 16 is 3.

00:02:40.290 --> 00:02:51.816
This is equal to 3
field goals per game.

00:02:51.816 --> 00:02:52.440
And we're done.

00:02:52.440 --> 00:02:55.280
How many field goals did
he make in each game?

00:02:55.280 --> 00:02:58.060
He scored 3.


Dividing numbers: intro to remainders

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:05.420
Let's take the number
7 and divide it by 3.

00:00:05.420 --> 00:00:08.029
And I'm going to
conceptualize dividing by 3

00:00:08.029 --> 00:00:13.160
as let me see how many groups
of 3 I can make out of the 7.

00:00:13.160 --> 00:00:19.770
So let me draw 7 things--
1, 2, 3, 4, 5, 6, 7.

00:00:19.770 --> 00:00:21.690
So let me try to
create groups of 3.

00:00:21.690 --> 00:00:25.370
So I can definitely create one
group of 3 right over here.

00:00:25.370 --> 00:00:27.170
I can definitely create
another group of 3.

00:00:27.170 --> 00:00:30.960
So I'm able to create
two groups of 3.

00:00:30.960 --> 00:00:33.240
And then I can't create
any more full groups of 3.

00:00:33.240 --> 00:00:35.982
I have essentially this thing
right over here left over.

00:00:35.982 --> 00:00:38.190
So this right over here, I
have this thing remaining.

00:00:38.190 --> 00:00:41.440
This right over
here is my remainder

00:00:41.440 --> 00:00:44.740
after creating as many
groups of 3 as I can.

00:00:44.740 --> 00:00:47.110
And so when you see
something like this,

00:00:47.110 --> 00:00:49.330
people will often
say 7 divided by 3.

00:00:49.330 --> 00:00:52.520
Well, I can create
two groups of 3.

00:00:52.520 --> 00:00:57.060
But it doesn't divide evenly, or
3 doesn't divide evenly into 7.

00:00:57.060 --> 00:00:59.160
I end up with
something left over.

00:00:59.160 --> 00:01:00.390
I have a leftover.

00:01:00.390 --> 00:01:03.450
I have a remainder of 1.

00:01:03.450 --> 00:01:08.360
So this is literally saying 7
divided by 3 is 2 remainder 1.

00:01:08.360 --> 00:01:09.420
And that makes sense.

00:01:09.420 --> 00:01:10.990
2 times 3 is 6.

00:01:10.990 --> 00:01:12.840
So it doesn't get
you all the way to 7.

00:01:12.840 --> 00:01:14.720
But then if you have
your extra remainder,

00:01:14.720 --> 00:01:20.000
6 plus that 1 remainder
gets you all the way to 7.

00:01:20.000 --> 00:01:21.430
Let's do another one.

00:01:21.430 --> 00:01:28.850
Let's imagine 15 divided by 4.

00:01:28.850 --> 00:01:35.115
Let me draw 15 objects-- 1, 2,
3, 4, 5, 6, 7, 8, 9, 10, 11,

00:01:35.115 --> 00:01:37.570
12, 13, 14, 15.

00:01:37.570 --> 00:01:41.200
Now, let me try to divide
it into groups of 4.

00:01:41.200 --> 00:01:44.100
So let's see, that's
one group of 4.

00:01:44.100 --> 00:01:46.930
That's another group of 4.

00:01:46.930 --> 00:01:49.330
And then that's
another group of 4.

00:01:49.330 --> 00:01:52.050
So I'm able to create
three groups of 4.

00:01:52.050 --> 00:01:54.930
But then I can't create
a fourth full group of 4.

00:01:54.930 --> 00:01:59.520
I am then left with this
remainder right over here.

00:01:59.520 --> 00:02:02.490
I have a remainder
right over here of 3.

00:02:02.490 --> 00:02:05.010
I have 3 left over.

00:02:05.010 --> 00:02:10.560
So we could say that 15
divided by 4 is 3 remainder 3.

00:02:10.560 --> 00:02:12.950
4 goes into 15 three times.

00:02:12.950 --> 00:02:14.180
But that only gets us to 12.

00:02:14.180 --> 00:02:15.149
4 times 3 is 12.

00:02:15.149 --> 00:02:17.440
To get all the way to 15, we
need to use our remainder.

00:02:17.440 --> 00:02:19.360
We have to get 3 more.

00:02:19.360 --> 00:02:22.641
So 15 divided by 4,
I have 3 left over.


Multiplying: 4 digits times 1 digit (with carrying)

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:12.540
Let's multiply 9 times 8,085.

00:00:12.540 --> 00:00:15.640
That should be a pretty fun
little calculation to do.

00:00:15.640 --> 00:00:17.710
So like always, let's
just rewrite this.

00:00:17.710 --> 00:00:19.726
So I'm going to write the 8,085.

00:00:19.726 --> 00:00:22.100
I'm going to write
the 9 right below it

00:00:22.100 --> 00:00:24.650
and write our little
multiplication symbol.

00:00:24.650 --> 00:00:26.420
And now, we're ready to compute.

00:00:26.420 --> 00:00:29.400
So first we can
tackle 9 times 5.

00:00:29.400 --> 00:00:32.880
Well, we know that
9 times 5 is 45.

00:00:32.880 --> 00:00:34.780
We can write the 5
in the ones place

00:00:34.780 --> 00:00:37.400
and carry the 5
to the tens place.

00:00:37.400 --> 00:00:41.030
So 9 times 5 is 45.

00:00:41.030 --> 00:00:45.637
Now we're ready to
move on to 9 times 8.

00:00:45.637 --> 00:00:47.220
And we're going to
calculate 9 times 8

00:00:47.220 --> 00:00:49.830
and then add the 4
that we just carried.

00:00:49.830 --> 00:00:56.720
So 9 times 8 is 72,
plus the 4 is 76.

00:00:56.720 --> 00:01:01.630
So we'll write the 6 right here
the tens place and carry the 7.

00:01:01.630 --> 00:01:05.340
Now we are ready to
calculate, and I'm

00:01:05.340 --> 00:01:06.920
looking for a suitable color.

00:01:06.920 --> 00:01:13.890
9 times 0 100's plus-- and this
is a 7 in the hundreds place,

00:01:13.890 --> 00:01:15.040
so that's actually 700.

00:01:15.040 --> 00:01:17.890
Or if we're just kind of
going with the computation, 9

00:01:17.890 --> 00:01:19.060
times 0 plus 7.

00:01:19.060 --> 00:01:24.830
Well, 9 times 0
is 0, plus 7 is 7.

00:01:24.830 --> 00:01:29.130
And then, finally, we
have-- and once again, I'm

00:01:29.130 --> 00:01:34.140
looking for a suitable
color-- 9 times 8.

00:01:34.140 --> 00:01:35.990
This is the last thing
we have to compute.

00:01:35.990 --> 00:01:39.580
We already know that
9 times 8 is 72.

00:01:39.580 --> 00:01:43.680
And we just write the 72 right
down here, and we're done.

00:01:43.680 --> 00:01:50.360
8,085 times 9 is 72,765.

00:01:50.360 --> 00:01:52.300
Let's do one more
example just to make sure

00:01:52.300 --> 00:01:54.456
that this is really clear
in your brain, at least

00:01:54.456 --> 00:01:55.580
the process for doing this.

00:01:55.580 --> 00:01:57.860
And I also want you to
think about why this works.

00:01:57.860 --> 00:02:08.998
So let's try 7 times 5,396.

00:02:08.998 --> 00:02:14.000
And I encourage you to pause it
and try it on your own as well.

00:02:14.000 --> 00:02:19.490
I'm going to rewrite
it-- 5,396 times 7.

00:02:19.490 --> 00:02:24.470
First, we'll think
about what 7 times 6 is.

00:02:24.470 --> 00:02:25.910
We know that's 42.

00:02:25.910 --> 00:02:27.790
We'll put the 2
in the ones place.

00:02:27.790 --> 00:02:29.940
4 we will carry.

00:02:29.940 --> 00:02:34.330
Then we need to concern
ourselves with 7 times 9.

00:02:34.330 --> 00:02:37.360
But then, we have to calculate
that and then add the 4.

00:02:37.360 --> 00:02:42.990
7 times 9 is 63, plus 4 is 67.

00:02:42.990 --> 00:02:47.800
So we put the 7 down
here and carry the 6.

00:02:47.800 --> 00:02:54.370
Then we have to worry
about 7 times 3 plus this 6

00:02:54.370 --> 00:02:55.970
that we had just
finished carrying.

00:02:55.970 --> 00:03:01.300
7 times 3 is 21, plus 6 is 27.

00:03:01.300 --> 00:03:05.720
So we'll write the 7 here in the
hundreds place and carry the 2.

00:03:05.720 --> 00:03:11.525
And then, finally, we have
7 times 5, which is 35.

00:03:11.525 --> 00:03:13.163
But we have to add the 2.

00:03:13.163 --> 00:03:16.640
35 plus 2 is 37.

00:03:16.640 --> 00:03:21.122
So 5,396 times 7 is 37,772.


Multiplying: 3 digits times 1 digit (with carrying)

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:08.540
Let's multiply 7 times
253 and see what we get.

00:00:08.540 --> 00:00:10.760
So just like in the last
example, what I like to do

00:00:10.760 --> 00:00:13.690
is I like to rewrite the
largest number first.

00:00:13.690 --> 00:00:16.079
So that's 253.

00:00:16.079 --> 00:00:18.260
And then write the
smaller number below it

00:00:18.260 --> 00:00:20.160
and align the
place value, the 7.

00:00:20.160 --> 00:00:22.840
It only has a ones place, so
I'll put the 7 right over here

00:00:22.840 --> 00:00:24.849
below the ones place in 253.

00:00:24.849 --> 00:00:27.140
And then put the multiplication
symbol right over here.

00:00:27.140 --> 00:00:29.850
So you could read
this as 253 times 7,

00:00:29.850 --> 00:00:33.040
which we know is the same
thing as 7 times 253.

00:00:33.040 --> 00:00:35.690
And now we are ready to compute.

00:00:35.690 --> 00:00:37.280
And there are many
ways of doing this,

00:00:37.280 --> 00:00:39.760
but this one you could
call the standard way.

00:00:39.760 --> 00:00:41.760
So what I do is I
start with my 7.

00:00:41.760 --> 00:00:44.420
And I multiply it times
each of the numbers up here,

00:00:44.420 --> 00:00:46.750
and I carry appropriately.

00:00:46.750 --> 00:00:49.840
So first I start with 7 times 3.

00:00:49.840 --> 00:00:53.242
Well, 7 times 3 we know is 21.

00:00:53.242 --> 00:00:54.200
Let me write that down.

00:00:54.200 --> 00:00:57.120
7 times 3 is equal to 21.

00:00:57.120 --> 00:00:58.620
You could do this
part in your head,

00:00:58.620 --> 00:00:59.670
but I just want to
make it clear where

00:00:59.670 --> 00:01:01.210
I'm getting these numbers from.

00:01:01.210 --> 00:01:03.040
What I would do in
the standard method

00:01:03.040 --> 00:01:06.000
is I would write the
1 into 21 down here,

00:01:06.000 --> 00:01:08.680
but then carry the
2 to the tens place.

00:01:08.680 --> 00:01:12.405
Now I want to figure
out what 7 times 5 is.

00:01:15.480 --> 00:01:18.930
We know from our multiplication
tables that 7 times 5

00:01:18.930 --> 00:01:20.650
is equal to 35.

00:01:20.650 --> 00:01:23.870
Now, we can't just somehow
put the 35 down here.

00:01:23.870 --> 00:01:26.370
We still have to deal with
this 2 that we carried.

00:01:26.370 --> 00:01:30.360
So we compute 7 times 5 is 35,
but then we also add that 2.

00:01:30.360 --> 00:01:33.880
So it's 35 plus 2 is 37.

00:01:33.880 --> 00:01:36.880
Now, we write the 7 right
over here in the tens

00:01:36.880 --> 00:01:38.710
place and carry the 3.

00:01:42.520 --> 00:01:47.230
Now we need to compute
what 7 times 2 is.

00:01:47.230 --> 00:01:52.920
We know that 7 times 2 is 14
from our multiplication tables.

00:01:52.920 --> 00:01:55.410
We can't just put
a 14 down here.

00:01:55.410 --> 00:01:56.550
We have this 3 to add.

00:01:56.550 --> 00:02:01.420
So 7 times 2 is
14, plus 3 is 17.

00:02:01.420 --> 00:02:03.830
So now we can write
the 17 down here,

00:02:03.830 --> 00:02:06.780
because 2 is the last number
that we had to deal with.

00:02:06.780 --> 00:02:08.169
And so we have our answer.

00:02:08.169 --> 00:02:11.839
7 times 253 is 1,771.


Multiplying: 2 digits times 1 digit (with carrying)

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.160
Let's try to compute 6 times 37.

00:00:04.160 --> 00:00:06.005
And I'll show you one
way of doing this.

00:00:06.005 --> 00:00:07.380
And then in future
videos, we can

00:00:07.380 --> 00:00:09.129
look at other ways of
doing this and think

00:00:09.129 --> 00:00:11.374
about why this is
actually working.

00:00:11.374 --> 00:00:13.040
So what I like to
do-- and this is often

00:00:13.040 --> 00:00:14.681
called the standard
method-- is take

00:00:14.681 --> 00:00:15.930
the larger of the two numbers.

00:00:15.930 --> 00:00:18.644
It doesn't matter if you're
doing 6 times 37 or 37 times 6.

00:00:18.644 --> 00:00:19.560
They equal each other.

00:00:19.560 --> 00:00:23.269
6 times 37 is the same
thing as 37 times 6.

00:00:23.269 --> 00:00:25.810
So what I like to do is I take
the larger of the two numbers,

00:00:25.810 --> 00:00:26.840
and I write it on top.

00:00:26.840 --> 00:00:27.985
So I'll write 37.

00:00:30.514 --> 00:00:32.180
And then the smaller
of the two numbers,

00:00:32.180 --> 00:00:33.846
which is 6, I'll write
it on the bottom.

00:00:33.846 --> 00:00:36.060
And I'll align it by
the correct place.

00:00:36.060 --> 00:00:37.410
This only has one digit.

00:00:37.410 --> 00:00:39.060
It's in the ones
place, obviously.

00:00:39.060 --> 00:00:44.090
So I can write the
6 right over here.

00:00:44.090 --> 00:00:45.930
And I'll write the
multiplication symbol

00:00:45.930 --> 00:00:46.780
like that.

00:00:46.780 --> 00:00:49.980
And this is just another way
of expressing 37 times 6, which

00:00:49.980 --> 00:00:52.940
is the same thing as 6 times 37.

00:00:52.940 --> 00:00:55.380
Now, what we do is
we go with this,

00:00:55.380 --> 00:00:56.922
the first place in
this lower number.

00:00:56.922 --> 00:00:58.254
And there's only one place here.

00:00:58.254 --> 00:00:59.926
It's only the number
6 right over here.

00:00:59.926 --> 00:01:02.300
And we're going to multiply
that times each of the digits

00:01:02.300 --> 00:01:03.390
up here.

00:01:03.390 --> 00:01:08.300
So first, we will
start with 6 times 7.

00:01:08.300 --> 00:01:11.710
So we're going to first
multiply 6 times 7.

00:01:11.710 --> 00:01:14.030
Well, you remember from
your multiplication tables,

00:01:14.030 --> 00:01:17.720
6 times 7 is equal to 42.

00:01:17.720 --> 00:01:19.200
Now, we don't just
write 42 here.

00:01:19.200 --> 00:01:20.700
At least, not in
the standard method

00:01:20.700 --> 00:01:22.360
we wouldn't write 42 here.

00:01:22.360 --> 00:01:25.140
We'd write the 2 in
42 in the ones place.

00:01:25.140 --> 00:01:26.830
So we'd write that
right over there.

00:01:26.830 --> 00:01:32.100
And then we'd carry the 4
in 42 up to the tens place.

00:01:32.100 --> 00:01:38.550
Now we need to think
about what 6 times 3 is.

00:01:38.550 --> 00:01:41.080
Well, once again,
we know 6 times 3.

00:01:41.080 --> 00:01:44.320
6 times 3 is equal to 18.

00:01:44.320 --> 00:01:46.370
But we can't just
write an 18 down here.

00:01:46.370 --> 00:01:48.160
We still have this
4 to deal with.

00:01:48.160 --> 00:01:54.000
So 6 times 3 is 18, but
we've got to then add the 4.

00:01:54.000 --> 00:01:56.303
So 6 times 3 is
18, plus 4 is 22.

00:01:59.645 --> 00:02:01.020
So it's 6 times
3, and then we're

00:02:01.020 --> 00:02:03.160
adding that 4 right over there.

00:02:03.160 --> 00:02:06.380
And that's how we get
our answer-- so 222.


Multiplying: 3 digits times 1 digit

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.545
Let's multiply 4 times 2,012.

00:00:03.545 --> 00:00:05.420
Actually, let's make it
a little bit simpler.

00:00:05.420 --> 00:00:09.600
Let's multiply 4 times
201 just to simplify

00:00:09.600 --> 00:00:10.800
things a little bit.

00:00:10.800 --> 00:00:13.240
So 4 times 201.

00:00:13.240 --> 00:00:15.060
So as we've seen
in previous videos,

00:00:15.060 --> 00:00:16.910
I like to write the
larger number on top.

00:00:16.910 --> 00:00:19.280
This is just one of
many ways of tackling

00:00:19.280 --> 00:00:20.590
a calculation like this.

00:00:20.590 --> 00:00:21.822
I'll write the 201.

00:00:21.822 --> 00:00:23.530
And then I'll write
the 4 right below it,

00:00:23.530 --> 00:00:25.620
and I'll write it right
below the ones place.

00:00:25.620 --> 00:00:28.564
And so I have 201 times 4.

00:00:28.564 --> 00:00:30.730
Now, just like we did when
we were multiplying a one

00:00:30.730 --> 00:00:33.590
digit times a two digit, we do
essentially the same process.

00:00:33.590 --> 00:00:36.870
We first multiply 4 times the 1.

00:00:36.870 --> 00:00:40.920
Well, 4 times 1 we
know is equal to 4.

00:00:40.920 --> 00:00:44.310
So we put a 4 right over
there in the ones place.

00:00:44.310 --> 00:00:47.660
Then we can multiply
our 4 times the digit

00:00:47.660 --> 00:00:49.500
that we have in the tens place.

00:00:49.500 --> 00:00:52.430
In this case, we have
a 0 in the tens place.

00:00:52.430 --> 00:00:56.230
So 4 times 0, well,
that's just 0.

00:00:56.230 --> 00:00:57.320
4 times 0 is 0.

00:00:57.320 --> 00:01:00.130
We put the 0 in the tens
place right over here.

00:01:00.130 --> 00:01:06.320
And then last, we have 4
times this 2 right over here.

00:01:06.320 --> 00:01:10.130
And so 4 times 2 is equal to 8.

00:01:10.130 --> 00:01:12.320
And we put the 8
right over here.

00:01:12.320 --> 00:01:15.090
And we get our answer-- 804.

00:01:15.090 --> 00:01:16.674
Now, why did this work?

00:01:16.674 --> 00:01:18.090
Well, remember,
when we multiplied

00:01:18.090 --> 00:01:20.010
4 times 1, that was
literally just 4.

00:01:20.010 --> 00:01:21.910
And we've got that
4 right over here.

00:01:21.910 --> 00:01:25.010
When we multiply 4
times 0, that's 0 tens.

00:01:25.010 --> 00:01:27.370
So we've got 0 tens
right over here.

00:01:27.370 --> 00:01:30.230
And when we multiplied 4 times
2, this was actually a 200.

00:01:30.230 --> 00:01:31.590
It's in the hundreds place.

00:01:31.590 --> 00:01:34.890
So 4 times 200 is 800.

00:01:34.890 --> 00:01:36.845
So what we're essentially
doing by writing it

00:01:36.845 --> 00:01:39.560
in the right place is
we're saying, 4 times 201,

00:01:39.560 --> 00:01:42.060
that's the same thing
as 4 times 200, which

00:01:42.060 --> 00:01:44.700
is 800, plus 4
times 0 tens, which

00:01:44.700 --> 00:01:48.010
is 0 tens, plus 4
times 1, which is 4.

00:01:48.010 --> 00:01:51.600
So 800 plus 0 plus 4 is 804.


Multiplying: 2 digits times 1 digit

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.660
Let's try to
calculate 3 times 32.

00:00:03.660 --> 00:00:06.380
And I like to rewrite it, and
this is one way of doing it.

00:00:06.380 --> 00:00:09.390
I like to rewrite it where I
have a larger number on top.

00:00:09.390 --> 00:00:11.180
So in this case it's 32.

00:00:11.180 --> 00:00:13.202
And I write the smaller
number right below it.

00:00:13.202 --> 00:00:15.160
And since the smaller
number is only one digit,

00:00:15.160 --> 00:00:18.670
it's only a ones digit, I
put that below the ones place

00:00:18.670 --> 00:00:19.550
on the larger number.

00:00:19.550 --> 00:00:21.440
So I'll put the 3
right over here.

00:00:21.440 --> 00:00:23.950
And of course, we can't forget
our multiplication symbol.

00:00:23.950 --> 00:00:26.490
And this is essentially a
way of saying the same thing.

00:00:26.490 --> 00:00:28.650
You could read
this as 32 times 3.

00:00:28.650 --> 00:00:32.600
But 32 times 3 is the exact
same value as 3 times 32.

00:00:32.600 --> 00:00:35.200
It doesn't matter what
order you multiply in.

00:00:35.200 --> 00:00:36.419
Now let's try to compute it.

00:00:36.419 --> 00:00:38.460
And once again, this is
only one way of doing it.

00:00:38.460 --> 00:00:39.750
There's many ways of doing it.

00:00:39.750 --> 00:00:42.890
And I want you to think
about why this works.

00:00:42.890 --> 00:00:45.790
We'll start with
this 3 down here,

00:00:45.790 --> 00:00:48.730
and we're going to multiply it
times each of the digits in 32.

00:00:48.730 --> 00:00:51.510
So we'll start with 3 times 2.

00:00:51.510 --> 00:00:54.627
Well, 3 times 2 from our
multiplication tables,

00:00:54.627 --> 00:00:56.460
and you can figure it
out even if you didn't

00:00:56.460 --> 00:00:58.660
know your multiplication
tables, is 6.

00:00:58.660 --> 00:01:02.400
So 3 times 2, I'll write 6 right
over here in the ones place.

00:01:02.400 --> 00:01:07.130
Now we're going to figure
out what 3 times 3 is.

00:01:07.130 --> 00:01:11.170
Well, once again, we
know that 3 times 3 is 9.

00:01:11.170 --> 00:01:14.100
And since I'm multiplying times
the tens place right over here,

00:01:14.100 --> 00:01:17.160
I'm going to put it in the
tens place right like this.

00:01:17.160 --> 00:01:17.810
So we're done.

00:01:17.810 --> 00:01:21.150
We got 32 times 3 is 96.

00:01:21.150 --> 00:01:23.800
And I really encourage you to
think about why this worked.

00:01:23.800 --> 00:01:25.942
And I'll give you a
little bit of a hint here.

00:01:25.942 --> 00:01:28.400
I'll give you a little bit of
a hint about why this worked.

00:01:28.400 --> 00:01:35.120
Remember, 3 times 32 is the
same thing as 3 times 30

00:01:35.120 --> 00:01:38.657
plus 3 times 2.

00:01:38.657 --> 00:01:40.240
And if you look at
it that way, that's

00:01:40.240 --> 00:01:42.100
essentially what
this process did.

00:01:42.100 --> 00:01:44.180
We did 3 times 2 is 6.

00:01:44.180 --> 00:01:46.670
3 times 30 is 90.

00:01:46.670 --> 00:01:50.380
You add them together,
you get 90 plus 6 is 96.


Multiplying by multiples of 10

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.717
Let's see if we can
figure out 3 times 60.

00:00:04.717 --> 00:00:07.050
Well, there's a couple of
ways you could think about it.

00:00:07.050 --> 00:00:09.870
You could literally view
this as 60 three times.

00:00:09.870 --> 00:00:14.495
So you could view this
as 60 plus 60 plus 60.

00:00:14.495 --> 00:00:16.620
And you might be able to
compute this in your head.

00:00:16.620 --> 00:00:21.620
60 plus 60 is 120,
plus another 60 is 180.

00:00:21.620 --> 00:00:22.770
And you'd be done.

00:00:22.770 --> 00:00:26.060
But another way to think
about this is that 3 times 60

00:00:26.060 --> 00:00:29.770
is the same thing as 3
times-- instead of thinking

00:00:29.770 --> 00:00:33.420
of it as 60, you could
think of 60 as 6 times 10.

00:00:33.420 --> 00:00:35.980
3 times 6 times 10.

00:00:35.980 --> 00:00:38.240
Now, when you're multiplying
three numbers like this,

00:00:38.240 --> 00:00:40.710
it doesn't matter what
order you multiply them in.

00:00:40.710 --> 00:00:43.910
So we could multiply
the 3 times 6 first

00:00:43.910 --> 00:00:48.470
and get 18 and then
multiply that times 10.

00:00:48.470 --> 00:00:51.720
And 18 times 10 is
just going to be 180.

00:00:51.720 --> 00:00:54.140
It's going to be 18
with another zero.

00:00:54.140 --> 00:01:00.110
So this is going to be 180.

00:01:00.110 --> 00:01:02.810
Now, the more practice you
get here, you'll realize,

00:01:02.810 --> 00:01:05.379
hey, I could have just
said 3 times 6 is 18,

00:01:05.379 --> 00:01:08.366
but I have to worry about
this 0 right over here.

00:01:08.366 --> 00:01:10.240
So I'm going to put one
more zero at the end.

00:01:10.240 --> 00:01:11.710
It's going to be 180.

00:01:11.710 --> 00:01:14.740
Same answer that we
got right over there.

00:01:14.740 --> 00:01:16.590
Let's do another one of these.

00:01:16.590 --> 00:01:23.286
So let's say we want
to multiply 50 times 7.

00:01:23.286 --> 00:01:25.660
And I encourage you to pause
the video and think about it

00:01:25.660 --> 00:01:28.480
yourself, and then unpause
it and see what I do.

00:01:28.480 --> 00:01:32.020
So 50, well, there's a couple of
ways you could think about it.

00:01:32.020 --> 00:01:34.920
One, you could literally
try to add 50 seven times.

00:01:34.920 --> 00:01:37.280
Adding 7 fifty times
would take forever,

00:01:37.280 --> 00:01:42.510
but you could literally say 50
plus 50 plus 50 plus 50-- let's

00:01:42.510 --> 00:01:46.370
see, that's four--
plus 50 plus 50.

00:01:46.370 --> 00:01:47.475
Let's see, that is six.

00:01:47.475 --> 00:01:48.850
I'll do one more
right over here.

00:01:48.850 --> 00:01:50.090
50 right over here.

00:01:50.090 --> 00:01:52.130
So this is 50 seven times.

00:01:52.130 --> 00:01:58.760
If you add together 50 plus
50 is 100, 150, 200, 250, 300,

00:01:58.760 --> 00:01:59.710
350.

00:01:59.710 --> 00:02:01.010
So you could do it that way.

00:02:01.010 --> 00:02:03.900
But you could imagine that
there is an easier way to do it.

00:02:03.900 --> 00:02:05.770
You just need to
realize that 50 is

00:02:05.770 --> 00:02:08.100
the same thing as 10 times 5.

00:02:08.100 --> 00:02:12.470
So we could write
this as 10 times 5,

00:02:12.470 --> 00:02:15.340
and then we're
multiplying that times 7.

00:02:15.340 --> 00:02:18.210
Once again, the order that
we multiply does not matter.

00:02:18.210 --> 00:02:20.760
So we can multiply the
5 times the 7 first.

00:02:20.760 --> 00:02:24.840
We know that that
is 35, and we're

00:02:24.840 --> 00:02:26.970
going to multiply that times 10.

00:02:26.970 --> 00:02:28.950
10 times 35, well,
we're just going

00:02:28.950 --> 00:02:31.140
to stick a zero at
the end of the 35.

00:02:31.140 --> 00:02:36.330
It's going to be equal to 350.

00:02:36.330 --> 00:02:39.050
Now I want to do that
zero in that same color.

00:02:39.050 --> 00:02:41.500
It's going to be 350.

00:02:41.500 --> 00:02:43.420
Now, you might
realize, hey, look,

00:02:43.420 --> 00:02:46.030
I could have just looked
at this 5 right over here,

00:02:46.030 --> 00:02:51.070
multiplied the 5 times the
7, and have gotten the 35.

00:02:51.070 --> 00:02:53.616
And then, not forgetting
that it's actually not a 5,

00:02:53.616 --> 00:02:54.115
it's a 50.

00:02:54.115 --> 00:02:56.500
So I have to
multiply by 10 again,

00:02:56.500 --> 00:02:58.530
or I have to throw
that 0 at the end of it

00:02:58.530 --> 00:03:00.580
to get that 0 right over here.

00:03:00.580 --> 00:03:03.610
So 50 times 7 is 350.


Comparing with multiplication: Ron and Hermione strength

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:03.510
Ron and Hermione are
practicing lifting feathers

00:00:03.510 --> 00:00:04.910
into the air with magic.

00:00:04.910 --> 00:00:08.280
Hermione can lift a feather
35 feet into the air, which

00:00:08.280 --> 00:00:11.250
is 5 times as high as
Ron can lift a feather.

00:00:11.250 --> 00:00:14.250
How high can Ron lift a
feather with his magic?

00:00:14.250 --> 00:00:18.320
So Hermione, this 35 feet
is 5 times more than Ron.

00:00:18.320 --> 00:00:20.180
So to figure out
what Ron can do,

00:00:20.180 --> 00:00:23.300
we just take what Hermione
can do and divide by 5.

00:00:23.300 --> 00:00:27.230
So what's 35 divided by 5?

00:00:27.230 --> 00:00:30.020
Well, 5 times 7 is 35.

00:00:30.020 --> 00:00:32.220
So 35 divided by 5 is 7.

00:00:32.220 --> 00:00:36.830
So Ron can lift a feather
7 feet in the air.

00:00:36.830 --> 00:00:37.955
So let's check our answer.

00:00:37.955 --> 00:00:39.880
Now, if you have trouble
with this-- and we

00:00:39.880 --> 00:00:41.800
could also look at
the hints for this.

00:00:41.800 --> 00:00:45.870
So they show that Hermione can
lift a feather 35 feet, which

00:00:45.870 --> 00:00:47.570
is 5 times higher than Ron.

00:00:47.570 --> 00:00:51.020
So this could be the length
of how high Ron can raise it.

00:00:51.020 --> 00:00:53.810
And so, if you were
to actually divide 35

00:00:53.810 --> 00:00:56.875
into 5 equal sections,
so 5 times question mark

00:00:56.875 --> 00:01:00.820
is 35, that's the same thing
as saying 35 divided by 5

00:01:00.820 --> 00:01:02.050
is question mark.

00:01:02.050 --> 00:01:08.690
And 35 divided by 5 is indeed 7.


Division word problem example 1

WEBVTT
Kind: captions
Language: en

00:00:00.557 --> 00:00:02.069
PROBLEM: "Omar rode his boat

00:00:02.069 --> 00:00:06.149
for a total of 50 miles over the past 5 days.

00:00:06.149 --> 00:00:09.108
And he rowed the same amount each day.

00:00:09.108 --> 00:00:10.991
How many miles did Omar

00:00:10.991 --> 00:00:13.472
row his boat each day?"

00:00:13.472 --> 00:00:15.568
So let’s just visualize what's going on.

00:00:15.568 --> 00:00:17.512
He is able to travel 50 miles.

00:00:17.512 --> 00:00:18.864
So let's make a –

00:00:18.864 --> 00:00:20.214
Let’s say this line represents

00:00:20.214 --> 00:00:22.084
the 50 miles that he travels.

00:00:22.084 --> 00:00:24.237
So this whole distance right over here

00:00:24.237 --> 00:00:27.778
is 50 miles.

00:00:27.778 --> 00:00:29.620
And he does it –

00:00:29.620 --> 00:00:32.947
They tell us that he does it over 5 days,

00:00:32.947 --> 00:00:35.505
and that each day, he does the same amount.

00:00:35.505 --> 00:00:38.514
So this 50 miles is if you were to add together

00:00:38.514 --> 00:00:40.556
all of what he did over the 5 days.

00:00:40.556 --> 00:00:41.479
And so, if you want to know

00:00:41.479 --> 00:00:43.491
how much he did each day,

00:00:43.491 --> 00:00:44.759
you essentially want to divide

00:00:44.759 --> 00:00:47.470
this 50 miles into 5 equal sections.

00:00:47.470 --> 00:00:48.797
And the length of each of those sections

00:00:48.797 --> 00:00:50.802
is the amount he did each day.

00:00:50.802 --> 00:00:52.204
So if we just visualize it –

00:00:52.204 --> 00:00:54.874
So that's one section – second section –

00:00:54.874 --> 00:00:57.641
third section – fourth – and fifth section.

00:00:57.641 --> 00:00:58.579
And actually, I didn't do that very well.

00:00:58.579 --> 00:01:00.411
It should look a little bit more equal than that.

00:01:00.411 --> 00:01:02.832
First, second –

00:01:02.832 --> 00:01:04.318
(And that's not going to – Let's see.)

00:01:04.318 --> 00:01:08.505
First, second, third, fourth, and fifth.

00:01:08.505 --> 00:01:09.985
And you don't have to actually do this.

00:01:09.985 --> 00:01:11.270
This is just to help visualize.

00:01:11.270 --> 00:01:14.385
So essentially, what we want to figure out is what is

00:01:14.385 --> 00:01:15.817
one of these distances?

00:01:15.817 --> 00:01:17.279
And as you can see in our visualization,

00:01:17.279 --> 00:01:19.537
this is really just taking our 50 miles

00:01:19.537 --> 00:01:22.522
and dividing it into 5 equal chunks.

00:01:22.522 --> 00:01:28.227
So we're essentially just taking 50,

00:01:28.227 --> 00:01:31.118
and we're going to divide it by 5.

00:01:31.118 --> 00:01:34.026
So 50 divided by 5

00:01:34.026 --> 00:01:37.319
is going to be equal to 10.

00:01:37.319 --> 00:01:39.580
So if he goes 50 miles over 5 days,

00:01:39.580 --> 00:01:41.346
and you divide by the 5 days,

00:01:41.346 --> 00:01:45.223
he goes 10 miles each day.

00:01:45.223 --> 00:01:47.162
And we're done.


Multiplication word problem example 1

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.100
Toby is a farmer.

00:00:02.100 --> 00:00:05.160
He plants 12 rows of
carrots in a field.

00:00:05.160 --> 00:00:07.670
Each row has 6 carrots.

00:00:07.670 --> 00:00:10.040
How many carrots did Toby plant?

00:00:10.040 --> 00:00:12.050
So let's try to
visualize this thing.

00:00:12.050 --> 00:00:14.035
So they tell us each
row has 6 carrots.

00:00:14.035 --> 00:00:15.780
So let's try to visualize a row.

00:00:15.780 --> 00:00:19.310
So that's 1 carrot,
that's 2 carrots, that's

00:00:19.310 --> 00:00:22.860
3 carrots, that's 4
carrots, that's 5 carrots,

00:00:22.860 --> 00:00:23.850
and that's 6 carrots.

00:00:23.850 --> 00:00:25.450
And I can even do,
just so it looks

00:00:25.450 --> 00:00:27.480
nice, the little leafy
part of the carrots.

00:00:27.480 --> 00:00:29.470
So we can visualize
these as carrots.

00:00:29.470 --> 00:00:30.830
So that's 1 row of carrots.

00:00:30.830 --> 00:00:33.130
They tell us each
row has 6 carrots.

00:00:33.130 --> 00:00:35.280
Now, there're 12 of these.

00:00:35.280 --> 00:00:37.050
He plants 12 rows of carrots.

00:00:37.050 --> 00:00:39.000
Let me see if I can
copy and paste this.

00:00:39.000 --> 00:00:42.310
So copy and paste.

00:00:42.310 --> 00:00:44.114
So that's 2 rows.

00:00:44.114 --> 00:00:45.780
I want to make sure
I have enough space.

00:00:45.780 --> 00:00:47.110
2 rows.

00:00:47.110 --> 00:00:49.720
That's 3 rows.

00:00:49.720 --> 00:00:52.530
That's 4 rows.

00:00:52.530 --> 00:00:55.800
This is 5 rows.

00:00:55.800 --> 00:00:57.957
This is 6 rows.

00:00:57.957 --> 00:01:00.290
And actually, I can just copy
and paste the whole thing.

00:01:00.290 --> 00:01:01.110
This is 6 rows.

00:01:01.110 --> 00:01:03.080
I just need to double
that to get to 12 rows.

00:01:03.080 --> 00:01:04.050
So let me do that.

00:01:04.050 --> 00:01:06.840
So copy and paste.

00:01:06.840 --> 00:01:07.430
There you go.

00:01:07.430 --> 00:01:09.820
So there's 12 rows
right over here.

00:01:09.820 --> 00:01:10.900
And let's number this.

00:01:10.900 --> 00:01:19.120
So they tell us each row has
6 carrots-- 1, 2, 3, 4, 5, 6.

00:01:19.120 --> 00:01:21.670
They tell us he
plants a total of 12

00:01:21.670 --> 00:01:31.830
rows-- 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12.

00:01:31.830 --> 00:01:33.490
How many carrots did Toby plant?

00:01:33.490 --> 00:01:35.620
Well, you could do
this exercise and then

00:01:35.620 --> 00:01:37.174
try to count all
of these carrots.

00:01:37.174 --> 00:01:38.590
But that seems a
little bit crazy.

00:01:38.590 --> 00:01:40.049
These are already
a lot of carrots.

00:01:40.049 --> 00:01:42.090
And especially if these
were even larger numbers,

00:01:42.090 --> 00:01:43.690
it would take you
forever to count it.

00:01:43.690 --> 00:01:50.610
Luckily for us, we have
multiplication as a tool

00:01:50.610 --> 00:01:51.910
that we can use.

00:01:51.910 --> 00:01:56.690
So if you have 12 rows and
each of those rows have 6,

00:01:56.690 --> 00:01:59.520
this is really 12 times 6.

00:01:59.520 --> 00:02:01.200
You're going to have 12 rows.

00:02:01.200 --> 00:02:03.930
In each of those,
you have 6 carrots.

00:02:03.930 --> 00:02:06.720
So you could say you
have 6 carrots 12 times.

00:02:06.720 --> 00:02:08.460
So it's going to be 12 times 6.

00:02:08.460 --> 00:02:11.740
If you remember your
multiplication tables up to 12,

00:02:11.740 --> 00:02:16.750
you'll remember that
12 times 6 is 72.

00:02:16.750 --> 00:02:18.530
So he has 72 carrots.

00:02:18.530 --> 00:02:21.149
Even if you only remember--
well, I'll just leave it there.

00:02:21.149 --> 00:02:22.690
If you remember your
multiplication--

00:02:22.690 --> 00:02:25.148
which you should, because that
is one of the things in life

00:02:25.148 --> 00:02:30.939
that will have long-lasting
benefits-- it's 72 carrots.


Comparing with multiplication: basic exercises

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:05.060
The number 36 is three
times as large as 12.

00:00:05.060 --> 00:00:08.910
Write this comparison as
a multiplication equation.

00:00:08.910 --> 00:00:12.300
So they're saying that the
number 36, we'll type 36,

00:00:12.300 --> 00:00:19.880
is equal to 3 times 12.

00:00:19.880 --> 00:00:21.646
And I will use this
little star thing.

00:00:21.646 --> 00:00:23.270
Let's see, if I were
to put an x there,

00:00:23.270 --> 00:00:24.561
would that have worked as well?

00:00:24.561 --> 00:00:27.430
No, they would have
viewed that as 3x.

00:00:27.430 --> 00:00:29.640
So to get this little star
thing, which is normally

00:00:29.640 --> 00:00:32.790
when you're typing on a computer
how you show multiplication,

00:00:32.790 --> 00:00:37.860
to get that you have to
press Shift and the number 8.

00:00:37.860 --> 00:00:40.300
And it also sometimes shows
up on a numeric keypad.

00:00:40.300 --> 00:00:42.730
But by doing that, I was
able to show multiplication.

00:00:42.730 --> 00:00:46.420
So this could be read
as 36 is equal to 3

00:00:46.420 --> 00:00:51.100
times 12, or 36 is 3
times as large as 12.

00:00:51.100 --> 00:00:53.610
Let's check our answer.

00:00:53.610 --> 00:00:54.160
Got it right.

00:00:54.160 --> 00:00:56.920
Let's do a couple more of these.

00:00:56.920 --> 00:01:00.340
We can compare the numbers
4 and 20 using addition

00:01:00.340 --> 00:01:04.860
by saying that 20
is 16 more than 4.

00:01:04.860 --> 00:01:05.390
Fair enough.

00:01:05.390 --> 00:01:06.310
That's right.

00:01:06.310 --> 00:01:09.740
We could also compare 4 and
20 using multiplication.

00:01:09.740 --> 00:01:12.060
Fill in the blank correctly
to compare the numbers

00:01:12.060 --> 00:01:14.100
4 and 20 using multiplication.

00:01:14.100 --> 00:01:18.880
So they're saying 20 is
blank times as large as four.

00:01:18.880 --> 00:01:24.380
Well, we know that 4 times 5
is 20, or that 5 times 4 is 20.

00:01:24.380 --> 00:01:27.410
So 20 is 5 times as large as 4.

00:01:27.410 --> 00:01:30.770
If I do 4 five times,
I'll get to 20.

00:01:30.770 --> 00:01:32.760
So let me check my answer.

00:01:32.760 --> 00:01:33.970
There you go.

00:01:33.970 --> 00:01:35.910
Let's do one more.

00:01:35.910 --> 00:01:39.120
The number 6 is three
times as large as 2.

00:01:39.120 --> 00:01:41.780
Write this comparison as
a multiplication equation.

00:01:41.780 --> 00:01:46.470
So once again, we could
write that 6 is equal to 2,

00:01:46.470 --> 00:01:48.740
and then I press
Shift and the number 8

00:01:48.740 --> 00:01:50.670
to get that times
symbol, that little

00:01:50.670 --> 00:01:52.830
asterisk
snowflake-looking thing.

00:01:52.830 --> 00:01:53.992
2 times 3.

00:01:53.992 --> 00:01:55.450
So I could write
it like that, or I

00:01:55.450 --> 00:01:57.260
could write 2 times
3 is equal to 6.

00:01:57.260 --> 00:01:59.810
Either of those would be
completely valid things to say.

00:01:59.810 --> 00:02:03.624
So here I'm saying 6
is-- well, actually,

00:02:03.624 --> 00:02:04.790
I'd like to say it this way.

00:02:04.790 --> 00:02:06.810
I'd like to say three
times as large as 2.

00:02:06.810 --> 00:02:08.710
I think that's a
little bit more fun.

00:02:08.710 --> 00:02:13.980
So this is exactly 6 is
3 times as large as 2.

00:02:13.980 --> 00:02:15.660
And we're done.


Number line 1

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.440
Where is 30 on the number line?

00:00:03.440 --> 00:00:05.110
So on the number
line right over here,

00:00:05.110 --> 00:00:08.090
we see that as we go from
0, the first slash isn't 1.

00:00:08.090 --> 00:00:08.590
It's 3.

00:00:08.590 --> 00:00:12.144
So every slash here is
going to go up by 3.

00:00:12.144 --> 00:00:14.060
So let's see if we can
figure out where 30 is.

00:00:14.060 --> 00:00:16.160
So this is already marked as 3.

00:00:16.160 --> 00:00:17.680
Now this is going to be 3 more.

00:00:17.680 --> 00:00:28.059
It's 6, 9, 12, 15,
18, 21, 24, 27, 30.

00:00:28.059 --> 00:00:29.850
Another way you could
have thought about it

00:00:29.850 --> 00:00:32.750
is if each of these marks
are 3, to get to 30,

00:00:32.750 --> 00:00:34.260
we have to go to
10 of these marks.

00:00:34.260 --> 00:00:40.730
So you go 1, 2, 3,
4, 5, 6, 7, 8, 9, 10.

00:00:40.730 --> 00:00:43.340
Let's do a few more.

00:00:43.340 --> 00:00:45.610
Where is 24 on the number line?

00:00:45.610 --> 00:00:48.300
Well, once again, each of
these marks is again 3.

00:00:48.300 --> 00:00:56.880
So it's going to be 3, 6,
9, 12, 15, 18, 21, and 24.

00:00:56.880 --> 00:00:58.390
Let's do one more.

00:00:58.390 --> 00:01:00.290
So now each mark is 4.

00:01:00.290 --> 00:01:06.750
So we're going to
go 4, 8, 12, 16, 20.

00:01:06.750 --> 00:01:08.250
Another way you
could think about it

00:01:08.250 --> 00:01:11.290
is if we're going up by 4,
you have to go up 5 times

00:01:11.290 --> 00:01:12.270
to get up to 20.

00:01:12.270 --> 00:01:15.650
So this is 1, 2, 3, 4, 5.

00:01:15.650 --> 00:01:18.725
Either way, this should
be the right answer.


The idea of division

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:04.092
We've got 24 triangle
things right over here.

00:00:04.092 --> 00:00:05.550
And what I want to
do in this video

00:00:05.550 --> 00:00:09.130
is to divide it into
different numbers of groups.

00:00:09.130 --> 00:00:10.600
So the first thing
I want to do is

00:00:10.600 --> 00:00:16.760
I want to divide this 24
triangle things into 3 groups

00:00:16.760 --> 00:00:20.020
and think about how many
do I end up per group.

00:00:20.020 --> 00:00:21.660
So let's try that out.

00:00:21.660 --> 00:00:24.370
So I'm going to divide
it into 3 equal groups.

00:00:24.370 --> 00:00:28.450
So that is one equal
group right over there.

00:00:28.450 --> 00:00:32.470
Then I have another equal
group right over here.

00:00:32.470 --> 00:00:36.340
And then I have a third
equal group right over here.

00:00:36.340 --> 00:00:42.280
So if I divide 24 into 3 equal
groups-- 1, 2, 3-- how many

00:00:42.280 --> 00:00:43.890
are going to be in each group?

00:00:43.890 --> 00:00:44.970
Well, we can count that.

00:00:44.970 --> 00:00:52.500
We have 1, 2, 3, 4, 5,
6, 7, 8 in each group.

00:00:52.500 --> 00:00:59.239
So we could say that 24
divided by 3 is equal to 8.

00:00:59.239 --> 00:01:00.780
Now, you might say,
hey, this is very

00:01:00.780 --> 00:01:02.488
similar to what we
saw in multiplication.

00:01:02.488 --> 00:01:06.830
In multiplication, we said
if we have 3 groups of 8,

00:01:06.830 --> 00:01:10.720
we could view that as
3 times 8 and get 24.

00:01:10.720 --> 00:01:12.035
And you are exactly right.

00:01:14.670 --> 00:01:18.940
Let me do those same colors--
we could also write that 3

00:01:18.940 --> 00:01:25.620
times 8-- so if I
have 3 groups of 8,

00:01:25.620 --> 00:01:32.040
that that is going
to be equal to 24.

00:01:32.040 --> 00:01:34.580
So when we started in this
video, we had 24 things.

00:01:34.580 --> 00:01:37.440
We want to divide it
into 3 equal groups.

00:01:37.440 --> 00:01:42.780
We got 8 in each group, or you
could say 3 equal groups of 8

00:01:42.780 --> 00:01:45.160
is equal to 24.

00:01:45.160 --> 00:01:47.620
But there's even other ways
of thinking about this.

00:01:47.620 --> 00:01:49.840
So let me clear this
up a little bit.

00:01:49.840 --> 00:01:51.390
So let me clear that.

00:01:54.450 --> 00:01:59.460
So in the first example, I
divided 24 into 3 equal groups.

00:01:59.460 --> 00:02:02.730
But you could also
view 24 divided by 3

00:02:02.730 --> 00:02:05.500
as dividing 24 into groups of 3.

00:02:05.500 --> 00:02:07.890
So let's think about
what that looks like.

00:02:07.890 --> 00:02:10.680
So if we divide it
into groups of 3,

00:02:10.680 --> 00:02:15.270
then, for example,
this is a group of 3.

00:02:15.270 --> 00:02:17.860
That is a group of 3.

00:02:17.860 --> 00:02:19.494
This is a group of 3.

00:02:19.494 --> 00:02:20.910
You might see where
this is going.

00:02:20.910 --> 00:02:23.050
That's a group of 3.

00:02:23.050 --> 00:02:25.430
That is another group of 3.

00:02:25.430 --> 00:02:27.570
And we're going to think
about how many groups of 3

00:02:27.570 --> 00:02:29.190
we're actually going to get.

00:02:29.190 --> 00:02:32.780
So this is another group of 3.

00:02:32.780 --> 00:02:35.750
And that's another group of 3.

00:02:35.750 --> 00:02:38.490
So how many groups
of 3 did we get?

00:02:38.490 --> 00:02:45.120
Let's see, we have 1, 2, 3,
4, 5, 6, 7, 8 groups of 3.

00:02:45.120 --> 00:02:50.580
So another way of
viewing 24 divided by 3

00:02:50.580 --> 00:02:54.450
is divide 24 into groups of 3.

00:02:54.450 --> 00:02:57.120
And then you will
have 8 groups of 3.

00:02:57.120 --> 00:02:58.620
And one way of
thinking about this--

00:02:58.620 --> 00:03:01.900
if you want to express
the same thing in terms

00:03:01.900 --> 00:03:03.550
of multiplication--
is if you have

00:03:03.550 --> 00:03:13.070
8 groups of 3, that is also
going to be equal to 24.

00:03:13.070 --> 00:03:18.140
Whether you have 3 groups
of 8 or 8 groups of 3,

00:03:18.140 --> 00:03:21.820
either way, you're
going to have 24.

00:03:21.820 --> 00:03:25.830
Now, let's make things
more interesting.

00:03:25.830 --> 00:03:30.650
What I want you to think about
is, based on what we just saw,

00:03:30.650 --> 00:03:37.640
what is 24 divided by 12?

00:03:37.640 --> 00:03:39.330
And I encourage you
to pause the video,

00:03:39.330 --> 00:03:42.050
draw out 24 triangles
like this, and try

00:03:42.050 --> 00:03:46.460
to figure out what
24 divided by 12 is.

00:03:46.460 --> 00:03:48.170
Well, I assume you've
paused the video.

00:03:48.170 --> 00:03:51.260
And there's two ways to
think about 24 divided by 12.

00:03:51.260 --> 00:03:55.860
You could say, well, let's
divide 24 into groups of 12

00:03:55.860 --> 00:03:57.720
and think about how
many groups we have.

00:03:57.720 --> 00:03:58.660
So we could do that.

00:03:58.660 --> 00:03:59.201
So let's see.

00:03:59.201 --> 00:04:04.530
This is one group of
12 right over here.

00:04:04.530 --> 00:04:10.000
That's one group of 12, and then
here is another group of 12.

00:04:10.000 --> 00:04:13.080
So how many groups
of 12 do we have?

00:04:13.080 --> 00:04:14.780
Well, we have 2 groups of 12.

00:04:14.780 --> 00:04:19.120
So we could say 24
divided by 12 is 2.

00:04:19.120 --> 00:04:23.010
But another just as reasonable
way of doing this is you

00:04:23.010 --> 00:04:27.190
could have said,
well, let me divide 24

00:04:27.190 --> 00:04:30.460
into 12 groups instead
of groups of 12.

00:04:30.460 --> 00:04:32.240
So if I want to divide
it into 12 groups,

00:04:32.240 --> 00:04:34.530
12 equal groups--
well, let's see.

00:04:34.530 --> 00:04:38.749
This is 1 equal group,
2 equal groups--

00:04:38.749 --> 00:04:40.040
actually let me do it this way.

00:04:40.040 --> 00:04:54.600
Well, let me do this-- 2 equal
groups, 3 4, 5, 6, 7, 8, 9, 10,

00:04:54.600 --> 00:04:56.840
11, 12.

00:04:56.840 --> 00:04:59.910
So once again, if you say, oh,
I'm going to divide 24 into 12

00:04:59.910 --> 00:05:02.480
equal groups, how many do
you have in each group?

00:05:02.480 --> 00:05:03.620
Well, you have 2.

00:05:03.620 --> 00:05:05.100
So once again, 24
could be viewed

00:05:05.100 --> 00:05:07.585
as 24 divided into
12 equal groups.

00:05:07.585 --> 00:05:09.210
And how many do you
have in each group?

00:05:09.210 --> 00:05:12.860
Or 24 divided into groups of
12, and how many groups would

00:05:12.860 --> 00:05:13.420
you have?

00:05:13.420 --> 00:05:16.020
And that's what we saw
in the last example.

00:05:16.020 --> 00:05:18.940
So now, let's make things
even more interesting.

00:05:18.940 --> 00:05:21.840
What I want you to think
about-- a couple of things.

00:05:21.840 --> 00:05:31.100
I want you think about
what 24 divided by 6 is.

00:05:31.100 --> 00:05:36.215
And I also want you to figure
out what 24 divided by--

00:05:36.215 --> 00:05:41.070
let me use that
same color-- 4 is.

00:05:41.070 --> 00:05:43.450
And once again, I encourage
you to pause the video,

00:05:43.450 --> 00:05:45.170
draw these triangles,
and figure it out.

00:05:45.170 --> 00:05:49.210
What is 24 divided by
6 and 24 divided by 4?

00:05:49.210 --> 00:05:51.360
So let's tackle 24
divided by 6 first.

00:05:51.360 --> 00:05:59.239
And let's try to divide
24 into 6 equal groups.

00:05:59.239 --> 00:05:59.780
So let's see.

00:05:59.780 --> 00:06:05.310
This could be 1 equal group,
2 equal groups-- in fact,

00:06:05.310 --> 00:06:09.490
each group here is a group of 4.

00:06:09.490 --> 00:06:11.300
And we have 6 rows.

00:06:11.300 --> 00:06:19.190
So 3 equal groups, 4, 5, and 6.

00:06:19.190 --> 00:06:23.150
And so if you divide
24 into 6 equal groups,

00:06:23.150 --> 00:06:24.690
how many do you
have in each group?

00:06:24.690 --> 00:06:27.780
Well, it's pretty
clear you have 4.

00:06:27.780 --> 00:06:29.406
You have 4 In each group.

00:06:29.406 --> 00:06:30.780
Another way we
could have thought

00:06:30.780 --> 00:06:39.710
about that is we
could have said,

00:06:39.710 --> 00:06:43.220
let me divide 24
into groups of 6.

00:06:43.220 --> 00:06:45.860
So if you divided
24 into groups of 6,

00:06:45.860 --> 00:06:49.000
you could have
viewed it like this.

00:06:49.000 --> 00:06:53.490
So that's 1 group
of 6 right there.

00:06:53.490 --> 00:06:57.090
That's another group
of 6 right over here.

00:06:57.090 --> 00:06:58.475
That's another group of 6.

00:06:58.475 --> 00:07:01.480
And I think you see how
many groups of 6 we have.

00:07:01.480 --> 00:07:03.056
How many groups of 6 do we have?

00:07:03.056 --> 00:07:04.484
We have 4.

00:07:04.484 --> 00:07:10.120
We have 4 groups of 6.

00:07:10.120 --> 00:07:13.370
Well, now let's think about
what 24 divided by 4 is.

00:07:13.370 --> 00:07:18.400
Well, if I view 24 divided by
4 as taking 24 and dividing it

00:07:18.400 --> 00:07:20.780
into 4 equal groups,
I've just drawn that.

00:07:20.780 --> 00:07:23.965
I have 4 equal groups, and
in each group I have 6.

00:07:27.740 --> 00:07:30.453
So notice 24 divided by 6 is 4.

00:07:30.453 --> 00:07:33.900
24 divided by 4 is 6.

00:07:33.900 --> 00:07:38.100
And that's because I could
view this as 4 groups of 6

00:07:38.100 --> 00:07:44.140
or say that 4 times
6 is equal to 24.

00:07:47.140 --> 00:07:50.890
Or you could just as
equivalently say that 6 times 4

00:07:50.890 --> 00:07:52.570
is 24.

00:07:52.570 --> 00:08:00.198
You could equivalently say
that 6 times 4 is equal to 24.


More on the concept of multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:03.940
If we have 2 groups and
in each group I have 4,

00:00:03.940 --> 00:00:06.880
so that's one group
of 4, and then

00:00:06.880 --> 00:00:10.310
here is my second
group of 4, we already

00:00:10.310 --> 00:00:20.230
know that we could
write this as 2 times 4,

00:00:20.230 --> 00:00:25.960
which is the same
thing as 4 plus 4.

00:00:25.960 --> 00:00:27.890
Notice I have two 4's here.

00:00:27.890 --> 00:00:30.540
I have 4 plus another 4.

00:00:33.340 --> 00:00:36.600
Well, if I have 4 plus 4,
or if I have 2 groups of 4,

00:00:36.600 --> 00:00:40.306
either way, I'm going to
have a total of 8 things.

00:00:40.306 --> 00:00:41.680
And you see that
right over here.

00:00:41.680 --> 00:00:47.380
We have 1, 2, 3, 4,
5, 6, 7, 8 things.

00:00:47.380 --> 00:00:50.050
What I want you to do
is pause the video now

00:00:50.050 --> 00:00:53.720
and try to group these
same 8 things, but to group

00:00:53.720 --> 00:00:56.010
it in other ways so
that we can represent 8

00:00:56.010 --> 00:00:57.920
as the product of whole numbers.

00:00:57.920 --> 00:01:02.020
Here I've represented 8
as the product of 2 and 4.

00:01:02.020 --> 00:01:04.660
2 times 4 is 8.

00:01:04.660 --> 00:01:06.660
See if you can represent
8 as the product

00:01:06.660 --> 00:01:09.180
of other whole numbers,
or as whole numbers

00:01:09.180 --> 00:01:13.120
in different ways, grouping
it in different ways.

00:01:13.120 --> 00:01:15.040
So I assume you've
paused the video.

00:01:15.040 --> 00:01:17.250
So let's try it out ourselves.

00:01:17.250 --> 00:01:18.730
So one thing we
could do, we could

00:01:18.730 --> 00:01:20.700
view this instead
of as 2 groups of 4,

00:01:20.700 --> 00:01:24.030
we can view 8 as 4 groups of 2.

00:01:24.030 --> 00:01:29.420
So that's 1 group of 2, 2
groups of 2, 3 groups of 2,

00:01:29.420 --> 00:01:31.890
4 groups of 2.

00:01:31.890 --> 00:01:42.150
So we could write that
4 times 2 is equal to 8.

00:01:42.150 --> 00:01:44.490
And we could view this as
the same thing as, literally,

00:01:44.490 --> 00:01:45.490
4 2's.

00:01:45.490 --> 00:01:48.520
We have one 1, 2, 3, 4 2's.

00:01:48.520 --> 00:01:49.850
Each of these have 2 in them.

00:01:49.850 --> 00:01:55.110
So we're going to
say 1, 2, 3, 4 2's.

00:01:55.110 --> 00:01:58.760
2 plus 2, plus 2,
plus 2 is equal to 8.

00:01:58.760 --> 00:02:00.130
These are both equivalent.

00:02:00.130 --> 00:02:04.560
4 times 2, literally
4 groups of 2.

00:02:04.560 --> 00:02:09.460
That's the same thing as taking
4 2's and adding them together.

00:02:09.460 --> 00:02:11.670
Notice, we have 2
2's right over here.

00:02:11.670 --> 00:02:14.340
We added them together, 1, 2.

00:02:14.340 --> 00:02:16.320
Here, we have 4 2's
and we're adding

00:02:16.320 --> 00:02:18.720
them together, 1, 2, 3, 4.

00:02:18.720 --> 00:02:21.620
We take our 4 2's and
we add them together.

00:02:21.620 --> 00:02:24.450
How else could we represent 8?

00:02:24.450 --> 00:02:28.240
Well, we literally could
view it as 8 groups of 1.

00:02:28.240 --> 00:02:29.750
So let's do that.

00:02:29.750 --> 00:02:31.920
So 8 groups of 1
would look like this.

00:02:31.920 --> 00:02:40.880
That's 1 group of 1,
2, 3, 4, 5, 6, 7, 8.

00:02:40.880 --> 00:02:45.660
So we could write this
down as 8 times 1.

00:02:45.660 --> 00:02:51.205
8 times 1 is, once
again, equal to 8.

00:02:51.205 --> 00:02:53.580
And if we wanted to write this
down as repeated addition,

00:02:53.580 --> 00:02:55.350
well, this is literally 8 1's.

00:02:55.350 --> 00:03:00.760
So 1 plus 1, plus 1,
plus 1, plus 1, plus 1.

00:03:00.760 --> 00:03:01.260
Let's see.

00:03:01.260 --> 00:03:05.320
That's 1, 2, 3, 4, 5, 6, 7, 8.

00:03:05.320 --> 00:03:09.630
1, 2, 3, 4, 5, 6, 7, 8.

00:03:09.630 --> 00:03:12.490
1 plus 1, plus 1, plus 1,
plus 1, plus 1, plus 1,

00:03:12.490 --> 00:03:15.180
plus 1 is equal to 8.

00:03:15.180 --> 00:03:16.980
Now, you might be
a little stumped.

00:03:16.980 --> 00:03:19.470
Well, what's another
way of getting to 8?

00:03:19.470 --> 00:03:25.170
Well, you could literally
view it as 1 group of 8.

00:03:25.170 --> 00:03:26.560
So let me view it that way.

00:03:26.560 --> 00:03:31.530
So this is just 1 entire
group of 8, the whole thing.

00:03:31.530 --> 00:03:35.030
The whole thing is a group of 8.

00:03:35.030 --> 00:03:40.110
So let me scroll over to
the right a little bit.

00:03:40.110 --> 00:03:44.240
We could write this
down as 1 times 8.

00:03:44.240 --> 00:03:48.770
And 1 times 8 is equal to 8.

00:03:48.770 --> 00:03:50.580
And how would we view that?

00:03:50.580 --> 00:03:52.660
Well, we only have one 8 now.

00:03:52.660 --> 00:03:54.830
We don't have to add that
one 8 to anything else.

00:03:54.830 --> 00:03:58.440
So if we wanted to do it the
way we've done the last few,

00:03:58.440 --> 00:04:01.560
we could literally write it
down as we just have one 8.

00:04:01.560 --> 00:04:06.950
Well, one 8 is clearly
going to just be equal to 8.

00:04:06.950 --> 00:04:08.740
So now let me ask
you another question.

00:04:08.740 --> 00:04:11.880
So far we've been focused
on each of these groups,

00:04:11.880 --> 00:04:16.480
but what if we actually
view this as 4 groups of 8.

00:04:25.940 --> 00:04:28.320
Then how many things are
we actually going to have?

00:04:28.320 --> 00:04:30.360
So let me make this very clear.

00:04:30.360 --> 00:04:37.940
So we have 1 group of eight 8,
2 groups of 8, 3 groups of 8,

00:04:37.940 --> 00:04:41.000
and 4 groups of 8.

00:04:41.000 --> 00:04:49.230
So we would view this
as 4 times 8, or which

00:04:49.230 --> 00:04:56.880
is going to be the same thing
as 8 plus 8, plus 8, plus 8.

00:04:56.880 --> 00:04:58.110
4 8's.

00:04:58.110 --> 00:04:59.694
What is this going
to be equal to?

00:04:59.694 --> 00:05:01.360
A And I encourage you
to pause the video

00:05:01.360 --> 00:05:03.535
and figure it out right now.

00:05:03.535 --> 00:05:05.160
Well, there's a couple
of ways that you

00:05:05.160 --> 00:05:06.567
could have thought about this.

00:05:06.567 --> 00:05:08.400
You could have literally
just counted these.

00:05:08.400 --> 00:05:12.570
Or you could say, well, let's
see, you can skip count by 8.

00:05:12.570 --> 00:05:16.840
8, 16, 24, 32.

00:05:16.840 --> 00:05:22.250
Or you could have said, 8
plus 8 is 16, plus 8 is 24,

00:05:22.250 --> 00:05:23.860
plus 8 is 32.

00:05:23.860 --> 00:05:28.270
Or you could have literally
just counted the triangles here.


Multiplication as groups of objects

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.179
So, I have several groups of
these ball-looking things.

00:00:03.179 --> 00:00:05.470
And let's think about how
many balls are in each group.

00:00:05.470 --> 00:00:12.230
We have 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11 12.

00:00:12.230 --> 00:00:15.430
And what I want to do is
think about the different ways

00:00:15.430 --> 00:00:20.110
of dividing these 12 balls into
different numbers of groups.

00:00:20.110 --> 00:00:24.750
So, for example, I could
view these 12 balls as one,

00:00:24.750 --> 00:00:31.110
so that's 1 group of 3, 2
groups of 3, 3 groups of 3,

00:00:31.110 --> 00:00:33.460
4 groups of 3.

00:00:33.460 --> 00:00:37.990
So I could view 12 as
being 4 groups of 3.

00:00:37.990 --> 00:00:41.870
And the way that
we would write that

00:00:41.870 --> 00:00:50.710
is that 12 is equal
to 4 groups of 3.

00:00:53.450 --> 00:00:58.480
Or another way of reading this
is that 12 is equal to 4 times

00:00:58.480 --> 00:00:59.240
3.

00:00:59.240 --> 00:01:03.930
If I have 1, 2, 3, 4 groups,
and in each of those groups

00:01:03.930 --> 00:01:06.660
I have 1, 2, 3,
objects, I'm going

00:01:06.660 --> 00:01:09.460
to have a total of 12 objects.

00:01:09.460 --> 00:01:11.940
But that's not the only
way we can get to 12.

00:01:11.940 --> 00:01:14.900
We could also view
it as 3 groups of 4.

00:01:14.900 --> 00:01:16.230
So, let's look at that.

00:01:16.230 --> 00:01:25.430
So we could have it as 1
group of 4, 2 groups of 4,

00:01:25.430 --> 00:01:29.090
3 groups of 4.

00:01:29.090 --> 00:01:33.890
So now we could view 12
as being 3 groups of 4.

00:01:33.890 --> 00:01:46.410
Or we could say-- let me get the
right tool out-- that 3 times 4

00:01:46.410 --> 00:01:48.030
is equal to 12.

00:01:48.030 --> 00:01:51.260
So, whether we're doing
4 times 3 or 3 times 4,

00:01:51.260 --> 00:01:53.140
they're both going
to be equal to 12.

00:01:53.140 --> 00:01:57.380
4 groups of 3 is
12, 3 groups of 4.

00:01:57.380 --> 00:01:58.990
But we don't have to stop there.

00:01:58.990 --> 00:02:02.610
We could also view
12 as, well, we

00:02:02.610 --> 00:02:04.740
could view it as 2 groups of 6.

00:02:04.740 --> 00:02:06.960
Let's look at that.

00:02:06.960 --> 00:02:10.130
So this is 1 group
of 6 right over here.

00:02:10.130 --> 00:02:12.140
So, that's one 1 of 6.

00:02:12.140 --> 00:02:15.430
That's another group of 6.

00:02:15.430 --> 00:02:20.750
So, once again, we could
view this as 2 times 6.

00:02:20.750 --> 00:02:23.910
2 times 6 will also get us a 12.

00:02:23.910 --> 00:02:27.500
Well, what about doing
it as 6 groups of 2?

00:02:27.500 --> 00:02:30.640
Well, we can draw that
out, too-- 6 groups of 2.

00:02:30.640 --> 00:02:32.410
So that's 1 group of 2.

00:02:32.410 --> 00:02:34.805
Let me do that in
a different color.

00:02:34.805 --> 00:02:36.740
Let me do it in
this purple color.

00:02:36.740 --> 00:02:42.050
We have 1 group of 2, 2
groups of 2, 3 groups of 2,

00:02:42.050 --> 00:02:47.620
4 groups of 2, 5 groups
of 2, and 6 groups of 2.

00:02:47.620 --> 00:02:49.720
So once again, this
is all different ways

00:02:49.720 --> 00:02:52.270
of writing 12, something
equivalent to 12.

00:02:52.270 --> 00:03:02.200
We could write 6 times 2--
6 groups of 2-- 6 times 2

00:03:02.200 --> 00:03:04.670
is also equal 12.

00:03:04.670 --> 00:03:06.750
But we don't have to stop there.

00:03:06.750 --> 00:03:12.470
We could also literally
view 12 as 1 group of 12.

00:03:12.470 --> 00:03:13.840
So how would that look?

00:03:13.840 --> 00:03:16.020
So 1 group of 12.

00:03:16.020 --> 00:03:22.820
So this whole thing is
just 1 group of 12 here.

00:03:22.820 --> 00:03:34.080
So we could literally say
1 times 12 is equal to 12.

00:03:34.080 --> 00:03:36.600
We have one entire group of 12.

00:03:36.600 --> 00:03:39.154
1 times 12 is equal to 12.

00:03:39.154 --> 00:03:41.070
And we could think of
it the other way around.

00:03:41.070 --> 00:03:45.470
We could view this
as 12 groups of 1.

00:03:45.470 --> 00:03:46.990
Let me draw that.

00:03:46.990 --> 00:03:48.200
So 12 groups of 1.

00:03:48.200 --> 00:03:50.798
This is 1 group of
1, 2 groups of 1,

00:03:50.798 --> 00:04:03.680
3, 4, 5, 6, 7, 8, 9, 10,
11, and 12-- 12 groups of 1.

00:04:03.680 --> 00:04:09.070
So we could also write 12.

00:04:09.070 --> 00:04:12.820
12 groups, and in
each one, I have 1.

00:04:12.820 --> 00:04:17.541
Well, that's still
going to get me to 12.


Intro to multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:03.670
I have these three star patches,
I guess you could call them,

00:00:03.670 --> 00:00:04.710
right over here.

00:00:04.710 --> 00:00:08.210
And so I could say, if I had
one group of three star patches,

00:00:08.210 --> 00:00:10.470
how many star patches do I have?

00:00:10.470 --> 00:00:15.420
So I literally have one
group of three star patches.

00:00:15.420 --> 00:00:17.680
Well, that means that I
have three star patches.

00:00:17.680 --> 00:00:18.600
1, 2, 3.

00:00:18.600 --> 00:00:20.720
This is my one group of three.

00:00:20.720 --> 00:00:22.720
Now let's make it a little
bit more interesting.

00:00:22.720 --> 00:00:25.310
Let's say that I had two groups.

00:00:25.310 --> 00:00:28.050
Let's say that I had
two groups of three.

00:00:28.050 --> 00:00:32.390
So that's one group, and
then here's a second group.

00:00:32.390 --> 00:00:34.350
Here's two groups of three.

00:00:34.350 --> 00:00:37.270
So how many total star
patches do I have now?

00:00:37.270 --> 00:00:40.650
Well, I have two
groups of three.

00:00:43.490 --> 00:00:46.990
Or another way of thinking
about it is this is 3 plus 3.

00:00:46.990 --> 00:00:54.700
This is equal to 3 plus
3, which is equal to 6.

00:00:54.700 --> 00:00:58.100
So we see 1 times 3--
one group of 3 is 3.

00:00:58.100 --> 00:01:02.460
Two groups of 3, which is
literally two 3's, is 6.

00:01:02.460 --> 00:01:04.450
Let's make it even
more interesting.

00:01:04.450 --> 00:01:06.015
Let's have three groups of 3.

00:01:09.600 --> 00:01:13.450
Now, what is this
going to be equal to?

00:01:13.450 --> 00:01:15.150
Well, it's three groups of 3.

00:01:15.150 --> 00:01:20.770
So I could write this as
three groups, 3 times 3.

00:01:20.770 --> 00:01:23.620
And how many of these star
patches do I now have?

00:01:23.620 --> 00:01:27.140
Well, this is going
to be 3 plus 3 plus 3.

00:01:27.140 --> 00:01:30.530
It's going to be
3 plus 3 plus 3.

00:01:30.530 --> 00:01:32.630
Notice I have three 3's.

00:01:32.630 --> 00:01:34.370
I have two 3's.

00:01:34.370 --> 00:01:36.170
I have one 3.

00:01:36.170 --> 00:01:42.192
So this is 3 plus 3
plus 3 is equal to 9.

00:01:42.192 --> 00:01:43.150
And you can count them.

00:01:43.150 --> 00:01:47.410
1, 2, 3, 4, 5, 6, 7, 8, 9, or
you could just count by 3's.

00:01:47.410 --> 00:01:49.980
3, 6, 9.

00:01:49.980 --> 00:01:51.860
And I think you see
where this is going.

00:01:51.860 --> 00:01:53.540
Let's keep incrementing it.

00:01:53.540 --> 00:01:55.830
Let's get four groups of 3.

00:01:55.830 --> 00:01:57.820
So let's think about
what 4 times 3 is.

00:01:57.820 --> 00:02:04.070
1, 2, 3, and 4.

00:02:04.070 --> 00:02:06.590
This right over here
is four groups of 3.

00:02:06.590 --> 00:02:08.910
We could write this
down as 4 times

00:02:08.910 --> 00:02:19.480
3, which is the same thing
as 3 plus 3 plus 3 plus 3.

00:02:24.090 --> 00:02:25.810
Notice I have four 3's.

00:02:25.810 --> 00:02:29.420
One 3, two 3's,
three 3's, four 3's.

00:02:29.420 --> 00:02:34.080
One 3, two 3's,
three 3's, four 3's.

00:02:34.080 --> 00:02:40.610
So we get 3, 6, 9, 12.

00:02:40.610 --> 00:02:42.160
So what I encourage
you to do now,

00:02:42.160 --> 00:02:44.940
now that the video is almost
over, is to keep going.

00:02:44.940 --> 00:02:51.160
I want you to figure out what
5 times 3 is, and 6 times

00:02:51.160 --> 00:03:01.090
3, and 7 times 3, and 8 times 3,
and 9 times 3, and 10 times 3.

00:03:01.090 --> 00:03:02.424
And I'll give you a little hint.

00:03:02.424 --> 00:03:04.381
You don't always have to
draw the star patches,

00:03:04.381 --> 00:03:05.640
but it's nice to visualize it.

00:03:05.640 --> 00:03:08.320
We saw 4 times 3 is
literally four 3's.

00:03:08.320 --> 00:03:11.780
Well, 5 times 3 is
going to be five 3's.

00:03:11.780 --> 00:03:17.390
So 2, 3, 4, 5.

00:03:17.390 --> 00:03:24.920
Which is equal to
3, 6, 9, 12, 15.

00:03:24.920 --> 00:03:27.740
So I encourage you to think
about what all of these

00:03:27.740 --> 00:03:31.990
are after this video is done.


ATP hydrolysis mechanism

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:02.150
- We've talked a lot about ATP

00:00:02.150 --> 00:00:05.120
being the energy currency of cells,

00:00:05.120 --> 00:00:08.450
but I want to dig a little bit
deeper into that in this video.

00:00:08.450 --> 00:00:13.450
And as we'll see when
we go from ATP to ADP,

00:00:13.690 --> 00:00:18.690
ADP + a Phosphate group, we
have a release of free energy.

00:00:19.080 --> 00:00:20.220
If we look at just the system,

00:00:20.220 --> 00:00:23.450
ATP's free energy is over
here, but once hydrolysis

00:00:23.450 --> 00:00:26.650
has taken place and now it's
ADP + a Phosphate group,

00:00:26.650 --> 00:00:31.650
the free energy has dropped
by roughly 30.5 KJ/mol

00:00:33.720 --> 00:00:37.450
our Delta G is -30.5 Kilojoules per mole

00:00:37.450 --> 00:00:39.550
and if you watch the
videos on Gibbs Free Energy

00:00:39.550 --> 00:00:42.310
this tells us that this
is a spontaneous reaction.

00:00:42.310 --> 00:00:46.020
Delta, Delta G is less than zero,

00:00:46.020 --> 00:00:50.750
which tells us that this
is going to be spontaneous.

00:00:50.750 --> 00:00:52.120
Now when I first learned this I was like,

00:00:52.120 --> 00:00:54.020
well if it's going to be
spontaneous why wouldn't all of

00:00:54.020 --> 00:00:55.550
the ATP just spontaneously
and all of the water

00:00:55.550 --> 00:00:57.620
turn into ADP and just release it's energy

00:00:57.620 --> 00:00:59.920
as heat or whatever else?

00:00:59.920 --> 00:01:02.560
And the key is, it has
to get over this hump.

00:01:02.560 --> 00:01:03.790
You have this activation energy,

00:01:03.790 --> 00:01:05.650
it has to go uphill a little bit

00:01:05.650 --> 00:01:08.650
if there's no enzyme to catalyze it.

00:01:08.650 --> 00:01:11.320
And the reason why we have
this uphill hump is the way

00:01:11.320 --> 00:01:13.950
that ATP gets broken
up is that you have to,

00:01:13.950 --> 00:01:16.520
if we're talking about a water
molecule doing the hydrolysis

00:01:16.520 --> 00:01:18.750
which is typically what
people think about hydrolysis,

00:01:18.750 --> 00:01:21.240
although it can sometimes be
done by a different molecule,

00:01:21.240 --> 00:01:22.590
if you think about the water molecule

00:01:22.590 --> 00:01:24.990
what needs to happen is, you need

00:01:24.990 --> 00:01:28.420
this lone pair of electrons on this oxygen

00:01:28.420 --> 00:01:31.690
to be able to do what we
call a nucleophilic attack

00:01:31.690 --> 00:01:34.550
on this Phosphorous in
this phosphate group

00:01:34.550 --> 00:01:38.020
and if that happens, it forms this bond

00:01:38.020 --> 00:01:43.020
and then these electrons can
be taken back by this oxygen

00:01:43.250 --> 00:01:46.750
which gives its negative
charge right over there.

00:01:46.750 --> 00:01:48.350
Now you might say, "Okay,
this makes a lot of sense",

00:01:48.350 --> 00:01:50.450
but you have to rememeber
this, electrons are negative

00:01:50.450 --> 00:01:53.120
and they're surrounded by
these negative charges.

00:01:53.120 --> 00:01:55.920
So they have to overcome
getting close to these things

00:01:55.920 --> 00:01:58.550
so as they're approaching
these negative charges,

00:01:58.550 --> 00:02:00.910
they want to repel each other,
so you have to overcome that.

00:02:00.910 --> 00:02:03.590
And the way that it's
overcome is a class of enzymes

00:02:03.590 --> 00:02:08.590
called ATPases, ATPase.

00:02:12.850 --> 00:02:14.850
And what they do is,
remember these enzymes

00:02:14.850 --> 00:02:18.820
are these big protein
complexes and the ATP molecule

00:02:18.820 --> 00:02:21.820
combined in the right place.

00:02:21.820 --> 00:02:24.690
And they essentially try and
surround the ATP molecule

00:02:24.690 --> 00:02:26.550
with some positive ions.

00:02:26.550 --> 00:02:28.750
So let's say there's a positive ion here.

00:02:28.750 --> 00:02:31.550
So it can keep these electrons
busy while the water,

00:02:31.550 --> 00:02:33.620
or whatever is doing
the nucleophilic attack,

00:02:33.620 --> 00:02:36.920
can not have to worry about
these electrons over here.

00:02:36.920 --> 00:02:38.850
So it might have a positive ion here.

00:02:38.850 --> 00:02:41.020
And remember, if we think
in three dimensions,

00:02:41.020 --> 00:02:43.190
this thing is all wrapped
around in different ways

00:02:43.190 --> 00:02:48.190
around the ATP molecule,
so this is the ATPase.

00:02:48.420 --> 00:02:50.920
And so by having the enzyme over here,

00:02:50.920 --> 00:02:52.650
you're lowering the activation energy.

00:02:52.650 --> 00:02:55.399
And so it might end up looking
something more like this.

00:02:55.399 --> 00:02:57.790
And so the reaction can actually occur.

00:02:57.790 --> 00:02:59.290
So the reason why you don't
just see this happening

00:02:59.290 --> 00:03:00.820
all the time without an enzyme is why you

00:03:00.820 --> 00:03:02.050
have to overcome this hump.

00:03:02.050 --> 00:03:03.590
But once you have the enzymes
they can actually allow

00:03:03.590 --> 00:03:05.890
the reaction to occur.

00:03:05.890 --> 00:03:09.720
So this attacks this, forms this
bond, and then you say well,

00:03:09.720 --> 00:03:12.450
"You have another
hydrogen right over here,

00:03:12.450 --> 00:03:15.850
you have another hydrogen
right over here."

00:03:15.850 --> 00:03:18.720
But this could get nabbed
up by another water molecule

00:03:18.720 --> 00:03:20.450
becoming a hydronium ion, which is

00:03:20.450 --> 00:03:22.350
actually what's going to happen.

00:03:22.350 --> 00:03:25.420
And then, this character takes
away this pair of electrons,

00:03:25.420 --> 00:03:27.090
becomes a negative charge,
and we're left with

00:03:27.090 --> 00:03:30.790
that Phosphate group has
been broken off and then

00:03:30.790 --> 00:03:34.950
we have our ADP and we
have our release of energy.

00:03:34.950 --> 00:03:38.150
Now, typically you don't
want this to release energy

00:03:38.150 --> 00:03:39.419
for no good reason.

00:03:39.419 --> 00:03:42.050
Most ATPases are going
to leverage that energy

00:03:42.050 --> 00:03:45.950
that release this reaction
to phosphorylate a molecule.

00:03:45.950 --> 00:03:47.720
And in this case you can
kind of think that this

00:03:47.720 --> 00:03:49.390
hydroxide has been phosphorylated,

00:03:49.390 --> 00:03:51.620
but it might phosphorylate something else.

00:03:51.620 --> 00:03:55.910
Or it might change the
confirmation of this ATPase

00:03:55.910 --> 00:03:57.590
so it can do some other type of work,

00:03:57.590 --> 00:04:00.120
transfer molecules against
their concentration gradient

00:04:00.120 --> 00:04:03.160
or chemoelectric gradient,
whatever it might be.

00:04:03.160 --> 00:04:04.820
You don't want to just release the energy.

00:04:04.820 --> 00:04:06.250
Sometimes it might happen
if you just want to

00:04:06.250 --> 00:04:07.920
generate some heat, but
usually you're going to do

00:04:07.920 --> 00:04:10.820
some useful work for the actual cells.

00:04:10.820 --> 00:04:12.050
Hopefully this gives you a little more

00:04:12.050 --> 00:04:15.190
appreciation for ATP and
the reaction by which

00:04:15.190 --> 00:04:17.640
the hydrolysis can occur.

00:04:17.640 --> 00:04:19.650
This is called hydrolysis
because, at least in this example

00:04:19.650 --> 00:04:21.820
we're using a water
molecule to break it up

00:04:21.820 --> 00:04:25.300
or to take off a phosphate group.