17 Product of primes near 50

WEBVTT
Kind: captions
Language: en

00:00:00.683 --> 00:00:01.835
- [Voiceover] What is the product

00:00:01.835 --> 00:00:06.835
of the smallest prime number
that is greater than 50

00:00:07.013 --> 00:00:11.030
and the greatest prime
number that is less than 50?

00:00:11.030 --> 00:00:13.468
So let's just think about
the numbers around 50.

00:00:13.468 --> 00:00:17.787
So I don't know, I'll start
with five below and five above.

00:00:17.787 --> 00:00:22.787
So we go 45, 46, 47, 48, 49, 50,

00:00:25.551 --> 00:00:30.551
51, 52, 53, 54, and 55

00:00:30.983 --> 00:00:34.667
and I could obviously keep
going in both directions.

00:00:34.667 --> 00:00:36.641
Now we want to take the product

00:00:36.641 --> 00:00:39.496
of the smallest prime number
that is greater than 50.

00:00:39.496 --> 00:00:42.492
So let's start at 50 and
let's go greater than that,

00:00:42.492 --> 00:00:44.163
think about which of these are prime.

00:00:44.163 --> 00:00:46.834
Now 51 is always a little
tricky one, it feels prime

00:00:47.141 --> 00:00:51.083
but it's good to know that
that's 17 times three,

00:00:51.083 --> 00:00:53.103
and one way to test a
divisibility by three

00:00:53.103 --> 00:00:53.815
is to add these digits.

00:00:53.815 --> 00:00:55.680
Five plus one is equal to six,

00:00:55.680 --> 00:00:57.027
and we have other Khan Academy videos

00:00:57.027 --> 00:00:58.420
on why that actually works.

00:00:58.420 --> 00:01:00.254
But six is divisible by three,

00:01:00.254 --> 00:01:01.879
so this is not a prime number.

00:01:01.879 --> 00:01:04.851
52 is clearly even, so
it's not a prime number.

00:01:04.851 --> 00:01:06.872
And so the smallest prime number above 50

00:01:06.872 --> 00:01:08.590
is going to be 53.

00:01:08.590 --> 00:01:11.097
53. Try things out.

00:01:11.097 --> 00:01:14.836
53. Its only factors are one and itself.

00:01:14.836 --> 00:01:16.229
So this is the smallest prime number

00:01:16.229 --> 00:01:17.390
that is greater than 50,

00:01:17.605 --> 00:01:19.363
and what is the greatest prime number

00:01:19.363 --> 00:01:20.154
that is less than 50?

00:01:20.154 --> 00:01:22.614
So now let's start at 50 and go backwards,

00:01:22.614 --> 00:01:23.937
and find the greatest prime number

00:01:23.937 --> 00:01:25.006
that is less than 50.

00:01:25.006 --> 00:01:26.793
49 is seven times seven,

00:01:26.793 --> 00:01:28.930
this is even, and this is six times eight,

00:01:28.930 --> 00:01:32.804
47 is a prime number, its only
factors are one and itself.

00:01:33.987 --> 00:01:38.612
So really, they want us to
take the product of 53 and 47.

00:01:38.612 --> 00:01:43.612
So, 53 times 47,

00:01:45.731 --> 00:01:50.291
and we are going to get
seven times three is 21,

00:01:50.291 --> 00:01:55.291
seven times five is 35 plus two is 37,

00:01:55.562 --> 00:02:00.363
and now we can do four
times, put a zero here

00:02:00.363 --> 00:02:02.574
because we're really
multiplying with a 40,

00:02:02.574 --> 00:02:05.964
four times three is 12,

00:02:05.964 --> 00:02:09.441
four times five is 20 plus one is 21,

00:02:09.441 --> 00:02:12.697
and we get, one plus zero is one,

00:02:12.697 --> 00:02:15.615
seven plus two is nine,
three plus one is four,

00:02:15.615 --> 00:02:19.615
and we have a two, we get 2,491.


3 Solving simple equation

WEBVTT
Kind: captions
Language: en

00:00:00.913 --> 00:00:06.503
If 3 times a number is equal to 3 halves

00:00:06.503 --> 00:00:09.723
what is the number? so let's just say

00:00:09.787 --> 00:00:11.637
that our number..unknown number

00:00:11.637 --> 00:00:13.807
let's just call that "x". So we could say

00:00:13.815 --> 00:00:23.675
3 times our number x is equal to 3 halves

00:00:23.692 --> 00:00:26.012
so now we just have to solve for x

00:00:26.034 --> 00:00:27.574
so if we want to isolate an x here on the

00:00:27.574 --> 00:00:29.194
left hand side, the most obvious thing

00:00:29.197 --> 00:00:31.257
to do is 3 times x, if we divide by 3,

00:00:31.257 --> 00:00:32.927
we're gonna left with just x

00:00:32.927 --> 00:00:34.757
We can't just do that to one side

00:00:34.757 --> 00:00:36.227
of the equation if we want to

00:00:36.227 --> 00:00:37.283
maintain this equality

00:00:37.283 --> 00:00:38.527
we have to do it to both sides

00:00:38.527 --> 00:00:39.401
so you divide both

00:00:39.401 --> 00:00:42.157
sides by 3 and we're going to be left with

00:00:42.157 --> 00:00:46.137
x is equal to 3 halves divided by 3

00:00:46.137 --> 00:00:49.687
that is the same thing as 3 halves times

00:00:49.687 --> 00:00:55.297
one-third which is going to be

00:00:55.297 --> 00:00:58.947
equal to one half. You can verify that

00:00:58.953 --> 00:01:02.953
3 times one-half is 3 halves


12 Missing angle alternate solution

WEBVTT
Kind: captions
Language: en

00:00:00.828 --> 00:00:02.068
- [Voiceover] So let's go
it through another way,

00:00:02.068 --> 00:00:03.380
'cause this is a really
interesting problem.

00:00:03.380 --> 00:00:05.493
Let's think about another way
that we could have tackled it.

00:00:05.493 --> 00:00:07.216
So once again, x is equal to 20,

00:00:07.216 --> 00:00:08.751
so this is a 20 degree angle,

00:00:11.227 --> 00:00:13.445
And y is equal to 30,

00:00:13.445 --> 00:00:16.097
so this is a 30 degree angle.

00:00:16.421 --> 00:00:18.385
What is the value of z?

00:00:18.788 --> 00:00:19.788
So the last way we did it,

00:00:19.788 --> 00:00:21.965
we constructed a line right over here

00:00:21.965 --> 00:00:23.509
to figure out the value of z.

00:00:23.509 --> 00:00:25.212
We were able to figure out
that angle, then that angle,

00:00:25.212 --> 00:00:26.852
then this angle, then z.

00:00:26.852 --> 00:00:28.424
Another way we could have done it,

00:00:28.424 --> 00:00:31.233
is we could have constructed a line

00:00:32.186 --> 00:00:34.523
right over here.

00:00:34.907 --> 00:00:36.766
And we could say, well look,

00:00:36.766 --> 00:00:40.229
the measure of this angle is a degrees,

00:00:40.801 --> 00:00:42.693
and let's say the measure of this angle

00:00:42.693 --> 00:00:44.348
is b degrees.

00:00:44.664 --> 00:00:45.864
So what do we know?

00:00:45.864 --> 00:00:49.372
Well, we would know that a plus b,

00:00:51.986 --> 00:00:53.719
plus z,

00:00:56.103 --> 00:00:58.308
they're all the interior
angles of this smaller triangle

00:00:58.308 --> 00:00:59.208
right over here,

00:00:59.208 --> 00:01:01.748
so they're going to add up to 180.

00:01:02.288 --> 00:01:04.555
And what else do we know?

00:01:04.984 --> 00:01:07.329
Well, a plus x is this
whole angle right over here,

00:01:07.329 --> 00:01:08.522
so let's write that down.

00:01:08.522 --> 00:01:09.801
So if we have,

00:01:09.801 --> 00:01:11.938
so a plus x,

00:01:14.115 --> 00:01:16.299
is this whole angle right over there,

00:01:16.299 --> 00:01:17.419
and then we have

00:01:17.419 --> 00:01:20.475
plus b plus x,

00:01:20.475 --> 00:01:22.222
which is this whole angle over here.

00:01:22.222 --> 00:01:23.166
Actually let me not,

00:01:23.166 --> 00:01:24.582
instead of writing x let me write,

00:01:24.582 --> 00:01:25.741
'cause we know x is 20.

00:01:25.741 --> 00:01:28.977
So a plus 20,

00:01:31.580 --> 00:01:34.512
plus b plus 20,

00:01:36.448 --> 00:01:37.792
is this whole angle right over there.

00:01:37.792 --> 00:01:39.552
Then you add that to y, which is 30,

00:01:40.291 --> 00:01:41.988
so plus 30,

00:01:42.447 --> 00:01:45.244
that's going to be equal to 180.

00:01:45.287 --> 00:01:46.911
Because that's this whole angle,

00:01:46.911 --> 00:01:48.608
plus that whole angle,
plus that whole angle.

00:01:48.608 --> 00:01:50.384
It's the interior, some
of the interior angles

00:01:50.384 --> 00:01:51.983
of this larger triangle.

00:01:51.983 --> 00:01:53.852
And so let me write that down.

00:01:53.855 --> 00:01:56.371
That's going to be equal to 180.

00:01:56.648 --> 00:01:57.727
And what's interesting here is

00:01:57.727 --> 00:01:59.748
we can solve for what a plus b is.

00:01:59.748 --> 00:02:02.578
And if we know what a plus b
is, then we can solve for z.

00:02:02.704 --> 00:02:03.733
So let's do that.

00:02:03.733 --> 00:02:07.536
So we're gonna get a plus b,

00:02:09.814 --> 00:02:11.954
and then let's see, we're gonna have 20,

00:02:12.518 --> 00:02:14.167
let me do this in a new color.

00:02:14.167 --> 00:02:17.970
We're going to have 20 plus 20

00:02:18.566 --> 00:02:19.982
plus 30.

00:02:19.982 --> 00:02:22.650
So 20 plus 20 plus 30 is 40 plus 30 is 70.

00:02:22.733 --> 00:02:25.087
So plus 70

00:02:25.087 --> 00:02:28.365
is equal to 180.

00:02:28.856 --> 00:02:31.183
Or subtract 70 from both sides,

00:02:31.183 --> 00:02:35.381
you get a plus b

00:02:35.381 --> 00:02:37.698
is equal to 110.

00:02:38.206 --> 00:02:39.914
And now we can just
substitute back up here.

00:02:39.914 --> 00:02:43.046
A plus b is equal to 110.

00:02:44.530 --> 00:02:47.774
So we have z plus 110 is equal to 180,

00:02:48.146 --> 00:02:51.786
or z would be equal to 180

00:02:51.786 --> 00:02:53.823
minus 110.

00:02:55.038 --> 00:02:56.888
I'm just subtracting 110 from both sides

00:02:56.888 --> 00:02:58.659
of this equation right over here.

00:02:59.623 --> 00:03:03.499
And we would get z is equal to 70.

00:03:03.974 --> 00:03:04.814
And we're done!

00:03:04.814 --> 00:03:06.632
So there's many ways to actually tackle

00:03:06.632 --> 00:03:10.562
this pretty interesting
problem right over here.


Subtracting vectors exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.766 --> 00:00:03.300
Voiceover:These are
vectors A and B, all right?

00:00:03.300 --> 00:00:05.333
That's vector A, that's vector B.

00:00:05.366 --> 00:00:07.366
Which of the following describes a valid

00:00:07.366 --> 00:00:11.766
way of obtaining vector A minus vector B?

00:00:11.766 --> 00:00:13.800
So let's look at our
choices right over here.

00:00:13.833 --> 00:00:16.633
Here they just depicted
vector B, just like it's

00:00:16.633 --> 00:00:19.300
depicted here, it's maybe
shifted over a little bit.

00:00:19.300 --> 00:00:21.700
And then over here,
they depicted vector A,

00:00:21.700 --> 00:00:25.100
the negative of vector
A, and it looks like

00:00:25.100 --> 00:00:27.633
they're taking the negative of vector A,

00:00:28.479 --> 00:00:31.566
and to that they're adding
vector B, and then they're

00:00:31.566 --> 00:00:36.266
claiming that this magenta
vector is A minus B.

00:00:37.623 --> 00:00:40.100
If we ignore what they
wrote over here, what

00:00:40.100 --> 00:00:43.433
they've done is they've
added negative A to B.

00:00:43.433 --> 00:00:48.366
So this is the negative
of vector A plus vector B.

00:00:48.366 --> 00:00:51.833
Or we could think of it as
vector B minus vector A.

00:00:51.833 --> 00:00:53.900
So this one right over here is not right.

00:00:53.900 --> 00:00:57.133
If we swap the signs,
if this was a negative,

00:00:57.300 --> 00:01:00.433
if we put negative A plus
B, then we'd say okay,

00:01:00.433 --> 00:01:02.300
this is accurate, but
they want us to figure out

00:01:02.300 --> 00:01:05.100
valid ways of obtaining A
minus B and this isn't that.

00:01:05.100 --> 00:01:06.767
This is B minus A.

00:01:06.767 --> 00:01:08.333
Let's look at the other choice.

00:01:08.633 --> 00:01:10.166
You have vector A.

00:01:10.166 --> 00:01:11.833
Let's start at the tail of vector A,

00:01:11.833 --> 00:01:13.566
get to the head of vector A.

00:01:13.566 --> 00:01:17.633
Over here we have the tail
of the negative of vector B,

00:01:17.633 --> 00:01:19.900
so they essentially flipped
it over when you take the

00:01:19.900 --> 00:01:23.366
negative of it, and so we're
adding negative B to A.

00:01:23.366 --> 00:01:26.166
If we take the tail of
where we started to the head

00:01:26.166 --> 00:01:29.633
of where we ended, yes,
this would be vector A

00:01:29.633 --> 00:01:32.500
plus negative vector B,
which is the same thing

00:01:32.500 --> 00:01:38.100
as vector A minus vector
B, so that one is right.

00:01:38.100 --> 00:01:42.733
That's a valid way of obtaining
vector A minus vector B.


Ben Eater eats the numbers

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Kind: captions
Language: en

00:00:00.586 --> 00:00:02.456
- [Sal] Esther, I'm in so much trouble!

00:00:03.176 --> 00:00:04.617
- [Esther] What happened, Sal?

00:00:05.387 --> 00:00:08.449
- [Sal] Ben Eater, the number
eater, ate all my numbers

00:00:08.449 --> 00:00:10.551
on my perfect number grid.

00:00:10.551 --> 00:00:14.544
- [Ben] Hahahaha! Those
were some good numbers!

00:00:14.544 --> 00:00:17.050
(chewing)

00:00:17.050 --> 00:00:20.499
I better get out of here
before I get into some trouble!

00:00:21.083 --> 00:00:22.502
- [Sal] Ben!

00:00:22.817 --> 00:00:24.387
Esther, what do I do?

00:00:24.387 --> 00:00:26.217
- [Esther] It's okay, Sal.

00:00:26.217 --> 00:00:29.400
We can work together to
find the missing numbers.

00:00:29.650 --> 00:00:31.650
Actually, I think our friends who are

00:00:31.650 --> 00:00:34.284
watching the video can help us with it.

00:00:34.284 --> 00:00:37.217
So, friend, go ahead and pause the video

00:00:37.217 --> 00:00:40.933
and see if you can figure
out the missing numbers.

00:00:42.121 --> 00:00:43.550
- [Sal] Okay, I'm assuming our friends

00:00:43.550 --> 00:00:45.151
have given a go at it.

00:00:45.151 --> 00:00:47.485
So how would you think about this, Esther?

00:00:47.485 --> 00:00:49.667
- [Esther] Let's see, hmm.

00:00:49.883 --> 00:00:53.984
Well, maybe I can use the
numbers that are before

00:00:53.984 --> 00:00:57.302
the missing number to
help us figure this out.

00:00:57.483 --> 00:01:00.150
So let's see, if we go from

00:01:00.150 --> 00:01:05.115
4, 5, 6, 7!

00:01:05.985 --> 00:01:07.533
- [Sal] That makes sense.

00:01:08.315 --> 00:01:09.817
What about this one?

00:01:09.817 --> 00:01:11.217
- [Esther] Hmm.

00:01:11.217 --> 00:01:14.083
Let's see, let's see if we
can figure out what would

00:01:14.083 --> 00:01:17.367
come after the number 13.

00:01:17.757 --> 00:01:18.921
- [Sal] Maybe we should get our friends

00:01:18.921 --> 00:01:20.483
to think about it first.

00:01:20.483 --> 00:01:21.897
- [Esther] Good idea.

00:01:22.650 --> 00:01:23.858
- [Sal] So I'm assuming they tried.

00:01:23.858 --> 00:01:24.883
So how do you think about it?

00:01:24.883 --> 00:01:27.187
What number comes after 13?

00:01:27.187 --> 00:01:30.567
- [Esther] Let's see, 13, 14!

00:01:31.327 --> 00:01:32.388
- [Sal] And that works out, because then

00:01:32.388 --> 00:01:35.075
14 is the number that comes before 15.

00:01:35.075 --> 00:01:36.587
- [Esther] Exactly.

00:01:36.587 --> 00:01:37.751
- [Sal] I see.

00:01:37.751 --> 00:01:40.183
Now look at this, this
is all the way out here.

00:01:40.183 --> 00:01:42.456
How are you going to figure
this one out, Esther?

00:01:42.456 --> 00:01:44.333
- [Esther] Hmm, let's see.

00:01:44.516 --> 00:01:46.350
Maybe I can look at the numbers that come

00:01:46.350 --> 00:01:48.267
after the missing number.

00:01:48.484 --> 00:01:53.451
So if we have 37 and 38 and 39,

00:01:53.451 --> 00:01:58.267
I think 36 comes before 37!

00:01:59.084 --> 00:02:00.916
- [Sal] Oh, that makes a lot of sense.

00:02:01.366 --> 00:02:03.724
And what about this one
that's right below it?

00:02:03.724 --> 00:02:07.225
- [Esther] Okay, let's see,
I think I'm gonna count

00:02:07.225 --> 00:02:09.017
in a different way.

00:02:09.017 --> 00:02:12.183
I'm gonna go from the top to the bottom.

00:02:12.183 --> 00:02:14.060
- [Sal] Wow, how are you going to do that?

00:02:14.060 --> 00:02:17.017
- [Esther] Let's see, I'll
start with number 6, and I'll

00:02:17.017 --> 00:02:21.970
go down to 16, go down to 26,

00:02:21.970 --> 00:02:23.790
go down to 36.

00:02:23.790 --> 00:02:25.049
- [Sal] I think I see a pattern here.

00:02:25.049 --> 00:02:27.284
All of these numbers have a 6.

00:02:27.284 --> 00:02:29.949
- [Esther] Exactly. So I think if we're

00:02:29.949 --> 00:02:33.834
in the missing number, that's gonna be 46.

00:02:34.283 --> 00:02:36.050
- [Sal] Oh, I see, that makes sense.

00:02:36.050 --> 00:02:40.754
Cos here this was, we're in 26, then 36,

00:02:40.986 --> 00:02:42.722
here we're in the 40s.

00:02:42.722 --> 00:02:44.550
So it's going to be 46.

00:02:44.550 --> 00:02:45.993
- [Esther] Exactly!

00:02:45.993 --> 00:02:47.118
- [Sal] That's really cool.

00:02:47.118 --> 00:02:48.484
Now what about these three down here?

00:02:48.484 --> 00:02:50.283
Maybe our friends can pause first and see

00:02:50.283 --> 00:02:52.067
if they can figure it out.

00:02:52.151 --> 00:02:53.700
- [Esther] Sounds good!

00:02:54.683 --> 00:02:57.617
- [Sal] Okay, how would
I tackle this, Esther?

00:02:57.617 --> 00:03:00.883
- [Esther] Let's see. Well,
there are three missing numbers

00:03:00.883 --> 00:03:05.400
all in a row, so let's look at
the numbers before and after.

00:03:05.817 --> 00:03:10.817
So we have 81, 82, 83, 84!

00:03:11.850 --> 00:03:14.051
- [Sal] Oh, that makes
sense cos we're in the 80s.

00:03:14.051 --> 00:03:16.919
And so just like we have 1, 2, 3, 4,

00:03:16.919 --> 00:03:20.502
now it's 81, 82, 83, 84.

00:03:20.818 --> 00:03:22.017
- [Esther] Exactly.

00:03:22.017 --> 00:03:26.801
And I know the number
that is next to 84 is 85.

00:03:27.184 --> 00:03:29.950
- [Sal] And we know that
because 5 comes after 4,

00:03:29.950 --> 00:03:32.951
so 85 comes after 84.

00:03:32.951 --> 00:03:33.883
- [Esther] Bingo!

00:03:33.883 --> 00:03:35.914
- [Sal] Wow. And this would be...

00:03:35.914 --> 00:03:37.708
- [Both] 86!

00:03:38.717 --> 00:03:40.251
- [Sal] Very exciting.

00:03:40.251 --> 00:03:41.884
- [Esther] Very fun.

00:03:41.884 --> 00:03:43.817
("Charge")

00:03:43.817 --> 00:03:46.083
- [Sal] Oh no, it's Ben
Eater the number eater again!

00:03:46.083 --> 00:03:48.924
- [Ben] Oh, more numbers to munch on!

00:03:48.924 --> 00:03:50.907
I'm getting hungry!


Visualizing vector addition examples

WEBVTT
Kind: captions
Language: en

00:00:00.593 --> 00:00:02.070
Voiceover:Sort the drawings according

00:00:02.070 --> 00:00:03.923
to the equation that correctly describes

00:00:03.923 --> 00:00:07.029
the relationship between
the vectors drawn.

00:00:07.029 --> 00:00:09.216
An equation may have any number of cards,

00:00:09.216 --> 00:00:11.500
and cards with no corresponding equation

00:00:11.500 --> 00:00:12.832
should be left unsorted.

00:00:12.832 --> 00:00:14.451
Let's see what they're talking about here.

00:00:14.451 --> 00:00:16.703
Right over here, they have depicted,

00:00:16.703 --> 00:00:19.564
let's see, they're adding
vector A to vector B.

00:00:19.564 --> 00:00:21.265
We see that, because vector B's tail

00:00:21.265 --> 00:00:23.721
is at the head of vector A.

00:00:23.721 --> 00:00:25.791
They're adding vector A to vector B.

00:00:25.791 --> 00:00:28.021
If you connect the tail of vector A

00:00:28.021 --> 00:00:30.039
to the head of Vector B,

00:00:30.039 --> 00:00:31.351
you get vector C.

00:00:31.351 --> 00:00:32.264
They're really showing that

00:00:32.264 --> 00:00:35.813
vector A plus vector B
is equal to vector C.

00:00:35.813 --> 00:00:37.550
Let's see if we can find that.

00:00:37.550 --> 00:00:41.646
A is showing that vectors A+B,

00:00:41.646 --> 00:00:44.811
vector A plus vector B
is equal to vector C.

00:00:44.980 --> 00:00:46.962
Let's look at the second choice.

00:00:46.962 --> 00:00:50.148
This choice, right over here is showing

00:00:50.148 --> 00:00:56.323
that vector B plus vector
C is equal to vector A.

00:00:56.323 --> 00:00:58.579
So B+C is equal to A.

00:00:58.870 --> 00:01:01.603
That is this choice, right over here.

00:01:01.603 --> 00:01:03.980
Diagram or graph B is showing us

00:01:03.980 --> 00:01:06.553
that vector B plus vector
C is equal to vector A.

00:01:06.814 --> 00:01:10.115
Now let's look at this
choice, right over here.

00:01:10.222 --> 00:01:17.603
This is showing us that B+A is equal to C.

00:01:18.756 --> 00:01:20.580
It doesn't matter which
way you order them.

00:01:20.795 --> 00:01:24.153
The way they've set it up,

00:01:24.153 --> 00:01:26.277
this is B+A is equal to C.

00:01:27.984 --> 00:01:33.477
That is not one of the choices,

00:01:33.477 --> 00:01:37.082
but you could, whether you're doing A+B,

00:01:37.082 --> 00:01:39.243
or B+A, it's going to be the same thing.

00:01:39.243 --> 00:01:41.069
I'm gonna throw it into this category,

00:01:41.069 --> 00:01:42.312
right over here.

00:01:42.312 --> 00:01:44.228
This is the same thing as A+B.

00:01:44.228 --> 00:01:47.872
Instead of putting the
tail of B at the head of A,

00:01:47.872 --> 00:01:49.908
they just put the tail
of A at the head of B.

00:01:49.908 --> 00:01:50.995
It doesn't matter what order

00:01:50.995 --> 00:01:53.687
you are adding those two vectors.

00:01:53.687 --> 00:01:55.906
You're still going to
get the same, vector C.

00:01:55.906 --> 00:01:57.752
Finally, this right over here,

00:01:57.752 --> 00:02:00.488
this is C+A is equal to B.

00:02:02.180 --> 00:02:07.204
This is C+A or A+C is equal to B.

00:02:07.204 --> 00:02:08.972
It doesn't matter if we
do vector A plus vector C

00:02:08.972 --> 00:02:11.267
or vector C plus vector A.

00:02:11.267 --> 00:02:13.012
Let's just make sure that

00:02:13.012 --> 00:02:17.469
we've got all of these right, and we have.


Sal talks to CEO of the College Board, David Coleman

WEBVTT
Kind: captions
Language: en

00:00:02.472 --> 00:00:04.286
- So I'm here with David Coleman

00:00:04.286 --> 00:00:05.919
who's the president of the College Board,

00:00:05.919 --> 00:00:07.286
and we have a pretty big

00:00:07.286 --> 00:00:09.573
announcement/collaboration together.

00:00:09.573 --> 00:00:11.422
Tell us more about what
we're doing, David.

00:00:11.422 --> 00:00:12.765
- I'm super excited,

00:00:12.765 --> 00:00:14.365
'cause today the College
Board is announcing

00:00:14.365 --> 00:00:15.783
a broader and renewed commitment

00:00:15.783 --> 00:00:16.989
to delivering opportunity.

00:00:16.989 --> 00:00:19.102
Which means going beyond
an assessment itself

00:00:19.102 --> 00:00:20.837
to make sure kids take advantage of

00:00:20.837 --> 00:00:22.154
the opportunities they earned.

00:00:22.154 --> 00:00:24.551
But also to dare to take seriously

00:00:24.551 --> 00:00:27.097
the corruptions that have
risen around the exam process.

00:00:27.244 --> 00:00:29.182
What I mean is the perception

00:00:29.182 --> 00:00:32.677
that costly test preparation
has made the ground uneven.

00:00:33.003 --> 00:00:35.074
And this exam at its finest
should be about merit,

00:00:35.074 --> 00:00:37.300
should be about hard work
that's equally available to all.

00:00:37.501 --> 00:00:39.822
And there is no better
statement of world class,

00:00:39.822 --> 00:00:41.132
high quality stuff for all,

00:00:41.132 --> 00:00:42.771
than partnering with Kahn Academy,

00:00:42.771 --> 00:00:45.817
to provide free SAT test
preparation for the world,

00:00:45.817 --> 00:00:47.418
which is what we commit to today.

00:00:47.418 --> 00:00:48.745
- And I just wanna, you know,

00:00:48.745 --> 00:00:50.944
I guess doubly underline
what you just said.

00:00:50.944 --> 00:00:53.081
I mean, we all remember
when we took the SAT

00:00:53.081 --> 00:00:55.009
there's you know, there are some folks

00:00:55.009 --> 00:00:57.699
taking these fairly costly prep courses

00:00:57.699 --> 00:01:00.224
based on the idea that it might help them,

00:01:00.224 --> 00:01:02.320
and maybe on the margin it did help them.

00:01:02.320 --> 00:01:05.016
And what we're saying together is,

00:01:05.616 --> 00:01:07.426
that ability to pay for that,

00:01:07.426 --> 00:01:09.314
should not be a factor anymore.

00:01:10.437 --> 00:01:11.412
- You're exactly right.

00:01:11.412 --> 00:01:12.603
And what should be a factor,

00:01:12.603 --> 00:01:13.984
is the quality of your practice.

00:01:14.491 --> 00:01:17.233
And your command of
mathematics for example.

00:01:17.233 --> 00:01:19.559
Which is as you know to
command a few things well.

00:01:19.559 --> 00:01:21.217
So as we redesign the exam,

00:01:21.217 --> 00:01:23.197
we're allowing kids to really focus

00:01:23.197 --> 00:01:24.716
on the math that matters most,

00:01:24.716 --> 00:01:26.534
that's the most flexible and widely used.

00:01:26.534 --> 00:01:28.918
And the only way to gain
real command of that

00:01:28.918 --> 00:01:30.084
is through practice.

00:01:30.084 --> 00:01:31.881
- And what I'm really excited about,

00:01:31.881 --> 00:01:34.268
you know when we first
started talking about this,

00:01:34.268 --> 00:01:35.627
this isn't just you know,

00:01:35.627 --> 00:01:36.735
"Hey, Kahn Academy,

00:01:36.735 --> 00:01:38.251
"why don't you just try
to see what you can do

00:01:38.251 --> 00:01:39.740
"and make some test-prep materials".

00:01:39.740 --> 00:01:42.518
it's really about a
very close collaboration

00:01:42.518 --> 00:01:43.837
where we work together.

00:01:43.837 --> 00:01:45.212
And not just to make a,

00:01:45.212 --> 00:01:46.948
"Hey, you know if you
can't afford something,

00:01:46.948 --> 00:01:47.963
"here's something nice".

00:01:47.963 --> 00:01:49.684
But to make something truly world class

00:01:49.684 --> 00:01:51.427
so that we can make something that's

00:01:51.427 --> 00:01:52.811
as good or better

00:01:52.811 --> 00:01:56.013
than any of the high-cost
courses or materials,

00:01:56.013 --> 00:01:57.182
but do it for free.

00:01:57.182 --> 00:01:59.294
- That's exactly our shared aspiration.

00:01:59.294 --> 00:02:01.429
And we wanna go beyond
narrow test preparation,

00:02:01.429 --> 00:02:04.390
to really preparing kids
for an understanding

00:02:04.390 --> 00:02:06.884
and skill and fluency
with math and literacy,

00:02:06.884 --> 00:02:08.208
that allows them to encounter

00:02:08.208 --> 00:02:09.821
a wide range of problems with confidence.

00:02:10.359 --> 00:02:12.397
- And I think that's
where we're very aligned.

00:02:12.397 --> 00:02:13.863
Because the one thing even when my,

00:02:13.863 --> 00:02:15.280
my cousins were,

00:02:15.280 --> 00:02:17.189
or even when a lot of my
friends were taking the SAT,

00:02:17.189 --> 00:02:18.655
they would take these courses and say,

00:02:18.655 --> 00:02:19.670
"Oh, when in doubt guess C,

00:02:19.670 --> 00:02:21.389
"and rule this answer out".
- [David] Exactly.

00:02:21.389 --> 00:02:22.934
- And I would always tell people no,

00:02:22.934 --> 00:02:26.093
if you really understand
the underlying mathematics

00:02:26.093 --> 00:02:27.459
or the underlying grammar,

00:02:28.428 --> 00:02:29.909
you don't have to play these games.

00:02:29.909 --> 00:02:32.381
That the answers will come to you,

00:02:32.381 --> 00:02:34.144
and so the way we're going to tackle this

00:02:34.144 --> 00:02:36.526
isn't about kind of gaming the system.

00:02:36.526 --> 00:02:39.357
It's really about getting
those core fundamental ideas

00:02:39.357 --> 00:02:41.702
so not only will you do well
on the SAT and other tests,

00:02:41.702 --> 00:02:43.245
you'll actually be prepared for college.

00:02:43.245 --> 00:02:45.830
- And the redesigned SAT
makes that clearer than ever.

00:02:45.830 --> 00:02:47.493
That's it's no longer about tricks,

00:02:47.493 --> 00:02:49.461
guessing is not penalized,

00:02:49.461 --> 00:02:50.837
you put your best answer forward.

00:02:50.837 --> 00:02:52.147
We're talking about reading with

00:02:52.147 --> 00:02:53.730
attention to evidence when you read.

00:02:53.730 --> 00:02:55.727
We're talking about a core of math that is

00:02:55.727 --> 00:02:57.422
flexible, as you say, for the future.

00:02:57.422 --> 00:02:59.262
So it's much less about tricks,

00:02:59.262 --> 00:03:00.576
about mysterious things,

00:03:00.576 --> 00:03:02.949
than an open exam that
celebrates good work.

00:03:03.239 --> 00:03:04.806
And the work you've done
in your high school.

00:03:04.806 --> 00:03:07.442
- Well, all of us, the whole
team here at Kahn Academy,

00:03:07.442 --> 00:03:09.670
you know we've always been
all about the student,

00:03:09.670 --> 00:03:12.461
helping the student kind of
take control of their learning.

00:03:12.638 --> 00:03:14.862
And we're honored that you guys would

00:03:15.861 --> 00:03:17.325
work with us on this, once again,

00:03:17.325 --> 00:03:19.774
very close partnership to make not just a

00:03:19.774 --> 00:03:22.206
nice to have tool for students,

00:03:22.206 --> 00:03:24.598
but a really interactive, deep experience,

00:03:24.598 --> 00:03:26.065
that is as good as,

00:03:26.065 --> 00:03:28.029
or frankly I think it's
going to be much better

00:03:28.029 --> 00:03:29.104
than anything else out there.

00:03:29.104 --> 00:03:31.270
- No, to be clear, this will
be the only place in the world,

00:03:31.270 --> 00:03:32.389
and free to the world,

00:03:32.389 --> 00:03:33.630
besides on our own website,

00:03:33.630 --> 00:03:35.677
that students will be able to encounter

00:03:35.677 --> 00:03:37.064
materials for the exam

00:03:37.064 --> 00:03:40.317
that are ensurely focused
on the core of the math

00:03:40.317 --> 00:03:41.957
and the literacy that matters most.

00:03:41.957 --> 00:03:43.967
We will partner with the
Kahn Academy to ensure

00:03:43.967 --> 00:03:46.339
the quality and validity of every item,

00:03:46.339 --> 00:03:47.588
every practice item,

00:03:47.588 --> 00:03:49.779
and make sure that this
work we do together

00:03:49.779 --> 00:03:51.220
is of the highest quality.

00:03:51.220 --> 00:03:52.963
And there will be no
other such partnerships,

00:03:52.963 --> 00:03:54.676
so I think it can be
the very best there is.

00:03:54.676 --> 00:03:55.966
- Super exciting.

00:03:55.966 --> 00:03:57.570
- Thank you.


Adding 8 + 7

WEBVTT
Kind: captions
Language: en

00:00:00.433 --> 00:00:01.366
Voiceover:Let's try to think about

00:00:01.366 --> 00:00:05.333
what 8 plus 7 is equal to.

00:00:05.700 --> 00:00:06.966
And we could imagine that this is like

00:00:06.966 --> 00:00:10.700
taking 8 tomatoes and adding 7 blueberries

00:00:10.700 --> 00:00:12.433
and then saying, "How many pieces of fruit

00:00:12.433 --> 00:00:14.000
"do you have in total?"

00:00:14.500 --> 00:00:16.166
I encourage you to pause the video and try

00:00:16.166 --> 00:00:17.666
this out on your own.

00:00:18.433 --> 00:00:20.666
So, I'm assuming you've given a try.

00:00:21.300 --> 00:00:23.466
So let's think about it a little bit.

00:00:23.833 --> 00:00:25.233
And the way I'm going
to think about it is,

00:00:25.233 --> 00:00:28.700
I'm going to keep adding
tomatoes until I fill up

00:00:28.700 --> 00:00:31.367
my first bucket of 10
and then I'm going to see

00:00:31.367 --> 00:00:34.387
how many more I have to add to get,

00:00:34.387 --> 00:00:37.000
to essentially, add a total of 7.

00:00:37.366 --> 00:00:45.200
So let's have 1 tomato,
2 tomatoes, 3 tomatoes,

00:00:46.034 --> 00:00:55.033
4 tomatoes, 5 tomatoes, 6 tomatoes, 7,

00:00:55.033 --> 00:00:57.533
and 8 tomatoes.

00:00:57.633 --> 00:00:59.166
Now, to those 8 tomatoes I'm going to

00:00:59.166 --> 00:01:00.933
add 7 blueberries.

00:01:01.767 --> 00:01:08.333
So that's going to be 1, 2 ...

00:01:08.566 --> 00:01:09.633
And actually, I'm going to pause there.

00:01:09.633 --> 00:01:11.866
Because I just finished a group of 10.

00:01:12.300 --> 00:01:14.833
This, right over here, I've only added 2

00:01:14.833 --> 00:01:15.700
of the blueberries.

00:01:15.700 --> 00:01:17.367
I still have some more blueberries to add,

00:01:17.367 --> 00:01:19.833
but just with adding those 2 blueberries,

00:01:19.833 --> 00:01:22.134
I have completed a group of 10.

00:01:22.966 --> 00:01:25.933
So I already have 10 pieces of fruit.

00:01:26.300 --> 00:01:27.767
So I can represent that.

00:01:27.767 --> 00:01:30.500
I know, since I have a group of 10 here,

00:01:30.500 --> 00:01:33.666
that I have 1 group of 10.

00:01:34.167 --> 00:01:35.933
So this is going to be a 2 digit number,

00:01:36.033 --> 00:01:38.400
where the digit on the
left is going to be a 1.

00:01:38.766 --> 00:01:40.233
But I'm not done adding blueberries.

00:01:40.233 --> 00:01:41.600
I've only added 2 blueberries.

00:01:41.966 --> 00:01:43.800
Now let me add a third.

00:01:44.166 --> 00:01:48.800
3 blueberries, 4
blueberries, 5 blueberries,

00:01:49.633 --> 00:01:53.933
6 blueberries, and 7 blue berries.

00:01:54.633 --> 00:01:57.233
So I have a group of 10 and then,

00:01:57.233 --> 00:01:59.546
above and beyond filling
out that group of 10,

00:01:59.546 --> 00:02:01.966
which is represented here,
because this is going

00:02:01.966 --> 00:02:03.600
to be a 2 digit number.

00:02:03.766 --> 00:02:05.333
This is the left digit.

00:02:05.366 --> 00:02:08.100
So this 1, since it's in
the tens place, it literally

00:02:08.100 --> 00:02:12.800
represents 1 group of 10,
which is right over there.

00:02:13.168 --> 00:02:15.333
And then how many more do I have?

00:02:15.766 --> 00:02:17.600
Well I have another ...

00:02:17.633 --> 00:02:20.200
I have another 5, 1's.

00:02:20.833 --> 00:02:22.933
5, 1's right over here.

00:02:23.366 --> 00:02:27.533
So we have 1, 10 and we have 5, 1's.

00:02:28.233 --> 00:02:30.000
So what is 8 plus 7?

00:02:30.233 --> 00:02:32.866
It is 15.

00:02:33.166 --> 00:02:34.901
And another way, just to remind ourselves

00:02:34.901 --> 00:02:37.000
what 15 represents.

00:02:37.100 --> 00:02:40.867
The 1 in 15 represents 1, 10.

00:02:41.100 --> 00:02:42.233
Let me write it down.

00:02:42.233 --> 00:02:45.866
It represents, it represents 1, 10.

00:02:46.766 --> 00:02:49.466
And the 5 represents 5, 1's.

00:02:50.100 --> 00:02:53.133
5, 1's.

00:02:54.033 --> 00:02:56.017
And so another way of thinking about it,

00:02:56.017 --> 00:03:00.207
1, 10 is the same thing as 10 and 5, 1's

00:03:00.207 --> 00:03:03.333
is the same thing as 5.

00:03:03.701 --> 00:03:06.200
So 8 plus 7 is 15.

00:03:06.433 --> 00:03:07.934
That was the whole point of this video.

00:03:08.100 --> 00:03:09.566
But we also want to think about what

00:03:09.566 --> 00:03:11.201
these numbers represent.

00:03:11.300 --> 00:03:15.266
15 literally represents 1, 10 plus 5, 1's.

00:03:15.300 --> 00:03:17.533
Or 10 plus 5.


Cousin Fal's shape collection

WEBVTT
Kind: captions
Language: en

00:00:00.605 --> 00:00:02.424
- [Voiceover] Oh look, it's my cousin Fal!

00:00:02.424 --> 00:00:03.952
Cousin Fal, what can I do for you?

00:00:03.952 --> 00:00:05.184
- [Voiceover] Hey Sal, what's up?

00:00:05.184 --> 00:00:07.128
- [Voiceover] Oh, well not
much, I'm just sitting here

00:00:07.128 --> 00:00:10.407
thinking about what
video I should do next.

00:00:10.428 --> 00:00:11.799
- [Voiceover] Well then
maybe you can help me.

00:00:11.799 --> 00:00:13.141
I'm starting a shape collection.

00:00:13.141 --> 00:00:15.341
- [Voiceover] Okay, so
how do you need help?

00:00:15.341 --> 00:00:16.949
- [Voiceover] Well look
at these shapes over here!

00:00:16.949 --> 00:00:18.393
I have no way to classify them!

00:00:19.036 --> 00:00:21.772
- [Voiceover] Okay, well
there's a bunch of ways

00:00:21.772 --> 00:00:23.164
that you could classify them.

00:00:23.164 --> 00:00:25.588
The first way, you could think about

00:00:25.588 --> 00:00:28.828
how many sides each of these shapes have,

00:00:28.828 --> 00:00:30.081
so that's one way

00:00:30.372 --> 00:00:31.428
to think about it,

00:00:31.428 --> 00:00:32.760
how many sides they have.

00:00:33.188 --> 00:00:34.262
- [Voiceover] How do I figure that out?

00:00:34.262 --> 00:00:37.114
- [Voiceover] Okay, well
let's look at each of these

00:00:37.412 --> 00:00:39.287
and think about how many sides they have.

00:00:39.667 --> 00:00:42.798
So this first shape right over here

00:00:42.798 --> 00:00:44.560
this is one side.

00:00:44.936 --> 00:00:47.140
We see this kind of
straight right over here,

00:00:47.140 --> 00:00:50.427
it's the straight line
between these two corners,

00:00:50.427 --> 00:00:51.719
so that's one side.

00:00:52.293 --> 00:00:54.846
Two sides, three sides,

00:00:55.611 --> 00:00:56.754
and four sides.

00:00:57.042 --> 00:00:59.671
So this shape right over
here has four sides.

00:00:59.930 --> 00:01:01.275
- [Voiceover] Okay, I think I get it.

00:01:01.275 --> 00:01:02.699
What about this shape right over here?

00:01:02.699 --> 00:01:04.559
- [Voiceover] Okay,well let's see.

00:01:04.842 --> 00:01:06.322
This one, well same idea,

00:01:06.322 --> 00:01:08.945
this one also looks like it has one,

00:01:08.945 --> 00:01:12.420
two, three,

00:01:13.297 --> 00:01:14.693
and four.

00:01:14.898 --> 00:01:16.522
Four sides.

00:01:16.904 --> 00:01:18.008
- [Voiceover] What about this one?

00:01:18.008 --> 00:01:19.592
- [Voiceover] Okay, well this one,

00:01:19.592 --> 00:01:20.865
actually this is interesting.

00:01:20.865 --> 00:01:22.594
This one actually had no corners,

00:01:22.594 --> 00:01:24.565
and it has no sides,

00:01:24.746 --> 00:01:27.176
there's no straight edges over here,

00:01:27.176 --> 00:01:28.808
this thing is all curved.

00:01:28.808 --> 00:01:30.512
So since it has no corners,

00:01:30.512 --> 00:01:32.409
it also has no sides,

00:01:32.409 --> 00:01:34.045
or at least the way that
I'm thinking about it.

00:01:34.225 --> 00:01:35.835
So I would say that this one over here

00:01:35.835 --> 00:01:37.620
has zero sides.

00:01:39.482 --> 00:01:40.906
Let me write that a
little bit neater for you.

00:01:40.906 --> 00:01:43.018
- [Voiceover] Okay yeah, I like
to write things really neat.

00:01:43.018 --> 00:01:44.859
- [Voiceover] Okay, zero sides.

00:01:44.859 --> 00:01:47.118
And it has zero corners.

00:01:47.194 --> 00:01:48.619
Actually let me write
them for all of them.

00:01:48.619 --> 00:01:51.016
This one has four sides,

00:01:51.501 --> 00:01:53.143
and we could also count the corners.

00:01:53.235 --> 00:01:55.486
We have one, that's
where the two sides meet.

00:01:55.645 --> 00:01:58.942
One, two, three, four corners.

00:02:01.791 --> 00:02:03.115
This shape over here,

00:02:03.115 --> 00:02:05.401
it also has four sides,

00:02:05.401 --> 00:02:06.569
we already counted that and it has

00:02:06.569 --> 00:02:10.117
one, two, three, four corners.

00:02:10.505 --> 00:02:12.945
So four sides and four
corners actually describes

00:02:12.945 --> 00:02:14.570
both of these shapes,

00:02:14.570 --> 00:02:17.220
this one has zero sides and zero corners--

00:02:17.305 --> 00:02:18.561
- [Voiceover] What about the green one?

00:02:18.561 --> 00:02:20.635
- [Voiceover] Okay, well let's just see.

00:02:20.635 --> 00:02:22.870
One side, two sides,

00:02:22.870 --> 00:02:25.448
three sides, and four sides,

00:02:25.448 --> 00:02:29.833
and it also has one,
two, three, four corners.

00:02:30.591 --> 00:02:33.028
So this is also four
sides and four corners

00:02:33.028 --> 00:02:35.259
is also true of that one there.

00:02:35.259 --> 00:02:36.443
- [Voiceover] Do all shapes have

00:02:36.443 --> 00:02:37.583
four sides and four corners?

00:02:37.583 --> 00:02:38.768
- [Voiceover] No no, not at all.

00:02:38.768 --> 00:02:39.834
I mean this one right over here had

00:02:39.834 --> 00:02:41.101
zero sides and zero corners,

00:02:41.101 --> 00:02:43.004
and actually this last
shape right over here,

00:02:43.004 --> 00:02:45.524
doesn't seem to have four
sides or four corners.

00:02:45.524 --> 00:02:46.734
If we count them,

00:02:46.734 --> 00:02:50.046
we have one side, two sides,

00:02:51.136 --> 00:02:52.604
and three sides.

00:02:52.834 --> 00:02:54.888
So this one has three sides,

00:02:54.888 --> 00:02:57.259
and if we count the corners we have

00:02:57.259 --> 00:03:00.766
one corner, two corners, and three,

00:03:01.642 --> 00:03:03.595
and three corners.

00:03:03.934 --> 00:03:05.645
- [Voiceover] Oh, thanks a
bunch Uncle-- Cousin Sal.

00:03:05.645 --> 00:03:07.374
But I always get confused whether

00:03:07.374 --> 00:03:09.688
you're my cousin or my uncle.

00:03:09.688 --> 00:03:11.220
But I'll call you Cousin Sal.

00:03:11.220 --> 00:03:13.921
But I read in a shape magazine

00:03:14.167 --> 00:03:15.982
that there was other
ways that I can classify.

00:03:15.982 --> 00:03:17.158
I can classify based on

00:03:17.158 --> 00:03:19.142
whether the sides have the same size.

00:03:19.142 --> 00:03:21.694
So which of these have all the sides

00:03:21.694 --> 00:03:22.823
are the same size?

00:03:22.823 --> 00:03:24.616
- [Voiceover] Okay, well calm down.

00:03:24.616 --> 00:03:26.030
Let's look at this.

00:03:26.030 --> 00:03:28.576
So if you look at this
one, it has four sides.

00:03:28.917 --> 00:03:31.676
But we see, that this
side right over here,

00:03:31.676 --> 00:03:34.646
this green side, is clearly shorter

00:03:34.646 --> 00:03:36.512
than this purple side.

00:03:36.725 --> 00:03:39.353
So all the four sides here
are not the same length.

00:03:39.845 --> 00:03:41.724
But if we look at this
shape right over here,

00:03:41.724 --> 00:03:45.712
this side at least looks like
the same length as this side.

00:03:46.300 --> 00:03:48.631
And that looks like the
same length as this side,

00:03:49.100 --> 00:03:50.796
and that looks like the
same length as that side.

00:03:50.796 --> 00:03:51.815
So it looks over here like

00:03:51.815 --> 00:03:53.675
all four sides are the same length.

00:03:54.119 --> 00:03:55.624
And that's also true for this shape.

00:03:55.624 --> 00:03:57.902
This side looks the same
length as that side,

00:03:57.902 --> 00:03:59.654
looks the same length as that side,

00:03:59.654 --> 00:04:01.426
looks the same length as that side.

00:04:01.832 --> 00:04:03.710
That's also true for
this three-sided shape.

00:04:03.710 --> 00:04:07.066
This side looks the same
length as that side,

00:04:07.480 --> 00:04:09.105
This is the same length as that one.

00:04:09.437 --> 00:04:10.925
- [Voiceover] Oh, thank
you so much Cousin Sal!

00:04:10.925 --> 00:04:13.437
I'm ready to go classify
my shape collection.

00:04:13.437 --> 00:04:14.846
This is awesome!

00:04:14.846 --> 00:04:17.561
- [Voiceover] Well I'm
glad that I could help!


Place value example with 42

WEBVTT
Kind: captions
Language: en

00:00:00.328 --> 00:00:02.273
Voiceover:What I want to
do in this video is take

00:00:02.273 --> 00:00:05.944
this quantity represented
by these greenish

00:00:05.944 --> 00:00:09.480
emerald-looking bars, where
each bar represents one,

00:00:09.480 --> 00:00:13.504
and try to represent them
using the digits that we know,

00:00:13.504 --> 00:00:16.770
the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

00:00:16.770 --> 00:00:18.696
And the way I'm going
to do it is I'm going

00:00:18.696 --> 00:00:21.259
to group this into tens
so I'm going to see

00:00:21.259 --> 00:00:23.848
how many tens I have and then essentially

00:00:23.848 --> 00:00:25.793
how many ones that I have left over,

00:00:25.793 --> 00:00:27.883
so let's try to group
this into groups of ten.

00:00:27.883 --> 00:00:34.321
So 1, 2, 3, 4, 5, 6, 7, 8, 9,

00:00:34.321 --> 00:00:35.840
Let me actually, I
messed that up, let see,

00:00:35.840 --> 00:00:42.003
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

00:00:42.003 --> 00:00:44.272
So that's one group of
ten right over there.

00:00:44.272 --> 00:00:49.960
Let me box that off so
that i know that is ten.

00:00:49.960 --> 00:00:53.912
So, that's one group
of ten, now let's see,

00:00:53.912 --> 00:00:56.268
let's see if I can get a few
more groups of ten in there.

00:00:56.268 --> 00:01:03.502
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

00:01:03.502 --> 00:01:08.319
That's my second group
of ten right over there.

00:01:08.319 --> 00:01:11.706
Second group of ten and now let's see,

00:01:11.706 --> 00:01:14.213
it looks like I got several
more groups of ten here.

00:01:14.213 --> 00:01:20.355
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

00:01:20.355 --> 00:01:23.083
So that's a third group of ten,

00:01:23.083 --> 00:01:25.187
and it looks like I might
be able to get one more,

00:01:25.187 --> 00:01:30.121
so that's my third group
of ten and then we have,

00:01:30.137 --> 00:01:36.210
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

00:01:36.210 --> 00:01:39.477
So that's my fourth group of ten.

00:01:39.477 --> 00:01:42.744
Fourth group of ten right over here.

00:01:42.744 --> 00:01:46.756
So I was able to fit in
at least just this is 1,

00:01:46.756 --> 00:01:50.216
This is a ten right over here,

00:01:50.216 --> 00:01:52.548
This is another ten right over here.

00:01:52.548 --> 00:01:56.576
That's the third ten, and
then that's the fourth ten.

00:01:56.576 --> 00:02:00.705
So we can say that's 10+10+10+10.

00:02:00.705 --> 00:02:06.667
Or all of this combined is equal to 4,

00:02:06.667 --> 00:02:13.277
1, 2, 3, 4, tens but I'm
not done with the number,

00:02:13.277 --> 00:02:19.349
I still have 1, 2, I
have 2 ones left over.

00:02:19.349 --> 00:02:23.560
So this is a one, so that's
a one right over there.

00:02:23.560 --> 00:02:28.351
And then this is another
one, so that's another one.

00:02:28.351 --> 00:02:32.996
So I have 2 ones, so this right over here,

00:02:32.996 --> 00:02:36.367
That is 2 ones, remember for each of these

00:02:36.367 --> 00:02:39.073
little green bars we
said represents a one.

00:02:39.073 --> 00:02:41.394
So this whole number is what?

00:02:41.394 --> 00:02:46.914
What's going to be the 4 tens,
1 ten, 2 ten, 3 tens, 4 tens.

00:02:46.914 --> 00:02:50.854
Plus the 2 ones, so
let me write that down.

00:02:50.854 --> 00:03:00.261
So it is 4 tens plus 2
ones, but what is that?

00:03:00.261 --> 00:03:03.008
Well if we use the number systems

00:03:03.008 --> 00:03:05.204
that we're used to
representing, we would say,

00:03:05.204 --> 00:03:07.244
"Well look, our number system gives us

00:03:07.244 --> 00:03:08.660
a way of representing that."

00:03:08.660 --> 00:03:13.693
Because the first place,
this is the ones' place

00:03:13.693 --> 00:03:16.553
So let me gets some of this blue color,

00:03:16.553 --> 00:03:18.344
So this is the ones' place.

00:03:18.344 --> 00:03:21.132
There we use one of the digits,
0, 1, 2, 3, 4, 5, 6, 7, 8, 9,

00:03:21.132 --> 00:03:22.988
To see how many ones we have.

00:03:22.988 --> 00:03:24.892
And we have 2 ones.

00:03:24.892 --> 00:03:27.690
And then the next place over to the left,

00:03:27.690 --> 00:03:30.529
that's where we say how many tens we have.

00:03:30.529 --> 00:03:33.582
That's where we say how many tens.

00:03:33.582 --> 00:03:37.453
And we know how many tens
we have, we have 4 tens.

00:03:37.453 --> 00:03:39.240
So the way that we would represent this

00:03:39.240 --> 00:03:40.835
in our number systems we would say

00:03:40.835 --> 00:03:44.177
that this was 4 tens and 2 ones.

00:03:44.177 --> 00:03:47.932
We call this right over
here, we call this 42.

00:03:47.932 --> 00:03:50.241
Another way that you can think about this,

00:03:50.241 --> 00:03:56.004
is 4 tens, that's 10+10+10+10,

00:03:56.004 --> 00:03:59.580
that's going to be 40, but once again,

00:03:59.580 --> 00:04:02.382
we're using our number
system to represent this,

00:04:02.382 --> 00:04:07.681
and then we have 2 ones but
I like to write it this way,

00:04:07.681 --> 00:04:09.787
just to remind ourselves that this 4 which

00:04:09.787 --> 00:04:14.983
represents 4 tens also represents 40.

00:04:14.983 --> 23:59:59.999
Because 4 tens is 40.


Monkeys for a party

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.366
Voiceover:Let's say that
you're throwing a party,

00:00:02.366 --> 00:00:04.433
and as we all know, any party needs

00:00:04.433 --> 00:00:06.266
a sufficient number of monkeys.

00:00:06.633 --> 00:00:08.733
Right now you have 10 monkeys.

00:00:08.767 --> 00:00:12.800
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

00:00:13.100 --> 00:00:16.733
But you would like to have 18 monkeys.

00:00:17.233 --> 00:00:19.900
How many more monkeys do you need

00:00:19.900 --> 00:00:23.000
in order to have 18 monkeys?

00:00:23.233 --> 00:00:24.833
I encourage you to pause the video and

00:00:24.833 --> 00:00:26.166
think about it on your own before

00:00:26.166 --> 00:00:27.333
I work through it.

00:00:28.100 --> 00:00:31.733
Remember, the goal here is to get to ...

00:00:32.566 --> 00:00:33.966
The goal here ...

00:00:33.966 --> 00:00:35.166
Let me write it this way.

00:00:35.166 --> 00:00:40.267
The goal here is to get to 18, 18 monkeys.

00:00:41.366 --> 00:00:43.466
Now what do we know about the number 18?

00:00:44.100 --> 00:00:47.568
Well the number 18, you have this 1

00:00:47.568 --> 00:00:49.400
right over here in the tens place.

00:00:49.566 --> 00:00:53.200
That literally represents 1, 10.

00:00:54.233 --> 00:00:57.866
And then this 8 represents 8, 1's.

00:00:59.869 --> 00:01:02.133
Plus 8, 1's.

00:01:02.833 --> 00:01:05.266
That's the same thing as saying 1, 10 ...

00:01:05.500 --> 00:01:08.466
1, 10, that's just a 10 right over there.

00:01:08.500 --> 00:01:10.333
And 8, 1's, that's just 8.

00:01:10.766 --> 00:01:13.333
18 is the same thing as 10 plus 8.

00:01:13.900 --> 00:01:17.167
If we start with 10
monkeys, which is how many

00:01:17.167 --> 00:01:19.300
we have started with, how many more

00:01:19.300 --> 00:01:22.266
do we need to get to 18?

00:01:22.700 --> 00:01:25.500
Well, if you start at 10, you need 8 more

00:01:25.500 --> 00:01:27.533
to get to 18.

00:01:28.033 --> 00:01:30.333
We can verify that visually, if we like.

00:01:30.566 --> 00:01:33.267
Let's see.

00:01:33.900 --> 00:01:36.066
Let me copy and paste 1 of these monkeys.

00:01:36.766 --> 00:01:37.800
Copy.

00:01:37.966 --> 00:01:39.066
This is 10.

00:01:39.100 --> 00:01:40.466
Now, let's get to 18.

00:01:41.166 --> 00:02:04.400
11, 12, 12, 13, 14, 15, 16, 17, and 18.

00:02:04.701 --> 00:02:06.000
And 18.

00:02:06.300 --> 00:02:07.800
How many more did we add?

00:02:08.033 --> 00:02:11.100
Well, once again, we started, we started

00:02:11.100 --> 00:02:13.000
with this 10 right over here.

00:02:13.033 --> 00:02:14.333
Let me box them off.

00:02:15.033 --> 00:02:18.066
We started with these 10 monkeys.

00:02:19.500 --> 00:02:21.800
That's 1 group of 10 right over there.

00:02:22.567 --> 00:02:24.866
We started with the 10.

00:02:25.433 --> 00:02:27.600
And then we added 8 more monkeys.

00:02:27.767 --> 00:02:31.666
1, 2, 3, 4, 5, 6, 7, 8.

00:02:31.833 --> 00:02:37.400
1, 2, 3, 4, 5, 6, 7, 8.

00:02:37.633 --> 00:02:39.000
So we started with 10.

00:02:39.167 --> 00:02:41.133
We added 8 more monkeys.

00:02:41.500 --> 00:02:47.800
We added 8 more monkeys
to get, to get to 18.

00:02:47.966 --> 00:02:49.633
And once again, how did we first think

00:02:49.633 --> 00:02:50.733
about that?

00:02:50.766 --> 00:02:54.600
Well, 18 is 1, 10 plus 8, 1's.

00:02:54.700 --> 00:02:57.133
Or 10 plus 8.


29 Toxic chemicals

WEBVTT
Kind: captions
Language: en

00:00:00.506 --> 00:00:04.651
- The cost of safely disposing
of the toxic chemicals

00:00:04.651 --> 00:00:07.224
is approximately five times

00:00:07.224 --> 00:00:10.881
what the company paid to purchase it.

00:00:10.881 --> 00:00:12.901
And maybe we have no error.

00:00:12.901 --> 00:00:15.377
So this first part reads
a little bit weird,

00:00:15.377 --> 00:00:16.171
but I think it's right.

00:00:16.171 --> 00:00:16.640
Let's see.

00:00:16.640 --> 00:00:20.362
The cost of safely disposing
of the toxic chemicals.

00:00:20.362 --> 00:00:22.442
Yeah, there's a lot of of's
there, but I think that's right.

00:00:22.442 --> 00:00:25.242
I mean, we could have said
the cost of disposing of,

00:00:25.242 --> 00:00:27.062
and this safely is just an adverb

00:00:27.062 --> 00:00:28.698
describing how we're disposing.

00:00:28.698 --> 00:00:29.752
We're safely disposing.

00:00:29.752 --> 00:00:30.668
So this seems reasonable.

00:00:30.668 --> 00:00:33.660
The cost of safely disposing
of the toxic chemicals.

00:00:33.660 --> 00:00:36.385
It's a little bit a lot of wordy.

00:00:36.385 --> 00:00:38.842
It's got the of's there,
but it doesn't seem wrong.

00:00:38.842 --> 00:00:41.202
The cost of safely disposing of

00:00:41.202 --> 00:00:43.592
the toxic chemicals is approximately.

00:00:43.592 --> 00:00:46.162
So we're talking about the cost.

00:00:46.162 --> 00:00:48.942
So it's a singular subject,

00:00:48.942 --> 00:00:51.895
and so it makes sense that we
would use is right over here.

00:00:51.895 --> 00:00:52.780
We'd use the singular verb.

00:00:52.780 --> 00:00:56.482
The cost is approximately five times

00:00:56.482 --> 00:01:00.862
what the company paid to purchase it.

00:01:00.862 --> 00:01:03.463
Now this it is a little bit fishy.

00:01:03.463 --> 00:01:06.522
What is this it referring to?

00:01:06.522 --> 00:01:09.866
Is this it referring to the cost?

00:01:09.866 --> 00:01:12.189
Is it referring to the cost?

00:01:12.189 --> 00:01:12.742
No.

00:01:12.742 --> 00:01:16.122
The it, I think, based on
the meaning of the sentence,

00:01:16.122 --> 00:01:18.702
they're referring to the chemicals.

00:01:18.702 --> 00:01:23.045
The company paid to
purchase the chemicals.

00:01:23.045 --> 00:01:26.620
So this right over here
should be the chemicals.

00:01:26.620 --> 00:01:30.204
The chemicals, or if they
want to use a pronoun,

00:01:30.204 --> 00:01:33.761
this is a plural noun right
over here, the chemicals,

00:01:33.761 --> 00:01:36.328
you would use to purchase them.

00:01:36.328 --> 00:01:37.761
You would use to purchase them.

00:01:37.761 --> 00:01:40.202
You wouldn't use it right over here.

00:01:40.202 --> 00:01:43.081
What did they purchase?

00:01:43.081 --> 00:01:47.341
They purchased the toxic chemicals.

00:01:47.341 --> 00:01:48.230
So it's plural.

00:01:48.230 --> 00:01:50.701
So you would either say
to purchase the chemicals

00:01:50.701 --> 00:01:52.181
or to purchase them.

00:01:52.181 --> 00:01:57.181
So let's just read it with them.

00:01:57.301 --> 00:02:00.501
The cost of safely disposing
of the toxic chemicals

00:02:00.501 --> 00:02:03.161
is approximately five
times what the company paid

00:02:03.161 --> 00:02:04.581
to purchase them.

00:02:04.581 --> 00:02:05.761
Yeah, that makes a little bit more sense,

00:02:05.761 --> 00:02:06.517
or if you want to be very clear,

00:02:06.517 --> 00:02:08.421
you could just write the chemicals there.

00:02:08.421 --> 00:02:11.201
The cost of safely disposing
of the toxic chemicals

00:02:11.201 --> 00:02:14.701
is approximately five
times what the company paid

00:02:14.701 --> 00:02:16.101
to purchase the chemicals.

00:02:16.101 --> 00:02:17.801
Yeah, that makes it, at least in my mind,

00:02:17.801 --> 00:02:20.821
that's how I would have written
it, making it very clear.


Introduction to place value

WEBVTT
Kind: captions
Language: en

00:00:00.903 --> 00:00:02.774
Voiceover:Let's say that
you wanna count the days

00:00:02.774 --> 00:00:04.178
since your last birthday

00:00:04.178 --> 00:00:06.441
because you just wanna
know how long its been.

00:00:06.441 --> 00:00:08.234
And so one day after your birthday

00:00:08.234 --> 00:00:10.433
you put a mark on a wall.

00:00:10.433 --> 00:00:13.244
Then the next day you put
another mark on the wall.

00:00:13.244 --> 00:00:15.305
The day after that you put
another mark on the wall.

00:00:15.305 --> 00:00:19.445
So out that day you say well
how many days has it been?

00:00:19.445 --> 00:00:20.974
Well you can say look there's
been one, two, three days.

00:00:20.974 --> 00:00:21.901
So one way to think about it

00:00:21.901 --> 00:00:24.911
is this set of symbols right over here

00:00:24.911 --> 00:00:26.112
represents the number three.

00:00:26.112 --> 00:00:27.500
But then you keep going.

00:00:27.500 --> 00:00:29.778
The fourth day, you put another mark.

00:00:29.778 --> 00:00:31.437
Fifth day, you put another mark.

00:00:31.437 --> 00:00:33.369
And then you keep going like that

00:00:33.369 --> 00:00:37.910
day after day each day
you add another mark.

00:00:37.910 --> 00:00:40.558
And this is actually the earliest way,

00:00:40.558 --> 00:00:43.100
the most basic way of
representing numbers.

00:00:43.100 --> 00:00:45.636
The number is represented
by the number of marks.

00:00:45.636 --> 00:00:47.300
So after bunch of days you get here

00:00:47.300 --> 00:00:49.170
and you're like on well
how many days has it been?

00:00:49.170 --> 00:00:50.981
Well you just recount everything.

00:00:50.981 --> 00:00:54.912
You say one, two, three,
four, five, six, seven, eight,

00:00:54.912 --> 00:01:02.445
nine, 10, 11, 12, 13, 14, 15, 16, 17 days.

00:01:02.445 --> 00:01:04.303
So well you know this
number representation

00:01:04.303 --> 00:01:06.174
it took me little bit of time

00:01:06.174 --> 00:01:07.501
to realize that this is 17

00:01:07.501 --> 00:01:09.974
but it seems to be working
so you just keep going.

00:01:09.974 --> 00:01:13.178
Day after day after day after day

00:01:13.178 --> 00:01:16.578
you just keep marking off the days

00:01:16.578 --> 00:01:19.702
on your wall just to sense
you're counting the days

00:01:19.702 --> 00:01:21.645
since your last birthday.

00:01:21.645 --> 00:01:24.305
But at some point you realize

00:01:24.305 --> 00:01:25.112
every time you wanna know

00:01:25.112 --> 00:01:26.647
how may days its been to count it

00:01:26.647 --> 00:01:28.107
is a little bit painful.

00:01:28.107 --> 00:01:30.110
And not only that, is this is taking up

00:01:30.110 --> 00:01:32.634
a lot of space on your wall.

00:01:32.634 --> 00:01:34.441
You wish that there was an easier way

00:01:34.441 --> 00:01:37.766
to represent whatever number this is.

00:01:37.766 --> 00:01:39.573
So first of all let's just think about

00:01:39.573 --> 00:01:41.367
what number this actually is.

00:01:41.367 --> 00:01:44.300
One, two, three, four,
five, six, seven, eight,

00:01:44.300 --> 00:01:51.245
nine, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22,

00:01:51.245 --> 00:01:56.910
23, 24, 25, 26, 27, 28, 29, 30, 31, 32,

00:01:56.910 --> 00:02:00.633
33, 34, 35, 36, 37.

00:02:00.633 --> 00:02:02.306
So you wish that there was a better way

00:02:02.306 --> 00:02:05.971
to represent this number
which we now call 37.

00:02:05.971 --> 00:02:07.107
And maybe when you're first trying

00:02:07.107 --> 00:02:10.103
you might have even called
it something called 37.

00:02:10.103 --> 00:02:13.036
You would just call it this, this number.

00:02:13.036 --> 00:02:15.111
This number of days since my birthday.

00:02:15.111 --> 00:02:15.969
And I say well look.

00:02:15.969 --> 00:02:18.636
What if there was an easier
way to group the numbers?

00:02:18.636 --> 00:02:21.246
You know I have 10 fingers on my hands.

00:02:21.246 --> 00:02:23.301
What if I were to group them into 10s?

00:02:23.301 --> 00:02:25.706
And then I would say just
how many groups of 10 I have

00:02:25.706 --> 00:02:28.245
and then how many ones
do I have left over.

00:02:28.245 --> 00:02:29.846
Maybe that would be an easier way

00:02:29.846 --> 00:02:34.104
to represent, to represent
this quantity here.

00:02:34.104 --> 00:02:35.641
And so let's do that.

00:02:35.641 --> 00:02:39.836
So one, two, three, four, five,
six, seven, eight, nine, 10.

00:02:39.836 --> 00:02:44.578
So that's a group of 10 right over there.

00:02:44.578 --> 00:02:48.846
And then you have one, two,
three, four, five, six,

00:02:48.846 --> 00:02:51.309
seven, eight, nine, 10.

00:02:51.309 --> 00:02:56.171
So this is another group
of 10 right over here.

00:02:56.171 --> 00:02:57.169
And then let's see.

00:02:57.169 --> 00:03:00.178
We have one, two, three,
four, five, six, seven,

00:03:00.178 --> 00:03:02.040
eight, nine, 10.

00:03:02.040 --> 00:03:06.777
So that is another group
of 10 right over there.

00:03:06.777 --> 00:03:09.445
And then finally you have
one, two, three, four,

00:03:09.445 --> 00:03:11.380
five, six, seven.

00:03:11.380 --> 00:03:13.112
So you don't get a whole group of 10

00:03:13.112 --> 00:03:14.711
so you don't circle em.

00:03:14.711 --> 00:03:17.708
So just by doing this very simple thing

00:03:17.708 --> 00:03:19.446
now all of a sudden it's much easier

00:03:19.446 --> 00:03:22.510
to realize how many days have passed.

00:03:22.510 --> 00:03:23.767
You don't have to count everything.

00:03:23.767 --> 00:03:24.706
You just have to say okay.

00:03:24.706 --> 00:03:26.436
One group of 10.

00:03:26.436 --> 00:03:27.844
Two groups of 10.

00:03:27.844 --> 00:03:29.039
Three groups of 10.

00:03:29.039 --> 00:03:31.834
Or you can say one, two, three 10s.

00:03:31.834 --> 00:03:33.844
And so that's essentially 30.

00:03:33.844 --> 00:03:35.846
And then I have another one, two, three,

00:03:35.846 --> 00:03:38.510
four, five, six, seven.

00:03:38.510 --> 00:03:40.770
And so you say oh I have 30 and then seven

00:03:40.770 --> 00:03:44.779
if you knew to use those
words which we now use.

00:03:44.779 --> 00:03:48.244
Now this is essentially
what our number system does

00:03:48.244 --> 00:03:50.775
using the 10 digits we know of.

00:03:50.775 --> 00:03:54.646
The 10 digits we know of
are zero, one, two, three,

00:03:54.646 --> 00:03:59.568
four, five, six, seven, eight, nine.

00:03:59.568 --> 00:04:02.041
Now what our number system allows us to do

00:04:02.041 --> 00:04:03.702
is using only these 10 digits

00:04:03.702 --> 00:04:06.779
we can essentially
represent any number we want

00:04:06.779 --> 00:04:07.966
in a very quick way,

00:04:07.966 --> 00:04:11.101
a very easy way for our
brains to understand it.

00:04:11.101 --> 00:04:14.779
So here if we want to represent three 10s,

00:04:14.779 --> 00:04:16.433
we would have put a three

00:04:16.433 --> 00:04:19.634
in what we would call the 10s place.

00:04:19.634 --> 00:04:22.978
We would put a three in the 10s place.

00:04:22.978 --> 00:04:25.108
And then we would put the ones,

00:04:25.108 --> 00:04:27.305
one, two, three, four, five, six, seven.

00:04:27.305 --> 00:04:30.776
We'd put the seven in the ones place.

00:04:30.776 --> 00:04:32.969
And so how do you know
which place is which?

00:04:32.969 --> 00:04:35.903
Well the first place
starting from the right,

00:04:35.903 --> 00:04:39.036
the first place is the ones place.

00:04:39.036 --> 00:04:41.907
And then you go one
space to the left of it,

00:04:41.907 --> 00:04:42.911
you get to the 10s place.

00:04:42.911 --> 00:04:44.233
And as we'll see you go one more space

00:04:44.233 --> 00:04:45.575
you go to the 100s space.

00:04:45.575 --> 00:04:47.774
But we'll cover that in a future video.

00:04:47.774 --> 00:04:50.845
So this essentially tells
us the exact same thing.

00:04:50.845 --> 00:04:53.576
This tells us the exact same thing

00:04:53.576 --> 00:04:55.970
as this does right over here.

00:04:55.970 --> 00:04:58.178
This tells us three 10s.

00:04:58.178 --> 00:04:59.509
One, two, three.

00:04:59.509 --> 00:05:00.633
Three 10s

00:05:00.633 --> 00:05:02.236
Three groups of 10.

00:05:02.236 --> 00:05:04.110
And then another set of ones.

00:05:04.110 --> 00:05:05.906
So we could rewrite this.

00:05:05.906 --> 00:05:12.909
This is equal to, this
is equal to three 10s

00:05:12.909 --> 00:05:19.503
three 10s plus, plus seven ones.

00:05:19.503 --> 00:05:20.644
Or another way to think about it

00:05:20.644 --> 00:05:22.311
what are the three 10s?

00:05:22.311 --> 00:05:23.778
Well if use the same number system

00:05:23.778 --> 00:05:27.571
to represent three 10s you
would write that down as 30.

00:05:27.571 --> 00:05:28.712
And then seven ones.

00:05:28.712 --> 00:05:30.709
Once again if you use
our same number system

00:05:30.709 --> 00:05:33.514
you would represent that as seven.

00:05:33.514 --> 00:05:36.773
So these are all different
ways of representing 37.

00:05:36.773 --> 00:05:38.710
And hopefully this
allows you to appreciate

00:05:38.710 --> 00:05:41.094
how neat our number system is.

00:05:41.094 --> 00:05:42.513
Where even a number like 37

00:05:42.513 --> 00:05:44.308
as soon as you just write
scratches on a wall,

00:05:44.308 --> 00:05:47.106
it becomes pretty hard to read.

00:05:47.106 --> 00:05:48.240
And you can imagine when you get to much

00:05:48.240 --> 00:05:51.904
much larger numbers like 1,052

00:05:51.904 --> 00:05:54.376
to have to count that
many marks every time.

00:05:54.376 --> 00:05:58.352
But our number system gives
us a way of dealing with it.


27 Theories of evolution

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:02.663
- Lynn Margulis's theory

00:00:02.663 --> 00:00:07.412
that evolution is a process
involving interdependency

00:00:07.412 --> 00:00:11.638
rather than competition among organisms

00:00:11.638 --> 00:00:15.278
differs dramatically from most biologists,

00:00:15.278 --> 00:00:16.723
or no error.

00:00:17.218 --> 00:00:18.052
So this is a little interesting.

00:00:18.052 --> 00:00:18.953
Let's just think about what

00:00:18.953 --> 00:00:20.668
the sentence is comparing.

00:00:20.668 --> 00:00:23.551
They're comparing Lynn Margulis's theory

00:00:23.551 --> 00:00:26.610
that evolution is a process
involving interdependency

00:00:26.610 --> 00:00:29.436
rather than competition among organisms.

00:00:29.436 --> 00:00:31.048
They're saying that that theory

00:00:31.048 --> 00:00:34.058
differs dramatically from most biologists.

00:00:34.443 --> 00:00:35.811
So this sentence

00:00:35.811 --> 00:00:37.780
as it's constructed right over here,

00:00:37.780 --> 00:00:39.171
it's comparing a theory,

00:00:39.679 --> 00:00:42.811
it's comparing Lynn Margulis's theory,

00:00:42.811 --> 00:00:44.899
it's comparing that to,

00:00:44.899 --> 00:00:46.931
they're saying the theory
differs dramatically

00:00:46.931 --> 00:00:48.734
from most biologists.

00:00:49.042 --> 00:00:49.767
Well this is weird.

00:00:49.767 --> 00:00:52.038
You're comparing a theory to people?

00:00:52.038 --> 00:00:54.596
You're comparing a theory to biologists?

00:00:54.596 --> 00:00:55.810
No, what you really want to do

00:00:55.810 --> 00:00:57.339
is compare the theory

00:00:57.339 --> 00:00:59.391
to the biologists' theory,

00:00:59.391 --> 00:01:01.470
to the biologists' ideas.

00:01:01.470 --> 00:01:02.740
So if you wanted to make this

00:01:02.740 --> 00:01:04.040
make sense,

00:01:04.040 --> 00:01:06.207
you would say, "differs dramatically from"

00:01:06.207 --> 00:01:08.150
not "most biologists,"

00:01:08.150 --> 00:01:12.542
you would want to say, "most biologists',

00:01:13.219 --> 00:01:15.743
most biologists'"

00:01:16.067 --> 00:01:17.134
and it's possessive,

00:01:17.134 --> 00:01:20.219
"most biologists' theories,"

00:01:21.635 --> 00:01:22.819
or ideas,

00:01:22.819 --> 00:01:24.151
so you would cross this out.

00:01:24.151 --> 00:01:28.359
Notice, you're comparing
Lynn Margulis's theory to,

00:01:29.005 --> 00:01:31.182
and you're saying, "Lynn Margulis's theory

00:01:31.182 --> 00:01:32.850
differs dramatically from,"

00:01:33.173 --> 00:01:34.924
not most biologists,

00:01:34.924 --> 00:01:36.789
but "differs dramatically from

00:01:36.789 --> 00:01:39.289
most biologists' theories."

00:01:39.289 --> 00:01:41.175
You want to compare Lynn Margulis's theory

00:01:41.175 --> 00:01:42.724
to the other biologists' theories.

00:01:42.724 --> 00:01:43.634
You don't want to compare

00:01:43.634 --> 00:01:45.351
a theory to the actual people.

00:01:45.351 --> 00:01:45.966
That's just--

00:01:45.966 --> 00:01:47.751
you're comparing weird--

00:01:47.751 --> 00:01:50.082
you're comparing, well,
it doesn't make sense.

00:01:50.082 --> 00:01:51.002
You're comparing an idea

00:01:51.002 --> 00:01:54.615
to actual physical, living human beings.

00:01:55.169 --> 00:01:57.617
So, definitely error in Part D.


Vector representations exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.245 --> 00:00:01.630
Voiceover:Which of the following

00:00:01.630 --> 00:00:03.791
can represent a vector?

00:00:03.791 --> 00:00:05.618
A vector is something that

00:00:05.618 --> 00:00:09.091
has both a magnitude and a direction.

00:00:09.091 --> 00:00:10.862
Let's see which one of these
could represent something

00:00:10.862 --> 00:00:12.762
that has a magnitude and a direction.

00:00:12.762 --> 00:00:16.001
The first choice right
over here is the number 5.

00:00:16.001 --> 00:00:19.403
5 could represent a magnitude,

00:00:19.403 --> 00:00:21.733
you could say, essentially,
how large something is.

00:00:21.733 --> 00:00:23.459
That's all the information has.

00:00:23.459 --> 00:00:25.747
It doesn't say 5 in a certain direction.

00:00:25.747 --> 00:00:28.157
This one by itself would not be a

00:00:28.157 --> 00:00:30.093
or it couldn't represent a vector.

00:00:30.093 --> 00:00:32.973
You need to specify a direction as well.

00:00:32.973 --> 00:00:36.095
The angle measure 5 degrees.

00:00:36.095 --> 00:00:37.957
Well the angle measure 5 degrees

00:00:37.957 --> 00:00:39.370
could represent a direction.

00:00:39.370 --> 00:00:40.604
If you say it's 5 degrees.

00:00:40.604 --> 00:00:43.806
Let's say that's the positive X axis

00:00:43.806 --> 00:00:45.876
that's the positive Y axis.

00:00:45.876 --> 00:00:47.930
If you said it's 5 degrees relative

00:00:47.930 --> 00:00:49.739
to the positive X axis.

00:00:49.739 --> 00:00:52.555
If you said something like this,

00:00:52.555 --> 00:00:54.318
so if you had 5 degrees over there,

00:00:54.318 --> 00:00:56.300
that would be specifying a direction,

00:00:56.300 --> 00:00:58.069
but it's not giving us a magnitude.

00:00:58.069 --> 00:01:00.186
It's not saying how far in that direction

00:01:00.186 --> 00:01:02.747
or how large in that direction.

00:01:02.747 --> 00:01:04.424
An angle measure of 5 degrees,

00:01:04.424 --> 00:01:06.304
you're giving us a direction,

00:01:06.304 --> 00:01:07.589
but no magnitude.

00:01:07.589 --> 00:01:09.400
Let's just cross these out.

00:01:09.400 --> 00:01:12.993
The  point (5,5). Now this is interesting.

00:01:13.362 --> 00:01:17.543
If we say that the point is
the end point of a vector

00:01:17.543 --> 00:01:20.925
that starts at the origin,

00:01:20.925 --> 00:01:22.647
so lets draw that out.

00:01:22.876 --> 00:01:25.884
That's the positive X
axis, positive Y axis.

00:01:25.884 --> 00:01:31.419
This is 1,2,3,4,5 in the
horizontal direction.

00:01:31.419 --> 00:01:35.747
1,2,3,4,5 in the positive
vertical direction.

00:01:35.747 --> 00:01:40.038
The .5,5 is going to be right over there.

00:01:40.038 --> 00:01:46.339
If you said that this
represents the head of a vector

00:01:46.339 --> 00:01:48.508
who's tail is at the origin.

00:01:50.646 --> 00:01:53.151
The vector would look something like this.

00:01:53.151 --> 00:01:56.903
Now, you're giving both a
magnitude and a direction.

00:01:56.903 --> 00:01:58.272
What's the magnitude?

00:01:58.272 --> 00:01:59.828
What's the distance between the origin

00:01:59.828 --> 00:02:01.724
and this point, right over here.

00:02:01.724 --> 00:02:04.532
You could solve using the Pythagorean theorem

00:02:04.532 --> 00:02:12.122
and the direction, it's
this general direction.

00:02:12.122 --> 00:02:16.906
This could specify a
magnitude and a direction.

00:02:16.906 --> 00:02:19.278
This could represent a vector.

00:02:19.478 --> 00:02:23.590
The outcome of 5+5,
obviously that's just 10.

00:02:23.590 --> 00:02:24.652
That's just a number.

00:02:24.652 --> 00:02:26.083
For the same reason, this first choice,

00:02:26.083 --> 00:02:27.118
if I just have a number,

00:02:27.118 --> 00:02:28.117
that could be a magnitude,

00:02:28.117 --> 00:02:29.845
but we don't have a direction as well.

00:02:29.845 --> 00:02:32.333
This could not represent and

00:02:32.333 --> 00:02:34.730
that one could not represent a vector.


Pieces of fruit

WEBVTT
Kind: captions
Language: en

00:00:00.648 --> 00:00:04.817
Voiceover:We have one, two, three,

00:00:04.817 --> 00:00:08.104
four, five blueberries here,

00:00:08.104 --> 00:00:10.535
so we have five blueberries,

00:00:10.535 --> 00:00:19.466
and we have one, two, three
cherries, three cherries.

00:00:19.466 --> 00:00:23.088
Now, if I wanted to figure
out the total number

00:00:23.088 --> 00:00:24.952
of fruit that I have here,

00:00:24.952 --> 00:00:28.327
I could say the total
number of pieces of fruit,

00:00:28.327 --> 00:00:33.138
what should I do? Should
I add five plus three?

00:00:33.138 --> 00:00:36.122
So, should I add five plus three,

00:00:36.122 --> 00:00:40.743
or should I subtract? Should
I do five minus three,

00:00:40.743 --> 00:00:44.705
to get the total number
of fruit that I have?

00:00:44.705 --> 00:00:46.983
I encourage you to pause the video now,

00:00:46.983 --> 00:00:50.522
and try to think about it on your own.

00:00:50.522 --> 00:00:52.627
If I want the total number of fruit,

00:00:52.627 --> 00:00:54.987
I'm starting with five blueberries,

00:00:54.987 --> 00:00:57.387
and now I have three more.

00:00:57.387 --> 00:00:59.288
I'm going to add three.

00:00:59.288 --> 00:01:01.833
I'm going to get a larger number of fruit.

00:01:01.833 --> 00:01:04.877
I start with five; I'm
going to have three more,

00:01:04.877 --> 00:01:07.955
so I am going to add three,

00:01:07.955 --> 00:01:10.131
and so, how many blueberries do I have?

00:01:10.131 --> 00:01:13.404
What's five plus three?
It's one, two, three,

00:01:13.404 --> 00:01:20.437
four, five, six, seven,
eight pieces of fruit.

00:01:20.437 --> 00:01:22.450
Eight pieces of fruit.

00:01:22.450 --> 00:01:25.750
Now, what would five
minus three have meant?

00:01:25.750 --> 00:01:29.360
What would five minus three,

00:01:29.360 --> 00:01:33.305
five minus three, what would that mean?

00:01:33.305 --> 00:01:36.996
You can imagine starting off
with five pieces of fruit,

00:01:36.996 --> 00:01:41.277
let me copy and paste
that, copy and paste that,

00:01:41.277 --> 00:01:43.955
so, if you start with
five pieces of fruit,

00:01:43.955 --> 00:01:46.116
and now if you're subtracting three,

00:01:46.116 --> 00:01:48.891
this means taking away
three of your fruit,

00:01:48.891 --> 00:01:51.889
so take away one, take away two,

00:01:51.889 --> 00:01:55.303
and take away three, so
how many are you left with?

00:01:55.303 --> 00:01:59.090
You're just left with these,
one, two, right over here,

00:01:59.090 --> 00:02:01.729
so five minus three is two.

00:02:01.729 --> 00:02:03.916
If someone says, "Hey,
I have five blueberries,

00:02:03.916 --> 00:02:05.562
"and then I have three more cherries,

00:02:05.562 --> 00:02:06.673
"how many pieces of fruit?"

00:02:06.673 --> 00:02:08.467
well, now I'm going to be adding.

00:02:08.467 --> 00:02:10.249
If someone says, "I have five blueberries,

00:02:10.249 --> 00:02:12.085
"and then I ate three of them,

00:02:12.085 --> 00:02:13.549
"how many do I have left?"

00:02:13.549 --> 00:02:16.284
well, then you have two left.


Number grid

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.384
Voiceover: The goal of this video
is to essentially write down

00:00:03.384 --> 00:00:07.726
all the numbers in order
from zero to a 100.

00:00:07.726 --> 00:00:09.954
But I’m going to do it in
an interesting way, a way

00:00:09.954 --> 00:00:11.993
that maybe will allow
us to see some patterns

00:00:11.993 --> 00:00:13.969
in the numbers themselves.

00:00:13.969 --> 00:00:20.118
So let me just start, so I’m
gonna start at zero, one, two,

00:00:20.118 --> 00:00:30.213
three, four, five, six,
seven. eight, and nine,

00:00:30.213 --> 00:00:32.233
and instead of, of course we
know the next number is 10,

00:00:32.233 --> 00:00:34.848
which I could write down
but instead of doing that

00:00:34.848 --> 00:00:37.703
I’m just going to copy
and paste all of this.

00:00:37.703 --> 00:00:43.876
So copy and paste all of that
and see what this does for us.

00:00:43.876 --> 00:00:46.693
So if I do that, how does that help us?

00:00:46.693 --> 00:00:48.547
Well we know the next number is 10,

00:00:48.547 --> 00:00:50.311
which one way to think about it is,

00:00:50.311 --> 00:00:52.652
it’s a one followed by a zero.

00:00:52.652 --> 00:00:53.786
What’s the number after that?

00:00:53.786 --> 00:00:56.715
Well it’s 11, which is
a one followed by a one.

00:00:56.715 --> 00:00:57.616
What’s the number after that?

00:00:57.616 --> 00:00:59.793
Well it’s a 12, which is
a one followed by a two,

00:00:59.793 --> 00:01:08.231
and then 13, 14, 15, 16, 17, 18, and 19.

00:01:08.231 --> 00:01:09.818
Well that was pretty neat.

00:01:09.818 --> 00:01:12.677
This next row of numbers
as I went from 10 to 19

00:01:12.677 --> 00:01:14.724
looked just like the first ones,

00:01:14.724 --> 00:01:16.766
so the 2nd number is the same in yellow,

00:01:16.766 --> 00:01:19.789
but then I added a purple
one to the front of it.

00:01:19.789 --> 00:01:23.025
And one way to think about
it is, each of these numbers,

00:01:23.025 --> 00:01:27.441
the purple one that I
added, that represents 10.

00:01:27.441 --> 00:01:31.130
So 11 could be viewed as 10 plus one,

00:01:31.130 --> 00:01:35.191
12 could be viewed as 10 plus two.

00:01:35.191 --> 00:01:37.768
Let’s see if this keeps working.

00:01:37.768 --> 00:01:40.607
So let’s take another row of numbers.

00:01:40.607 --> 00:01:43.419
So let’s take another
row , my original row,

00:01:43.419 --> 00:01:45.910
and what do I get to after 19?

00:01:45.910 --> 00:01:51.174
Well of course after 19 we get to 20.

00:01:51.174 --> 00:02:03.507
So 20, two zero and then 21,
22, 23, 24, 25, 26, 27, 28, 29.

00:02:03.507 --> 00:02:06.218
I think you might start
to see the pattern here.

00:02:06.218 --> 00:02:08.346
What are we gonna do for the next row?

00:02:08.346 --> 00:02:10.816
Well now we’re in the 30s.

00:02:10.816 --> 00:02:14.862
So the first number 30 is 30 plus zero,

00:02:14.862 --> 00:02:19.379
30 plus one, 30 plus two,
30 plus three which is 33,

00:02:19.379 --> 00:02:26.220
34, 35, 36, 37, 38, 39.

00:02:26.220 --> 00:02:29.027
So just doing that I think
you already see the pattern.

00:02:29.027 --> 00:02:33.019
The number on the right we
keep going from zero, one, two,

00:02:33.019 --> 00:02:34.295
three, four, five, six, seven,

00:02:34.295 --> 00:02:36.682
eight, nine, and then
the number on the left,

00:02:36.682 --> 00:02:40.046
if we’re between 10 and 19,
you’ll always have a one.

00:02:40.046 --> 00:02:42.386
If you’re between 20 and 29,
you’ll always have a two.

00:02:42.386 --> 00:02:46.626
If you’re between 30 and 39,
you’ll always have a three.

00:02:46.626 --> 00:02:48.247
Now how can I complete this

00:02:48.247 --> 00:02:51.026
going all the way to 99 pretty fast?

00:02:51.026 --> 00:02:53.588
Well let’s do that, so
that’s going to be my 40s,

00:02:53.588 --> 00:02:56.522
I haven’t written it out yet.

00:02:56.522 --> 00:03:02.036
That’s going to be my 50s,
this is going to be my 60s,

00:03:04.205 --> 00:03:13.484
70s, 80s, and then I have
my 90s right over here.

00:03:13.484 --> 00:03:18.954
And so this one we already said
this is going to be my 40s,

00:03:18.954 --> 00:03:24.074
this is going to be my
50s, this is going to be,

00:03:24.074 --> 00:03:27.785
I’m trying to make sure I
use all my colors, my 60s,

00:03:27.785 --> 00:03:33.315
this is going to be my 70s,
and then I have my 80s,

00:03:33.315 --> 00:03:39.452
and then of course I have
my, let me do a color that,

00:03:39.452 --> 00:03:42.540
I’ll re-use magenta, I’ll have my 90s.

00:03:42.540 --> 00:03:44.971
So what if I woulda just take this,

00:03:44.971 --> 00:03:48.283
so let me just take this right over here,

00:03:48.283 --> 00:03:50.197
and copy and paste that.

00:03:50.197 --> 00:03:52.916
So copy, and then let me paste it.

00:03:52.916 --> 00:03:57.585
So now I have 41, 51, 61, 71, 81, and 91.

00:03:58.215 --> 00:04:03.196
Now I can do it over here,
42, 52, 62, 72, 82, and 92.

00:04:03.196 --> 00:04:07.126
And I can do it for each of
these, for each of these now.

00:04:07.126 --> 00:04:12.378
So that’s, now, 44,
54, 64, 74, 84, and 94.

00:04:12.378 --> 00:04:16.487
And so when I do that
we see the full pattern.

00:04:16.487 --> 00:04:19.930
We see the full pattern
and I’m almost done.

00:04:19.930 --> 00:04:28.738
I’m almost done filling in
my numbers from zero to 99.

00:04:28.738 --> 00:04:34.461
49, 59,69,79,89,99 and if
we just wanna feel good,

00:04:34.461 --> 00:04:40.979
we could throw in, we
could throw in a 100,

00:04:40.979 --> 00:04:44.902
a 100 right over there.

00:04:44.902 --> 00:04:47.109
And you’ll see the pattern
still holds, we went from one,

00:04:47.109 --> 00:04:50.477
two, three, four, five,
six, seven, eight, nine,

00:04:50.477 --> 00:04:55.123
and now we got to 10 followed by a zero.

00:04:55.123 --> 00:04:57.470
That was pretty neat.


Subtracting 14 - 6

WEBVTT
Kind: captions
Language: en

00:00:00.241 --> 00:00:02.200
Voiceover:Let's see if we can compute what

00:00:02.200 --> 00:00:06.692
14 - 2 is, and then
figure out what 14 - 4 is,

00:00:06.692 --> 00:00:09.357
and then figure out what 14 - 6 is.

00:00:09.357 --> 00:00:10.997
And I encourage you to pause this video

00:00:10.997 --> 00:00:14.149
and try to figure these out
before I work through them.

00:00:14.580 --> 00:00:16.422
Voiceover:So I'm assuming you've given

00:00:16.422 --> 00:00:18.117
it a try, now let's think about it.

00:00:18.117 --> 00:00:21.512
The number 14 just by how it's written,

00:00:21.512 --> 00:00:24.105
we know that it's going
to be 1 group of ten.

00:00:24.105 --> 00:00:28.353
That's what this 1 tells
us plus another 4 ones.

00:00:28.353 --> 00:00:31.250
So let's verify that we
have 14 objects down here.

00:00:31.250 --> 00:00:37.088
This is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

00:00:37.088 --> 00:00:39.888
So we definitely have a group of ten.

00:00:39.888 --> 00:00:43.619
We have 1 group of ten here,
so that's my group of ten.

00:00:44.496 --> 00:00:48.473
So let me box that off,
so I have my group of ten.

00:00:48.473 --> 00:00:50.183
And then how many ones do I have?

00:00:50.183 --> 00:00:54.165
I have 1, 2, 3, 4, I have four ones.

00:00:54.165 --> 00:00:58.321
So this is indeed, 14.

00:00:58.321 --> 00:01:00.823
This one that we have right over here,

00:01:00.823 --> 00:01:05.143
this represents this
ten, it's 1 group of ten.

00:01:05.143 --> 00:01:09.386
This is 1 group of ten,
let me write that down.

00:01:09.386 --> 00:01:11.954
Group of ten.

00:01:11.954 --> 00:01:18.665
And then we have 4 ones,
4, I guess we could say,

00:01:18.665 --> 00:01:23.191
4 ones like that, so that is indeed 14.

00:01:23.191 --> 00:01:27.705
Now let's look at each of
these, what is 14-2 going to be?

00:01:27.705 --> 00:01:30.828
Well we could take away
2, so take away 1...

00:01:30.828 --> 00:01:33.918
take away 2, and how
many are we left with?

00:01:33.918 --> 00:01:36.710
Well we still have our one group of ten,

00:01:36.710 --> 00:01:40.250
so it's going to be 1,
followed by how many ones

00:01:40.250 --> 00:01:44.387
do we have left over, well we have 2 ones.

00:01:44.387 --> 00:01:46.956
We have 2 ones left over.

00:01:46.956 --> 00:01:51.828
1, 2, so we are going to be left with 12.

00:01:51.828 --> 00:01:57.510
Now what about 14 - 4?
So let me clear this out.

00:02:01.463 --> 00:02:03.475
Actually let me do it like this.

00:02:03.491 --> 00:02:10.513
So if I clear that, so
what is 14 - 4 going to be?

00:02:10.513 --> 00:02:16.742
So now I'm going take away 1, 2, 3, and 4.

00:02:16.742 --> 00:02:19.802
I have essentially taken away all 4 ones.

00:02:19.802 --> 00:02:21.197
So what am I left with?

00:02:21.197 --> 00:02:23.917
Well I still have my 1 group of ten,

00:02:23.917 --> 00:02:28.642
I still have my 1 group of ten
and I have 0 ones left over.

00:02:28.642 --> 00:02:32.168
No ones anymore, so now I have 0 ones.

00:02:32.168 --> 00:02:35.794
And so 14 - 4 is 10, and that makes sense.

00:02:35.794 --> 00:02:39.590
14 is 10 + 4 and now we're subtracting 4,

00:02:39.590 --> 00:02:42.410
to get 10, let me write this down.

00:02:42.410 --> 00:02:48.252
This is the same thing, 14 is 10 + 4.

00:02:48.252 --> 00:02:52.343
Voiceover:10 + 4, this is 14

00:02:52.343 --> 00:02:55.130
and then we're going to subtract 4,

00:02:55.130 --> 00:02:58.381
Then we're subtracting 4,
so if you have 10 + 4 - 4.

00:02:58.381 --> 00:03:01.762
Well the 4 - 4 is going
to be 0, you're going

00:03:01.762 --> 00:03:05.259
to be left with 10 + 0 or just 10.

00:03:05.259 --> 00:03:07.386
We can do the same thing up here.

00:03:07.386 --> 00:03:16.254
This is equal to 14 is 10 +
4 and then we subtracted 2.

00:03:16.254 --> 00:03:18.822
And then we subtracted 2.

00:03:18.822 --> 00:03:21.810
So what you're going to be
left with 10 + what's 4 - 2?

00:03:21.810 --> 00:03:24.543
It's 2, so that's what
we got right over here.

00:03:24.543 --> 00:03:28.301
Let me make this very
clear, this simplifies to 2.

00:03:28.301 --> 00:03:30.482
This 2 right over here.

00:03:30.482 --> 00:03:33.726
This simplifies to 0,
this 0 right over here.

00:03:33.726 --> 00:03:38.266
Now let's do the last one, what is 14 - 6?

00:03:38.266 --> 00:03:41.298
Well we're going to take away 1, 2, 3, 4,

00:03:41.298 --> 00:03:46.439
then 5, and 6, so now we've
broken into our group of ten.

00:03:46.439 --> 00:03:49.122
So this is going to be a one-digit number.

00:03:49.122 --> 00:03:54.035
And we are left with
1, 2, 3, 4, 5, 6, 7, 8.

00:03:54.035 --> 00:04:02.364
So this is going to be equal
to 8 and we're all done.


Adding vectors exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.833 --> 00:00:03.033
Voiceover:Vectors A and B are drawn

00:00:03.033 --> 00:00:05.800
in the interactive graph below.

00:00:06.033 --> 00:00:08.966
Click to graph the terminal point of the

00:00:08.966 --> 00:00:11.966
vector obtained by adding A and B,

00:00:11.966 --> 00:00:14.366
who's initial point is the origin.

00:00:14.366 --> 00:00:17.300
So we want to add A and B.

00:00:17.300 --> 00:00:18.433
So let's see.

00:00:18.433 --> 00:00:22.533
So vector A, we're going to
start at the tail of vector A.

00:00:22.766 --> 00:00:25.333
And so vector A takes us right over here.

00:00:25.500 --> 00:00:28.500
And then if we were to put
vector B, if we put it's tail

00:00:28.500 --> 00:00:31.833
at the head of vector A,
and so essentially shift

00:00:31.833 --> 00:00:35.300
vector B over like this,
we see that vector B,

00:00:35.300 --> 00:00:37.866
we're going to go in the X
direction negative three,

00:00:37.900 --> 00:00:39.933
and in the Y direction positive three.

00:00:40.300 --> 00:00:42.300
So over here you would
go in the X direction

00:00:42.300 --> 00:00:44.633
negative three, and in the Y direction you

00:00:44.633 --> 00:00:48.133
would go positive three, just like that.

00:00:48.366 --> 00:00:52.766
So this right over here would
be the head of A plus B,

00:00:52.766 --> 00:00:56.566
Notice vector A and then if we
were to shift vector B over,

00:00:56.566 --> 00:01:00.133
right over here, vector B
would be right over here,

00:01:00.366 --> 00:01:04.033
and so A plus B would start
at the tail of vector A

00:01:04.033 --> 00:01:08.400
and go to the head of
our shifted vector B.

00:01:08.500 --> 00:01:09.566
You could have done it
the other way around.

00:01:09.566 --> 00:01:10.900
You could have said, "Oh, B plus A.

00:01:10.900 --> 00:01:14.500
"Let me just take this vector
and shift it over here,

00:01:14.500 --> 00:01:17.100
"so this vector goes
four in the positive X.

00:01:17.100 --> 00:01:19.233
"direction and one in the positive Y."

00:01:19.233 --> 00:01:21.233
Four in the positive
X, one in the positive

00:01:21.233 --> 00:01:23.233
Y would have gotten you to the same point.

00:01:23.233 --> 00:01:24.100
It doesn't matter whether you're

00:01:24.100 --> 00:01:28.014
adding A plus B or B plus A.


Measuring a golden statue

WEBVTT
Kind: captions
Language: en

00:00:00.531 --> 00:00:02.551
- Hello, your Majesty,
what can I help you with?

00:00:02.551 --> 00:00:06.417
Oh Sal, I have such an emergency.

00:00:06.417 --> 00:00:09.390
I have commissioned a gold statue

00:00:09.390 --> 00:00:12.895
of my father's head, and I'm not sure

00:00:12.895 --> 00:00:15.459
if it's the right size.

00:00:15.459 --> 00:00:19.640
Oh, well what is the right size?

00:00:19.640 --> 00:00:23.204
Well as you know, in our kingdom we have

00:00:23.204 --> 00:00:25.258
a tradition of measuring things

00:00:25.258 --> 00:00:28.161
in terms of blue fuzzy squares.

00:00:28.161 --> 00:00:29.182
Yep, I'm aware of that.

00:00:29.182 --> 00:00:30.881
I also like to measure things

00:00:30.881 --> 00:00:32.982
in terms of blue fuzzy squares.

00:00:32.982 --> 00:00:37.982
And by law, all gold
statues in our kingdom

00:00:38.521 --> 00:00:43.521
have to be exactly four
blue fuzzy squares tall.

00:00:45.044 --> 00:00:47.262
Okay, that seems like a reasonable law.

00:00:47.262 --> 00:00:48.277
So what's the issue?

00:00:48.277 --> 00:00:50.075
Well I'm not sure if this statue

00:00:50.075 --> 00:00:53.558
is four blue fuzzy squares tall.

00:00:53.558 --> 00:00:55.880
Well okay, let's calm down.

00:00:55.880 --> 00:00:57.552
I think we can work through this.

00:00:57.552 --> 00:00:59.670
I have a bunch of blue fuzzy squares here.

00:00:59.670 --> 00:01:02.270
Why don't we put them next to the statue

00:01:02.270 --> 00:01:05.753
and see if we are exactly four

00:01:05.753 --> 00:01:07.576
blue fuzzy squares tall.

00:01:07.576 --> 00:01:09.793
Oh, that would be just wonderful.

00:01:09.793 --> 00:01:11.905
Here you go, here you go, your Highness.

00:01:12.997 --> 00:01:15.534
Why don't you stack them up?

00:01:15.534 --> 00:01:18.160
Oh so let me do this, okay.

00:01:18.160 --> 00:01:23.160
So that's one, two, and three blue fuzz--

00:01:25.184 --> 00:01:27.111
Oh my God, this is an emergency!

00:01:27.111 --> 00:01:29.550
This is only three blue
fuzzy squares tall!

00:01:29.550 --> 00:01:31.093
It is not tall enough!

00:01:31.093 --> 00:01:32.405
Oh no, what am I going to do?

00:01:32.405 --> 00:01:35.101
Oh wait, hold on, hold on.

00:01:35.101 --> 00:01:38.040
I don't think you have to panic just yet

00:01:38.040 --> 00:01:41.570
because when I look at
how you stacked them up,

00:01:41.570 --> 00:01:42.933
and it's actually quite amazing

00:01:42.933 --> 00:01:44.930
because you somehow made these

00:01:44.930 --> 00:01:46.438
blue fuzzy squares levitate.

00:01:46.438 --> 00:01:48.354
Oh, of course, all blue fuzzy squares

00:01:48.354 --> 00:01:49.608
need to levitate.

00:01:49.608 --> 00:01:51.918
Okay, well yeah, that's neat.

00:01:51.918 --> 00:01:53.067
But if you really wanted to measure,

00:01:53.067 --> 00:01:53.721
you need to make sure that you

00:01:53.721 --> 00:01:56.073
don't have any gaps over here.

00:01:56.073 --> 00:01:57.337
So why don't you try again?

00:01:57.337 --> 00:02:00.158
So let's take these
blue fuzzy squares out,

00:02:00.158 --> 00:02:02.005
and why don't you try to measure again.

00:02:02.005 --> 00:02:04.915
Oh, okay, now that makes
all the sense in the world.

00:02:04.915 --> 00:02:07.052
Okay, so let me try again.

00:02:07.052 --> 00:02:12.052
So that's one, two, notice
I'm not letting them,

00:02:12.834 --> 00:02:13.994
no gaps here.

00:02:13.994 --> 00:02:16.323
Yep, yep, okay, although
it's already, well--

00:02:16.323 --> 00:02:17.533
Well let me keep going Sal,

00:02:17.533 --> 00:02:18.723
stop interrupting me.

00:02:18.723 --> 00:02:21.149
Okay, okay, keep going, your Highness.

00:02:21.149 --> 00:02:26.149
Three, four, and five.

00:02:28.592 --> 00:02:31.168
Oh no, oh no, no, no, this is all wrong!

00:02:31.168 --> 00:02:32.990
It's going to be a disaster!

00:02:32.990 --> 00:02:34.743
It needs to be four, not five

00:02:34.743 --> 00:02:35.928
blue fuzzy squares tall!

00:02:35.928 --> 00:02:38.192
Wait, wait, your
Highness, don't panic yet.

00:02:38.192 --> 00:02:39.306
I don't think this is quite right

00:02:39.306 --> 00:02:43.148
because you didn't start at the bottom

00:02:43.148 --> 00:02:45.192
of the statue, you
started below the bottom

00:02:45.192 --> 00:02:46.631
of the statue, and you didn't end

00:02:46.631 --> 00:02:48.407
at the top of the statue, you ended

00:02:48.407 --> 00:02:50.266
above the top of the statue.

00:02:50.266 --> 00:02:52.831
So it isn't five blue fuzzy squares tall.

00:02:52.831 --> 00:02:55.129
Oh Sal, I'm just so confused.

00:02:55.129 --> 00:02:56.709
Can you just do it for me?

00:02:57.478 --> 00:02:59.069
Well I think it's better
if you do it yourself.

00:02:59.069 --> 00:03:03.272
Just keep it in mind, start measuring

00:03:03.272 --> 00:03:04.887
at the bottom of the statue,

00:03:04.887 --> 00:03:07.244
and stop measuring at
the top of the statue.

00:03:07.244 --> 00:03:08.566
Just make sure that you don't have

00:03:08.566 --> 00:03:09.972
any spaces in between.

00:03:09.972 --> 00:03:12.944
Okay, I'm getting tired, but I'll give it

00:03:12.944 --> 00:03:14.987
one last try.

00:03:14.987 --> 00:03:17.409
Yeah, I think that's a good idea.

00:03:17.409 --> 00:03:21.565
It's always about making
a good effort on things.

00:03:21.565 --> 00:03:23.724
Okay, here I go.

00:03:24.231 --> 00:03:28.734
This is the bottom, so
I'll start stacking there.

00:03:28.734 --> 00:03:32.333
This is the top, so I
will stop stacking there.

00:03:32.333 --> 00:03:37.333
Okay, one, two, and I don't want any gaps,

00:03:38.718 --> 00:03:43.718
so two, three, and four.

00:03:47.274 --> 00:03:50.652
Oh, it actually looks just about right.

00:03:50.652 --> 00:03:52.428
It looks just about right.

00:03:52.428 --> 00:03:57.428
We are exactly four
blue fuzzy squares tall.

00:03:59.011 --> 00:03:59.655
Oh this is just going to be

00:03:59.655 --> 00:04:03.069
a wonderful addition to the palace.

00:04:03.069 --> 00:04:04.345
Thank you so much.

00:04:04.345 --> 00:04:07.166
Well I'm glad to help you, your Highness.


Teens as sums with 10

WEBVTT
Kind: captions
Language: en

00:00:00.318 --> 00:00:04.563
Let's count together from zero to 19.

00:00:04.563 --> 00:00:11.224
So, we have zero, one, two, three, four, five,

00:00:11.224 --> 00:00:16.520
six, seven, eight, nine.

00:00:16.920 --> 00:00:19.449
Now, we go to the two-digit numbers.

00:00:19.602 --> 00:00:35.938
Then, we go to 10, 11, 12, 13, 14, 15, 16, 17, 18 and 19.

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Now, why did I do this?

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The whole point of this is to see a pattern here,

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especially in these two-digit numbers.

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What do we see is true

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about all of these numbers?

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You could think of them they're being in their teens.

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They all, you might see, have a one as their first digit,

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and that one is sitting in what we'll call the tens place.

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and literaly says that each of these numbers

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that one represents a 10.

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What do I mean by that?

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If we go up to 10 over here,

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this is telling us that this one represents 10,

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and this zero represents zero ones,

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so 10 is 10 plus zero.

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You say, "Oh, well, what's the big deal?

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"We already knew that any number plus zero

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"is that number again."

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I think you'll see the pattern

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as we go into these other numbers.

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So, 11 is equal to -

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so the yellow one represents 10,

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and this pink one represents one,

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and then 12 - 12 - this yellow one represents 10,

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and this pink two represents two.

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You could think of it as

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this yellow one says this is one 10,

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so we have a 10 right over there.

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This is two ones, and you have two right over there.

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We can keep going.

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I think you see the pattern.

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13 is 10 plus three. 13 is 10 plus three.

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14 is equal to 10 plus four, plus four.

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15 is equal to 10 plus five.

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I'd encourage you to pause the video now,

00:02:12.742 --> 00:02:14.233
and try to finish the rest.

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Try to write 16, 17, 18 and 19

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as 10 plus some other number,

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and really just make sure

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that you understand the pattern that's happening.

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Let's do that.

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So, 16 - so I can even write them all like this.

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They're all going to be 10 plus something,

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so 16 is 10 plus six.

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17 is 10 plus seven.

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18 is 10 plus eight,

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and 19 is 10 plus nine.

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When you write it out like this,

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it becomes pretty clear

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that this one in each of these numbers

00:02:49.474 --> 00:02:54.056
represents 10, and the digit on the right -

00:02:54.779 --> 00:02:57.738
that just represents that many ones.