Subtraction 2

WEBVTT
Kind: captions
Language: en

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Let's review a little bit
about what we know so

00:00:02.850 --> 00:00:04.820
far about subtraction.

00:00:04.820 --> 00:00:09.210
So if I say 5 minus 3,
what does that mean?

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Well, there's a couple of
ways to think about it.

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I could have 5-- let's
say I had to 5 berries.

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So 1, 2, 3, 4, 5.

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So I could have 5 berries, and
when I say minus 3 you're

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subtracting 3 from it.

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I can view that as saying
I'm going to take away

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3 of these berries.

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So if I take away that berry,
that berry, and that berry.

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So I took away 1, 2, 3 berries.

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How many berries
do I have left?

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Well the only berries I have
left are right here-- 1, 2.

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So I have 2 berries
left just like that.

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Now the other way, the other
way that I could visualize or

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think about 5 minus 3,
I'll do it over here.

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5 minus 3-- is to think
about what the difference

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between 5 and 3 is.

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So let me draw this.

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So let's say I have 5 berries.

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1, 2, 3, 4, 5.

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And let's say that
you have 3 berries.

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Here's a slightly
different color.

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You have 3 berries.

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So another way to think about
5 minus 3 is how many more

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berries do I have
than you have?

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And if you look right here,
well, you see, this berry

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is another-- you have
also one berry there.

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We both have one berry there,
we both have one berry there.

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But I've got 1, 2 berries
that you don't have.

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So once again, I have 2 more
berries than you have.

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Now we can also think of
this from the number

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line point of view.

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So let me draw a number
line just like that.

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It's my number line.

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We've learned on the
addition videos we can

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keep going off forever.

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And actually, we could even go
to the left of 0 and go into

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negative numbers, which
we'll see in future videos.

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But I'll start at 0.

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0, 1, 2, 3, 4, 5--
I'll just go up to 7.

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So if we do 5 minus 3, if we
view 3 as being taken away from

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5, 5 minus 3 means start at 5.

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If I did 5 plus 3 I would jump
3 spots to the right because

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that's increasing the
number of things I have.

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But since I'm subtracting 3,
I want to decrease by 3.

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So I decrease by 1, 2, 3 and
I get to 2 just like that.

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Now if we visualize it from
this point of view, let me

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draw another number line.

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I want to show you.

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I mean this is, I'm taking away
3 and here I'm saying, how

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many more is 5 than 3?

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Even though they're the exact
same answer, but there are two

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different ways to
think about it.

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Let me draw a number
line here again.

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Let me draw the
same number line.

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I have 0, 1, 2, 3, 4, 5, 6, 7.

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So if I were to plot where 5
is on this number line, so

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this is the 5 right there.

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I'll put a little pink
square around it.

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5 is right there.

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Now 3, let me do 3 in
this yellow color.

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3 is right here on
the number line.

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So in this way of thinking
about 5 minus 3 you're saying,

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what is the difference--
let me write that down.

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Here we're saying, what is the
difference between 5 and 3?

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And to figure out the
difference you actually have to

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say, how much do you have to
add to 3 to get to 5?

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So the difference here, how
different is 5 than 3?

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Well you have to go up 1
and then up 2 to get to 5.

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So the difference between 5,
which is all the way over here,

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and 3, which is just that
far, is 2, just like that.

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That right there is 2.

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Let me draw that
in another box.

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So that's 2 right here.

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I want to make this difference
between subtraction and

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difference-- I want to make it
at least reasonably clear to

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you because these are two
different ways of viewing

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subtraction, but it ends up
being the exact same operation.

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You're going to get the same
answer regardless of which

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way you think about it.

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Now, I could view-- let me
do different numbers now.

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Let me do 7 minus 4.

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So I could view this as,
maybe I have a 7 foot

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long piece of wood.

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It's 7 feet long.

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If I put a ruler up against
it I would have 0,

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1, 2, 3, 4, 5, 6, 7.

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So I have a 7 foot
long piece of wood.

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And then I could saw
off 4 of those feet.

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So if I were to saw off
4 of these feet-- so

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I saw off 1, 2, 3, 4.

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How much wood do I have left?

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So all of this stuff right
here, I'm eliminating.

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I'm sawing it off.

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I'm sawing it off of the wood.

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Maybe I should do that in
a darker color to show

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that I'm sawing it off.

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So all of this stuff is
going to disappear.

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I'm grinding it away.

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I'm sawing that off.

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So I'm just left with-- after I
saw the 4 inches or feet or

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whatever of the wood, I'm left
with 1, 2, 3 inches of wood.

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So this is 3.

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So 7 minus 4 is equal to 3.

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This is viewing subtraction
as literally taking away.

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I sawed off the wood,
so I took away wood.

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Now I could think of it in a
slightly different way of

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thinking about it, but give
you the exact same answer.

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We could say 7 minus 4.

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So once again, I could have
the 7 inch long piece

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of wood like that.

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So if I put a ruler here
that's 1, 2, 3, 4, 5, 6, 7.

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So once again, a 7 inch
long piece of wood.

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And now instead of taking 4
away of it I'm comparing it--

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so that's a 7-- I'm comparing
it to a 4 inch long

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piece of wood.

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So I have another 4 inch long
piece of wood right there.

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That's my 4 inch long piece of
wood. that's 7, this is 4.

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You could view 7 minus 4 as
taking 4 inches away from

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the long piece of wood.

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Or you could view seven minus 4
as the difference between the 4

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inch piece of wood and the
7 inch piece of wood.

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So in this case, what's
the difference?

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To go from the 4 inch piece of
wood to the 7 inch piece of

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wood I would have to grow by 3
inches, or I would have to add

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a 3 inch piece of wood somehow.

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Or the wood would somehow
have to grow by 3 inches in

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order to become 7 inches.

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So these are 2 completely
equivalent ways to

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view subtraction.

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That's all a little bit of a
review from the last video.

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Now what I also want to do in
this video is start tackling

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slightly larger problems.

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But you'll see that really,
the number line applies just

00:07:06.170 --> 00:07:09.070
equally as well as to kind
of the simpler problems

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that we've done before.

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Let's do 17 minus 9.

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So just like everything
else, there's two ways

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we could've done it.

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You know, the more slow way is
you could draw 17 objects.

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Let's say I have 17 chips.

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1, 2, 3, 4, 5, 6, 7, 8, 9 10,
11, 12, 13, 14, 15, 16, 17.

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And I'm going to take
away 9 of them.

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So I'm going to take away
1, 2, 3, 4, 5, 6, 7, 8, 9.

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How many am I left with?

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I'm left with 1, 2,
3, 4 5, 6, 7, 8.

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So 17 minus 9 is equal to 8.

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But that took a long time and
you could imagine, if this

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number was a lot bigger it
would've taken me forever to

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draw all of these circles and
then scratch out things.

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And it would've wasted
paper and time.

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And we have other things to do.

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So another way you could do it,
and maybe this would be easier

00:08:10.050 --> 00:08:11.930
for you to visualize, is
to draw the number line.

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You always don't
have to start at 0.

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So if we draw the number line,
if we say that's 18, 17, 16,

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15, 14, 13, 12, 11, 10, 9, 8,
7-- you could imagine, I could

00:08:32.210 --> 00:08:34.550
keep going to the left
all the way to 0.

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But I start at 17.

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I could start at 17 and
take away 9 from it.

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So I go 1, 2, 3,
4, 5, 6, 7, 8, 9.

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And once again, we
are left at 8.

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Now this was, at least in my
head, a little bit cleaner

00:08:55.660 --> 00:08:57.110
and faster than this one.

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But in either case, you don't
want to do this every time you

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have to subtract 9 from 17 or
want to find the difference

00:09:02.160 --> 00:09:03.990
between 17 and 9.

00:09:03.990 --> 00:09:05.000
And then to realize that's 8.

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So this is something that
eventually you want

00:09:06.910 --> 00:09:08.080
to internalize.

00:09:08.080 --> 00:09:10.720
You'll want to know by heart
that, oh, 17 minus 9?

00:09:10.720 --> 00:09:12.200
I know that is 8.

00:09:12.200 --> 00:09:15.320
And by the way, 17 minus 8?

00:09:15.320 --> 00:09:17.270
What's 17 minus 8?

00:09:17.270 --> 00:09:19.710
Well, that is 9.

00:09:19.710 --> 00:09:21.820
And now why does all
of this make sense?

00:09:21.820 --> 00:09:27.270
Because 8 plus 9
is equal to 17.

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So 17 minus 9 is 8.

00:09:31.860 --> 00:09:35.330
Or 17 minus 8 is 9.

00:09:35.330 --> 00:09:39.110
When I say 17 minus 8, I'm
essentially saying that is

00:09:39.110 --> 00:09:42.550
equal to some number that if I
were to add to 8 will equal 17.

00:09:42.550 --> 00:09:43.580
Well, that's 9.

00:09:43.580 --> 00:09:46.720
When I say 17 minus 9 that's
saying, there's some number,

00:09:46.720 --> 00:09:49.250
that if I were to add
it to 9, I'll get 17.

00:09:49.250 --> 00:09:50.170
And that's 8.

00:09:50.170 --> 00:09:53.620
So all of these, all of these
statements, are kind of

00:09:53.620 --> 00:09:54.760
saying the same thing.

00:09:54.760 --> 00:09:56.500
That 8 plus 9 are 17.

00:09:56.500 --> 00:09:58.660
Or the difference
between 17 and 9 is 8.

00:09:58.660 --> 00:10:03.120
Or the difference
between 17 and 8 is 9.

00:10:03.120 --> 00:10:04.560
Hopefully I'm not
confusing you.

00:10:04.560 --> 00:10:08.660
So for most of these
subtraction problems where the

00:10:08.660 --> 00:10:12.620
answer is a one-digit answer,
you should eventually have them

00:10:12.620 --> 00:10:15.980
memorized, but in your head
it's good to be imagining

00:10:15.980 --> 00:10:16.640
this number line.

00:10:16.640 --> 00:10:18.890
Let's do a couple
more of these.

00:10:18.890 --> 00:10:22.470
And then, once we have these
memorized or at least be able

00:10:22.470 --> 00:10:25.880
to do a number line if we
forget, I'll show you have to

00:10:25.880 --> 00:10:29.460
do any subtraction problem,
arbitrarily for super

00:10:29.460 --> 00:10:30.860
large numbers.

00:10:30.860 --> 00:10:37.050
So let's say we're going
to do 13 minus 5.

00:10:37.050 --> 00:10:39.660
So once again, I'm not going
to do the whole circles

00:10:39.660 --> 00:10:40.540
or the berries this time.

00:10:40.540 --> 00:10:43.460
I'm just going to draw
the number line.

00:10:43.460 --> 00:10:45.720
Just draw the number
line like that.

00:10:45.720 --> 00:10:58.450
Let's start at 14, 13, 12, 11,
10, 9, 8, 7, 6, 5-- and you

00:10:58.450 --> 00:11:00.460
just can keep going
lower and lower.

00:11:00.460 --> 00:11:02.510
You can go to 0 or you
can even go past 0.

00:11:02.510 --> 00:11:03.830
We'll talk about
that in the future.

00:11:03.830 --> 00:11:05.810
But we start at 13.

00:11:05.810 --> 00:11:08.960
We're starting at 13.

00:11:08.960 --> 00:11:10.620
And we're going to
take 5 away from it.

00:11:10.620 --> 00:11:13.920
So this is the subtraction
view of subtraction;

00:11:13.920 --> 00:11:15.200
we're taking away.

00:11:15.200 --> 00:11:21.340
1, 2, 3, 4, 5 and we land at 8.

00:11:21.340 --> 00:11:26.010
So 13 minus 5-- let me
do this in a new color.

00:11:26.010 --> 00:11:29.810
13 minus 5 is equal to 8.

00:11:29.810 --> 00:11:31.580
Now another way we could
have thought about that,

00:11:31.580 --> 00:11:33.980
I plotted where 13 is.

00:11:33.980 --> 00:11:36.060
I can plot where 5 is.

00:11:36.060 --> 00:11:37.850
I could say look, this is 5.

00:11:37.850 --> 00:11:39.760
5 is right here on
my number line.

00:11:39.760 --> 00:11:42.736
What do I have to add
to 5 to get to 13?

00:11:42.736 --> 00:11:43.280
So let's see.

00:11:43.280 --> 00:11:49.250
I would have to go 1,
2, 3, 4, 5, 6, 7, 8.

00:11:49.250 --> 00:11:52.180
I have to add 8 to
5 to get to 13.

00:11:52.180 --> 00:11:55.790
5 plus 8 is equal to 13.

00:11:55.790 --> 00:11:59.940
So that tells me that 13
minus 5 is equal to 8.

00:11:59.940 --> 00:12:06.210
This also tells me that 13
minus 8 is equal to 5.

00:12:06.210 --> 00:12:08.690
All of these, are on some
level, telling me the

00:12:08.690 --> 00:12:09.650
exact same thing.

00:12:09.650 --> 00:12:11.635
But the difference
between 13 and 5 is 8.

00:12:11.635 --> 00:12:14.410
The difference between
13 and 8 is 5.

00:12:14.410 --> 00:12:17.080
5 plus 8 is 13.

00:12:17.080 --> 00:12:18.780
So hopefully you have the hang
of that and if you haven't done

00:12:18.780 --> 00:12:22.940
so already, it'll be good
to practice all of these.

00:12:22.940 --> 00:12:26.240
Taking a teen number and then
subtracting any of the

00:12:26.240 --> 00:12:28.470
one-digit numbers from
those teen numbers.

00:12:28.470 --> 00:12:32.110
That's in general, very,
very good practice for you.


Addition 4

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:02.910
Let's do a bunch more of
these addition problems.

00:00:02.910 --> 00:00:12.420
So let's say I have
9,367 plus 2,459.

00:00:12.420 --> 00:00:14.700
So we can do this the exact
same way we've done in

00:00:14.700 --> 00:00:15.730
the last few videos.

00:00:15.730 --> 00:00:17.620
We start in the 1's place
or you can even think of

00:00:17.620 --> 00:00:19.340
it as the 1's column.

00:00:19.340 --> 00:00:22.590
So you're going to add the
seven 1's plus the 9 1's.

00:00:22.590 --> 00:00:26.260
So you're going to have 7
plus 9, which we hopefully

00:00:26.260 --> 00:00:29.080
know by now is 16.

00:00:29.080 --> 00:00:32.260
So what we do is we write
the 6 in the 1's place

00:00:32.260 --> 00:00:35.480
and we carry the 1.

00:00:35.480 --> 00:00:38.850
Let me switch-- if this 1 is
going to be the same thing

00:00:38.850 --> 00:00:41.370
as that 1 right there.

00:00:41.370 --> 00:00:43.520
And this might look like a
little bit of a mystery or

00:00:43.520 --> 00:00:45.460
magic, and the whole reason
we did that is that

00:00:45.460 --> 00:00:46.805
this is the 10's place.

00:00:46.805 --> 00:00:50.870
And when you write 16 you
have six 1's and one 10.

00:00:50.870 --> 00:00:53.610
If you view this as money,
what's the best way to

00:00:53.610 --> 00:00:56.360
get $16 in a world where
there weren't $5 bills?

00:00:56.360 --> 00:00:59.140
Where you only had $1
bills, $10 bills, $10

00:00:59.140 --> 00:01:00.070
bills, and so on.

00:01:00.070 --> 00:01:02.440
Only multiples of 10.

00:01:02.440 --> 00:01:03.950
And we don't have any $5 bills.

00:01:03.950 --> 00:01:08.010
In that world you would
represent 16 as one $10

00:01:08.010 --> 00:01:11.930
bill just like that.

00:01:11.930 --> 00:01:13.750
And then six $1 bills.

00:01:16.820 --> 00:01:20.060
So that's two $1 bills.

00:01:20.060 --> 00:01:22.460
That's two more $1 bills.

00:01:22.460 --> 00:01:25.570
And then that's two
more $1 bills.

00:01:28.550 --> 00:01:30.700
The whole reason why I'm
drawing it this way or I'm even

00:01:30.700 --> 00:01:33.570
using this analogy or drawing
the dollar bills is to show

00:01:33.570 --> 00:01:34.890
you what these places mean.

00:01:37.900 --> 00:01:42.240
When I say that this right
here is the 10's place, I'm

00:01:42.240 --> 00:01:46.160
essentially telling you how
many $10 bills do I have?

00:01:46.160 --> 00:01:50.965
If I've $16 and I'm doing it
as efficiently as I can in

00:01:50.965 --> 00:01:52.380
a world without $5 bills.

00:01:52.380 --> 00:01:54.780
I only have $1's, $10's,
and $100's and $1,000

00:01:54.780 --> 00:01:55.850
bills and so forth.

00:01:55.850 --> 00:01:58.630
And this is the 1's.

00:01:58.630 --> 00:02:01.190
So when I write it this way I'm
literally telling you, I have

00:02:01.190 --> 00:02:05.050
one $10 bill and I
have six $1 bills.

00:02:05.050 --> 00:02:07.470
That's what $16 is.

00:02:07.470 --> 00:02:11.700
And so when I have 7 plus 9 is
equal to 16 I say that I have

00:02:11.700 --> 00:02:15.780
six $1 bills and I
have one $10 bill.

00:02:15.780 --> 00:02:17.760
And I add that one $10
bill to everything

00:02:17.760 --> 00:02:19.250
else in the ten space.

00:02:19.250 --> 00:02:21.770
And the tens place is
essentially telling you how

00:02:21.770 --> 00:02:23.560
many-- that's the tens.

00:02:23.560 --> 00:02:27.500
I could write it like that or
I could write the 10's place.

00:02:27.500 --> 00:02:31.850
When I have 67-- 67 means
I have six $10 bills

00:02:31.850 --> 00:02:33.470
plus another seven $1's.

00:02:33.470 --> 00:02:36.400
So that's six 10's, five 10's.

00:02:36.400 --> 00:02:38.690
So I add up everything
in the tens place.

00:02:38.690 --> 00:02:41.170
So 1 plus 6 plus 5.

00:02:41.170 --> 00:02:43.160
Let me do that in a new color.

00:02:43.160 --> 00:02:47.800
1 plus 6 plus 5 is equal
to-- 1 plus 6 is 7.

00:02:47.800 --> 00:02:50.000
7 plus 5 is 12.

00:02:50.000 --> 00:02:53.670
So I write the 2 in the 10's
place because remember, this

00:02:53.670 --> 00:02:58.710
is twelve $10 bills because
we're in the 10's place.

00:02:58.710 --> 00:03:02.250
So I have two in the 10's place
and I put the 1-- I carried

00:03:02.250 --> 00:03:05.240
this 1 right here into
the 100's place.

00:03:05.240 --> 00:03:09.950
Because if I have twelve
$10 bills, I have $120.

00:03:09.950 --> 00:03:12.070
I have one $100 bill.

00:03:12.070 --> 00:03:14.010
And I have two $10 bills.

00:03:14.010 --> 00:03:17.030
I'll stop going to the dollar
bill analogy just so we can

00:03:17.030 --> 00:03:18.870
make sure we understand
the process.

00:03:18.870 --> 00:03:20.690
But I think you
see how it works.

00:03:20.690 --> 00:03:22.800
You start at the right, you
add the two numbers up.

00:03:22.800 --> 00:03:26.930
If it's a two-digit answer you
carry the left most digit

00:03:26.930 --> 00:03:27.980
up to the next column.

00:03:27.980 --> 00:03:29.490
And you just keep doing that.

00:03:29.490 --> 00:03:30.720
So let's do this
one right here.

00:03:30.720 --> 00:03:32.580
1 plus 3 is 4.

00:03:32.580 --> 00:03:35.270
Let me write this down
in another color.

00:03:35.270 --> 00:03:36.980
1 plus 3 plus 4.

00:03:39.930 --> 00:03:41.260
1 plus 3 is 4.

00:03:41.260 --> 00:03:43.050
Plus 4 is 8.

00:03:43.050 --> 00:03:45.680
So 1 plus 3 plus 4 is 8.

00:03:45.680 --> 00:03:46.630
Nothing to carry.

00:03:46.630 --> 00:03:48.290
It was a one-digit number.

00:03:48.290 --> 00:03:50.930
And then finally,
I have 9 plus 2.

00:03:53.700 --> 00:03:57.070
That's equal to 11, so I
write the 1 down there.

00:03:57.070 --> 00:03:59.300
I write this 1 and then if
there was anything left here I

00:03:59.300 --> 00:04:03.880
would carry the 10's or the
other 1-- the 1 in the 10's

00:04:03.880 --> 00:04:05.570
place in 11-- I would carry it.

00:04:05.570 --> 00:04:07.265
But there's no where to
carry it to, so I write

00:04:07.265 --> 00:04:09.080
it down just like that.

00:04:09.080 --> 00:04:16.400
So 9.367 plus 2,459 is 11,826.

00:04:16.400 --> 00:04:18.530
And I just put that comma
there because it's

00:04:18.530 --> 00:04:19.834
easier for me to read.

00:04:19.834 --> 00:04:21.160
Let me do a bunch
more of these.

00:04:21.160 --> 00:04:24.110
Let's do a really, really
daunting problem.

00:04:24.110 --> 00:04:25.770
Let's do something
in the millions.

00:04:25.770 --> 00:04:29.435
Just to show you that
you can do any problem.

00:04:29.435 --> 00:04:37.860
So let's say we have 2,349,015.

00:04:37.860 --> 00:04:39.180
Let's throw a 0 in there.

00:04:39.180 --> 00:04:40.970
We have nothing in the
hundreds place there.

00:04:40.970 --> 00:04:44.920
And I want to add that to-- let
me switch colors just for fun.

00:04:44.920 --> 00:04:55.980
I want to add that to 7,--
let's put a 0 there-- 15,999.

00:04:55.980 --> 00:04:56.980
Let's add these two numbers.

00:04:56.980 --> 00:05:00.420
It seems like a hard problem,
but if we just focus on each

00:05:00.420 --> 00:05:03.770
of the places I think you'll
find that it's not too bad.

00:05:03.770 --> 00:05:07.730
So we start off with 5 plus 9.

00:05:07.730 --> 00:05:09.970
That's equal to 14.

00:05:09.970 --> 00:05:13.210
Write the 4 down
here, carry the 1.

00:05:13.210 --> 00:05:14.490
Then you go into
the 10's place.

00:05:14.490 --> 00:05:16.880
1 plus 1 is 2.

00:05:16.880 --> 00:05:19.260
2 plus 9-- let me
switch colors.

00:05:19.260 --> 00:05:20.740
1 plus 1 is 2.

00:05:20.740 --> 00:05:23.130
2 plus 9 is 11.

00:05:23.130 --> 00:05:25.050
Carry the 1.

00:05:25.050 --> 00:05:26.530
Now we're in the 100's place.

00:05:26.530 --> 00:05:28.970
1 plus 0 is 1.

00:05:28.970 --> 00:05:31.070
Plus 9 is 10.

00:05:31.070 --> 00:05:34.935
So we write the 0 from
the 10, carry the 1.

00:05:34.935 --> 00:05:37.630
Let me switch colors again.

00:05:37.630 --> 00:05:39.740
1 plus 9 is 10.

00:05:39.740 --> 00:05:43.360
10 plus 5 is 15.

00:05:43.360 --> 00:05:45.580
Now we're in the
10,000's place.

00:05:45.580 --> 00:05:47.730
1 plus 4 is 5.

00:05:47.730 --> 00:05:51.110
And 5 plus 1 is 6.

00:05:51.110 --> 00:05:52.900
And there's nothing to carry.

00:05:52.900 --> 00:05:55.030
Now we're in the
100,000's place.

00:05:55.030 --> 00:05:57.770
3-- we have nothing to carry,
so we just have the three

00:05:57.770 --> 00:06:00.570
100,000's plus zero 100,000's.

00:06:00.570 --> 00:06:02.960
Well, that's just
three 300,000.

00:06:02.960 --> 00:06:05.130
And then finally, we're
in the millions place.

00:06:05.130 --> 00:06:09.550
2,000,000 plus 7,000,000
is 9,000,000.

00:06:09.550 --> 00:06:10.360
Just like that.

00:06:10.360 --> 00:06:12.170
So this was a super
crazy number.

00:06:12.170 --> 00:06:17.960
2,349,015 plus 7,015,999.

00:06:17.960 --> 00:06:21.170
Just by keeping track of our
places and carrying the

00:06:21.170 --> 00:06:23.700
two-digit numbers or the second
digit in the two-digit numbers

00:06:23.700 --> 00:06:25.210
as necessary, we were able
to figure out that the

00:06:25.210 --> 00:06:31.570
answer is 9,365,014.

00:06:31.570 --> 00:06:33.400
So hopefully this gives
you a pretty good sense.

00:06:33.400 --> 00:06:35.670
And let me just do one more,
just to really make sure that

00:06:35.670 --> 00:06:41.120
we really understand how all of
this carrying business works.

00:06:41.120 --> 00:06:58.490
So let's do 15,999,001
plus 6,888,999.

00:06:58.490 --> 00:07:00.850
Let's just see how this
one's going to turn out.

00:07:00.850 --> 00:07:03.140
This seems like a like
a difficult problem.

00:07:03.140 --> 00:07:06.330
But once again, if we just
focus and don't get lost,

00:07:06.330 --> 00:07:09.350
we're going to get the
right answer hopefully.

00:07:09.350 --> 00:07:12.210
So 1 plus 9 is 10.

00:07:12.210 --> 00:07:14.410
Write the 0, carry the 1.

00:07:14.410 --> 00:07:17.420
1 plus 0 plus 9 is 10.

00:07:17.420 --> 00:07:20.180
Write the 0, carry the 1.

00:07:20.180 --> 00:07:22.360
1 plus 0 plus 9.

00:07:22.360 --> 00:07:24.170
That's 10 again.

00:07:24.170 --> 00:07:26.980
Write the 0, carry the 1.

00:07:26.980 --> 00:07:30.790
Now 1 plus 9 is 10, plus 8.

00:07:30.790 --> 00:07:32.810
10 plus 8 is 18.

00:07:32.810 --> 00:07:35.460
Write the 8, carry the 1.

00:07:35.460 --> 00:07:37.940
1 plus 9 is 10.

00:07:37.940 --> 00:07:40.410
Plus eight is 18.

00:07:40.410 --> 00:07:42.270
Write the 8, carry the 1.

00:07:42.270 --> 00:07:43.890
1 plus 9 is 10.

00:07:43.890 --> 00:07:45.080
Plus 8 is 18.

00:07:45.080 --> 00:07:47.980
Write the 8, carry the 1.

00:07:47.980 --> 00:07:50.260
Now we're in the
1,000,000's place.

00:07:50.260 --> 00:07:53.700
1,000,000 plus 5,000,000
is 6,000,000.

00:07:53.700 --> 00:07:56.310
Plus 6,000,000 is 12,000,000.

00:07:56.310 --> 00:08:00.390
Write the 200,000,000 and then
carry the 1 because 12,000,000

00:08:00.390 --> 00:08:03.730
is 2,000,000 plus 10,000,000.

00:08:03.730 --> 00:08:05.420
10,000,000 plus 10,000,000.

00:08:05.420 --> 00:08:08.460
This is one 10,000,000 plus
another one 10,000,000.

00:08:08.460 --> 00:08:10.880
That's 1 plus 1 is 2.

00:08:10.880 --> 00:08:12.070
And then we are done.

00:08:12.070 --> 00:08:23.450
15,999,001 plus 6,888,999
is 22,888,000.

00:08:23.450 --> 00:08:27.680
So you just saw, we're just
doing 7 and 8 digit number

00:08:27.680 --> 00:08:29.590
additions, but you could apply
this-- if I had a number with

00:08:29.590 --> 00:08:32.540
100 digits in it, you could
do the exact same thing.

00:08:32.540 --> 00:08:36.140
You just have to start at the
right, go each column by each

00:08:36.140 --> 00:08:38.550
column, and then if you end up
with a two-digit answer when

00:08:38.550 --> 00:08:42.100
you add the two one-digit
numbers, you just

00:08:42.100 --> 00:08:43.360
carry the 10's place.

00:08:43.360 --> 00:08:45.720
You just doing that and
work your way left.

00:08:45.720 --> 00:08:48.940
And if you make no errors,
you'll get the right answer.


Addition 3

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:04.660
What I want to do in this video
is one, just do a bunch of

00:00:04.660 --> 00:00:08.010
addition examples so that we
really get some good practice

00:00:08.010 --> 00:00:09.865
and we really get warmed
up with addition.

00:00:09.865 --> 00:00:12.940
And what I even more want to
show you is that we now have

00:00:12.940 --> 00:00:18.085
all the tools we need to really
tackle any addition problem.

00:00:18.085 --> 00:00:20.580
So let's just get warmed up
with some one-digit addition

00:00:20.580 --> 00:00:22.780
problems, but these are the
ones that always give me a

00:00:22.780 --> 00:00:23.850
little bit of a headache.

00:00:23.850 --> 00:00:26.525
Let's start with a really,
relatively straightforward.

00:00:26.525 --> 00:00:28.980
I want to say 2 plus 4.

00:00:28.980 --> 00:00:30.240
Well, we know what that is.

00:00:30.240 --> 00:00:32.870
I don't think we need to draw
the number line at this

00:00:32.870 --> 00:00:35.340
point, but you can if you
need to remember this.

00:00:35.340 --> 00:00:37.220
2 plus 4 is 6.

00:00:37.220 --> 00:00:38.430
Not too bad.

00:00:38.430 --> 00:00:42.020
What about 9 plus 3?

00:00:42.020 --> 00:00:44.000
We saw that in the last video.

00:00:44.000 --> 00:00:45.680
9 plus 1 is 10.

00:00:45.680 --> 00:00:47.260
Plus another 1 is 11.

00:00:47.260 --> 00:00:48.600
Plus 3 is 12.

00:00:48.600 --> 00:00:49.890
9 plus 3 is 12.

00:00:49.890 --> 00:00:51.610
And it's probably
not a bad idea.

00:00:51.610 --> 00:00:53.880
It's good to visualize what's
happening here, but it's also

00:00:53.880 --> 00:00:56.380
not a bad idea to be able
to do these very fast.

00:00:56.380 --> 00:00:58.500
To be able to memorize, at
least what the one-digit

00:00:58.500 --> 00:00:59.990
addition problems end up being.

00:00:59.990 --> 00:01:01.640
Let's do a couple
of harder ones.

00:01:01.640 --> 00:01:03.250
6 plus 7.

00:01:03.250 --> 00:01:07.180
This one I used to find
difficult to remember.

00:01:07.180 --> 00:01:10.920
But 6 plus 7 is 13.

00:01:10.920 --> 00:01:13.250
Draw out the number lines
and the lemons and limes

00:01:13.250 --> 00:01:14.630
if you don't believe me.

00:01:14.630 --> 00:01:16.670
6 four 7 is 13.

00:01:16.670 --> 00:01:21.640
8 plus 6 or 6 plus 8
is going to be 14.

00:01:21.640 --> 00:01:24.650
And that's the same
thing as 7 plus 7-- is

00:01:24.650 --> 00:01:27.470
also going to be 14.

00:01:27.470 --> 00:01:29.370
And if you think about
it, we got the same

00:01:29.370 --> 00:01:30.380
number here as there.

00:01:30.380 --> 00:01:32.150
And it makes sense, right?

00:01:32.150 --> 00:01:36.270
Because we took one away from
8, but we had one more than 6.

00:01:36.270 --> 00:01:39.050
So it's like you just shifted
the 1 from the 8 to the 6.

00:01:39.050 --> 00:01:40.360
That's why we got
the same answer.

00:01:40.360 --> 00:01:42.500
If that confused
you, ignore it.

00:01:42.500 --> 00:01:44.000
Let's just do a couple
of more of these.

00:01:44.000 --> 00:01:48.870
So 8 plus 8 is 16.

00:01:48.870 --> 00:01:51.920
These are things that hopefully
you'll be able to do super fast

00:01:51.920 --> 00:01:53.710
in the not too far off future.

00:01:53.710 --> 00:01:56.690
5 plus 6.

00:01:56.690 --> 00:01:59.950
Well, that's 11.

00:01:59.950 --> 00:02:02.000
Let me just do a couple
of more really fast.

00:02:02.000 --> 00:02:07.200
So let's say 7 plus 9
is going to be 16.

00:02:07.200 --> 00:02:08.550
You might want to draw
the number line if

00:02:08.550 --> 00:02:09.180
you don't believe me.

00:02:09.180 --> 00:02:13.640
And that's going to be the same
thing as 8 plus 8, is also 16.

00:02:13.640 --> 00:02:17.590
And the 9 plus 9 is 18.

00:02:17.590 --> 00:02:21.450
And then 9 plus 8 is 17.

00:02:21.450 --> 00:02:22.800
And that's just a
little bit of warm up.

00:02:22.800 --> 00:02:25.090
We didn't do all of the
possible combinations of

00:02:25.090 --> 00:02:27.410
one-digit numbers, but these
are some of the ones that

00:02:27.410 --> 00:02:29.500
give people a little
bit more headache.

00:02:29.500 --> 00:02:32.050
So now that we've done that
let's tackle some larger digit

00:02:32.050 --> 00:02:36.640
numbers that we started
doing in the previous video.

00:02:36.640 --> 00:02:38.070
Maybe I'll leave
that there for now.

00:02:38.070 --> 00:02:40.100
So let's do a couple of them.

00:02:40.100 --> 00:02:44.500
Let's do 22 plus 3.

00:02:47.040 --> 00:02:48.760
So we go to the ones place.

00:02:48.760 --> 00:02:51.770
2 plus 3 is 5.

00:02:51.770 --> 00:02:53.400
We didn't have to
carry anything.

00:02:53.400 --> 00:02:56.390
And then in the tens place we
just have this 2 sitting here.

00:02:56.390 --> 00:02:57.390
So we just take that 2.

00:02:57.390 --> 00:02:59.240
2 plus nothing-- it's two 10's.

00:02:59.240 --> 00:03:00.660
It's two dimes.

00:03:00.660 --> 00:03:01.810
So then we put that down there.

00:03:01.810 --> 00:03:03.210
So we get 25.

00:03:03.210 --> 00:03:07.580
2 dimes and 5 pennies, or $0.25
depending-- a lot of people,

00:03:07.580 --> 00:03:11.330
money makes it easier to
understand things or maybe

00:03:11.330 --> 00:03:13.930
to be motivated to
understand things.

00:03:13.930 --> 00:03:15.570
All right, let's
do another one.

00:03:15.570 --> 00:03:24.540
What is 38 plus 17?

00:03:28.550 --> 00:03:30.770
So we look at just
the ones place.

00:03:30.770 --> 00:03:32.460
What is 8 plus 7?

00:03:32.460 --> 00:03:34.670
We haven't done that one
yet; I'll do it up here.

00:03:34.670 --> 00:03:39.110
8 plus 7 is equal to--
it's going to be one

00:03:39.110 --> 00:03:40.560
more than 8 plus 6.

00:03:40.560 --> 00:03:43.580
8 plus 6 is 14, than 8
plus 7 is going to be

00:03:43.580 --> 00:03:44.330
one more than that.

00:03:44.330 --> 00:03:47.020
So it's going to
be equal to 15.

00:03:47.020 --> 00:03:50.090
So in this problem we
write the 5 here.

00:03:50.090 --> 00:03:51.810
Let me write this in
a different color.

00:03:51.810 --> 00:03:54.550
So the 5 in the 15 we'd
write right down there

00:03:54.550 --> 00:03:55.650
in the ones place.

00:03:55.650 --> 00:03:59.430
And we would carry the 1
because that's one dime.

00:03:59.430 --> 00:04:02.560
That's one 10.

00:04:02.560 --> 00:04:04.840
You know, this 15,
this is 10 plus 5.

00:04:04.840 --> 00:04:08.960
So this 1 really means
one 10 or one dime.

00:04:08.960 --> 00:04:11.630
So we put that 1 up there
in the tens place.

00:04:11.630 --> 00:04:14.210
We have 1 plus 3 is 4.

00:04:14.210 --> 00:04:15.990
Plus 1 is 5.

00:04:15.990 --> 00:04:17.710
So you get 55.

00:04:17.710 --> 00:04:20.000
1 plus 3 plus 1 is this 5.

00:04:20.000 --> 00:04:23.070
38 plus 17 is 55.

00:04:23.070 --> 00:04:26.150
Or five 10's and five 1's.

00:04:26.150 --> 00:04:28.340
That's the same thing as 55.

00:04:28.340 --> 00:04:30.510
Let's do a couple
more problems.

00:04:30.510 --> 00:04:32.480
I think you'll see that we
have the tools to tackle

00:04:32.480 --> 00:04:35.250
anything, any problem.

00:04:35.250 --> 00:04:37.680
Let's say we have 47.

00:04:37.680 --> 00:04:41.500
Let me switch colors just
so it stays interesting.

00:04:41.500 --> 00:04:44.065
47 plus 9.

00:04:46.680 --> 00:04:48.460
We just look at the ones place.

00:04:48.460 --> 00:04:50.120
7 plus 9.

00:04:50.120 --> 00:04:51.270
We know what that is already.

00:04:51.270 --> 00:04:52.750
We did that problem already.

00:04:52.750 --> 00:04:54.960
7 plus 9 is 16.

00:04:54.960 --> 00:04:59.090
So you write the 6 in the ones
place and carry the one.

00:04:59.090 --> 00:05:01.060
And now it's in the tens place.

00:05:01.060 --> 00:05:03.050
Because this is one
10 right there.

00:05:03.050 --> 00:05:07.350
So 1 dime plus 4
dimes is 5 dims.

00:05:07.350 --> 00:05:09.850
So it's 5 dimes and 6 pennies.

00:05:09.850 --> 00:05:11.240
It's 56.

00:05:11.240 --> 00:05:12.740
Let's do slightly
harder problems.

00:05:12.740 --> 00:05:14.750
Let me scroll down a little
bit so we have some

00:05:14.750 --> 00:05:18.740
space to work with.

00:05:18.740 --> 00:05:20.010
We always need that.

00:05:20.010 --> 00:05:22.620
All right, let's do
something hard.

00:05:22.620 --> 00:05:28.540
99 plus 88.

00:05:28.540 --> 00:05:30.520
That's a hard one.

00:05:30.520 --> 00:05:32.855
And you just have to look at
the parts of the problem and

00:05:32.855 --> 00:05:34.010
you'll see how it'll
all work out.

00:05:34.010 --> 00:05:36.060
You say, what's 9 plus 8?

00:05:36.060 --> 00:05:37.640
We did that up here.

00:05:37.640 --> 00:05:40.670
9 plus 8 we know already is 17.

00:05:40.670 --> 00:05:42.380
That's a good one to remember.

00:05:42.380 --> 00:05:44.760
9 plus 8 is 17, but it's
always good to be able

00:05:44.760 --> 00:05:46.700
to visualize it as well.

00:05:46.700 --> 00:05:49.200
so 9 plus 8 is 17.

00:05:49.200 --> 00:05:51.020
Carry the 1.

00:05:51.020 --> 00:05:54.150
And then we have
1 plus 9 is 10.

00:05:54.150 --> 00:05:57.870
10 plus 8 is 18.

00:05:57.870 --> 00:05:59.220
Now this is interesting.

00:05:59.220 --> 00:06:00.780
We want to write 18 down.

00:06:00.780 --> 00:06:04.630
So we write our 8 down there.

00:06:04.630 --> 00:06:07.520
We have 1 plus 9 plus 8.

00:06:07.520 --> 00:06:10.560
1 plus 9 plus 8 is equal to 18.

00:06:10.560 --> 00:06:13.960
We wrote the 8 down there and
we say, let's carry the 1.

00:06:13.960 --> 00:06:17.470
We carry the 1, but we carry
it into the hundreds place.

00:06:17.470 --> 00:06:20.340
This was the ones place,
the tens place, now we're

00:06:20.340 --> 00:06:22.190
in the hundreds place.

00:06:22.190 --> 00:06:23.730
But there's nothing else
in the hundreds place.

00:06:23.730 --> 00:06:25.250
So it just drops straight down.

00:06:25.250 --> 00:06:28.190
So you could almost just
write the 18 just like that.

00:06:28.190 --> 00:06:33.240
So 99 plus 88 is 187.

00:06:33.240 --> 00:06:35.270
Let's keep doing some examples.

00:06:35.270 --> 00:06:37.750
You could see, it's
all the same pattern.

00:06:37.750 --> 00:06:40.290
We could add two ten-digit
numbers to each other as

00:06:40.290 --> 00:06:43.270
long as we're just careful
about carrying our digits.

00:06:43.270 --> 00:06:48.340
So let's do 700-- let
me switch colors.

00:06:48.340 --> 00:06:50.570
Well do some
three-digit numbers.

00:06:50.570 --> 00:06:52.225
Let's do a four-digit number.

00:06:52.225 --> 00:06:53.430
Let's not mess around.

00:06:53.430 --> 00:06:54.880
Let's do a four-digit number.

00:06:54.880 --> 00:07:07.910
So let's do 4,368 plus 572.

00:07:07.910 --> 00:07:09.120
Let's see what happens here.

00:07:12.160 --> 00:07:12.950
I'll write it down here.

00:07:12.950 --> 00:07:14.490
8 plus 2.

00:07:14.490 --> 00:07:15.970
We know that that
is equal to 10.

00:07:15.970 --> 00:07:17.960
You can do the number
line if you need to.

00:07:17.960 --> 00:07:19.780
8 plus 2 is equal to 10.

00:07:19.780 --> 00:07:22.860
Put the 0 on the ones
place, carry the 1.

00:07:22.860 --> 00:07:24.180
Now we're in the tens place.

00:07:24.180 --> 00:07:25.280
This is really one 10.

00:07:25.280 --> 00:07:26.630
This is six 10's.

00:07:26.630 --> 00:07:27.800
This is seven 10's.

00:07:27.800 --> 00:07:29.120
Or you could think of
them as dimes if we're

00:07:29.120 --> 00:07:30.710
thinking about change.

00:07:30.710 --> 00:07:34.160
So 1 dime plus 6
dimes is 7 dimes.

00:07:34.160 --> 00:07:37.550
7 dimes plus 7 dimes is 14.

00:07:37.550 --> 00:07:38.590
Let me write if like this.

00:07:38.590 --> 00:07:45.160
We could write 1 plus 6 plus 7
is equal to-- 1 plus 6 is 7.

00:07:45.160 --> 00:07:47.850
7 plus 7 is 14.

00:07:47.850 --> 00:07:50.830
So this right here is
going to be equal to 14.

00:07:50.830 --> 00:07:52.870
Carry the 1.

00:07:52.870 --> 00:07:54.975
Now we have-- let me do
it in another color.

00:07:54.975 --> 00:07:56.800
I'll do it in pink.

00:07:56.800 --> 00:08:00.150
We have 1 plus 3.

00:08:00.150 --> 00:08:02.020
We're in the
hundreds place now.

00:08:02.020 --> 00:08:04.130
Plus 5.

00:08:04.130 --> 00:08:06.145
1 plus 3 plus 5.

00:08:06.145 --> 00:08:10.020
Well, 1 plus 3 is 4.

00:08:10.020 --> 00:08:11.750
Plus 5 is 9.

00:08:11.750 --> 00:08:14.990
4 plus 5 is 9, so this is
going to be equal to 9.

00:08:14.990 --> 00:08:16.390
Nothing to carry.

00:08:16.390 --> 00:08:18.720
We only had something
in our ones place.

00:08:18.720 --> 00:08:20.670
9 is just 9 pennies.

00:08:20.670 --> 00:08:21.720
It's no dimes.

00:08:21.720 --> 00:08:23.380
It's just 9 pennies.

00:08:23.380 --> 00:08:25.350
And then we go to the
thousands place.

00:08:25.350 --> 00:08:27.750
Nothing to add to the
thousands place.

00:08:27.750 --> 00:08:31.040
So you just take this 4,000--
you see a 4 here, but since

00:08:31.040 --> 00:08:35.040
it's in the fourth digit to
the left, this means 4,000.

00:08:35.040 --> 00:08:38.200
So this 4,000 right here, we
don't have any other thousands

00:08:38.200 --> 00:08:40.940
to add it to, so we just
bring it straight down.

00:08:40.940 --> 00:08:42.570
So you bring the 4 down there.

00:08:42.570 --> 00:08:50.860
So 4,368 plus 572 is 4,000--
we'll put a comma there

00:08:50.860 --> 00:08:52.140
to make it easy to read.

00:08:52.140 --> 00:08:52.540
4,940.


Addition 2

WEBVTT
Kind: captions
Language: en

00:00:00.370 --> 00:00:03.970
In the last video we got some
practice adding what we could

00:00:03.970 --> 00:00:05.820
consider smaller numbers.

00:00:05.820 --> 00:00:10.820
For example, if we added 3 plus
2, we could imagine that if

00:00:10.820 --> 00:00:14.830
maybe I had 3 lemons-- 1, 2, 3.

00:00:14.830 --> 00:00:20.010
And if I were to add to those
3 lemons maybe 3 lime--

00:00:20.010 --> 00:00:21.700
is it lime or limes?

00:00:21.700 --> 00:00:26.900
Well, 2 green lemons or 2
more tart pieces of fruit.

00:00:26.900 --> 00:00:30.220
How much tart, sour
fruit to do I have now?

00:00:30.220 --> 00:00:34.730
Well, we learned in the last
video we have 1, 2, 3,

00:00:34.730 --> 00:00:37.480
4, 5 pieces of fruit.

00:00:37.480 --> 00:00:39.450
So 3 plus 2 is equal to 5.

00:00:39.450 --> 00:00:41.880
And we also saw that that's
the exact same thing

00:00:41.880 --> 00:00:44.320
as if we add 2 plus 3.

00:00:44.320 --> 00:00:46.840
And I think that makes sense
because this is the same thing

00:00:46.840 --> 00:00:49.720
as starting with, maybe you
have 2 lemons and you

00:00:49.720 --> 00:00:52.220
add 3 lime to it.

00:00:52.220 --> 00:00:54.700
You're still going to end
up with 5 pieces of fruit.

00:00:54.700 --> 00:00:57.710
1, 2, 3, 4, or 5.

00:00:57.710 --> 00:00:58.520
Just like that.

00:00:58.520 --> 00:01:00.780
So it doesn't matter what
order you add in, you're

00:01:00.780 --> 00:01:02.070
still going to get 5.

00:01:02.070 --> 00:01:05.690
And this way of thinking about
addition I view as the counting

00:01:05.690 --> 00:01:07.070
way of thinking about addition.

00:01:07.070 --> 00:01:09.250
The other thing we saw in the
last video is the number

00:01:09.250 --> 00:01:11.860
line version and they're
essentially the same thing.

00:01:11.860 --> 00:01:13.190
So we could draw a line.

00:01:13.190 --> 00:01:15.520
And all a number line
is, it lists all of

00:01:15.520 --> 00:01:17.250
the numbers in order.

00:01:17.250 --> 00:01:18.920
It lists all of the numbers
and you can actually go as

00:01:18.920 --> 00:01:19.820
high as you need to go.

00:01:19.820 --> 00:01:22.550
You could go up to a million,
gazillion, trillion.

00:01:22.550 --> 00:01:25.020
We won't do that; I wouldn't
have space or time in

00:01:25.020 --> 00:01:25.710
this video to do it.

00:01:25.710 --> 00:01:27.200
And you actually can go
as low as possible.

00:01:27.200 --> 00:01:30.580
We'll start at 0, assuming-- in
future videos I'll tell you

00:01:30.580 --> 00:01:32.670
about numbers smaller than 0.

00:01:32.670 --> 00:01:35.500
And maybe you can think about
what that might mean tonight.

00:01:35.500 --> 00:01:38.520
But let's start at 0,
and 0 means nothing.

00:01:38.520 --> 00:01:44.300
If I have 0 lemons, it
means I have no lemons.

00:01:44.300 --> 00:01:56.270
So 0, 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11--

00:01:56.270 --> 00:01:57.020
let's go pretty high.

00:01:57.020 --> 00:01:57.360
12.

00:01:57.360 --> 00:01:59.940
That way I can reuse
the number line.

00:01:59.940 --> 00:02:03.910
13, 14, I could keep on going,
but maybe fourteen will

00:02:03.910 --> 00:02:05.360
be enough for this video.

00:02:05.360 --> 00:02:07.230
But let's use a number
line for these addition

00:02:07.230 --> 00:02:08.900
problems up here.

00:02:08.900 --> 00:02:12.300
So in last video, just as a bit
of review, you can view 3 plus

00:02:12.300 --> 00:02:15.440
2 as starting at 3 and
then adding 2 to it.

00:02:15.440 --> 00:02:17.750
Or going 2 greater than 3.

00:02:17.750 --> 00:02:20.770
And just going greater or
adding on the number line is

00:02:20.770 --> 00:02:23.320
just moving to the right
or moving up by 2.

00:02:23.320 --> 00:02:25.380
So let's move up by 2.

00:02:25.380 --> 00:02:27.290
I'll do that in
this orange color.

00:02:27.290 --> 00:02:29.510
So let's go up by 2.

00:02:29.510 --> 00:02:31.880
So we started at 3
and we go up by 1.

00:02:31.880 --> 00:02:33.830
And then we go up by
2, or we're jumping,

00:02:33.830 --> 00:02:35.970
and we end up at 5.

00:02:35.970 --> 00:02:37.610
Which is exactly
what we got before.

00:02:37.610 --> 00:02:40.990
If we have 3 lemons, we add
one lemon, we have 4 lemons.

00:02:40.990 --> 00:02:44.870
We add another lemon, we have 5
lemons or lime or tart pieces

00:02:44.870 --> 00:02:46.310
of fruit, whatever you
might want to say.

00:02:46.310 --> 00:02:48.620
And when you look at this
version of it where you

00:02:48.620 --> 00:02:52.800
switched the order, we started
at 2 and we're adding

00:02:52.800 --> 00:02:54.060
3 objects to it.

00:02:54.060 --> 00:02:57.880
In this case, they
were lemons or limes.

00:02:57.880 --> 00:02:59.370
So we're going to add 3 to it.

00:02:59.370 --> 00:03:02.910
1, 2, 3.

00:03:02.910 --> 00:03:06.020
And just like we expected,
we got the same thing.

00:03:06.020 --> 00:03:08.030
We got 5 again.

00:03:08.030 --> 00:03:10.510
Now what I want to do in video
and hopefully this was just a

00:03:10.510 --> 00:03:13.630
bit of review, is I want to
tackle harder problems.

00:03:13.630 --> 00:03:16.970
I want to tackle slightly
larger numbers.

00:03:16.970 --> 00:03:18.620
And in this video I want
to just give you practice

00:03:18.620 --> 00:03:20.390
dealing with the
slightly larger numbers.

00:03:20.390 --> 00:03:21.840
And then in the next video
we're going to dig a little

00:03:21.840 --> 00:03:24.860
deeper and think about
what numbers even mean.

00:03:24.860 --> 00:03:27.590
But let's just get some
practice understanding how do

00:03:27.590 --> 00:03:32.880
you actually do the addition
problems with larger numbers?

00:03:32.880 --> 00:03:36.590
Let me write it in a nice,
soothing purple color.

00:03:36.590 --> 00:03:44.410
Let's say I wanted
to add 9 plus 3.

00:03:44.410 --> 00:03:46.320
Well, there's a couple
of ways we could do it.

00:03:46.320 --> 00:03:48.610
We could draw circles again.

00:03:48.610 --> 00:03:49.810
We could say, let's see.

00:03:49.810 --> 00:03:51.100
Maybe I'll draw stars.

00:03:51.100 --> 00:04:01.950
1, 2, 3, 4-- my stars are
degradingg-- 5, 6, 7, 8, 9.

00:04:01.950 --> 00:04:05.130
That's 9 stars and then
I add 3 stars to it.

00:04:05.130 --> 00:04:09.480
So I add 1, 2, 3 stars.

00:04:09.480 --> 00:04:11.550
And then if you were to count
the total number of stars you

00:04:11.550 --> 00:04:16.490
would say, let me do that in a
different color-- 1, 2, 3, 4,

00:04:16.490 --> 00:04:21.120
5, 6, 7, 8, 9, 10, 11, 12.

00:04:21.120 --> 00:04:23.000
I now have 12 stars.

00:04:23.000 --> 00:04:29.500
So you would say that 9
plus 3 is equal to 12.

00:04:29.500 --> 00:04:33.130
If you looked at the number
line you're starting at 9,

00:04:33.130 --> 00:04:37.020
maybe you have 9 stars and
you add one star, two

00:04:37.020 --> 00:04:39.580
star, three stars to that.

00:04:39.580 --> 00:04:43.020
And you end up with 12
stars, which is the exact

00:04:43.020 --> 00:04:44.760
answer we got before.

00:04:44.760 --> 00:04:49.350
So you can do the same process
when you start adding larger

00:04:49.350 --> 00:04:52.030
numbers, even though that now--
and I want to you to notice,

00:04:52.030 --> 00:04:55.740
the difference now is our
answer has two digits in it.

00:04:55.740 --> 00:04:58.480
We'll talk more about digits
in a future video, but all

00:04:58.480 --> 00:05:00.240
a digit is is a numeral.

00:05:00.240 --> 00:05:01.620
It has a 1 and a 2.

00:05:01.620 --> 00:05:03.130
That's what 12 is.

00:05:03.130 --> 00:05:05.180
I won't dig too deep
into that right now.

00:05:05.180 --> 00:05:07.760
I think you're pretty
familiar with the number 12.

00:05:07.760 --> 00:05:09.800
But what I want to do is, now
what happens when you start

00:05:09.800 --> 00:05:13.210
adding more-- when you
start adding two-digit

00:05:13.210 --> 00:05:16.190
numbers like this?

00:05:16.190 --> 00:05:30.640
For example, if I were to
add 27 plus let's say--

00:05:30.640 --> 00:05:34.140
I don't know-- plus 15.

00:05:34.140 --> 00:05:37.650
Now, if you had a lot of time
on your hands and you didn't

00:05:37.650 --> 00:05:41.640
care about how people judged
you, you could draw out 27

00:05:41.640 --> 00:05:44.880
circles and then draw out
another 15 circles and then

00:05:44.880 --> 00:05:47.090
count the total number
of circles you had.

00:05:47.090 --> 00:05:48.400
And that would give
you an answer.

00:05:48.400 --> 00:05:50.120
Or you could draw
a number line.

00:05:50.120 --> 00:05:53.190
You could draw a number line
that went all the way to know

00:05:53.190 --> 00:05:55.060
whatever 27 plus 15 is.

00:05:55.060 --> 00:05:56.580
So it's going to be this
really, really large

00:05:56.580 --> 00:05:59.100
number, but that would
take you forever.

00:05:59.100 --> 00:06:01.130
So what I'm going to do is show
you a way to do this type of

00:06:01.130 --> 00:06:04.520
problem where you really just
have to know your addition,

00:06:04.520 --> 00:06:06.720
almost have it memorized, or at
least, if you don't have it

00:06:06.720 --> 00:06:08.620
memorized, be able to do
something like this for

00:06:08.620 --> 00:06:10.090
relatively small numbers.

00:06:10.090 --> 00:06:11.990
And by doing it for the
relatively small numbers,

00:06:11.990 --> 00:06:15.360
you can do the harder
problems like this.

00:06:15.360 --> 00:06:18.160
So what you do, this
is the fun part.

00:06:18.160 --> 00:06:21.120
You add, and I'll talk
more about what this

00:06:21.120 --> 00:06:22.010
means in the future.

00:06:22.010 --> 00:06:24.010
You look at each of the digits.

00:06:24.010 --> 00:06:26.760
So we call this place, the
rightmost place, we call

00:06:26.760 --> 00:06:28.030
that the 1's place.

00:06:30.860 --> 00:06:32.410
And why do we call
that the 1's place?

00:06:32.410 --> 00:06:37.810
Because 27 is 20 and seven 1's.

00:06:37.810 --> 00:06:39.930
It's 20 plus 7.

00:06:39.930 --> 00:06:42.310
It's 20 plus seven 1's.

00:06:42.310 --> 00:06:45.360
You could view it as
it's 20 plus 7 pennies.

00:06:45.360 --> 00:06:49.830
And this place right here
is called the 10's place.

00:06:49.830 --> 00:06:51.180
Now why is it called
the 10's place?

00:06:51.180 --> 00:06:52.630
I mean there's a 2 right there.

00:06:52.630 --> 00:06:54.790
It's the place that's
called the 10's place.

00:06:54.790 --> 00:06:57.740
So putting a 2 here
means two 10's.

00:06:57.740 --> 00:07:00.370
The number 20, that's two 10's.

00:07:00.370 --> 00:07:03.790
If I have one dime and you gave
me another dime, I now have

00:07:03.790 --> 00:07:06.160
two dimes, and that's $0.20.

00:07:06.160 --> 00:07:08.120
So that's what the
10's place is.

00:07:08.120 --> 00:07:10.160
I don't want to confuse you, I
just want to show you how to

00:07:10.160 --> 00:07:11.430
do these problems right now.

00:07:11.430 --> 00:07:13.660
We'll dig a little bit
deeper in future videos.

00:07:13.660 --> 00:07:15.270
I just want to give
you that idea.

00:07:15.270 --> 00:07:18.470
But the way to do these
problems is you look at the

00:07:18.470 --> 00:07:21.300
numbers in the 1's place
and add those up first.

00:07:21.300 --> 00:07:23.570
So you say, OK, I'm not
going to worry about this

00:07:23.570 --> 00:07:24.520
whole thing right now.

00:07:24.520 --> 00:07:27.680
Let me just add
the 7 and the 5.

00:07:27.680 --> 00:07:30.830
So I'm going to add
the 7 and the 5.

00:07:30.830 --> 00:07:33.190
And if you don't know what that
is-- hopefully you'll be able

00:07:33.190 --> 00:07:36.020
to do that in your head fairly
shortly-- you could look

00:07:36.020 --> 00:07:37.100
at the number line.

00:07:37.100 --> 00:07:39.040
Let's look at the
number line here.

00:07:39.040 --> 00:07:43.160
So if you add 7, if you take
7, and you add 5 to it.

00:07:43.160 --> 00:07:47.580
1, 2, 3, 4, 5.

00:07:47.580 --> 00:07:48.780
We end up at 12.

00:07:48.780 --> 00:07:50.865
Or if you started at
5 and added 7, you'd

00:07:50.865 --> 00:07:52.420
also end up at 12.

00:07:52.420 --> 00:07:53.460
So let's write that down.

00:07:53.460 --> 00:07:58.940
We know that 7 plus
5 is equal to 12.

00:07:58.940 --> 00:08:02.690
So what we do is we say 7
plus 5 is equal to-- and

00:08:02.690 --> 00:08:04.410
now this is a new thing.

00:08:04.410 --> 00:08:07.070
It might be a little bit of
a mystery, magical thing

00:08:07.070 --> 00:08:08.050
for you right now.

00:08:08.050 --> 00:08:12.100
And in future videos I'll
explain to you why this works.

00:08:12.100 --> 00:08:13.350
We want to write the 12.

00:08:13.350 --> 00:08:17.300
7 plus 5 is 12, but we
just write the 2 here

00:08:17.300 --> 00:08:19.260
and we carry the 1.

00:08:19.260 --> 00:08:19.770
12.

00:08:19.770 --> 00:08:20.870
1 2.

00:08:20.870 --> 00:08:25.040
Well, we wrote the 2 there,
but we put the 1 up here.

00:08:25.040 --> 00:08:28.080
And the reason-- I'll give
you a simple reason for

00:08:28.080 --> 00:08:28.760
doing that right now.

00:08:28.760 --> 00:08:30.570
I'll give you a better
reason in the future.

00:08:30.570 --> 00:08:35.440
Is that you only had space to
put one digit here and 12 is a

00:08:35.440 --> 00:08:38.030
two-digit number, so we had
to think of some other

00:08:38.030 --> 00:08:39.280
place to put that 1.

00:08:39.280 --> 00:08:42.080
If you really want to think
about it even more, 12 is the

00:08:42.080 --> 00:08:45.750
same thing as 10 plus 2.

00:08:45.750 --> 00:08:48.260
That's the same thing as 12.

00:08:48.260 --> 00:08:52.290
So if we say 7 plus 5 that's
the same thing as 12, which is

00:08:52.290 --> 00:08:59.050
the same thing as two 1's--
two pennies, plus one dime.

00:08:59.050 --> 00:09:01.000
Plus one 10.

00:09:01.000 --> 00:09:03.980
So we put that one dime
in the 10's place.

00:09:03.980 --> 00:09:08.360
So we really just said 7 plus
5 is one 10 plus two 1's.

00:09:08.360 --> 00:09:11.005
Or one dime plus two pennies.

00:09:11.005 --> 00:09:15.120
If that confuses you just say,
well, I just write the 1's

00:09:15.120 --> 00:09:18.300
digit of the 2 there
and I carry the 1.

00:09:18.300 --> 00:09:20.680
And then you do the exact same
thing in the 10's place.

00:09:20.680 --> 00:09:22.870
You add the 1 plus
the 2 plus the 1.

00:09:22.870 --> 00:09:26.210
So 1 plus 2, let's do
that on a number line.

00:09:26.210 --> 00:09:26.930
This is fun.

00:09:26.930 --> 00:09:27.680
So let's see.

00:09:27.680 --> 00:09:29.120
1 plus 2.

00:09:29.120 --> 00:09:31.155
Let me do it in a
vibrant color.

00:09:31.155 --> 00:09:32.480
Let me do it in this magenta.

00:09:32.480 --> 00:09:33.890
So we start at 1.

00:09:33.890 --> 00:09:35.420
We're going to add 2 to it.

00:09:35.420 --> 00:09:36.680
1 plus 2.

00:09:36.680 --> 00:09:38.890
We take that 1 from our 12.

00:09:38.890 --> 00:09:41.340
1 plus 2, so you go up 1, 2.

00:09:41.340 --> 00:09:42.780
You end up at 3.

00:09:42.780 --> 00:09:44.340
Then you're going to
add up another 1.

00:09:44.340 --> 00:09:45.660
So you add another 1.

00:09:45.660 --> 00:09:48.490
You're going to end up at 4.

00:09:48.490 --> 00:09:50.550
So you ended up at 42.

00:09:50.550 --> 00:09:53.050
And this was pretty neat
because we didn't have to

00:09:53.050 --> 00:09:55.020
draw a number line
all the way to 42.

00:09:55.020 --> 00:09:57.100
And we'd didn't have
to draw 42 objects.

00:09:57.100 --> 00:10:01.100
Just by knowing what 7 plus 5
was and by knowing what 1 plus

00:10:01.100 --> 00:10:03.720
2 plus 1 was, we were able
to figure out that

00:10:03.720 --> 00:10:07.040
27 and 15 is 42.

00:10:07.040 --> 00:10:08.080
Let's do another example.

00:10:08.080 --> 00:10:10.580
Maybe I'll do a little bit
of a simpler example.

00:10:10.580 --> 00:10:19.440
Let's say I had 78 plus 3.

00:10:19.440 --> 00:10:21.250
We do the exact
same thing before.

00:10:21.250 --> 00:10:23.140
We just look at the
1's place only.

00:10:23.140 --> 00:10:26.180
So we look at 8 plus 3.

00:10:26.180 --> 00:10:27.250
What's 8 plus 3?

00:10:27.250 --> 00:10:29.140
Hopefully we can do that in
our heads at this point.

00:10:29.140 --> 00:10:30.390
But let's just think about it.

00:10:30.390 --> 00:10:33.690
8 plus 1 is equal to 9.

00:10:33.690 --> 00:10:36.230
8 plus 2 is equal to 10.

00:10:36.230 --> 00:10:38.420
8 plus 3 is going
to be equal to 11.

00:10:38.420 --> 00:10:40.580
You could do that on the number
line if it makes it easier

00:10:40.580 --> 00:10:42.150
to visualize for you.

00:10:42.150 --> 00:10:45.000
So 8 plus 3 is equal to 11.

00:10:45.000 --> 00:10:50.490
So what we do here, we just
have 8 plus 3 is equal to 11.

00:10:50.490 --> 00:10:54.810
Put this 1 right here, put that
there, and carry the other 1.

00:10:54.810 --> 00:10:58.610
Because 11 is one 10, one
dime plus one penny.

00:10:58.610 --> 00:10:59.620
That's 11.

00:10:59.620 --> 00:11:01.790
And then we add the 10's place.

00:11:01.790 --> 00:11:06.450
1 dime plus 7 dimes
is equal to 8 dimes.

00:11:06.450 --> 00:11:10.720
So 78 plus 3 is equal to 81.

00:11:10.720 --> 00:11:12.080
And now there's one thing
I want to show you.

00:11:12.080 --> 00:11:15.320
You don't always have to
carry numbers like that.

00:11:15.320 --> 00:11:17.640
Only if the answer to
one of these has more

00:11:17.640 --> 00:11:19.250
than one digit in it.

00:11:19.250 --> 00:11:21.080
11 is a two-digit number.

00:11:21.080 --> 00:11:28.200
So for example, if
I have 56 plus 2.

00:11:28.200 --> 00:11:31.590
Here I could just
say 6 plus 2 is 8.

00:11:31.590 --> 00:11:34.430
Hopefully we're getting
good practice at this.

00:11:34.430 --> 00:11:36.530
So 6 plus 2 is 8.

00:11:36.530 --> 00:11:39.800
And then, I don't have anything
to add this 5 to, so I just

00:11:39.800 --> 00:11:41.150
bring the 5 down here.

00:11:41.150 --> 00:11:44.150
So 56 plus 2 is 58.

00:11:44.150 --> 00:11:45.270
Just like that.

00:11:45.270 --> 00:11:46.815
And this is one you actually
could have drawn on

00:11:46.815 --> 00:11:47.320
the number line.

00:11:47.320 --> 00:11:48.820
It wouldn't have been too hard.

00:11:48.820 --> 00:11:51.480
So if you were to draw the
number line like that, you

00:11:51.480 --> 00:11:53.790
know 0 would be way off
to the left some place.

00:11:53.790 --> 00:11:55.890
But let's say I had 50.

00:11:55.890 --> 00:12:00.000
49, you could keep going to the
left, but you have 51, 52,

00:12:00.000 --> 00:12:01.820
actually let me start a little
higher than that because I'm

00:12:01.820 --> 00:12:04.060
going to run out of space.

00:12:04.060 --> 00:12:11.880
Let me start at maybe 55, 56,
57, 58, 59, and I could go in

00:12:11.880 --> 00:12:14.030
both directions, keep going.

00:12:14.030 --> 00:12:17.350
But if we start at 56 right
there and we add 2, we

00:12:17.350 --> 00:12:19.700
go up one, we go up two.

00:12:19.700 --> 00:12:21.060
We end up at 58.

00:12:21.060 --> 00:12:23.960
So just like that we're
able to do that problem.

00:12:23.960 --> 00:12:26.070
I'll see you in the next video.


ck12.org: More empirical rule and z-score practice

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.620
It never hurts to get
a bit more practice.

00:00:02.620 --> 00:00:05.600
So this is problem number five
from the normal distribution

00:00:05.600 --> 00:00:11.560
chapter from ck12.org's
AP statistics FlexBook.

00:00:11.560 --> 00:00:16.030
So they're saying, the 2007 AP
statistics examination scores

00:00:16.030 --> 00:00:20.750
were not normally distributed
with a mean of 2.8

00:00:20.750 --> 00:00:23.964
and a standard
deviation of 1.34.

00:00:23.964 --> 00:00:25.630
They cite some College
Board stuff here.

00:00:25.630 --> 00:00:27.170
I didn't copy and paste that.

00:00:27.170 --> 00:00:29.170
What is the approximate z-score?

00:00:29.170 --> 00:00:31.530
Remember, z-score
is just how many

00:00:31.530 --> 00:00:33.980
standard deviations you
are away from the mean.

00:00:33.980 --> 00:00:35.950
What is the approximate
z-score that

00:00:35.950 --> 00:00:39.337
corresponds to an
exam score of 5?

00:00:39.337 --> 00:00:40.920
So we really just
have to figure out--

00:00:40.920 --> 00:00:42.628
this is a pretty
straightforward problem.

00:00:42.628 --> 00:00:45.720
We just need to figure out how
many standard deviations is

00:00:45.720 --> 00:00:48.340
5 from the mean?

00:00:48.340 --> 00:00:53.370
Well, you just take
5 minus 2.8, right?

00:00:53.370 --> 00:00:54.400
The mean is 2.8.

00:00:54.400 --> 00:00:56.121
Let me be very
clear, mean is 2.8.

00:00:56.121 --> 00:00:56.870
They give us that.

00:00:56.870 --> 00:00:58.800
Didn't even have
to calculate it.

00:00:58.800 --> 00:01:00.230
So the mean is 2.8.

00:01:00.230 --> 00:01:03.760
So 5 minus 2.8 is equal to 2.2.

00:01:03.760 --> 00:01:06.374
So we're 2.2 above the mean.

00:01:06.374 --> 00:01:08.540
And if we want that in terms
of standard deviations,

00:01:08.540 --> 00:01:10.770
we just divide by our
standard deviation.

00:01:10.770 --> 00:01:14.860
You divide by 1.34.

00:01:14.860 --> 00:01:17.290
Divide by 1.34.

00:01:17.290 --> 00:01:20.710
I'll take out the
calculator for this.

00:01:20.710 --> 00:01:31.280
So we have 2.2 divided
by 1.34 is equal to 1.64.

00:01:31.280 --> 00:01:34.966
So this is equal to 1.64.

00:01:34.966 --> 00:01:37.590
And that's choice C. So this was
actually very straightforward.

00:01:37.590 --> 00:01:40.620
We just have to see how far
away we are from the mean

00:01:40.620 --> 00:01:42.929
if we get a score of
5-- which hopefully you

00:01:42.929 --> 00:01:44.720
will get if you're
taking the AP statistics

00:01:44.720 --> 00:01:46.242
exam after watching
these videos.

00:01:46.242 --> 00:01:48.450
And then you divide by the
standard deviation to say,

00:01:48.450 --> 00:01:50.850
how many standard deviations
away from the mean

00:01:50.850 --> 00:01:52.230
is the score of 5?

00:01:52.230 --> 00:01:53.545
It's 1.64.

00:01:53.545 --> 00:01:55.670
I think the only tricky
thing here might have been,

00:01:55.670 --> 00:01:58.400
you might have been tempted
to pick choice E, which says,

00:01:58.400 --> 00:02:01.300
the z-score cannot be calculated
because the distribution is not

00:02:01.300 --> 00:02:01.800
normal.

00:02:01.800 --> 00:02:04.700
And I think the reason why you
might have had that temptation

00:02:04.700 --> 00:02:07.430
is because we've
been using z-scores

00:02:07.430 --> 00:02:10.300
within the context of
a normal distribution.

00:02:10.300 --> 00:02:12.860
But a z-score literally
just means how many

00:02:12.860 --> 00:02:15.950
standard deviations you
are away from the mean.

00:02:15.950 --> 00:02:18.290
It could apply to
any distribution

00:02:18.290 --> 00:02:21.820
that you could calculate a mean
and a standard deviation for.

00:02:21.820 --> 00:02:23.910
So E is not the correct answer.

00:02:23.910 --> 00:02:27.045
A z-score can apply to a
non-normal distribution.

00:02:27.045 --> 00:02:29.170
So the answer is C. And I
guess that's a good point

00:02:29.170 --> 00:02:31.094
of clarification to
get out of the way.

00:02:31.094 --> 00:02:33.260
And I thought I would do
two problems in this video,

00:02:33.260 --> 00:02:35.460
just because that
one was pretty short.

00:02:35.460 --> 00:02:36.900
So problem number six.

00:02:36.900 --> 00:02:39.350
The height of fifth grade
boys in the United States

00:02:39.350 --> 00:02:41.480
is approximately
normally distributed--

00:02:41.480 --> 00:02:45.690
that's good to know-- with
a mean height of 143.5

00:02:45.690 --> 00:02:46.410
centimeters.

00:02:46.410 --> 00:02:50.960
So it's a mean of
143.5 centimeters

00:02:50.960 --> 00:02:56.635
and a standard deviation
of about 7.1 centimeters.

00:03:01.700 --> 00:03:04.620
What is the probability that
a randomly chosen fifth grade

00:03:04.620 --> 00:03:09.134
boy would be taller
than 157.7 centimeters?

00:03:09.134 --> 00:03:10.800
So let's just draw
out this distribution

00:03:10.800 --> 00:03:13.755
like we've done in a
bunch of problems so far.

00:03:13.755 --> 00:03:15.600
They're just asking
us one question,

00:03:15.600 --> 00:03:19.320
so we can mark this
distribution up a good bit.

00:03:19.320 --> 00:03:21.410
Let's say that's
our distribution.

00:03:21.410 --> 00:03:28.270
And the mean here, the
mean they told us is 143.5.

00:03:28.270 --> 00:03:30.414
They're asking us
taller than 157.7.

00:03:30.414 --> 00:03:32.080
So we're going in the
upwards direction.

00:03:32.080 --> 00:03:35.360
So one standard
deviation above the mean

00:03:35.360 --> 00:03:37.740
will take us right there.

00:03:37.740 --> 00:03:40.510
And we just have to add 7.1
to this number right here.

00:03:40.510 --> 00:03:42.700
We're going up by 7.1.

00:03:42.700 --> 00:03:45.980
So 143.5 plus 7.1 is what?

00:03:45.980 --> 00:03:49.440
150.6.

00:03:49.440 --> 00:03:51.047
That's one standard deviation.

00:03:51.047 --> 00:03:52.880
If we were to go another
standard deviation,

00:03:52.880 --> 00:03:54.950
we'd go 7.1 more.

00:03:54.950 --> 00:03:57.500
What's 7.1 plus 150.6?

00:03:57.500 --> 00:04:02.950
It's 157.7, which
just happens to be

00:04:02.950 --> 00:04:04.220
the exact number they ask for.

00:04:04.220 --> 00:04:06.240
They're asking for
the probability

00:04:06.240 --> 00:04:08.304
of getting a height
higher than that.

00:04:08.304 --> 00:04:10.470
So they want to know, what's
the probability that we

00:04:10.470 --> 00:04:12.830
fall under this area right here?

00:04:12.830 --> 00:04:15.980
Or essentially more than
two standard deviations

00:04:15.980 --> 00:04:16.630
from the mean.

00:04:16.630 --> 00:04:18.670
Or above two
standard deviations.

00:04:18.670 --> 00:04:21.420
We can't count this
left tail right there.

00:04:21.420 --> 00:04:24.480
So we can use the
empirical rule.

00:04:24.480 --> 00:04:26.630
If we do our standard
deviations to the left,

00:04:26.630 --> 00:04:29.830
that's one standard deviation,
two standard deviations.

00:04:29.830 --> 00:04:32.010
We know what this whole area is.

00:04:32.010 --> 00:04:35.660
Let me pick a different
color so that I don't.

00:04:35.660 --> 00:04:39.170
So we know what this
area is, the area

00:04:39.170 --> 00:04:40.780
within two standard deviations.

00:04:40.780 --> 00:04:42.020
The empirical rule tells us.

00:04:42.020 --> 00:04:46.820
Or even better, the
68, 95, 99.7 rule

00:04:46.820 --> 00:04:48.830
tells us that this
area-- because it's

00:04:48.830 --> 00:04:55.300
within two standard
deviations-- is 95%, or 0.95.

00:04:55.300 --> 00:04:59.740
Or it's 95% of the area under
the normal distribution.

00:04:59.740 --> 00:05:02.400
Which tells us that what's
left over-- this tail

00:05:02.400 --> 00:05:04.880
that we care about and
this left tail right here--

00:05:04.880 --> 00:05:08.340
has to make up the
rest of it, or 5%.

00:05:08.340 --> 00:05:12.216
So those two combined
have to be 5%.

00:05:12.216 --> 00:05:13.570
And these are symmetrical.

00:05:13.570 --> 00:05:14.590
We've done this before.

00:05:14.590 --> 00:05:16.330
This is actually a little
redundant from other problems

00:05:16.330 --> 00:05:17.250
we've done.

00:05:17.250 --> 00:05:20.010
But if these are added, combined
5%, and they're the same,

00:05:20.010 --> 00:05:22.580
then each of these are 2.5%.

00:05:22.580 --> 00:05:24.792
Each of these are 2.5%.

00:05:24.792 --> 00:05:26.250
So the answer to
the question, what

00:05:26.250 --> 00:05:29.160
is the probability that a
randomly chosen fifth grade boy

00:05:29.160 --> 00:05:32.820
would be taller then
157.7 centimeters.

00:05:32.820 --> 00:05:34.320
Well, that's literally
just the area

00:05:34.320 --> 00:05:35.927
under this right green part.

00:05:35.927 --> 00:05:37.510
Maybe I'll do it in
a different color.

00:05:37.510 --> 00:05:39.660
This magenta part that
I'm coloring right now.

00:05:39.660 --> 00:05:40.920
That's just that area.

00:05:40.920 --> 00:05:43.600
We just figured out it's 2.5%.

00:05:43.600 --> 00:05:47.780
So there's a 2.5% chance we'd
randomly find a fifth grade

00:05:47.780 --> 00:05:51.260
boy who's taller than
157.7 centimeters,

00:05:51.260 --> 00:05:53.650
assuming this is the mean,
the standard deviation,

00:05:53.650 --> 00:05:56.680
and we are dealing with
a normal distribution.


k12.org exercise: Standard normal distribution and the empirical

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:05.160
We're now on problem number 4
from the Normal Distribution

00:00:05.160 --> 00:00:10.504
chapter from ck12.org's
FlexBook on AP Statistics.

00:00:10.504 --> 00:00:12.170
You can go to their
site to download it.

00:00:12.170 --> 00:00:13.660
It's all for free.

00:00:13.660 --> 00:00:15.830
So problem number 4, and
it's, at least in my mind,

00:00:15.830 --> 00:00:17.010
pretty good practice.

00:00:17.010 --> 00:00:22.500
For a normal, or a standard
normal distribution,

00:00:22.500 --> 00:00:25.570
place the following in order
from smallest to largest.

00:00:25.570 --> 00:00:28.035
So let's see, percentage of
data below 1, negative 1.

00:00:28.035 --> 00:00:28.600
OK.

00:00:28.600 --> 00:00:31.470
Let's draw our standard
normal distribution.

00:00:31.470 --> 00:00:33.230
So a standard
normal distribution

00:00:33.230 --> 00:00:36.230
is one where the
mean is-- sorry,

00:00:36.230 --> 00:00:37.850
I drew the standard
deviation-- is

00:00:37.850 --> 00:00:41.350
one where the mean, mu for
mean, is where the mean is

00:00:41.350 --> 00:00:45.940
equal to 0, and the standard
deviation is equal to 1.

00:00:45.940 --> 00:00:50.540
So let me draw that standard
normal distribution.

00:00:50.540 --> 00:00:54.590
Let's see, so let me draw
the axis right like that.

00:00:54.590 --> 00:00:59.245
Let me see if I can draw
a nice-looking bell curve.

00:00:59.245 --> 00:01:02.030
So there's the bell
curve right there.

00:01:02.030 --> 00:01:03.290
You get the idea.

00:01:03.290 --> 00:01:06.920
And this is a standard normal
distribution, so the mean,

00:01:06.920 --> 00:01:09.310
or you can kind view the
center point right here.

00:01:09.310 --> 00:01:11.110
It's not skewed.

00:01:11.110 --> 00:01:13.990
This says the mean is
going to be 0 right there,

00:01:13.990 --> 00:01:15.440
and the standard deviation is 1.

00:01:15.440 --> 00:01:19.120
So if we go 1 standard
deviation to the right,

00:01:19.120 --> 00:01:20.939
that is going to be 1.

00:01:20.939 --> 00:01:22.480
If you go 2 standard
deviations, it's

00:01:22.480 --> 00:01:25.800
going to be 2, 3 standard
deviations, 3, just like that.

00:01:25.800 --> 00:01:29.540
1 standard deviation to the
left is going to be minus 1.

00:01:29.540 --> 00:01:31.580
2 standard deviations
to the left

00:01:31.580 --> 00:01:34.290
will be minus 2, and
so on, and so forth.

00:01:34.290 --> 00:01:36.660
Minus 3 will be 3 standard
deviations to the left

00:01:36.660 --> 00:01:38.680
because the standard
deviation is 1.

00:01:38.680 --> 00:01:41.900
So let's see if we can
answer this question.

00:01:41.900 --> 00:01:45.460
So what's the percentage
of data below 1?

00:01:48.710 --> 00:01:50.560
Part a, that's this
stuff right here.

00:01:50.560 --> 00:01:54.967
So everything below 1, so
it's all of-- well, not

00:01:54.967 --> 00:01:56.300
just that little center portion.

00:01:56.300 --> 00:01:57.540
It keeps going.

00:01:57.540 --> 00:02:01.464
Everything below 1,
percentage of data below 1.

00:02:01.464 --> 00:02:02.880
So this is another
situation where

00:02:02.880 --> 00:02:05.080
we should use the
empirical rule.

00:02:07.700 --> 00:02:09.970
Never hurts to
get more practice.

00:02:09.970 --> 00:02:12.200
Empirical rule, or
maybe the better way

00:02:12.200 --> 00:02:18.190
to remember the empirical rule
is just the 68, 95, 99.7 rule.

00:02:18.190 --> 00:02:20.089
And I call that a
better way because it

00:02:20.089 --> 00:02:21.380
essentially gives you the rule.

00:02:21.380 --> 00:02:22.530
These are just the
numbers that you

00:02:22.530 --> 00:02:23.840
have to essentially memorize.

00:02:23.840 --> 00:02:26.867
And if you have a calculator
or a normal distribution table,

00:02:26.867 --> 00:02:27.950
you don't have to do this.

00:02:27.950 --> 00:02:29.510
But sometimes in
class, or people

00:02:29.510 --> 00:02:31.360
want you to estimate
percentages,

00:02:31.360 --> 00:02:34.619
and so you can impress people
if you can do this in your head.

00:02:34.619 --> 00:02:36.910
So let's see if we can use
the empirical rule to answer

00:02:36.910 --> 00:02:40.850
this question, the area under
the bell curve all the way up

00:02:40.850 --> 00:02:44.710
to 1, or everything
to the left of 1.

00:02:44.710 --> 00:02:48.380
So the empirical rule tells
us that this middle area

00:02:48.380 --> 00:02:51.550
between 1 standard
deviation to the left

00:02:51.550 --> 00:02:56.280
and 1 standard deviation to the
right, that right there is 68%.

00:02:56.280 --> 00:02:58.030
We saw that in the
previous video as well.

00:02:58.030 --> 00:03:00.010
That's what the
empirical rule tells us.

00:03:00.010 --> 00:03:02.690
Now, if that's 68%, we
saw in the last video

00:03:02.690 --> 00:03:04.380
that everything else
combined, it all

00:03:04.380 --> 00:03:09.300
has to add up to 1 or to 100%,
that this left-hand tail-- let

00:03:09.300 --> 00:03:11.030
me draw it a little
bit-- this part

00:03:11.030 --> 00:03:15.380
right here plus this part
right here has to add up,

00:03:15.380 --> 00:03:19.220
when you add it to 68, has
to add up to 1 or to 100%.

00:03:19.220 --> 00:03:22.150
So those two combined are 32%.

00:03:22.150 --> 00:03:24.300
32 plus 68 is 100.

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Now, this is symmetrical.

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These two things
are the exact same.

00:03:27.400 --> 00:03:31.090
So if they add up to 32,
this right here is 16%,

00:03:31.090 --> 00:03:33.575
and this right here is 16%.

00:03:33.575 --> 00:03:35.200
Now, the question,
they want us to know

00:03:35.200 --> 00:03:36.741
the area of everything--
let me do it

00:03:36.741 --> 00:03:39.860
in a new color--
everything less than 1,

00:03:39.860 --> 00:03:42.080
the percentage of data
below 1, so everything

00:03:42.080 --> 00:03:43.650
to the left of this point.

00:03:43.650 --> 00:03:45.070
So it's the 68%.

00:03:45.070 --> 00:03:48.370
It's right there,
so it's 68%, which

00:03:48.370 --> 00:03:50.860
is this middle area within
1 standard deviation,

00:03:50.860 --> 00:03:54.080
plus this left
branch right there.

00:03:54.080 --> 00:03:57.840
So 68 plus 16%, which is what?

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That's equal to 84%.

00:04:00.660 --> 00:04:02.715
So part a is 84%.

00:04:02.715 --> 00:04:05.090
They're going to want us to
put this in order eventually,

00:04:05.090 --> 00:04:06.710
but it's good to just
solve because that's really

00:04:06.710 --> 00:04:07.293
the hard part.

00:04:07.293 --> 00:04:10.000
Once we know the numbers,
ordering is pretty easy.

00:04:10.000 --> 00:04:14.080
Part b, the percentage
of data below minus 1.

00:04:14.080 --> 00:04:16.480
So minus 1 is right there.

00:04:16.480 --> 00:04:18.760
So they really just want
us to figure out this area

00:04:18.760 --> 00:04:22.070
right here, the percentage
of data below minus 1.

00:04:22.070 --> 00:04:23.310
Well, that's going to be 16%.

00:04:23.310 --> 00:04:25.690
We just figured that out.

00:04:25.690 --> 00:04:28.360
And you could have already
known just without even knowing

00:04:28.360 --> 00:04:30.920
the empirical, just looking
at a normal distribution,

00:04:30.920 --> 00:04:34.469
that this entire area, that
part b is a subset of part a,

00:04:34.469 --> 00:04:36.010
so it's going to be
a smaller number.

00:04:36.010 --> 00:04:37.920
So if you just have
to order things,

00:04:37.920 --> 00:04:39.390
you could have made
that intuition,

00:04:39.390 --> 00:04:43.122
but it's good to do practice
with the empirical rule.

00:04:43.122 --> 00:04:45.080
Now part c, they want to
know, what's the mean?

00:04:45.080 --> 00:04:46.371
Well, that's the easiest thing.

00:04:46.371 --> 00:04:48.670
The mean of a standard
normal distribution,

00:04:48.670 --> 00:04:50.710
by definition, is 0.

00:04:50.710 --> 00:04:53.120
So number c is 0.

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d, the standard deviation.

00:04:56.190 --> 00:04:58.580
Well, by definition,
the standard deviation

00:04:58.580 --> 00:05:01.350
for the standard normal
distribution is 1.

00:05:01.350 --> 00:05:03.990
So this is 1 right here.

00:05:03.990 --> 00:05:05.740
This is easier than I
thought it would be.

00:05:05.740 --> 00:05:10.320
Part e, the percentage
of data above 2.

00:05:10.320 --> 00:05:15.050
So they want the
percentage of data above 2.

00:05:15.050 --> 00:05:20.239
So we know from the
68, 95, 99.7 rule

00:05:20.239 --> 00:05:21.780
that if we want to
know how much data

00:05:21.780 --> 00:05:24.480
is within 2 standard
deviations-- so let

00:05:24.480 --> 00:05:25.580
me do it in a new color.

00:05:28.170 --> 00:05:31.440
Let me do it in a more
vibrant color, green.

00:05:31.440 --> 00:05:33.617
If we're looking from
this point to this point--

00:05:33.617 --> 00:05:35.450
so it's within 2 standard
deviations, right,

00:05:35.450 --> 00:05:37.654
the standard
deviation here is 1--

00:05:37.654 --> 00:05:39.570
if we're looking within
2 standard deviations,

00:05:39.570 --> 00:05:45.530
that whole area right there,
by the empirical rule, is 95%,

00:05:45.530 --> 00:05:47.570
within 2 standard deviations.

00:05:47.570 --> 00:05:49.770
This is 95%.

00:05:49.770 --> 00:05:52.710
Which tells us that
everything else combined--

00:05:52.710 --> 00:05:54.780
so if you take
this yellow portion

00:05:54.780 --> 00:05:57.260
right here and
this yellow portion

00:05:57.260 --> 00:05:59.680
right here, so everything
beyond 2 standard deviations

00:05:59.680 --> 00:06:01.920
in either direction-- that
has to be the remainder.

00:06:01.920 --> 00:06:03.420
So you know everything
in the middle

00:06:03.420 --> 00:06:05.930
was 95 within 2
standard deviations.

00:06:05.930 --> 00:06:10.370
So that has to be 5%, if you
add that tail and that tail

00:06:10.370 --> 00:06:12.640
together, everything to
the left and right of 2

00:06:12.640 --> 00:06:14.160
standard deviations.

00:06:14.160 --> 00:06:16.360
Well, I've made the
argument before, everything

00:06:16.360 --> 00:06:17.110
is symmetrical.

00:06:17.110 --> 00:06:18.750
This and this are equal.

00:06:18.750 --> 00:06:21.280
So this right here
is 2 and 1/2%,

00:06:21.280 --> 00:06:24.322
and this right here
is also 2 and 1/2%.

00:06:24.322 --> 00:06:25.780
So they're asking
us the percentage

00:06:25.780 --> 00:06:29.090
of data above 2, that's this
tail, just this tail right

00:06:29.090 --> 00:06:32.950
here, the percentage of data
more than 2 standard deviations

00:06:32.950 --> 00:06:34.200
away from the mean.

00:06:34.200 --> 00:06:35.660
So that's 2 and 1/2%.

00:06:35.660 --> 00:06:37.790
Let me do it in a darker
color-- 2 and 1/2%.

00:06:40.134 --> 00:06:41.800
Now, they're asking
us, let's see, place

00:06:41.800 --> 00:06:45.140
the following in order
from smallest to largest.

00:06:45.140 --> 00:06:46.890
So there's a little
bit of ambiguity here.

00:06:46.890 --> 00:06:50.700
Because if they're saying the
percentage of data below 1,

00:06:50.700 --> 00:06:53.250
do they want us to
say, well, it's 84%.

00:06:53.250 --> 00:06:56.700
So should we consider
the answer to part a, 84?

00:06:56.700 --> 00:06:59.240
Or should we consider-- if
they said the fraction of data

00:06:59.240 --> 00:07:01.562
below 1, I would say 0.84.

00:07:01.562 --> 00:07:03.520
So it depends on how they
want to interpret it.

00:07:03.520 --> 00:07:04.103
Same way here.

00:07:04.103 --> 00:07:07.690
The percentage of data below
minus 1, I could say the answer

00:07:07.690 --> 00:07:08.350
is 16.

00:07:08.350 --> 00:07:11.840
16 is the percentage
below minus 1.

00:07:11.840 --> 00:07:14.760
But the actual number, if
I said the fraction of data

00:07:14.760 --> 00:07:16.620
below minus 1, I would say 0.16.

00:07:16.620 --> 00:07:19.120
So this actually would be very
different in how we order it.

00:07:19.120 --> 00:07:21.036
Similarly, if someone
me asked me the percent,

00:07:21.036 --> 00:07:22.530
I'd say, oh, that's 2.5.

00:07:22.530 --> 00:07:25.560
But the actual number is 0.025.

00:07:25.560 --> 00:07:28.182
That's the actual fraction
or the actual decimal.

00:07:28.182 --> 00:07:29.890
So I mean, this is
just ordering numbers,

00:07:29.890 --> 00:07:32.310
so I shouldn't fixate
on this too much.

00:07:32.310 --> 00:07:34.860
But let's just say that
they're going with the decimal.

00:07:34.860 --> 00:07:36.980
So if we wanted
to do it that way,

00:07:36.980 --> 00:07:39.680
they want to do it from smallest
to largest, the smallest

00:07:39.680 --> 00:07:42.330
number we have here is c, right?

00:07:42.330 --> 00:07:43.620
That's 0.

00:07:43.620 --> 00:07:48.920
And then the next smallest
is e, which is 0.025.

00:07:48.920 --> 00:07:54.540
Then the next smallest
is b, which is 0.16.

00:07:54.540 --> 00:07:58.160
And then the next one after
that is a, which is 0.84.

00:07:58.160 --> 00:08:00.840
And then the largest would
be the standard deviation, d.

00:08:00.840 --> 00:08:02.299
So the answer is c, e, b, a , d.

00:08:02.299 --> 00:08:04.090
And obviously, the
order would be different

00:08:04.090 --> 00:08:05.910
if the answer to this,
instead of saying it's 0.84,

00:08:05.910 --> 00:08:07.970
if you said it was 84 because
it's asking for the percentage.

00:08:07.970 --> 00:08:09.179
So a little bit of ambiguity.

00:08:09.179 --> 00:08:10.969
If we had a question
like this on the exam,

00:08:10.969 --> 00:08:12.690
I would clarify that
with the teacher.

00:08:12.690 --> 00:08:15.310
But hopefully you
found this useful.