Similarity postulates

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Kind: captions
Language: en

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Let's say we have triangle ABC.

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It looks something like this.

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I want to think about the
minimum amount of information.

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I want to come up with
a couple of postulates

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that we can use to determine
whether another triangle is

00:00:17.330 --> 00:00:20.140
similar to triangle ABC.

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So we already know
that if all three

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of the corresponding
angles are congruent

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to the corresponding
angles on ABC, then

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we know that we're dealing
with congruent triangles.

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So for example, if
this is 30 degrees,

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this angle is 90 degrees, and
this angle right over here

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is 60 degrees.

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And we have another
triangle that

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looks like this, it's
clearly a smaller triangle,

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but it's corresponding angles.

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So this is 30 degrees.

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This is 90 degrees,
and this is 60 degrees,

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we know that XYZ in this case,
is going to be similar to ABC.

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So we would know from this
because corresponding angles

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are congruent, we would
know that triangle ABC

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is similar to triangle XYZ.

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And you've got to get the
order right to make sure

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that you have the right
corresponding angles.

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Y corresponds to
the 90-degree angle.

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X corresponds to
the 30-degree angle.

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A corresponds to
the 30-degree angle.

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So A and X are the
first two things.

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B and Y, which are the 90
degrees, are the second two,

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and then Z is the last one.

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So that's what we know already,
if you have three angles.

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But do you need three angles?

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If we only knew two of the
angles, would that be enough?

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Well, sure because if you know
two angles for a triangle,

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you know the third.

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So for example, if I
have another triangle

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that looks like this--
let me draw it like this--

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and if I told you that only
two of the corresponding angles

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are congruent.

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So maybe this angle right here
is congruent to this angle,

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and that angle right there
is congruent to that angle.

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Is that enough to say that
these two triangles are similar?

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Well, sure.

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Because in a triangle, if
you know two of the angles,

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then you know what the
last angle has to be.

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If you know that this is 30
and you know that that is 90,

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then you know that this
angle has to be 60 degrees.

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Whatever these two angles
are, subtract them from 180,

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and that's going
to be this angle.

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So in general, in order
to show similarity,

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you don't have to show three
corresponding angles are

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congruent, you really
just have to show two.

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So this will be the first of
our similarity postulates.

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We call it angle-angle.

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If you could show that two
corresponding angles are

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congruent, then we're dealing
with similar triangles.

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So for example, just to
put some numbers here,

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if this was 30 degrees, and
we know that on this triangle,

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this is 90 degrees
right over here,

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we know that this
triangle right over here

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is similar to that one there.

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And you can really just
go to the third angle

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in this pretty
straightforward way.

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You say this third
angle is 60 degrees,

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so all three angles
are the same.

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That's one of our
constraints for similarity.

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Now, the other thing we
know about similarity

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is that the ratio
between all of the sides

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are going to be the same.

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So for example, if we have
another triangle right

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over here-- let me
draw another triangle--

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I'll call this
triangle X, Y, and Z.

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And let's say that we know that
the ratio between AB and XY,

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we know that AB over XY-- so
the ratio between this side

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and this side-- notice we're not
saying that they're congruent.

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We're looking at
their ratio now.

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We're saying AB
over XY, let's say

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that that is equal
to BC over YZ.

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That is equal to BC over YZ.

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And that is equal to AC over XZ.

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So once again, this
is one of the ways

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that we say, hey,
this means similarity.

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So if you have all three
corresponding sides,

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the ratio between all
three corresponding sides

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are the same, then
we know we are

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dealing with similar triangles.

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So this is what we call
side-side-side similarity.

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And you don't want
to get these confused

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with side-side-side congruence.

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So these are all of our
similarity postulates

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or axioms or things that
we're going to assume

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and then we're
going to build off

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of them to solve problems
and prove other things.

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Side-side-side, when we're
talking about congruence,

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means that the corresponding
sides are congruent.

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Side-side-side for
similarity, we're

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saying that the ratio
between corresponding sides

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are going to be the same.

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So for example, let's say
this right over here is 10.

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No.

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Let me think of a bigger number.

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Let's say this is 60, this
right over here is 30,

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and this right over here
is 30 square roots of 3,

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and I just made those
numbers because we will soon

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learn what typical ratios
are of the sides of 30-60-90

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triangles.

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And let's say this
one over here is

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6, 3, and 3 square roots of 3.

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Notice AB over XY
30 square roots

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of 3 over 3 square roots
of 3, this will be 10.

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What is BC over XY?

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30 divided by 3 is 10.

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And what is 60 divided
by 6 or AC over XZ?

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Well, that's going to be 10.

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So in general, to go from
the corresponding side

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here to the
corresponding side there,

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we always multiply
by 10 on every side.

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So we're not saying
they're congruent

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or we're not saying
the sides are

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the same for this
side-side-side for similarity.

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We're saying that we're
really just scaling them up

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by the same amount,
or another way

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to think about it, the ratio
between corresponding sides

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are the same.

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Now, what about
if we had-- let's

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start another triangle
right over here.

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Let me draw it like this.

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Actually, I want to leave this
here so we can have our list.

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So let's draw
another triangle ABC.

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So this is A, B,
and C. And let's say

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that we know that this side,
when we go to another triangle,

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we know that XY is AB
multiplied by some constant.

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So I can write it over here.

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XY is equal to some
constant times AB.

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Actually, let me make XY
bigger, so actually, it

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doesn't have to be.

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That constant could be
less than 1 in which case

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it would be a smaller value.

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But let me just do it that way.

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So let me just make XY
look a little bit bigger.

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So let's say that this
is X and that is Y.

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So let's say that we
know that XY over AB

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is equal to some constant.

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Or if you multiply
both sides by AB,

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you would get XY is some
scaled up version of AB.

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So maybe AB is 5, XY is 10,
then our constant would be 2.

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We scaled it up
by a factor of 2.

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And let's say we also
know that angle ABC is

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congruent to angle XYZ.

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I'll add another
point over here.

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So let me draw another
side right over here.

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So this is Z. So
let's say we also

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know that angle ABC
is congruent to XYZ,

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and let's say we know that
the ratio between BC and YZ

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is also this constant.

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The ratio between
BC and YZ is also

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equal to the same constant.

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So an example where this 5
and 10, maybe this is 3 and 6.

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The constant we're
kind of doubling

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the length of the side.

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So is this triangle XYZ
going to be similar?

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Well, if you think about it, if
XY is the same multiple of AB

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as YZ is a multiple of BC,
and the angle in between

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is congruent, there's
only one triangle

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we can set up over here.

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We're only constrained to
one triangle right over here,

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and so we're
completely constraining

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the length of this side,
and the length of this side

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is going to have to be that
same scale as that over there.

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And so we call that
side-angle-side similarity.

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So once again, we
saw SSS and SAS

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in our congruence
postulates, but we're

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saying something
very different here.

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We're saying that
in SAS, if the ratio

00:08:55.240 --> 00:08:57.240
between corresponding
sides of the true triangle

00:08:57.240 --> 00:09:01.820
are the same, so AB and XY of
one corresponding side and then

00:09:01.820 --> 00:09:04.660
another corresponding side,
so that's that second side,

00:09:04.660 --> 00:09:08.390
so that's between BC and YZ,
and the angle between them

00:09:08.390 --> 00:09:11.640
are congruent, then we're
saying it's similar.

00:09:11.640 --> 00:09:14.180
For SAS for congruency, we
said that the sides actually

00:09:14.180 --> 00:09:15.060
had to be congruent.

00:09:15.060 --> 00:09:18.570
Here we're saying that the ratio
between the corresponding sides

00:09:18.570 --> 00:09:20.630
just has to be the same.

00:09:20.630 --> 00:09:24.830
So for example SAS, just to
apply it, if I have-- let

00:09:24.830 --> 00:09:26.950
me just show some examples here.

00:09:26.950 --> 00:09:33.100
So let's say I have a
triangle here that is 3, 2, 4,

00:09:33.100 --> 00:09:38.450
and let's say we have
another triangle here

00:09:38.450 --> 00:09:42.410
that has length
9, 6, and we also

00:09:42.410 --> 00:09:45.460
know that the angle in
between are congruent so

00:09:45.460 --> 00:09:47.480
that that angle is
equal to that angle.

00:09:47.480 --> 00:09:50.470
What SAS in the
similarity world tells

00:09:50.470 --> 00:09:52.470
you is that these
triangles are definitely

00:09:52.470 --> 00:09:55.680
going to be similar
triangles, that we're actually

00:09:55.680 --> 00:09:58.260
constraining because there's
actually only one triangle

00:09:58.260 --> 00:09:59.750
we can draw a right over here.

00:09:59.750 --> 00:10:01.310
It's the triangle
where all the sides

00:10:01.310 --> 00:10:03.930
are going to have to be
scaled up by the same amount.

00:10:03.930 --> 00:10:06.169
So there's only one long
side right here that we

00:10:06.169 --> 00:10:07.710
could actually draw,
and that's going

00:10:07.710 --> 00:10:10.840
to have to be scaled
up by 3 as well.

00:10:10.840 --> 00:10:12.590
This is the only
possible triangle.

00:10:12.590 --> 00:10:14.548
If you constrain this
side you're saying, look,

00:10:14.548 --> 00:10:17.614
this is 3 times that side, this
is 3 three times that side,

00:10:17.614 --> 00:10:19.280
and the angle between
them is congruent,

00:10:19.280 --> 00:10:21.810
there's only one
triangle we could make.

00:10:21.810 --> 00:10:23.920
And we know there is a
similar triangle there

00:10:23.920 --> 00:10:26.264
where everything is scaled
up by a factor of 3,

00:10:26.264 --> 00:10:27.680
so that one triangle
we could draw

00:10:27.680 --> 00:10:30.334
has to be that one
similar triangle.

00:10:30.334 --> 00:10:32.000
So this is what we're
talking about SAS.

00:10:32.000 --> 00:10:34.374
We're not saying that this
side is congruent to that side

00:10:34.374 --> 00:10:36.100
or that side is
congruent to that side,

00:10:36.100 --> 00:10:39.140
we're saying that they're
scaled up by the same factor.

00:10:41.700 --> 00:10:44.450
If we had another triangle
that looked like this,

00:10:44.450 --> 00:10:49.380
so maybe this is 9, this is
4, and the angle between them

00:10:49.380 --> 00:10:51.900
were congruent, you
couldn't say that they're

00:10:51.900 --> 00:10:55.960
similar because this side is
scaled up by a factor of 3.

00:10:55.960 --> 00:10:58.100
This side is only scaled
up by a factor of 2.

00:10:58.100 --> 00:11:00.460
So this one right over
there you could not

00:11:00.460 --> 00:11:03.240
say that it is
necessarily similar.

00:11:03.240 --> 00:11:08.320
And likewise if you had a
triangle that had length 9 here

00:11:08.320 --> 00:11:11.920
and length 6 there,
but you did not

00:11:11.920 --> 00:11:13.920
know that these two
angles are the same,

00:11:13.920 --> 00:11:16.110
once again, you're not
constraining this enough,

00:11:16.110 --> 00:11:18.910
and you would not know that
those two triangles are

00:11:18.910 --> 00:11:21.060
necessarily similar
because you don't

00:11:21.060 --> 00:11:22.930
know that middle
angle is the same.

00:11:22.930 --> 00:11:24.430
Now, you might be
saying, well there

00:11:24.430 --> 00:11:26.646
was a few other
postulates that we had.

00:11:26.646 --> 00:11:31.324
We had AAS when we
dealt with congruency,

00:11:31.324 --> 00:11:32.990
but if you think about
it, we've already

00:11:32.990 --> 00:11:34.740
shown that two
angles by themselves

00:11:34.740 --> 00:11:36.490
are enough to show similarity.

00:11:36.490 --> 00:11:38.780
So why worry about an
angle, an angle, and a side

00:11:38.780 --> 00:11:40.080
or the ratio between a side?

00:11:40.080 --> 00:11:41.700
So why even worry about that?

00:11:41.700 --> 00:11:44.390
And we also had
angle-side-angle in congruence,

00:11:44.390 --> 00:11:46.940
but once again, we already
know the two angles are enough,

00:11:46.940 --> 00:11:48.890
so we don't need to
throw in this extra side,

00:11:48.890 --> 00:11:51.060
so we don't even need
this right over here.

00:11:51.060 --> 00:11:53.580
So these are going to be
our similarity postulates,

00:11:53.580 --> 00:11:55.760
and I want to remind
you, side-side-side,

00:11:55.760 --> 00:11:58.694
this is different than the
side-side-side for congruence.

00:11:58.694 --> 00:12:01.110
We're talking about the ratio
between corresponding sides.

00:12:01.110 --> 00:12:03.190
We're not saying that
they're actually congruent.

00:12:03.190 --> 00:12:06.274
And here, side-angle-side,
it's different

00:12:06.274 --> 00:12:07.940
than the side-angle-side
for congruence.

00:12:07.940 --> 00:12:09.390
It's this kind of
related, but here we're

00:12:09.390 --> 00:12:11.400
talking about the ratio
between the sides, not

00:12:11.400 --> 00:12:13.272
the actual measures.


Sal during Education Week (Sep 2011) 1

WEBVTT
Kind: captions
Language: en

00:00:00.187 --> 00:00:02.288
Female: ... week in the NBC family.

00:00:02.288 --> 00:00:03.462
And here at CNBC,

00:00:03.462 --> 00:00:06.343
we want to give our viewers the
chance to learn more about finance,

00:00:06.343 --> 00:00:09.234
hoping you understand where
exactly your money is going.

00:00:09.234 --> 00:00:10.943
Joining us now in the studio is Sal Khan,

00:00:10.943 --> 00:00:11.422
(music)

00:00:11.422 --> 00:00:13.837
founder and executive
director of the Khan Academy.

00:00:13.837 --> 00:00:17.689
He is also the mind behind dozens
of videos featured on CNBC.com,

00:00:17.689 --> 00:00:19.872
and he'll explain financial concepts.

00:00:19.872 --> 00:00:20.917
So what exactly are you doing?

00:00:20.917 --> 00:00:22.217
What's your initiative?

00:00:22.217 --> 00:00:23.370
Sal: Khan Academy itself,

00:00:23.370 --> 00:00:26.190
it literally just started with
me trying to tutor cousins,

00:00:26.190 --> 00:00:29.282
and I put my videos on
YouTube, and they got popular,

00:00:29.282 --> 00:00:30.766
and so I quit my old job,

00:00:30.766 --> 00:00:33.649
and I started doing videos on algebra
and calculus and all the rest.

00:00:33.649 --> 00:00:37.331
And I even did a few on
economics and finance.

00:00:37.331 --> 00:00:38.671
CNBC contacted me and said,

00:00:38.671 --> 00:00:40.829
"Hey, we want you to do more of that

00:00:40.829 --> 00:00:42.730
"and explain what is a credit default swap

00:00:42.730 --> 00:00:45.911
"or what is the difference between
preferred equity and regular equity,

00:00:45.911 --> 00:00:48.160
"and what is the present
value of something?"

00:00:48.160 --> 00:00:50.528
Female: Is it true anecdotally
that you got letters from bankers

00:00:50.528 --> 00:00:52.790
who said that you saved
them from getting fired

00:00:52.790 --> 00:00:53.910
so they can learn more
about [unintelligible]

00:00:53.910 --> 00:00:55.376
Sal: Yeah, that happened again last week.

00:00:55.376 --> 00:00:55.951
I was at a ...

00:00:55.951 --> 00:00:56.667
Female: ... confidence-building.

00:00:56.667 --> 00:00:57.980
Sal: It is not very confidence-building,

00:00:57.980 --> 00:00:59.656
but at least they had that (laughs).

00:00:59.656 --> 00:01:00.341
(laughter)

00:01:00.341 --> 00:01:01.401
At least they had ...

00:01:01.401 --> 00:01:03.805
No, I had someone who worked
in mortgage-backed securities

00:01:03.805 --> 00:01:04.997
during the financial crisis.
Male Isn't it that the truth

00:01:04.997 --> 00:01:06.718
that people out there who,

00:01:06.718 --> 00:01:08.499
people think they know.

00:01:08.499 --> 00:01:10.118
They think they're supposed
to know these things,

00:01:10.118 --> 00:01:11.542
and most people really don't.

00:01:11.542 --> 00:01:13.130
Sal: No.

00:01:13.130 --> 00:01:15.741
And what I say is the
more jargon someone uses,

00:01:15.741 --> 00:01:18.297
the less likely they actually
know the underpinnings of what.

00:01:18.297 --> 00:01:20.387
And so yeah, I mean I think
there are a lot of people

00:01:20.387 --> 00:01:21.964
that say the mortgage-backed
security industry

00:01:21.964 --> 00:01:25.065
who are signing the documents, getting
them stamped by their rating agency,

00:01:25.065 --> 00:01:26.330
Male: Obviously, yeah.

00:01:26.330 --> 00:01:28.024
Sal: No one knew what was
going on in the big picture.

00:01:28.024 --> 00:01:29.391
Female: Last week, you were talking about

00:01:29.391 --> 00:01:32.329
how there is the need for more
understanding and education about ETFs.

00:01:32.329 --> 00:01:36.995
How much interest do you see in people
educating themselves about ETFs?

00:01:36.995 --> 00:01:37.820
Sal: Yeah, ETF is one of them.

00:01:37.820 --> 00:01:39.530
I didn't do a video on that,
but it's one of those things.

00:01:39.530 --> 00:01:40.155
A lot of people will say,

00:01:40.155 --> 00:01:43.546
Male: They don't really
understand what it is.

00:01:43.546 --> 00:01:44.340
Sal: "But how does that it work?

00:01:44.340 --> 00:01:45.054
"Is it a mutual fund?

00:01:45.054 --> 00:01:45.713
"Is it close-ended?

00:01:45.713 --> 00:01:46.408
"Is it open-ended?

00:01:46.408 --> 00:01:47.323
"Is it something in between?"

00:01:47.323 --> 00:01:48.426
and so that's what I try to do,

00:01:48.426 --> 00:01:50.766
is just to break it down,
draw it out, and explain it.

00:01:50.766 --> 00:01:52.945
Male: So your site is ad-free?

00:01:52.945 --> 00:01:53.375
Khan: It is ad-free.

00:01:53.375 --> 00:01:54.692
We're not for-profit,

00:01:54.692 --> 00:01:58.617
and so we get the Gates
Foundation, Google have funded us,

00:01:58.617 --> 00:02:00.922
and so yeah, kids can
learn algebra without ads.

00:02:00.922 --> 00:02:03.056
Male: And the economics
of a cupcake factory.

00:02:03.056 --> 00:02:04.880
Sal: And the economics of a cupcake ...

00:02:04.880 --> 00:02:05.382
Sal: Yes.

00:02:05.382 --> 00:02:08.063
Male: But how do you get the ideas then?

00:02:08.063 --> 00:02:08.739
if somebody comes and say,

00:02:08.739 --> 00:02:10.265
"I want to open a cup cake factory,

00:02:10.265 --> 00:02:12.996
"but i have no idea what the
economics of said factory would be,"

00:02:12.996 --> 00:02:14.476
and therefore, you make a video?

00:02:14.476 --> 00:02:16.107
Female: What's the [unintelligible]

00:02:16.107 --> 00:02:18.322
Sal: Some of those things,
like the cupcake factory,

00:02:18.322 --> 00:02:21.230
my old job as an analyst at a hedge fund,

00:02:21.230 --> 00:02:24.118
I had another company in mind,
but I just changed their industry

00:02:24.118 --> 00:02:26.388
when I talked about
predatory prices and cupcakes

00:02:26.388 --> 00:02:28.801
because it seemed much
more benign (laughs)

00:02:28.801 --> 00:02:30.180
Male: It's very mysterious.

00:02:30.180 --> 00:02:30.636
Male: Very mysterious.

00:02:30.636 --> 00:02:31.544
Sal: Very mysterious.

00:02:31.544 --> 00:02:33.027
Female: Well, thank
you for coming on, Sal.

00:02:33.027 --> 00:02:33.478
Sal: Yeah.

00:02:33.478 --> 00:02:34.082
Thanks for having me.

00:02:34.082 --> 00:02:36.711
Male: khanacademy.org, you have
some crazy classes on here.

00:02:36.711 --> 00:02:37.953
Sal: Indeed, yes.

00:02:37.953 --> 00:02:39.346
Female: What's the craziest?

00:02:39.346 --> 00:02:40.130
Male: Yeah, very cool stuff.

00:02:40.130 --> 00:02:43.243
Sal: Photosynthesis, in
case you're interested.

00:02:43.243 --> 00:02:44.457
(laughter)

00:02:44.457 --> 00:02:46.129
Male: We wouldn't know about that
because we don't have any sunlight.

00:02:46.129 --> 00:02:46.807
Sal: No, no.

00:02:46.807 --> 00:02:47.713
(laughter)

00:02:47.713 --> 00:02:48.413
Male: No skylight here.

00:02:48.413 --> 00:02:50.897
Female: I'll get my plants
to watch it (laughs).

00:02:50.897 --> 00:02:51.743
Male: Sal, thank you very much.

00:02:51.743 --> 00:02:52.822
Sal: Yeah, yeah, thank you.

00:02:52.822 --> 00:02:53.990
All right, learning a lot.

00:02:53.990 --> 00:02:57.003
Sue: Rockefeller Plaza has been
transformed for the summit,

00:02:57.003 --> 00:02:58.983
and that's where we find
Michelle Caruso-Cabrera

00:02:58.983 --> 00:03:00.670
along with two leading innovators
(music)

00:03:00.670 --> 00:03:01.710
in their respective fields.

00:03:01.710 --> 00:03:02.318
Take it away, Michelle.

00:03:02.318 --> 00:03:02.851
(music)

00:03:02.851 --> 00:03:03.761
Michelle: Here are some of the coolest

00:03:03.761 --> 00:03:06.340
and most interesting people
I've met in a really long time

00:03:06.340 --> 00:03:10.734
joining me here at Education
Nation down here at 30 Rock.

00:03:10.734 --> 00:03:13.433
Nt Etuk is the founder of DimensionU.

00:03:13.433 --> 00:03:17.107
This is an educational video
game company that he started.

00:03:17.107 --> 00:03:18.633
He used to be at Citigroup,

00:03:18.633 --> 00:03:20.616
and he is rolling out a
lot of exciting things.

00:03:20.616 --> 00:03:22.392
And Sal Khan joins us as well.

00:03:22.392 --> 00:03:23.983
He is the founder of the Khan Academy.

00:03:23.983 --> 00:03:26.906
This is an educational video and ...

00:03:26.906 --> 00:03:28.584
so not video games but
videos and software company.

00:03:28.584 --> 00:03:29.918
Sal: Yes, yes, yeah.

00:03:29.918 --> 00:03:31.683
Michelle: You both have the same idea,

00:03:31.683 --> 00:03:33.786
which is that you want to
go directly to the customer,

00:03:33.786 --> 00:03:35.603
so you're not selling to schools.

00:03:35.603 --> 00:03:37.928
You're not selling to
parents necessarily, right?

00:03:37.928 --> 00:03:41.339
Nt: Well, actually, we do have a
component that we sell to schools,

00:03:41.339 --> 00:03:42.645
and it's been proven ...

00:03:42.645 --> 00:03:44.607
well, first of all, we do
multiplayer educational video games,

00:03:44.607 --> 00:03:46.577
so imagine Halo minus the violence

00:03:46.577 --> 00:03:49.323
but allowing a kid in Seoul, Korea to
play against a kid in New York City

00:03:49.323 --> 00:03:51.723
while they're learning math,
literacy, science, history.

00:03:51.723 --> 00:03:53.641
And so we started out selling to schools.

00:03:53.641 --> 00:03:56.400
We have large portions of New York
City Public Schools, et cetera.

00:03:56.400 --> 00:03:58.662
And now, we're going to the consumer side.

00:03:58.662 --> 00:04:00.317
Michelle: I guess my
thought was, what I mean,

00:04:00.317 --> 00:04:01.944
Nt: Yes.

00:04:01.944 --> 00:04:04.615
Michelle: Because historically,
educational video games have been ...

00:04:04.615 --> 00:04:06.458
Michelle: ... boring, right?

00:04:06.458 --> 00:04:08.031
And textbooks and so are workbooks.

00:04:08.031 --> 00:04:10.151
So if you can do video games,
then they're, like, "Hey!"

00:04:10.151 --> 00:04:11.110
Michelle: That they really want to play.

00:04:11.110 --> 00:04:11.930
Nt: That they want to play.

00:04:11.930 --> 00:04:13.677
You can incentivize them to
learn, and it's fantastic.

00:04:13.677 --> 00:04:14.159
Michelle: Wow.

00:04:14.159 --> 00:04:16.238
Sal: Yeah, it's the same idea.

00:04:16.238 --> 00:04:17.229
One thing, we're not selling.

00:04:17.229 --> 00:04:19.649
We're not for profit, so that's why
we're a little bit small (laughs).

00:04:19.649 --> 00:04:20.956
Michelle: OK.

00:04:20.956 --> 00:04:24.937
Sal: But our target is the
students and at any age.

00:04:24.937 --> 00:04:25.960
We have six-year-olds.

00:04:25.960 --> 00:04:26.924
We have homeschoolers.

00:04:26.924 --> 00:04:28.030
We have people using supplement tools ...

00:04:28.030 --> 00:04:29.384
Michelle: And they're
getting them where, YouTube?

00:04:29.384 --> 00:04:30.530
Or how are they getting the product?

00:04:30.530 --> 00:04:32.617
Sal: It's off of YouTube, khanacademy.org.

00:04:32.617 --> 00:04:33.933
YouTube, it's just the videos.

00:04:33.933 --> 00:04:36.551
Most of our resources are on
making a software platform.

00:04:36.551 --> 00:04:38.078
It has a little bit of games in there,

00:04:38.078 --> 00:04:40.747
but so students can practice everything
that they might learn in the videos too,

00:04:40.747 --> 00:04:42.012
and it goes all the way to calculus.

00:04:42.012 --> 00:04:42.723
Nt: Yeah.

00:04:42.723 --> 00:04:44.741
Michelle: Does it matter if
it's profit, not for profit?

00:04:44.741 --> 00:04:45.958
Does it?

00:04:45.958 --> 00:04:49.315
Nt: From my perspective, for profit
is how you keep sustainability going.

00:04:49.315 --> 00:04:51.327
I think that there is a
great way to actually make

00:04:51.327 --> 00:04:53.426
a tremendous amount of money
while doing good at the same time,

00:04:53.426 --> 00:04:55.394
and so that's what we do.

00:04:55.394 --> 00:04:57.648
Michelle: How is rollout going?

00:04:57.648 --> 00:04:59.109
You are now selling to schools.

00:04:59.109 --> 00:05:00.369
Are you getting schools involved?

00:05:00.369 --> 00:05:03.611
And is it easier to sell to charter
schools and things like that?

00:05:03.611 --> 00:05:05.830
Are they more adaptable as
opposed to public schools

00:05:05.830 --> 00:05:08.339
that have a longer purchase cycle?

00:05:08.339 --> 00:05:11.510
Sal: Most of our users, we have
2.5 million students a month,

00:05:11.510 --> 00:05:13.387
those are mostly just
independently using it,

00:05:13.387 --> 00:05:14.680
but we have started pilots.

00:05:14.680 --> 00:05:18.969
The Los Altos School District in Silicon
Valley, they're district-wide with it.

00:05:18.969 --> 00:05:19.792
They're public school.

00:05:19.792 --> 00:05:21.514
They're not a charter school
or anything like that.

00:05:21.514 --> 00:05:24.384
We are piloting with some charter
schools, some private schools, public,

00:05:24.384 --> 00:05:25.872
and we're just trying to see
how we can use this thing

00:05:25.872 --> 00:05:27.364
that's fundamentally
for independent learners

00:05:27.364 --> 00:05:28.845
but how we can use it
in the classrooms too.

00:05:28.845 --> 00:05:31.200
Michelle: Are either one of you
providing backbone analytics ...

00:05:31.200 --> 00:05:31.585
Nt: Yes.

00:05:31.585 --> 00:05:33.031
Michelle: ... to the schools so
that you know that it works?

00:05:33.031 --> 00:05:33.830
Very, very important.

00:05:33.830 --> 00:05:36.261
So how long are the students playing?

00:05:36.261 --> 00:05:37.379
What are they getting right and wrong?

00:05:37.379 --> 00:05:38.675
But we actually can go even deeper.

00:05:38.675 --> 00:05:41.726
We can understand if a student
plays primarily by themselves

00:05:41.726 --> 00:05:42.685
or whether they play with others,

00:05:42.685 --> 00:05:45.619
so are they a social learner or
are they an individual learner?

00:05:45.619 --> 00:05:47.995
And then you know how you can
help by tutoring or whatever else,

00:05:47.995 --> 00:05:50.335
so a whole bunch of
analyticals that gets it right.

00:05:50.335 --> 00:05:51.182
Michelle: Sal, is there ...?

00:05:51.182 --> 00:05:51.856
Sal: Yeah, I know.

00:05:51.856 --> 00:05:53.331
The model that we're
trying in the classrooms,

00:05:53.331 --> 00:05:55.859
the teachers aren't
giving lectures anymore.

00:05:55.859 --> 00:05:56.997
They're not grading homework.

00:05:56.997 --> 00:05:58.369
The students are all
working at their own pace,

00:05:58.369 --> 00:06:01.347
and the teacher gets a dashboard and
highlights the students that are stuck.

00:06:01.347 --> 00:06:03.402
So the teachers not able to do
these one-on-one interventions,

00:06:03.402 --> 00:06:04.754
we get the peers to teach each other.

00:06:04.754 --> 00:06:06.014
Nt: Yeah, exactly.

00:06:06.014 --> 00:06:06.705
Sal: Yeah, yeah, yeah, yeah.

00:06:06.705 --> 00:06:09.574
Michelle: But what about schools
that don't have resources?

00:06:09.574 --> 00:06:12.205
Are they less likely to get this stuff?

00:06:12.205 --> 00:06:12.848
Nt: From our perspective,

00:06:12.848 --> 00:06:14.796
that's one of the reasons why
we went to the consumer side,

00:06:14.796 --> 00:06:17.386
much like we're doing a free model ...

00:06:17.386 --> 00:06:17.890
Michelle: For the consumer.

00:06:17.890 --> 00:06:19.208
Nt: For the consumer side.

00:06:19.208 --> 00:06:21.823
Nt: But the kids at home
can get it for free.

00:06:21.823 --> 00:06:24.030
Then we actually make money
through micro-transactions,

00:06:24.030 --> 00:06:25.749
somewhat like Zynga, et cetera,

00:06:25.749 --> 00:06:27.924
so that sort of thing as we roll that out.

00:06:27.924 --> 00:06:29.551
It's been very interesting.

00:06:29.551 --> 00:06:30.537
Michelle: How about you, Sal?

00:06:30.537 --> 00:06:32.384
Sal: Yeah, it's a ...

00:06:32.384 --> 00:06:34.481
It's, it's ....

00:06:34.481 --> 00:06:35.234
Yeah, we do ...

00:06:35.234 --> 00:06:35.772
We ...

00:06:35.772 --> 00:06:36.504
I'm sorry if I ...

00:06:36.504 --> 00:06:38.339
Michelle: Don't worry about it (laughs).

00:06:38.339 --> 00:06:39.675
Don't worry about it.

00:06:39.675 --> 00:06:41.365
Should people be hopeful about education?

00:06:41.365 --> 00:06:42.009
Is there ...?

00:06:42.009 --> 00:06:42.457
Sal: Yes.

00:06:42.457 --> 00:06:46.208
Michelle: We've talked so long
about how we've been failing
for years in this country.

00:06:46.208 --> 00:06:47.154
Michelle: You're optimistic?

00:06:47.154 --> 00:06:47.685
Nt: Very.

00:06:47.685 --> 00:06:48.139
Here's the problem.

00:06:48.139 --> 00:06:51.304
The problem right now with the innovation
conversation education is it's tiring.

00:06:51.304 --> 00:06:53.268
We could have had this
conversation 300 years ago.

00:06:53.268 --> 00:06:54.107
Michelle: Yeah.

00:06:54.876 --> 00:06:57.386
Nt: What we have is
Google, iPod, iPod, iPad,

00:06:57.386 --> 00:06:59.327
all of these sorts of
technology to come in,

00:06:59.327 --> 00:07:02.817
and we can get rid of the
traditional distribution structures,

00:07:02.817 --> 00:07:05.472
go directly to the kids or
go directly to the teachers

00:07:05.472 --> 00:07:07.608
Michelle: A real product
that they want to use.

00:07:07.608 --> 00:07:08.679
Male: Yeah.

00:07:08.679 --> 00:07:09.385
Sal: Yeah, same thing,

00:07:09.385 --> 00:07:11.025
and I just remembered your
question was about technology,

00:07:11.025 --> 00:07:11.645
(laughter)

00:07:11.645 --> 00:07:13.386
And I think right now, there
is a healthy skepticism

00:07:13.386 --> 00:07:15.462
of why do you use
technology in the classroom?

00:07:15.462 --> 00:07:17.975
And I think what we're talking about
is now you have a reason to use it.

00:07:17.975 --> 00:07:19.021
And once you have a good reason,

00:07:19.021 --> 00:07:20.645
it's actually not that expensive anymore.

00:07:20.645 --> 00:07:21.867
You have a $300 laptop.

00:07:21.867 --> 00:07:23.903
You could use it for
five years at $60 a year.

00:07:23.903 --> 00:07:25.423
Michelle: As opposed to ...

00:07:25.423 --> 00:07:26.162
And 10 kids could use it.

00:07:26.162 --> 00:07:27.345
Michelle: As opposed to

00:07:27.345 --> 00:07:28.700
textbooks, which could
be far more expensive.

00:07:28.700 --> 00:07:29.790
Sal: Far more, far more expensive.

00:07:29.790 --> 00:07:30.701
Michelle: Guys, thank you so much ...

00:07:30.701 --> 00:07:31.053
Nt: Thank you.

00:07:31.053 --> 00:07:32.311
Michelle: ... for this wonderful
conversation I had with you.

00:07:32.311 --> 00:07:32.955
Sal: All right, great.

00:07:32.955 --> 23:59:59.999
Tyler, back to you.


Khan Academy on 20 20 (Sep 2011)

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:03.860
Female: Into the globe, let's now go
to California where Sal Khan is using

00:00:03.860 --> 00:00:08.447
YouTube in a way that might
just blow up the classroom.

00:00:08.447 --> 00:00:09.946
Sal: We're trying to educate the world.

00:00:09.946 --> 00:00:11.879
Female: Tutoring millions online.

00:00:11.879 --> 00:00:13.675
Sal: That's what's cool about an equation.

00:00:13.675 --> 00:00:15.535
You can always check to see
if you got the right answer.

00:00:15.535 --> 00:00:18.939
Right now we're getting
2 to 300,000 views a day.

00:00:18.939 --> 00:00:20.876
We claim to be the largest
university in the world.

00:00:20.876 --> 00:00:26.437
Female: More than 2400
lessons and counting on
everything from basic addition.

00:00:26.437 --> 00:00:28.230
Sal: 8 + 1 = 9

00:00:28.230 --> 00:00:29.179
Female: To quadratic formulas.

00:00:29.179 --> 00:00:31.604
Sal: Negative 1 times 2 is -2.

00:00:31.604 --> 00:00:33.101
Female: Photosynthesis.

00:00:33.101 --> 00:00:34.803
Sal: It needs carbon dioxide.

00:00:34.803 --> 00:00:36.324
Female: To U.S. history.

00:00:36.324 --> 00:00:37.874
Sal: We've had enough
of you, Great Britain.

00:00:37.874 --> 00:00:40.929
We are now declaring ourselves
as an independent country.

00:00:40.929 --> 00:00:47.617
Female: Any subject just search and click,
all viewed more than 76 million times.

00:00:47.617 --> 00:00:52.675
Sal: They can pause, repeat,
rewind without feeling like
they're wasting my time,

00:00:52.675 --> 00:00:56.544
without feeling embarrassed
because maybe they weren't
paying attention the first time

00:00:56.544 --> 00:01:00.764
or maybe they forgot something
from last year or last week and now
it's in a destressed environment.

00:01:00.764 --> 00:01:03.524
Female: It's a long way
from where it all began.

00:01:03.524 --> 00:01:07.076
Sal: My friend said, "Hey why don't you
put some of your lectures on YouTube?"

00:01:07.076 --> 00:01:11.665
I said, "No, that's for dogs
on skateboards. That's not
for serious mathematics."

00:01:11.665 --> 00:01:13.835
Female: And yet that was the
light bulb moment for you.

00:01:13.835 --> 00:01:15.202
Sal: I gave it a shot.

00:01:15.202 --> 00:01:17.142
Female: It's now called Khan Academy.

00:01:17.142 --> 00:01:18.811
Bill Gates: What Sal
Khan has done is amazing.

00:01:18.811 --> 00:01:22.761
Female: He's been called Bill Gates'
favorite teacher because Gates' own sons

00:01:22.761 --> 00:01:24.685
were logging on to Khan's lessons.

00:01:24.685 --> 00:01:28.937
Bill Gates: I see Sal Khan as a
pioneer. It's the start of a revolution.

00:01:28.937 --> 00:01:32.614
Female: The Gates Foundation
gave him 1.5 million dollars.

00:01:32.614 --> 00:01:39.277
Google tossed in 2 million. Enough
to hire a small staff and translate
his lessons into ten languages.

00:01:39.277 --> 00:01:43.269
Sal: If it was just a constant,
we would be able to use the
[unintelligible] but we can't do that.

00:01:43.269 --> 00:01:47.165
Female: Sal is the only instructor
and he knows what he's talking about.

00:01:47.165 --> 00:01:52.136
The MIT and Harvard grad got three
degrees simultaneously in math,

00:01:52.136 --> 00:01:55.134
electrical engineering
and computer science.

00:01:55.134 --> 00:01:57.545
But Sal has a dirty little secret.

00:01:57.545 --> 00:02:01.605
Sal Khan, one of the best teachers
in the world, hated school?

00:02:01.605 --> 00:02:07.334
It didn't register with me. I would
go to class. Half the time I was
bored. Half the time I was lost.

00:02:07.334 --> 00:02:09.748
I had trouble paying
attention for 60 minutes.

00:02:09.748 --> 00:02:13.558
Female: Which may be why Sal
is an education revolutionary.

00:02:13.558 --> 00:02:18.628
Sal: Everything I have listed is true
of both 401Ks and traditional IRAs.

00:02:18.628 --> 00:02:22.151
Female: You could make a lot
of money off of your brain, but
you're giving it away for free.

00:02:22.151 --> 00:02:25.680
Sal: Hopefully on my death
bed, and hopefully that's
many decades in the future.

00:02:25.680 --> 00:02:27.475
Female: A hundred years from now.

00:02:27.475 --> 00:02:29.353
Sal: A hundred years from
now I'll be able to say,

00:02:29.353 --> 00:02:33.072
"There's a billion kids who learned from
those lectures, or five million kids."

00:02:33.072 --> 00:02:34.778
Female: We have a little surprise for you.

00:02:34.778 --> 00:02:38.284
We connected Sal with just
one of his millions of fans.

00:02:38.284 --> 00:02:41.358
Jasper is a medical
student in the Netherlands.

00:02:41.358 --> 00:02:42.623
Male: Oh, yeah.

00:02:42.623 --> 00:02:45.532
Female: Who has spent a lot of time
with you. You just don't know it.

00:02:45.532 --> 00:02:49.457
Male: I watched some
videos and they were great.

00:02:49.457 --> 00:02:53.074
They really helped to
understand what I had to do.

00:02:53.074 --> 00:02:56.894
Sal: That's awesome, see? I could
sleep and help teach people the
Kreb's cycle. It's exciting.

00:02:56.894 --> 00:03:01.537
One person can produce something on
their own without any extra resources

00:03:01.537 --> 00:03:05.941
almost as a hobby and you
can reach millions of people.

00:03:05.941 --> 00:03:11.820
Female: It's a place of big and
small. Whether you're storm chasing,
[unintelligible] revolution,

00:03:11.820 --> 00:03:15.820
or just trying to learn
biology, it's a world connected.


KA on John Stossel  Documentary

WEBVTT
Kind: captions
Language: en

00:00:06.990 --> 00:00:09.340
JOHN STOSSEL: Do your
kids have a good teacher?

00:00:09.340 --> 00:00:10.495
How do you know?

00:00:10.495 --> 00:00:13.180
Maybe the teacher
next door is better.

00:00:13.180 --> 00:00:15.985
Maybe there's a better
teacher in another state.

00:00:15.985 --> 00:00:19.680
Maybe there's a world's
best teacher, or several.

00:00:19.680 --> 00:00:22.910
Wouldn't it be great if your
kids could have that teacher?

00:00:22.910 --> 00:00:26.810
Well today, yes you can.

00:00:26.810 --> 00:00:27.980
SPEAKER 1: Yay, I got it.

00:00:30.870 --> 00:00:33.600
JOHN STOSSEL: These kids are
this excited about a math

00:00:33.600 --> 00:00:34.280
website.

00:00:34.280 --> 00:00:35.280
SPEAKER 2: It's amazing.

00:00:35.280 --> 00:00:39.622
SPEAKER 3: Negative 4
minus 4, and we are done.

00:00:39.622 --> 00:00:41.205
SPEAKER 1: It taught
me lot of things.

00:00:41.205 --> 00:00:44.150
JOHN STOSSEL: Five years ago,
hedge fund analyst Sal Khan

00:00:44.150 --> 00:00:46.914
created videos like these
to tutor his cousin.

00:00:46.914 --> 00:00:48.330
SALMAN KHAN: That
worked out well.

00:00:48.330 --> 00:00:50.750
So then I started tutoring
her brothers and more cousins,

00:00:50.750 --> 00:00:51.380
and all the rest.

00:00:51.380 --> 00:00:53.588
And I had to do the same
lecture over and over again.

00:00:53.588 --> 00:00:55.340
So I had a friend
who said, hey Sal,

00:00:55.340 --> 00:00:58.470
why don't you put some of
your lectures on YouTube?

00:00:58.470 --> 00:01:00.460
I decided to give it a shot.

00:01:00.460 --> 00:01:02.970
SPEAKER 3: Welcome to this
presentation on basic addition.

00:01:02.970 --> 00:01:05.457
JOHN STOSSEL: Soon, thousands
watched his lectures.

00:01:05.457 --> 00:01:07.040
SALMAN KHAN: I started
getting letters

00:01:07.040 --> 00:01:09.430
from people and
comments on YouTube.

00:01:09.430 --> 00:01:12.670
And they're not
like, hey, I think

00:01:12.670 --> 00:01:14.740
this kind of might have
helped my math exam.

00:01:14.740 --> 00:01:18.615
They're like, I failed
calculus the first time.

00:01:18.615 --> 00:01:19.990
I've started
watching the videos,

00:01:19.990 --> 00:01:21.625
now I'm acing the class.

00:01:21.625 --> 00:01:23.710
JOHN STOSSEL: The YouTube
numbers kept rising,

00:01:23.710 --> 00:01:26.905
and he got letters from
the Middle East, Africa.

00:01:26.905 --> 00:01:29.550
BILL GATES: What Sal
Khan has done is amazing.

00:01:29.550 --> 00:01:31.750
JOHN STOSSEL: Now Khan
is funded by Bill Gates,

00:01:31.750 --> 00:01:34.050
and he offers web lectures
on everything from history

00:01:34.050 --> 00:01:36.870
to economics to
computer science.

00:01:36.870 --> 00:01:39.015
His videos are viewed
millions of times.

00:01:39.015 --> 00:01:41.390
SALMAN KHAN: Not only is it
reaching millions of students

00:01:41.390 --> 00:01:44.270
right now, but even if, God
forbid, I got hit by a bus

00:01:44.270 --> 00:01:46.500
when I walked
outside, it'll still

00:01:46.500 --> 00:01:47.990
be able to reach
millions and maybe

00:01:47.990 --> 00:01:49.323
eventually billions of students.

00:01:49.323 --> 00:01:51.750
JOHN STOSSEL: You just happen
to be good at teaching?

00:01:51.750 --> 00:01:54.386
SALMAN KHAN: Well, I'll
take that as a compliment.

00:01:54.386 --> 00:01:55.760
JOHN STOSSEL: It
is a compliment.

00:01:55.760 --> 00:01:57.250
He's a great teacher.

00:01:57.250 --> 00:01:58.760
SPEAKER 3: I hope that helps.

00:01:58.760 --> 00:02:00.166
See you in the next video.

00:02:00.166 --> 00:02:02.290
SPEAKER 4: It's really
helping us learn a lot more.

00:02:02.290 --> 00:02:03.748
JOHN STOSSEL: It's
exciting that he

00:02:03.748 --> 00:02:07.180
gets kids so excited about math.

00:02:07.180 --> 00:02:11.160
In most parts of life, things
have gotten much better-- cars,

00:02:11.160 --> 00:02:13.020
computers, cell phones.

00:02:13.020 --> 00:02:13.920
Education?

00:02:13.920 --> 00:02:14.439
Not so much.

00:02:14.439 --> 00:02:15.230
SALMAN KHAN: Right.

00:02:15.230 --> 00:02:18.000
You rewind 80 or
90 or 100 years,

00:02:18.000 --> 00:02:21.120
and you would have
the local band--

00:02:21.120 --> 00:02:23.420
if you had a party, that
was the only gig in town.

00:02:23.420 --> 00:02:26.246
JOHN STOSSEL: Each village might
have a storyteller or singer.

00:02:26.246 --> 00:02:27.120
SALMAN KHAN: Exactly.

00:02:27.120 --> 00:02:28.840
But then once you have
mass media coming out,

00:02:28.840 --> 00:02:29.800
then people say, wait.

00:02:29.800 --> 00:02:32.660
Why don't we take the best
musician, the best actor,

00:02:32.660 --> 00:02:35.770
the best storyteller,
whatever way, and record it?

00:02:35.770 --> 00:02:40.750
And put it out on radio, put
it out on records, or whatever?

00:02:40.750 --> 00:02:42.275
And I think, in
theory, that could

00:02:42.275 --> 00:02:43.870
have happened with
education before.

00:02:43.870 --> 00:02:45.036
JOHN STOSSEL: But it hasn't.

00:02:45.036 --> 00:02:47.570
Even for basic math,
multiplication tables,

00:02:47.570 --> 00:02:49.790
I thought they'd be
using video games.

00:02:49.790 --> 00:02:50.590
SALMAN KHAN: Yeah.

00:02:50.590 --> 00:02:51.506
JOHN STOSSEL: Why not?

00:02:51.506 --> 00:02:53.820
SALMAN KHAN: There's a huge
bureaucracy, most of which

00:02:53.820 --> 00:02:56.157
wants to say no to
change the system.

00:02:56.157 --> 00:02:57.990
JOHN STOSSEL: It's the
blob, people call it.

00:02:57.990 --> 00:02:59.115
SALMAN KHAN: It's the blob.

00:02:59.115 --> 00:02:59.670
And it is.

00:02:59.670 --> 00:03:02.270
And I think what's fun
from our point of view

00:03:02.270 --> 00:03:05.586
is that we are able to reach
students outside of the blob.

00:03:05.586 --> 00:03:07.460
JOHN STOSSEL: This
California school district

00:03:07.460 --> 00:03:10.570
started using Khan's videos
in fifth grade classrooms.

00:03:10.570 --> 00:03:12.900
The teachers were
skeptical, but now they're

00:03:12.900 --> 00:03:14.707
impressed at what it
does for the kids.

00:03:14.707 --> 00:03:16.790
KAMI THORARDSON: They're
happy to walk in the door

00:03:16.790 --> 00:03:17.200
every morning.

00:03:17.200 --> 00:03:18.340
They're excited about math.

00:03:18.340 --> 00:03:20.170
It isn't, oh, we're doing math.

00:03:20.170 --> 00:03:22.211
It's like, oh my gosh, we
have math this morning.

00:03:22.211 --> 00:03:24.240
That's great.

00:03:24.240 --> 00:03:27.062
SALMAN KHAN: We assume that
most people on their own

00:03:27.062 --> 00:03:28.645
don't want to do,
don't want to learn,

00:03:28.645 --> 00:03:30.770
or don't want to get
engaged in mathematics.

00:03:30.770 --> 00:03:30.870
And--

00:03:30.870 --> 00:03:31.880
JOHN STOSSEL: Yeah, they didn't.

00:03:31.880 --> 00:03:33.755
SALMAN KHAN: I think
they're just frustrated,

00:03:33.755 --> 00:03:35.490
because most of them
are in classrooms

00:03:35.490 --> 00:03:37.474
that are not catering to them.

00:03:37.474 --> 00:03:39.140
JOHN STOSSEL: At
first, teachers worried

00:03:39.140 --> 00:03:41.960
that the online instruction
could replace them.

00:03:41.960 --> 00:03:42.514
But--

00:03:42.514 --> 00:03:44.180
SANDRA McGONAGLE: I
think it's so wrong,

00:03:44.180 --> 00:03:45.680
as my teachers would tell you.

00:03:45.680 --> 00:03:49.040
They have taught more math than
they've ever taught before.

00:03:49.040 --> 00:03:51.754
JOHN STOSSEL: Now teachers
can tutor kids one on one.

00:03:51.754 --> 00:03:53.295
SPEAKER 5: Colleen,
I notice you were

00:03:53.295 --> 00:03:57.186
having some issues
with fractions.

00:03:57.186 --> 00:03:58.920
SPEAKER 6: You can
go at your own pace.

00:03:58.920 --> 00:04:01.620
JOHN STOSSEL: And because kids
can go at their own pace--

00:04:01.620 --> 00:04:02.995
KELLY RAFFERTY:
I've got students

00:04:02.995 --> 00:04:05.000
who are still working
on easy multiplication.

00:04:05.000 --> 00:04:05.820
And then I've got
students who are

00:04:05.820 --> 00:04:07.360
working in high school math.

00:04:07.360 --> 00:04:09.690
JOHN STOSSEL: Some kids
enjoy Khan's lessons so much,

00:04:09.690 --> 00:04:10.482
they study at home.

00:04:10.482 --> 00:04:12.648
KELLY RAFFERTY: Some of
them are doing two and three

00:04:12.648 --> 00:04:14.930
hours a night at home when
I'm asking for 15 minutes.

00:04:14.930 --> 00:04:17.374
SPEAKER 7: When I'm at home
and I have some time, I just

00:04:17.374 --> 00:04:17.915
log on, and--

00:04:17.915 --> 00:04:19.809
SPEAKER 8: It's way
more fun to do math.

00:04:19.809 --> 00:04:22.100
JOHN STOSSEL: So finally,
after all these years of kids

00:04:22.100 --> 00:04:25.810
being bored in school and not
learning math, that's over?

00:04:25.810 --> 00:04:27.624
SALMAN KHAN: I
think it might be.

00:04:27.624 --> 00:04:28.540
JOHN STOSSEL: Hope so.

00:04:28.540 --> 00:04:30.810
If it happens, it'll
be thanks to those

00:04:30.810 --> 00:04:33.660
of online classes, or
the charter schools,

00:04:33.660 --> 00:04:35.620
or other experiments
that break out

00:04:35.620 --> 00:04:38.850
of the union-dominated
government monopoly.

00:04:38.850 --> 00:04:41.560
Let 1,000 flowers bloom.

00:04:41.560 --> 00:04:44.100
It's competition that's
given us better medicine,

00:04:44.100 --> 00:04:48.040
transportation,
technology, everything.

00:04:48.040 --> 00:04:51.770
Don't our kids deserve that too?

00:04:51.770 --> 00:04:53.710
That's our show.


Similar triangle basics

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:04.830
When we compare triangle
ABC to triangle XYZ,

00:00:04.830 --> 00:00:07.450
it's pretty clear that
they aren't congruent,

00:00:07.450 --> 00:00:11.300
that they have very different
lengths of their sides.

00:00:11.300 --> 00:00:13.130
But there does seem
to be something

00:00:13.130 --> 00:00:16.120
interesting about the
relationship between these two

00:00:16.120 --> 00:00:17.020
triangles.

00:00:17.020 --> 00:00:20.500
One, all of their corresponding
angles are the same.

00:00:20.500 --> 00:00:25.620
So the angle right here, angle
BAC, is congruent to angle YXZ.

00:00:25.620 --> 00:00:29.640
Angle BCA is congruent
to angle YZX,

00:00:29.640 --> 00:00:33.990
and angle ABC is
congruent to angle XYZ.

00:00:33.990 --> 00:00:37.550
So all of their corresponding
angles are the same.

00:00:37.550 --> 00:00:41.000
And we also see that
the sides are just

00:00:41.000 --> 00:00:43.600
scaled-up versions
of each other.

00:00:43.600 --> 00:00:50.380
So to go from the length of XZ
to AC, we can multiply by 3.

00:00:50.380 --> 00:00:52.320
We multiplied by 3 there.

00:00:52.320 --> 00:00:57.790
To go from the length of XY
to the length of AB, which

00:00:57.790 --> 00:01:01.960
is the corresponding side,
we are multiplying by 3.

00:01:01.960 --> 00:01:03.760
We have to multiply by 3.

00:01:03.760 --> 00:01:09.160
And then to go from the length
of YZ to the length of BC,

00:01:09.160 --> 00:01:13.702
we also multiplied by 3.

00:01:13.702 --> 00:01:16.830
So essentially,
triangle ABC is just

00:01:16.830 --> 00:01:19.830
a scaled-up version
of triangle XYZ.

00:01:19.830 --> 00:01:21.520
If they were the
same scale, they

00:01:21.520 --> 00:01:23.110
would be the exact
same triangles.

00:01:23.110 --> 00:01:25.162
But one is just a bigger,
a blown-up version

00:01:25.162 --> 00:01:25.870
of the other one.

00:01:25.870 --> 00:01:28.610
Or this is a miniaturized
version of that one over there.

00:01:28.610 --> 00:01:31.060
If you just multiply
all the sides by 3,

00:01:31.060 --> 00:01:32.700
you get to this triangle.

00:01:32.700 --> 00:01:34.230
And so we can't
call them congruent,

00:01:34.230 --> 00:01:37.320
but this does seem to be a
bit of a special relationship.

00:01:37.320 --> 00:01:40.720
So we call this special
relationship similarity.

00:01:40.720 --> 00:01:49.489
So we can write that triangle
ABC is similar to triangle--

00:01:49.489 --> 00:01:51.780
and we want to make sure we
get the corresponding sides

00:01:51.780 --> 00:01:55.126
right-- ABC is going
to be similar to XYZ.

00:01:59.810 --> 00:02:01.450
And so, based on
what we just saw,

00:02:01.450 --> 00:02:03.550
there's actually kind
of three ideas here.

00:02:03.550 --> 00:02:06.720
And they're all equivalent ways
of thinking about similarity.

00:02:06.720 --> 00:02:09.210
One way to think
about it is that one

00:02:09.210 --> 00:02:11.190
is a scaled-up
version of the other.

00:02:11.190 --> 00:02:17.850
So scaled-up or -down
version of the other.

00:02:17.850 --> 00:02:19.400
When we talked about
congruency, they

00:02:19.400 --> 00:02:20.680
had to be exactly the same.

00:02:20.680 --> 00:02:23.260
You could rotate it, you could
shift it, you could flip it.

00:02:23.260 --> 00:02:24.760
But when you do all
of those things,

00:02:24.760 --> 00:02:26.840
they would have to
essentially be identical.

00:02:26.840 --> 00:02:29.690
With similarity, you can
rotate it, you can shift it,

00:02:29.690 --> 00:02:30.460
you can flip it.

00:02:30.460 --> 00:02:33.980
And you can also scale it up
and down in order for something

00:02:33.980 --> 00:02:35.090
to be similar.

00:02:35.090 --> 00:02:44.310
So for example, let's
say triangle CDE,

00:02:44.310 --> 00:02:50.100
if we know that triangle CDE
is congruent to triangle FGH,

00:02:50.100 --> 00:02:52.450
then we definitely know
that they are similar.

00:02:52.450 --> 00:02:54.195
They are scaled up
by a factor of 1.

00:02:54.195 --> 00:02:57.680
Then we know, for a
fact, that CDE is also

00:02:57.680 --> 00:03:00.740
similar to triangle FGH.

00:03:00.740 --> 00:03:02.840
But we can't say it
the other way around.

00:03:02.840 --> 00:03:05.350
If triangle ABC
is similar to XYZ,

00:03:05.350 --> 00:03:07.430
we can't say that it's
necessarily congruent.

00:03:07.430 --> 00:03:09.140
And we see, for this
particular example,

00:03:09.140 --> 00:03:11.930
they definitely
are not congruent.

00:03:11.930 --> 00:03:13.965
So this is one way to
think about similarity.

00:03:13.965 --> 00:03:15.590
The other way to
think about similarity

00:03:15.590 --> 00:03:19.000
is that all of the corresponding
angles will be equal.

00:03:19.000 --> 00:03:22.520
So if something is similar, then
all of the corresponding angles

00:03:22.520 --> 00:03:23.890
are going to be congruent.

00:03:29.935 --> 00:03:31.990
I always have trouble
spelling this.

00:03:31.990 --> 00:03:41.835
It is 2 Rs, 1 S. Corresponding
angles are congruent.

00:03:44.370 --> 00:03:50.220
So if we say that triangle ABC
is similar to triangle XYZ,

00:03:50.220 --> 00:03:58.026
that is equivalent to saying
that angle ABC is congruent--

00:03:58.026 --> 00:03:59.650
or we could say that
their measures are

00:03:59.650 --> 00:04:01.136
equal-- to angle XYZ.

00:04:05.790 --> 00:04:13.405
That angle BAC is going to
be congruent to angle YXZ.

00:04:17.950 --> 00:04:24.880
And then finally,
angle ACB is going

00:04:24.880 --> 00:04:28.940
to be congruent to angle XZY.

00:04:35.080 --> 00:04:37.240
So if you have two triangles,
all of their angles

00:04:37.240 --> 00:04:39.490
are the same, then you could
say that they're similar.

00:04:39.490 --> 00:04:41.700
Or if you find two
triangles and you're

00:04:41.700 --> 00:04:43.474
told that they are
similar triangles,

00:04:43.474 --> 00:04:45.640
then you know that all of
their corresponding angles

00:04:45.640 --> 00:04:46.650
are the same.

00:04:46.650 --> 00:04:49.170
And the last way
to think about it

00:04:49.170 --> 00:04:52.120
is that the sides are all
just scaled-up versions

00:04:52.120 --> 00:04:53.180
of each other.

00:04:53.180 --> 00:04:58.450
So the sides scaled
by the same factor.

00:05:02.770 --> 00:05:05.191
In the example we did here,
the scaling factor was 3.

00:05:05.191 --> 00:05:06.190
It doesn't have to be 3.

00:05:06.190 --> 00:05:10.750
It just has to be the same
scaling factor for every side.

00:05:10.750 --> 00:05:13.560
If we scaled this side up by 3
and we only scaled this side up

00:05:13.560 --> 00:05:15.410
by 2, then we would
not be dealing

00:05:15.410 --> 00:05:17.110
with a similar triangle.

00:05:17.110 --> 00:05:19.230
But if we scaled all
of these sides up by 7,

00:05:19.230 --> 00:05:21.730
then that's still a similar,
as long as you have all of them

00:05:21.730 --> 00:05:25.240
scaled up or scaled down
by the exact same factor.

00:05:25.240 --> 00:05:28.964
So one way to think
about it is-- I

00:05:28.964 --> 00:05:30.630
want to still visualize
those triangles.

00:05:30.630 --> 00:05:33.010
Let me redraw them right over
here a little bit simpler.

00:05:33.010 --> 00:05:35.010
Because I'm not talking
in now in general terms,

00:05:35.010 --> 00:05:36.650
not even for that specific case.

00:05:36.650 --> 00:05:43.600
So if we say that this is A, B,
and C, and this right over here

00:05:43.600 --> 00:05:49.150
is X, Y, and Z. I
just redrew them

00:05:49.150 --> 00:05:52.110
so I can refer them
when we write down here.

00:05:52.110 --> 00:05:54.665
If we're saying that these
two things right over here

00:05:54.665 --> 00:05:57.680
are similar, that means
that corresponding sides

00:05:57.680 --> 00:05:59.530
are scaled-up versions
of each other.

00:05:59.530 --> 00:06:05.920
So we could say that
the length of AB

00:06:05.920 --> 00:06:08.110
is equal to some
scaling factor--

00:06:08.110 --> 00:06:10.640
and this thing could be
less than 1-- some scaling

00:06:10.640 --> 00:06:14.770
factor times the length of
XY, the corresponding sides.

00:06:14.770 --> 00:06:16.920
And I know that AB
corresponds to XY because

00:06:16.920 --> 00:06:20.180
of the order in which I wrote
this similarity statement.

00:06:20.180 --> 00:06:24.100
So some scaling factor times XY.

00:06:24.100 --> 00:06:30.000
We know that the
length of BC needs

00:06:30.000 --> 00:06:36.455
to be that same scaling
factor times the length of YZ.

00:06:44.270 --> 00:06:49.140
And then we know
the length of AC

00:06:49.140 --> 00:06:53.210
is going to be equal to that
same scaling factor times XZ.

00:06:53.210 --> 00:06:56.840
So that's XZ, and this
could be a scaling factor.

00:06:56.840 --> 00:07:01.470
So if ABC is larger
than XYZ, then

00:07:01.470 --> 00:07:03.390
these k's will be larger than 1.

00:07:03.390 --> 00:07:05.640
If they're the exact same
size, if they're essentially

00:07:05.640 --> 00:07:08.630
congruent triangles,
then these k's will be 1.

00:07:08.630 --> 00:07:12.420
And if XYZ is bigger
than ABC, then

00:07:12.420 --> 00:07:15.361
these [? scaling ?] factors
will be less than 1.

00:07:15.361 --> 00:07:17.860
But another way to write these
same statements-- notice, all

00:07:17.860 --> 00:07:19.235
I'm saying is
corresponding sides

00:07:19.235 --> 00:07:21.150
are scaled-up versions
of each other.

00:07:21.150 --> 00:07:24.710
This first statement right here,
if you divide both sides by XY,

00:07:24.710 --> 00:07:31.720
you get AB over XY is equal
to our scaling factor.

00:07:31.720 --> 00:07:33.470
And then the second
statement right

00:07:33.470 --> 00:07:37.320
over here, if you divide
both sides by YZ--

00:07:37.320 --> 00:07:39.400
let me do it in that
same color-- you

00:07:39.400 --> 00:07:47.557
get BC divided by YZ is
equal to that scaling factor.

00:07:47.557 --> 00:07:49.390
And remember, in the
example we just showed,

00:07:49.390 --> 00:07:52.512
that scaling factor was 3.

00:07:52.512 --> 00:07:54.970
But now we're saying in the
more general terms, similarity,

00:07:54.970 --> 00:07:57.370
as long as you have the
same scaling factor.

00:07:57.370 --> 00:07:59.280
And then finally, if
you divide both sides

00:07:59.280 --> 00:08:04.210
here by the length between X
and Z, or segment XZ's length,

00:08:04.210 --> 00:08:11.390
you get AC over XZ is
equal to k, as well.

00:08:11.390 --> 00:08:13.620
Or another way to think
about it is the ratio

00:08:13.620 --> 00:08:14.980
between corresponding sides.

00:08:14.980 --> 00:08:18.990
Notice, this is the
ratio between AB and XY.

00:08:18.990 --> 00:08:27.460
The ratio between BC and YZ,
the ratio between AC and XZ,

00:08:27.460 --> 00:08:29.590
that the ratio between
corresponding sides

00:08:29.590 --> 00:08:31.990
all gives us the same constant.

00:08:31.990 --> 00:08:37.059
Or you could rewrite
this as AB over XY

00:08:37.059 --> 00:08:48.050
is equal to BC over YZ is equal
to AC over XZ, which would be

00:08:48.050 --> 00:08:51.214
equal to some scaling
factor, which is equal to k.

00:08:51.214 --> 00:08:52.630
So if you have
similar triangles--

00:08:52.630 --> 00:08:54.580
let me draw an arrow
right over here.

00:08:54.580 --> 00:08:56.962
Similar triangles means that
they're scaled-up versions,

00:08:56.962 --> 00:08:59.170
and you can also flip and
rotate and do all the stuff

00:08:59.170 --> 00:09:00.450
with congruency.

00:09:00.450 --> 00:09:02.500
And you can scale
them up or down.

00:09:02.500 --> 00:09:04.390
Which means all of the
corresponding angles

00:09:04.390 --> 00:09:06.940
are congruent, which
also means that the ratio

00:09:06.940 --> 00:09:08.970
between corresponding
sides is going

00:09:08.970 --> 00:09:11.580
to be the same constant for
all the corresponding sides.

00:09:11.580 --> 00:09:14.360
Or the ratio between
corresponding sides

00:09:14.360 --> 00:09:15.942
is constant.


Challenging perimeter problem

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.940
Here's an interesting
problem involving perimeter

00:00:02.940 --> 00:00:06.890
from the 2000 American
Invitational Mathematics Exam.

00:00:06.890 --> 00:00:08.830
So it says the diagram
shows a rectangle that

00:00:08.830 --> 00:00:11.390
has been dissected into
nine overlapping squares.

00:00:11.390 --> 00:00:14.798
So one, two, three, four,
five, six, seven, eight,

00:00:14.798 --> 00:00:17.400
nine overlapping squares.

00:00:17.400 --> 00:00:20.380
Given that the width and
the height of the rectangle

00:00:20.380 --> 00:00:23.180
are positive integers with
greatest common divisor 1-- so

00:00:23.180 --> 00:00:25.180
they're talking about
the width and the height

00:00:25.180 --> 00:00:25.665
of the rectangle.

00:00:25.665 --> 00:00:28.039
And the reason why they're
saying greatest common divisor

00:00:28.039 --> 00:00:31.280
1 is they're saying that they
don't have a common divisor

00:00:31.280 --> 00:00:35.100
that you can divide them both by
to get a more simplified ratio.

00:00:35.100 --> 00:00:37.480
And to think about
that is, we might

00:00:37.480 --> 00:00:42.080
be faced with two choices,
one where maybe it's

00:00:42.080 --> 00:00:47.540
this side over here is-- let me
draw it like this-- 5 and 15.

00:00:47.540 --> 00:00:50.720
But over here, our greatest
common divisor isn't 1.

00:00:50.720 --> 00:00:52.950
They're both divisible
by 5 over here.

00:00:52.950 --> 00:00:54.450
So what you'd want
to do is say, no.

00:00:54.450 --> 00:00:58.880
Instead of 5 and 15,
they need to be 1 and 3.

00:00:58.880 --> 00:01:00.990
Now you have the
same ratio of sides,

00:01:00.990 --> 00:01:03.160
but now their greatest
common divisor is 1.

00:01:03.160 --> 00:01:05.440
You kind of have it in the
simplified form or the most

00:01:05.440 --> 00:01:10.170
simplified form if you divide
both this height and this width

00:01:10.170 --> 00:01:10.910
by 5.

00:01:10.910 --> 00:01:14.020
So that's why they're saying
greatest common divisor 1.

00:01:14.020 --> 00:01:17.070
And then they say, find the
perimeter of the rectangle.

00:01:17.070 --> 00:01:19.861
So let's see what
we can do here.

00:01:19.861 --> 00:01:22.360
And I encourage you to pause
it and try to do it on your own

00:01:22.360 --> 00:01:25.910
before I bumble my way
through this problem.

00:01:25.910 --> 00:01:27.687
So let's start in the beginning.

00:01:27.687 --> 00:01:30.270
Let's start at this square right
over here, the center square.

00:01:30.270 --> 00:01:32.500
And they did tell us
that they're all squares.

00:01:32.500 --> 00:01:37.180
So let's say that that square
right over here has a length

00:01:37.180 --> 00:01:38.980
x and a height x.

00:01:38.980 --> 00:01:41.200
It's an x by x square.

00:01:41.200 --> 00:01:42.420
So let me write it.

00:01:42.420 --> 00:01:45.580
So this is an x,
and that is an x.

00:01:45.580 --> 00:01:49.250
So this is an x by x
square right over there.

00:01:49.250 --> 00:01:52.370
And then you have this
square right over here.

00:01:52.370 --> 00:01:54.520
And we don't know
its measurements.

00:01:54.520 --> 00:01:57.670
So let's say that this square
right over here is y by y.

00:02:00.320 --> 00:02:05.110
So it has y width, and
it also has y height.

00:02:05.110 --> 00:02:08.009
Now, what is this
square over here?

00:02:08.009 --> 00:02:10.620
Well, this is an x
plus y by an x plus

00:02:10.620 --> 00:02:14.730
y square, because the width
of these two squares combined

00:02:14.730 --> 00:02:17.260
made the width of
this larger square.

00:02:17.260 --> 00:02:19.384
So what I'm going
to do is-- actually,

00:02:19.384 --> 00:02:21.050
this might be an
easier way to write it.

00:02:21.050 --> 00:02:22.633
Since these are all
squares, I'm going

00:02:22.633 --> 00:02:26.320
to write the dimension of
that square inside the square.

00:02:26.320 --> 00:02:28.580
So this is going to
be an x by x square.

00:02:28.580 --> 00:02:30.470
Kind of a
non-conventional notation,

00:02:30.470 --> 00:02:32.790
but it'll help us keep
things a little bit neat.

00:02:32.790 --> 00:02:34.737
This is going to
be a y by y square.

00:02:34.737 --> 00:02:36.070
So I'm not saying the area is y.

00:02:36.070 --> 00:02:38.230
I'm saying it's y by y.

00:02:38.230 --> 00:02:43.300
This over here is going
to be an x plus y times--

00:02:43.300 --> 00:02:46.350
an x plus y is going to
be each of its dimensions.

00:02:46.350 --> 00:02:50.500
So it's going to be x plus
y height and x plus y width.

00:02:50.500 --> 00:02:58.270
Then this one over here-- well,
if this dimension is x plus y

00:02:58.270 --> 00:03:00.080
and this dimension
right over here

00:03:00.080 --> 00:03:04.455
is x, then this whole side or
any of the sides of the square

00:03:04.455 --> 00:03:05.790
are going to be the sum of that.

00:03:05.790 --> 00:03:10.353
So x plus x plus y is 2x plus y.

00:03:13.087 --> 00:03:14.670
You can imagine that
I'm just labeling

00:03:14.670 --> 00:03:16.530
the left side of each
of these squares.

00:03:16.530 --> 00:03:19.210
The left side of this
square has length y.

00:03:19.210 --> 00:03:20.450
Left side of this one, x.

00:03:20.450 --> 00:03:22.050
This one has x plus y.

00:03:22.050 --> 00:03:24.740
And then this is 2x plus y.

00:03:24.740 --> 00:03:27.120
And then we can go
do this one up here.

00:03:27.120 --> 00:03:30.600
Well, if this distance
right over here is 2x plus y

00:03:30.600 --> 00:03:34.200
and this distance right
over here is x plus y,

00:03:34.200 --> 00:03:37.300
you add them together to
get the entire dimension

00:03:37.300 --> 00:03:39.110
of one side of the square.

00:03:39.110 --> 00:03:43.285
So it's going to be 3x plus 2y.

00:03:46.260 --> 00:03:48.850
I've just added the 2x
plus the x and the y

00:03:48.850 --> 00:03:52.720
plus the y to get 3x plus 2y
is the length of one dimension

00:03:52.720 --> 00:03:54.080
or one side of this square.

00:03:54.080 --> 00:03:55.406
And they're all the same.

00:03:55.406 --> 00:03:56.780
Now let's go to
this next square.

00:03:56.780 --> 00:03:59.900
Well, if this length is
3x plus 2y and this length

00:03:59.900 --> 00:04:03.530
is 2x plus y, then this
entire length right over here

00:04:03.530 --> 00:04:09.140
is going to be 5x plus 3y.

00:04:09.140 --> 00:04:13.590
5x plus 3y is going to be that
entire length right over there.

00:04:13.590 --> 00:04:16.370
And we can also go to
this side right over here

00:04:16.370 --> 00:04:20.790
where we have this length--
let me do that same color.

00:04:20.790 --> 00:04:22.660
This length is 3x plus 2y.

00:04:22.660 --> 00:04:24.200
This is x plus y.

00:04:24.200 --> 00:04:25.250
And this is y.

00:04:25.250 --> 00:04:28.280
So if you add 3x plus
2y plus x plus y plus y,

00:04:28.280 --> 00:04:32.800
you get 4x plus-- what
is that-- 4y, right?

00:04:32.800 --> 00:04:34.240
2y, 3y, 4y.

00:04:38.480 --> 00:04:40.860
And then we can express
this character's dimensions

00:04:40.860 --> 00:04:45.760
in terms of x and y because
this is going to be 5x plus 3y.

00:04:45.760 --> 00:04:47.720
Then you're going
to have 2x plus y.

00:04:47.720 --> 00:04:49.160
And then you're going to have x.

00:04:49.160 --> 00:04:50.326
So you add the x's together.

00:04:50.326 --> 00:04:55.810
5x plus 2x is 7x, plus x is 8x.

00:04:55.810 --> 00:04:58.930
And then you add the
y's together, 3y plus y,

00:04:58.930 --> 00:05:00.470
and then you don't
have a y there.

00:05:00.470 --> 00:05:02.430
So that's going to be plus 4y.

00:05:02.430 --> 00:05:04.670
That's the dimensions
of this square.

00:05:04.670 --> 00:05:07.420
And then finally, we have
this square right over here.

00:05:07.420 --> 00:05:12.030
Its dimensions are going to
be the y plus the 4x plus 4y.

00:05:12.030 --> 00:05:14.510
So that's 4x plus 5y.

00:05:18.640 --> 00:05:24.890
And then if we think
about the dimensions

00:05:24.890 --> 00:05:31.620
of this actual
rectangle over here,

00:05:31.620 --> 00:05:35.130
if we think about its
height right over there,

00:05:35.130 --> 00:05:41.010
that's going to be 5x
plus 3y plus 8x plus 4y.

00:05:41.010 --> 00:05:44.410
So 5 plus 8 is 13.

00:05:44.410 --> 00:05:50.870
So it's 13x plus 3 plus 4 is 7y.

00:05:50.870 --> 00:05:51.887
So that's its height.

00:05:51.887 --> 00:05:53.470
But we can also think
about its height

00:05:53.470 --> 00:05:54.844
by going on the
other side of it.

00:05:54.844 --> 00:05:57.082
And maybe this will give
us some useful constraints

00:05:57.082 --> 00:05:58.540
because this is
going to have to be

00:05:58.540 --> 00:06:00.610
the same length
as this over here.

00:06:00.610 --> 00:06:05.680
And so if we add 4x
plus 4x, we get 8x.

00:06:05.680 --> 00:06:08.660
And then if we add 4y
plus 5y, we get 9y.

00:06:11.172 --> 00:06:13.380
So these are going to have
to be equal to each other,

00:06:13.380 --> 00:06:15.260
so that's an
interesting constraint.

00:06:15.260 --> 00:06:24.022
So we have 13x plus 7y is going
to have to equal 8x plus 9y.

00:06:31.150 --> 00:06:32.310
And we can simplify this.

00:06:32.310 --> 00:06:36.130
If you subtract 8x from
both sides, you get 5x.

00:06:36.130 --> 00:06:38.990
And if you subtract
7y from both sides,

00:06:38.990 --> 00:06:40.510
you get 5x is equal to 2y.

00:06:44.560 --> 00:06:50.010
Or you could say x
is equal to 2/5 y.

00:06:50.010 --> 00:06:51.760
In order for these to
show up as integers,

00:06:51.760 --> 00:06:52.820
we have to pick integers here.

00:06:52.820 --> 00:06:55.236
But let's see if we have any
other interesting constraints

00:06:55.236 --> 00:06:57.630
if we look at the bottom
and the top of this,

00:06:57.630 --> 00:07:00.260
if this gives us any
more information.

00:07:00.260 --> 00:07:04.750
So if we add 5x plus 3y plus
3x plus 2y plus 4x plus 4y,

00:07:04.750 --> 00:07:08.580
this top dimension-- 5
plus 3 is 8 plus 4 is 12.

00:07:08.580 --> 00:07:09.085
You get 12x.

00:07:14.400 --> 00:07:16.880
And then you get
3y plus 2y plus 4y.

00:07:16.880 --> 00:07:19.330
So that's 5 plus 4.

00:07:19.330 --> 00:07:20.480
That's 9y.

00:07:20.480 --> 00:07:21.480
Plus 9y.

00:07:21.480 --> 00:07:22.760
That's this top dimension.

00:07:22.760 --> 00:07:27.460
And if you go down here,
you have 8 plus 4 is 12x.

00:07:27.460 --> 00:07:28.610
Let me do that same color.

00:07:31.810 --> 00:07:36.179
And then you have 4
plus 5 is 9y, plus 9y.

00:07:36.179 --> 00:07:38.720
So these actually ended up to
be the same in terms of x and ,

00:07:38.720 --> 00:07:40.636
so they're not giving
us any more information,

00:07:40.636 --> 00:07:41.700
no more constraints.

00:07:41.700 --> 00:07:46.240
Obviously, 12x plus 9y is going
to be equal to 12x plus 9y.

00:07:46.240 --> 00:07:48.500
So our only constraint
on this problem

00:07:48.500 --> 00:07:51.470
is what we got by setting
this left-hand side

00:07:51.470 --> 00:07:56.610
to this right-hand side, is that
x needs to be equal to 2/5 y.

00:07:56.610 --> 00:07:58.150
So let's just pick
some numbers so

00:07:58.150 --> 00:07:59.749
that we get nice
integers for x and y

00:07:59.749 --> 00:08:01.290
and then we figure
out the perimeter.

00:08:01.290 --> 00:08:04.300
We want to make sure
that the dimensions don't

00:08:04.300 --> 00:08:05.850
have any common divisors.

00:08:05.850 --> 00:08:14.290
So if we pick y to be 5, so
let's pick y to be equal to 5,

00:08:14.290 --> 00:08:16.150
then looking at this
constraint, what is x?

00:08:16.150 --> 00:08:17.040
Well, then x is 2.

00:08:17.040 --> 00:08:19.350
It's going to be 2/5 times 5.

00:08:19.350 --> 00:08:21.800
So then x is equal to 2.

00:08:21.800 --> 00:08:24.350
So let's see what we get for
the dimensions of this rectangle

00:08:24.350 --> 00:08:24.960
then.

00:08:24.960 --> 00:08:26.610
So the height of
this rectangle is

00:08:26.610 --> 00:08:33.789
going to be 13 times 2 is
26, plus 7 times 5 is 35.

00:08:33.789 --> 00:08:36.970
So 26 plus 35 gets us what.

00:08:36.970 --> 00:08:38.460
That gets us to 61.

00:08:38.460 --> 00:08:40.652
This is equal to 61.

00:08:40.652 --> 00:08:42.159
Did I do that right?

00:08:42.159 --> 00:08:45.120
You see the 55 plus 6 is 61.

00:08:45.120 --> 00:08:48.070
And then when you
look at its width,

00:08:48.070 --> 00:08:52.220
you have 12x, which
is 24, plus 9y.

00:08:52.220 --> 00:08:56.490
y Is 5, so plus 45.

00:08:56.490 --> 00:08:59.120
24 plus 45 is what?

00:08:59.120 --> 00:09:01.990
That is 69.

00:09:01.990 --> 00:09:06.817
And 61 and 69 do not share any
common divisors other than 1.

00:09:06.817 --> 00:09:08.900
So it looks like we're
done, or we're almost done.

00:09:08.900 --> 00:09:10.566
We know the dimensions
of the rectangle.

00:09:10.566 --> 00:09:13.740
It is a 61 by 69 rectangle.

00:09:13.740 --> 00:09:15.580
And if you want its
actual perimeter,

00:09:15.580 --> 00:09:16.930
you just add them all up.

00:09:16.930 --> 00:09:19.910
So the perimeter
here is going to be--

00:09:19.910 --> 00:09:21.380
we could have a drum roll now.

00:09:21.380 --> 00:09:24.160
The perimeter is
going to be 61 plus 69

00:09:24.160 --> 00:09:32.810
plus 61 plus 69, which is equal
to-- well, 61 plus 69 is 130.

00:09:32.810 --> 00:09:35.400
That's another 130 right there.

00:09:35.400 --> 00:09:40.580
130 plus 130 is 260.

00:09:40.580 --> 00:09:42.330
So it actually wasn't
too bad of a problem

00:09:42.330 --> 00:09:45.350
if we just started at the middle
and just built up from there,

00:09:45.350 --> 00:09:49.480
built up the dimensions in terms
of the dimensions of these two

00:09:49.480 --> 00:09:52.080
smallest squares.

00:09:52.080 --> 00:09:55.410
And then we were able
to find the perimeter.


Koch snowflake fractal

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.122
So let's say that this is
an equilateral triangle.

00:00:03.122 --> 00:00:05.080
And what I want to do is
make another shape out

00:00:05.080 --> 00:00:06.205
of this equilateral triangle.

00:00:06.205 --> 00:00:07.621
And I'm going to
do that by taking

00:00:07.621 --> 00:00:09.370
each of the sides
of this triangle,

00:00:09.370 --> 00:00:11.855
and divide them into
three equal sections.

00:00:15.010 --> 00:00:18.736
So my equilateral triangle
wasn't drawn super ideally.

00:00:18.736 --> 00:00:20.110
But I think you'll
get the point.

00:00:20.110 --> 00:00:21.810
And in the middle section
I want to construct

00:00:21.810 --> 00:00:23.105
another equilateral triangle.

00:00:29.510 --> 00:00:32.680
So it's going to look
something like this.

00:00:32.680 --> 00:00:35.090
And then right
over here I'm going

00:00:35.090 --> 00:00:37.740
to put another
equilateral triangle.

00:00:37.740 --> 00:00:40.270
And so now I went from
that equilateral triangle

00:00:40.270 --> 00:00:43.250
to something that's looking
like a star, or a Star of David.

00:00:43.250 --> 00:00:44.670
And then I'm going
to do it again.

00:00:44.670 --> 00:00:46.700
So each of the sides
now I'm going to divide

00:00:46.700 --> 00:00:48.410
into three equal sides.

00:00:48.410 --> 00:00:50.000
And in that middle
segment I'm going

00:00:50.000 --> 00:00:51.295
to put an equilateral triangle.

00:00:54.130 --> 00:00:56.370
So in the middle
segment I'm going

00:00:56.370 --> 00:00:59.350
to put an equilateral triangle.

00:00:59.350 --> 00:01:01.950
So I'm going to do it for
every one of the sides.

00:01:01.950 --> 00:01:05.600
So let me do it right there.

00:01:05.600 --> 00:01:06.760
And then right there.

00:01:06.760 --> 00:01:10.030
I think you get the idea,
but I want to make it clear.

00:01:10.030 --> 00:01:15.260
So let me just, so then like
that, and then like that,

00:01:15.260 --> 00:01:20.060
like that, and then almost
done for this iteration.

00:01:20.060 --> 00:01:21.840
This pass.

00:01:21.840 --> 00:01:23.260
And it'll look like that.

00:01:23.260 --> 00:01:24.330
Then I could do it again.

00:01:24.330 --> 00:01:26.663
Each of the segments I can
divide into three equal sides

00:01:26.663 --> 00:01:28.260
and draw another
equilateral triangle.

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So I could just there,
there, there, there.

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I think you see
where this is going.

00:01:33.260 --> 00:01:37.534
And I could keep going
on forever and forever.

00:01:37.534 --> 00:01:38.950
So what I want to
do in this video

00:01:38.950 --> 00:01:40.930
is think about
what's going on here.

00:01:40.930 --> 00:01:43.000
And what I'm actually
drawing, if we just

00:01:43.000 --> 00:01:45.000
keep on doing this
forever and forever,

00:01:45.000 --> 00:01:47.990
every side, every iteration,
we look at each side,

00:01:47.990 --> 00:01:49.520
we divide into
three equal sides.

00:01:49.520 --> 00:01:52.177
And then the next iteration,
or three equal segments,

00:01:52.177 --> 00:01:53.760
the next iteration,
the middle segment

00:01:53.760 --> 00:01:55.799
we turn to another
equilateral triangle.

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This shape that we're
describing right here

00:01:57.590 --> 00:01:59.950
is called a Koch snowflake.

00:01:59.950 --> 00:02:03.300
And I'm sure I'm
mispronouncing the Koch part.

00:02:03.300 --> 00:02:05.480
A Koch snowflake,
and it was first

00:02:05.480 --> 00:02:08.229
described by this gentleman
right over here, who

00:02:08.229 --> 00:02:11.270
is a Swedish mathematician,
Niels Fabian Helge von

00:02:11.270 --> 00:02:14.610
Koch, who I'm sure
I'm mispronouncing it.

00:02:14.610 --> 00:02:17.740
And this was one of the
earliest described fractals.

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So this is a fractal.

00:02:19.690 --> 00:02:22.120
And the reason why it
is considered a fractal

00:02:22.120 --> 00:02:24.890
is that it looks the same,
or it looks very similar,

00:02:24.890 --> 00:02:26.630
on any scale you look at it.

00:02:26.630 --> 00:02:29.591
So when you look at it at this
scale, so if you look at this,

00:02:29.591 --> 00:02:31.090
it like you see a
bunch of triangles

00:02:31.090 --> 00:02:32.320
with some bumps on it.

00:02:32.320 --> 00:02:35.130
But then if you were to
zoom in right over there,

00:02:35.130 --> 00:02:38.190
then you would still see
that same type of pattern.

00:02:38.190 --> 00:02:39.926
And then if you were
to zoom in again,

00:02:39.926 --> 00:02:41.300
you would see it
again and again.

00:02:41.300 --> 00:02:43.320
So a fractal is anything
that at on any scale,

00:02:43.320 --> 00:02:46.770
on any level of zoom, it kind
of looks roughly the same.

00:02:46.770 --> 00:02:48.270
So that's why it's
called a fractal.

00:02:48.270 --> 00:02:49.790
Now what's particularly
interesting,

00:02:49.790 --> 00:02:53.700
and why I'm putting it at this
point in the geometry playlist,

00:02:53.700 --> 00:02:56.417
is that this actually has
an infinite perimeter.

00:02:56.417 --> 00:02:58.500
If you were to keep doing
it, if you were actually

00:02:58.500 --> 00:03:00.690
to make the Koch
snowflake, where

00:03:00.690 --> 00:03:04.130
you keep an infinite number
of times on every smaller

00:03:04.130 --> 00:03:05.840
little triangle
here, you keep adding

00:03:05.840 --> 00:03:09.754
another equilateral
triangle on its side.

00:03:09.754 --> 00:03:11.670
And to show that it has
an infinite perimeter,

00:03:11.670 --> 00:03:13.890
let's just consider
one side over here.

00:03:13.890 --> 00:03:15.960
So let's say that
this side, so let's

00:03:15.960 --> 00:03:18.024
say we're starting
right when we started

00:03:18.024 --> 00:03:19.940
with that original
triangle, that's that side.

00:03:19.940 --> 00:03:21.565
Let's say it has length s.

00:03:21.565 --> 00:03:23.565
And then we divide it
into three equal segments.

00:03:26.170 --> 00:03:29.910
So those are going to
be s/3, s/3-- actually,

00:03:29.910 --> 00:03:31.670
let me write it this way.

00:03:31.670 --> 00:03:36.170
s/3, s/3, and s/3.

00:03:36.170 --> 00:03:38.730
And in the middle segment, you
make an equilateral triangle.

00:03:41.840 --> 00:03:44.462
So each of these sides
are going to be s/3.

00:03:44.462 --> 00:03:47.290
s/3, s/3.

00:03:47.290 --> 00:03:51.472
And now the length of this new
part-- I can't call it a line

00:03:51.472 --> 00:03:53.180
anymore, because it
has this bump in it--

00:03:53.180 --> 00:03:57.580
the length of this part right
over here, this side, now

00:03:57.580 --> 00:04:01.510
doesn't have just a length
of s, it is now s/3 times 4.

00:04:01.510 --> 00:04:03.160
Before it was s/3 times 3.

00:04:03.160 --> 00:04:07.730
Now you have 1, 2, 3, 4
segments that are s/3.

00:04:07.730 --> 00:04:10.400
So now, after one
time, after one pace,

00:04:10.400 --> 00:04:15.160
after one time of doing
this adding triangles,

00:04:15.160 --> 00:04:17.480
our new side, after
we add that bump,

00:04:17.480 --> 00:04:20.760
is going to be four times s/3.

00:04:20.760 --> 00:04:23.730
Or it equals 4/3 s.

00:04:23.730 --> 00:04:28.670
So if our original
perimeter when it was just

00:04:28.670 --> 00:04:31.100
a triangle is p sub 0.

00:04:31.100 --> 00:04:33.930
After one pass, after
we add one set of bumps,

00:04:33.930 --> 00:04:39.840
then our perimeter is going to
be 4/3 times the original one.

00:04:39.840 --> 00:04:42.630
Because each of the sides are
going to be 4/3 bigger now.

00:04:42.630 --> 00:04:44.610
So this was made
up of three sides.

00:04:44.610 --> 00:04:46.870
Now each of those sides
are going to be 4/3 bigger.

00:04:46.870 --> 00:04:49.400
So the new perimeter's
going to be 4/3 times that.

00:04:49.400 --> 00:04:51.680
And then when we take
a second pass on it,

00:04:51.680 --> 00:04:54.740
it's going to be 4/3
times this first pass.

00:04:54.740 --> 00:04:57.650
So every pass you take,
it's getting 4/3 bigger.

00:04:57.650 --> 00:05:00.380
Or it's getting, I guess,
a 1/3 bigger on every,

00:05:00.380 --> 00:05:03.437
it's getting 4/3
the previous pass.

00:05:03.437 --> 00:05:05.520
And so if you do that an
infinite number of times,

00:05:05.520 --> 00:05:09.220
if you multiply any
number by 4/3 an

00:05:09.220 --> 00:05:11.180
infinite number of
times, you're going

00:05:11.180 --> 00:05:13.820
to get an infinite number
of infinite length.

00:05:13.820 --> 00:05:16.274
So P infinity.

00:05:16.274 --> 00:05:18.690
The perimeter, if you do this
an infinite number of times,

00:05:18.690 --> 00:05:20.370
is infinite.

00:05:20.370 --> 00:05:21.920
Now that by itself
is kind of cool,

00:05:21.920 --> 00:05:24.461
just to think about something
that has an infinite perimeter.

00:05:24.461 --> 00:05:28.192
But what's even neater is that
it actually has a finite area.

00:05:28.192 --> 00:05:29.900
And when I say a finite
area, it actually

00:05:29.900 --> 00:05:32.164
covers a bounded
amount of space.

00:05:32.164 --> 00:05:34.080
And I could actually
draw a shape around this,

00:05:34.080 --> 00:05:36.120
and this thing will
never expand beyond that.

00:05:36.120 --> 00:05:37.620
And to think about
it, I'm not going

00:05:37.620 --> 00:05:39.240
to do a really
formal proof, just

00:05:39.240 --> 00:05:42.520
think about it, what happens
on any one of these sides.

00:05:42.520 --> 00:05:44.630
So on that first pass we
have that this triangle

00:05:44.630 --> 00:05:46.010
gets popped out.

00:05:46.010 --> 00:05:49.340
And then, if you think about it,
if you just draw what happens,

00:05:49.340 --> 00:05:51.681
the next iteration you draw
these two triangles right

00:05:51.681 --> 00:05:52.180
over there.

00:05:52.180 --> 00:05:54.270
And these two characters
right over there.

00:05:54.270 --> 00:05:56.330
And then you put some
triangles over here,

00:05:56.330 --> 00:05:58.520
and here, and here,
and here, and here.

00:05:58.520 --> 00:05:59.700
So on and so forth.

00:05:59.700 --> 00:06:01.960
But notice, you can keep
adding more and more.

00:06:01.960 --> 00:06:04.630
You can add essentially an
infinite number of these bumps,

00:06:04.630 --> 00:06:07.380
but you're never going to
go past this original point.

00:06:07.380 --> 00:06:10.490
And the same thing is going
to be true on this side

00:06:10.490 --> 00:06:11.540
right over here.

00:06:11.540 --> 00:06:14.240
It's also going to be true
of this side over here.

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Also going to be true
at this side over here.

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Also going to be true
this side over there.

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And then also going to be
true that side over there.

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So even if you do this an
infinite number of times,

00:06:24.750 --> 00:06:27.450
this shape, this Koch
snowflake will never

00:06:27.450 --> 00:06:30.430
have a larger area than
this bounding hexagon.

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Or which will never have a
larger area than a shape that

00:06:33.190 --> 00:06:34.342
looks something like that.

00:06:34.342 --> 00:06:35.925
I'm just kind of
drawing an arbitrary,

00:06:35.925 --> 00:06:38.070
well I want to make it
outside of the hexagon,

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I could put a circle
outside of it.

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So this thing I drew in blue, or
this hexagon I drew in magenta,

00:06:44.690 --> 00:06:47.050
those clearly have a fixed area.

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And this Koch snowflake
will always be bounded.

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Even though you can add these
bumps an infinite number

00:06:52.070 --> 00:06:53.050
of times.

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So a bunch of really
cool things here.

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One, it's a fractal.

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You can keep zooming in
and it'll look the same.

00:06:58.880 --> 00:07:01.750
The other thing, infinite,
infinite perimeter,

00:07:01.750 --> 00:07:04.787
and finite, finite area.

00:07:04.787 --> 00:07:06.120
Now you might say, wait Sal, OK.

00:07:06.120 --> 00:07:07.590
This is a very abstract thing.

00:07:07.590 --> 00:07:10.510
Things like this don't actually
exist in the real world.

00:07:10.510 --> 00:07:12.850
And there's a fun
thought experiment

00:07:12.850 --> 00:07:14.880
that people talk about
in the fractal world,

00:07:14.880 --> 00:07:17.540
and that's finding the
perimeter of England.

00:07:17.540 --> 00:07:19.390
Or you can actually
do it with any island.

00:07:19.390 --> 00:07:21.020
And so England looks
something like--

00:07:21.020 --> 00:07:22.750
and I'm not an
expert on, let's say

00:07:22.750 --> 00:07:24.600
it looks something
like that-- so at first

00:07:24.600 --> 00:07:26.099
you might approximate
the perimeter.

00:07:26.099 --> 00:07:28.330
And you might measure
this distance,

00:07:28.330 --> 00:07:32.240
you might measure this
distance, plus this distance,

00:07:32.240 --> 00:07:34.970
plus this distance, plus that
distance, plus that distance,

00:07:34.970 --> 00:07:36.051
plus that distance.

00:07:36.051 --> 00:07:38.050
And you're like look, it
has a finite perimeter.

00:07:38.050 --> 00:07:39.660
It clearly has a finite area.

00:07:39.660 --> 00:07:42.070
But you're like, look, that
has a finite perimeter.

00:07:42.070 --> 00:07:43.610
But you're like, no,
wait that's not as good.

00:07:43.610 --> 00:07:45.610
You have to approximate it a
little bit better than that.

00:07:45.610 --> 00:07:47.068
Instead of doing
it that rough, you

00:07:47.068 --> 00:07:48.762
need to make a bunch
of smaller lines.

00:07:48.762 --> 00:07:50.470
You need to make a
bunch of smaller lines

00:07:50.470 --> 00:07:52.400
so you can hug the coast
a little bit better.

00:07:52.400 --> 00:07:55.647
And you're like, OK, that's
a much better approximation.

00:07:55.647 --> 00:07:57.730
But then, let's say you're
at some piece of coast,

00:07:57.730 --> 00:08:02.620
if we zoom in enough,
the actual coast line

00:08:02.620 --> 00:08:04.492
is going to look
something like this.

00:08:04.492 --> 00:08:05.950
The actual coast
line will have all

00:08:05.950 --> 00:08:08.240
of these little divots in it.

00:08:08.240 --> 00:08:10.960
And essentially, when you did
that first, when did this pass,

00:08:10.960 --> 00:08:13.292
you were just measuring that.

00:08:13.292 --> 00:08:15.750
And you're like, that's not
the perimeter of the coastline.

00:08:15.750 --> 00:08:17.791
You're going to have to
do many, many more sides.

00:08:17.791 --> 00:08:22.300
You're going to do something
like this, to actually get

00:08:22.300 --> 00:08:25.747
the perimeter of the coast line.

00:08:25.747 --> 00:08:27.330
And you're just like,
hey, now that is

00:08:27.330 --> 00:08:29.179
a good approximation
for the perimeter.

00:08:29.179 --> 00:08:31.720
But if you were to zoom in on
that part of the coastline even

00:08:31.720 --> 00:08:34.539
more, it'll actually turn out
that it won't look exactly

00:08:34.539 --> 00:08:35.429
like that.

00:08:35.429 --> 00:08:37.230
It'll actually come
in and out like this.

00:08:37.230 --> 00:08:39.230
Maybe it'll look
something like that.

00:08:39.230 --> 00:08:41.689
So instead of having these
rough lines that just measure it

00:08:41.689 --> 00:08:43.605
like that, you're going
to say, oh wait, no, I

00:08:43.605 --> 00:08:46.280
need to go a little bit closer
and hug it even tighter.

00:08:46.280 --> 00:08:48.390
And you can really keep
on doing that until you

00:08:48.390 --> 00:08:50.560
get to the actual atomic level.

00:08:50.560 --> 00:08:55.300
So the actual coastline of
an island, or a continent,

00:08:55.300 --> 00:08:58.640
or anything, is actually
somewhat kind of fractalish.

00:08:58.640 --> 00:09:00.760
And it is, you can
kind of think of it

00:09:00.760 --> 00:09:02.477
as having an almost
infinite perimeter.

00:09:02.477 --> 00:09:04.060
Obviously at some
point you're getting

00:09:04.060 --> 00:09:06.640
to kind of the atomic level,
so it won't quite be the same.

00:09:06.640 --> 00:09:08.450
But it's kind of
the same phenomenon.

00:09:08.450 --> 00:09:11.420
It's an interesting thing
to actually think about.