Dividing line segments according to ratio

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:01.880
- [Instructor] We're told point A

00:00:01.880 --> 00:00:05.100
is at negative one comma four and point C

00:00:05.100 --> 00:00:07.420
is at four common negative six.

00:00:07.420 --> 00:00:12.420
Find the coordinates of point
B on segment, line segment AC

00:00:12.660 --> 00:00:17.660
such that the ratio of AB
to AC is three to five.

00:00:18.030 --> 00:00:20.703
So pause this video and see
if you can figure that out.

00:00:21.670 --> 00:00:23.490
All right, now let's work
through this together

00:00:23.490 --> 00:00:27.400
and to help us visualize,
let's plot these points.

00:00:27.400 --> 00:00:31.490
So first, let us plot point A

00:00:31.490 --> 00:00:33.460
which is at negative one comma four.

00:00:33.460 --> 00:00:37.940
So negative one comma
one, two, three, four

00:00:37.940 --> 00:00:41.830
so that right over there is point A

00:00:41.830 --> 00:00:43.960
and then let's think about point C

00:00:43.960 --> 00:00:46.580
which is at four common negative six.

00:00:46.580 --> 00:00:50.180
So one, two, three,
four, comma negative six,

00:00:50.180 --> 00:00:52.600
negative one, negative
two, negative three,

00:00:52.600 --> 00:00:57.600
negative four, negative five,
negative six, just like that

00:00:58.030 --> 00:01:03.030
and so the segment AC, I
get my ruler tool out here.

00:01:03.170 --> 00:01:07.447
Segment AC is going to look like that

00:01:09.680 --> 00:01:12.580
and the ratio between
the distance of A to B

00:01:12.580 --> 00:01:15.870
and A to C is three to five or another way

00:01:15.870 --> 00:01:18.260
to think about it is B is going to be

00:01:18.260 --> 00:01:22.810
three fifths along the way from A to C.

00:01:22.810 --> 00:01:24.540
Now the way that I think about it is

00:01:24.540 --> 00:01:28.140
in order to be three fifths
along the way from A to C

00:01:28.140 --> 00:01:31.510
you have to be three fifths
along the way in the X direction

00:01:31.510 --> 00:01:34.780
and three fifths along the
way in the Y direction.

00:01:34.780 --> 00:01:37.160
So let's think about
the X direction first.

00:01:37.160 --> 00:01:41.160
We are going from X equals
negative one to X equals four

00:01:41.160 --> 00:01:43.340
to go from this point to that point.

00:01:43.340 --> 00:01:48.340
Our change in X is one,
two, three, four, five

00:01:48.740 --> 00:01:50.930
and so if we wanna go
three fifths of that,

00:01:50.930 --> 00:01:52.980
we went a total of five,
three fifths of that

00:01:52.980 --> 00:01:55.070
is going just three.

00:01:55.070 --> 00:01:57.650
So that is going to be B is X coordinate

00:01:57.650 --> 00:02:00.350
and then we can look on
the Y coordinate side.

00:02:00.350 --> 00:02:04.130
To go from A to C, we are
going from four to negative six

00:02:04.130 --> 00:02:08.360
so we're going down by
one, two, three, four,

00:02:08.360 --> 00:02:13.270
five, six, seven, eight, nine, 10

00:02:13.270 --> 00:02:17.000
and so three fifths of 10 would be six.

00:02:17.000 --> 00:02:19.580
So B's coordinate is going to be

00:02:19.580 --> 00:02:24.580
one, two, three, four, five, six down.

00:02:24.680 --> 00:02:26.640
So just like that, we
were able to figure out

00:02:26.640 --> 00:02:30.450
the X and the Y coordinates for point B,

00:02:30.450 --> 00:02:32.860
which would be right over here

00:02:32.860 --> 00:02:34.790
and you could look at
this directly and say,

00:02:34.790 --> 00:02:37.550
look, B is going to be
have the coordinates.

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This looks like this is
two comma negative two

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which we were able to do the graph paper.

00:02:44.010 --> 00:02:46.310
So another way you could think
about it even algebraically

00:02:46.310 --> 00:02:48.850
is the coordinates of B,
we could think about it

00:02:48.850 --> 00:02:51.820
as starting with the coordinates of A

00:02:51.820 --> 00:02:54.500
so negative one comma four,

00:02:54.500 --> 00:02:56.610
but we're gonna move
three fifths along the way

00:02:56.610 --> 00:02:58.640
in each of these dimensions towards C.

00:02:58.640 --> 00:03:02.950
So it's going to be
plus three fifths times

00:03:02.950 --> 00:03:05.860
how far we've gone in the extraction.

00:03:05.860 --> 00:03:07.880
So in the extraction to go from A to C,

00:03:07.880 --> 00:03:09.720
you were going from negative one to four

00:03:09.720 --> 00:03:13.090
and so that distance is
four minus negative one

00:03:13.090 --> 00:03:15.690
and this of course is
going to be equal to five

00:03:15.690 --> 00:03:19.320
and then on the Y dimension,
this is going to be our

00:03:19.320 --> 00:03:23.760
A's Y coordinate plus three fifths times

00:03:24.690 --> 00:03:27.740
the distance that we
travel in the Y direction

00:03:27.740 --> 00:03:30.100
and here we're going from
four to negative six.

00:03:30.100 --> 00:03:34.870
So we say negative six minus
four, that is negative 10

00:03:34.870 --> 00:03:37.620
and so the coordinates of
B are gonna be negative one

00:03:37.620 --> 00:03:40.560
plus three fifths times five
is going to be plus three

00:03:40.560 --> 00:03:44.880
and then four plus three
fifths times negative 10,

00:03:44.880 --> 00:03:48.390
well, three three fifths
negative 10 is negative six.

00:03:48.390 --> 00:03:53.310
and so that gets us two comma
negative two and we are done,

00:03:53.310 --> 00:03:55.823
which is exactly what
we got right over there.


Classifying figures with coordinates

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:03.050
- [Instructor] We're told
that parallelogram ABCD

00:00:03.050 --> 00:00:04.650
has the following vertices,

00:00:04.650 --> 00:00:07.300
and they give us the coordinates
of the different vertices.

00:00:07.300 --> 00:00:12.300
And they say is parallelogram
ABCD a rectangle, and why?

00:00:12.370 --> 00:00:14.410
So pause this video, and
try to think about this

00:00:14.410 --> 00:00:16.760
on your own before we
work through it together.

00:00:17.700 --> 00:00:20.020
All right, now let's
work through it together.

00:00:20.020 --> 00:00:20.853
So in general,

00:00:20.853 --> 00:00:24.490
if you know that something
is already a parallelogram

00:00:24.490 --> 00:00:27.180
and you wanted to determine
whether it's a rectangle,

00:00:27.180 --> 00:00:30.120
it's really a question of
whether the adjacent sides

00:00:30.120 --> 00:00:32.130
intersect at a right angle.

00:00:32.130 --> 00:00:33.040
So for example,

00:00:33.040 --> 00:00:36.330
a parallelogram might
look something like this.

00:00:36.330 --> 00:00:38.160
What we know about a parallelogram is

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that the opposite sides are parallel,

00:00:39.950 --> 00:00:41.750
so this side is parallel to that side

00:00:41.750 --> 00:00:45.010
and that this side is
parallel to this side.

00:00:45.010 --> 00:00:47.500
And all rectangles are parallelograms,

00:00:47.500 --> 00:00:49.630
but not all parallelograms are rectangles.

00:00:49.630 --> 00:00:52.830
In order for a parallelogram
to be a rectangle,

00:00:52.830 --> 00:00:56.120
these sides need to
intersect at right angles.

00:00:56.120 --> 00:00:57.600
And clearly, the way I drew this one,

00:00:57.600 --> 00:00:59.230
it doesn't look like that.

00:00:59.230 --> 00:01:01.060
But let's see if we can
figure that out based

00:01:01.060 --> 00:01:04.030
on the coordinates that
they have given us.

00:01:04.030 --> 00:01:05.410
And to help us visualize,

00:01:05.410 --> 00:01:06.610
let me just put some coordinates.

00:01:06.610 --> 00:01:09.460
Let me draw some axes here.

00:01:09.460 --> 00:01:12.100
So that's my x-axis,

00:01:12.100 --> 00:01:16.780
and then this is my y-axis.

00:01:16.780 --> 00:01:18.040
Let's see, the coordinates,

00:01:18.040 --> 00:01:19.930
let's see, we have twos, fours, sixes,

00:01:19.930 --> 00:01:21.660
so let me actually count by, and eights,

00:01:21.660 --> 00:01:23.560
let me count by twos here.

00:01:23.560 --> 00:01:27.010
So we have two, four,

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six, and eight,

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and then we have negative
two, negative four,

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negative six, negative eight.

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We have two, four, six, and eight,

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and then we'd have negative
two, negative four,

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negative six, and negative eight.

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So each hash mark is another two.

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I'm counting by twos here.

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And so let's plot these points,

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and I'll do it in a different
color, so we can keep track.

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So A is negative six comma negative four.

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So negative two, negative
four, negative six,

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and then negative four
would go right over here.

00:02:00.470 --> 00:02:02.900
That is point A.

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Then we have point B, which
is negative two comma six.

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So negative two comma six,

00:02:10.640 --> 00:02:14.640
so that's going to go
up two, four, and six.

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So that is point B right over there.

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Then we have point C, which
is at eight comma two.

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So eight comma two

00:02:24.050 --> 00:02:25.750
right over there,

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that is point C.

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And then last but not
least, we have point D,

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which is at four comma negative eight,

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four comma negative eight

00:02:35.300 --> 00:02:37.400
right over there, point D.

00:02:37.400 --> 00:02:39.400
And so our quadrilateral,

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or we actually know it's a
parallelogram, looks like this.

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So you have segment AB like that.

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You have segment BC that looks like that.

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Segment CD looks like this.

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And segment AD looks like this.

00:02:55.180 --> 00:02:57.461
And we know already that
it's a parallelogram,

00:02:57.461 --> 00:02:59.510
so we know that segment AB
is parallel to segment DC

00:03:02.084 --> 00:03:02.917
and segment BC is parallel to segment AD.

00:03:06.900 --> 00:03:08.400
But what we really need
to do is figure out

00:03:08.400 --> 00:03:11.170
whether they are
intersecting at right angles.

00:03:11.170 --> 00:03:13.720
And to do that using the
coordinates to figure that out,

00:03:13.720 --> 00:03:14.950
we have to figure out the slopes

00:03:14.950 --> 00:03:17.290
of these different line segments.

00:03:17.290 --> 00:03:21.780
And so let's figure out
first the slope of AB.

00:03:21.780 --> 00:03:25.080
So the slope of

00:03:25.080 --> 00:03:27.850
segment AB is going to be equal

00:03:27.850 --> 00:03:30.060
to our change in y over change in x.

00:03:30.060 --> 00:03:33.990
So our change in y is going
to be six minus negative four,

00:03:33.990 --> 00:03:37.820
six minus negative four over

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negative two minus negative six,

00:03:41.060 --> 00:03:43.720
negative two minus negative six.

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And so this is going to
be equal to six plus four,

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which is 10,

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over negative two minus negative six.

00:03:51.650 --> 00:03:54.330
That's the same thing as
negative two plus six.

00:03:54.330 --> 00:03:55.710
So that's going to be over four,

00:03:55.710 --> 00:03:57.830
which is the same thing as 5/2.

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All right, that's interesting.

00:03:59.440 --> 00:04:04.180
What is the slope of segment BC?

00:04:04.180 --> 00:04:07.440
The slope of segment BC

00:04:07.440 --> 00:04:09.350
is going to be equal to,

00:04:09.350 --> 00:04:11.470
once again, change in y over change in x.

00:04:11.470 --> 00:04:15.250
Our y-coordinates' change
in y is two minus six,

00:04:15.250 --> 00:04:17.370
two minus six

00:04:17.370 --> 00:04:20.170
over eight minus negative two,

00:04:20.170 --> 00:04:22.200
eight minus negative two,

00:04:22.200 --> 00:04:25.810
which is equal to negative four over,

00:04:25.810 --> 00:04:27.770
and then eight minus negative
two is the same thing

00:04:27.770 --> 00:04:29.620
as eight plus two over 10,

00:04:29.620 --> 00:04:32.430
which is the same thing as negative 2/5.

00:04:32.430 --> 00:04:34.360
Now in other videos in your algebra class,

00:04:34.360 --> 00:04:37.300
you might have learned
that the slopes of lines

00:04:37.300 --> 00:04:40.410
that intersect at right
angles or the slopes of lines

00:04:40.410 --> 00:04:43.530
that form a right angle at
their point of intersection,

00:04:43.530 --> 00:04:46.590
that they are going to be
the opposite reciprocals.

00:04:46.590 --> 00:04:48.180
And you can actually see
that right over here.

00:04:48.180 --> 00:04:49.590
These are opposite reciprocals.

00:04:49.590 --> 00:04:53.010
If you take the reciprocal of
this top slope, you'd get 2/5.

00:04:53.010 --> 00:04:54.720
And then you take the opposite of it or,

00:04:54.720 --> 00:04:56.350
in this case, the negative of it,

00:04:56.350 --> 00:04:58.910
you are going to get negative 2/5.

00:04:58.910 --> 00:05:03.540
So these are perpendicular lines.

00:05:03.540 --> 00:05:05.790
So this lets us know

00:05:05.790 --> 00:05:10.250
that AB is perpendicular,

00:05:10.250 --> 00:05:13.390
segment AB is perpendicular to segment BC.

00:05:13.390 --> 00:05:16.030
So we know that this is the case.

00:05:16.030 --> 00:05:17.820
And we could keep on doing that.

00:05:17.820 --> 00:05:19.580
But in a parallelogram,

00:05:20.920 --> 00:05:23.680
if one set of segments
intersect at a right angle,

00:05:23.680 --> 00:05:26.020
all of them are going to
intersect at a right angle.

00:05:26.020 --> 00:05:28.490
And we could show that more
rigorously in other places,

00:05:28.490 --> 00:05:30.270
but this is enough evidence for me to know

00:05:30.270 --> 00:05:33.340
that this is indeed
going to be a rectangle.

00:05:33.340 --> 00:05:35.940
If you want, you could
continue to do this analysis,

00:05:35.940 --> 00:05:38.080
and you will see that
this is perpendicular,

00:05:38.080 --> 00:05:41.670
this is perpendicular, and
that is perpendicular as well.

00:05:41.670 --> 00:05:43.510
But let's see which of
these choices match up

00:05:43.510 --> 00:05:45.390
to what we just deduced.

00:05:45.390 --> 00:05:46.890
So choice A says yes,

00:05:46.890 --> 00:05:48.700
and yes would be that it is a rectangle

00:05:48.700 --> 00:05:52.360
because AB is equal.

00:05:52.360 --> 00:05:54.430
So the length of segment AB is equal

00:05:54.430 --> 00:05:56.593
to the length of segment AD,

00:05:58.300 --> 00:06:00.370
and the length of segment BC is equal

00:06:00.370 --> 00:06:02.640
to the length of segment CD.

00:06:02.640 --> 00:06:04.580
So that might be true.

00:06:04.580 --> 00:06:06.400
I haven't validated it.

00:06:06.400 --> 00:06:08.720
But just because this is true

00:06:08.720 --> 00:06:12.010
and because we do know that
ABCD is a parallelogram,

00:06:12.010 --> 00:06:13.940
that wouldn't let me know

00:06:13.940 --> 00:06:16.200
that we are actually
dealing with a rectangle.

00:06:16.200 --> 00:06:18.440
For example, you can have a parallelogram

00:06:18.440 --> 00:06:21.290
where even all the sides are congruent.

00:06:21.290 --> 00:06:24.620
So you could have a parallelogram
that looks like this.

00:06:24.620 --> 00:06:26.310
And obviously, if all of
the sides are congruent,

00:06:26.310 --> 00:06:27.330
you're dealing with a rhombus,

00:06:27.330 --> 00:06:29.800
but a rhombus is still
not necessarily going

00:06:29.800 --> 00:06:30.933
to be a rectangle.

00:06:31.790 --> 00:06:33.930
And so I would rule this top one out.

00:06:33.930 --> 00:06:36.100
This second choice says yes,

00:06:36.100 --> 00:06:39.030
and it says because BC
is perpendicular to AB.

00:06:39.030 --> 00:06:41.590
Yeah, we saw that by
seeing that their slopes

00:06:41.590 --> 00:06:44.400
are the opposite
reciprocals of each other.

00:06:44.400 --> 00:06:46.900
And of course, we know that
ABCD is a parallelogram.

00:06:46.900 --> 00:06:48.720
So I am liking this choice.

00:06:48.720 --> 00:06:51.470
And these other ones claim
that this is not a rectangle,

00:06:51.470 --> 00:06:53.230
but we already deduced
that it is a rectangle.

00:06:53.230 --> 00:06:55.593
So we could rule these out as well.


Using related volumes

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:01.550
- [Instructor] We're told that all

00:00:01.550 --> 00:00:04.760
of the following figures
have the same height.

00:00:04.760 --> 00:00:08.450
All of the figures except
for B have square bases.

00:00:08.450 --> 00:00:10.040
So that's a square base, that's a square,

00:00:10.040 --> 00:00:12.380
that's a square, and that's a square.

00:00:12.380 --> 00:00:16.700
All of the figures
except for C are prisms.

00:00:16.700 --> 00:00:19.890
Yeah, C is a pyramid right over here.

00:00:19.890 --> 00:00:23.300
All of the figures except for D are right.

00:00:23.300 --> 00:00:25.670
You can see D right over
here is a little bit skewed,

00:00:25.670 --> 00:00:27.780
or you can view it as oblique.

00:00:27.780 --> 00:00:32.780
All of the figures except for
E have the same base area.

00:00:32.830 --> 00:00:37.700
The base of figure E is a
dilation of the base of figure A

00:00:37.700 --> 00:00:39.313
by a scale factor of 1.5.

00:00:40.808 --> 00:00:43.210
All right, so it tells a figure A

00:00:43.210 --> 00:00:46.270
has a volume of 28 cubic centimeters.

00:00:46.270 --> 00:00:49.120
What are the volumes of the other figures?

00:00:49.120 --> 00:00:50.140
So I'll pause this video

00:00:50.140 --> 00:00:52.090
and see if you can have a go with that.

00:00:53.480 --> 00:00:56.250
All right, now let's work
through this together.

00:00:56.250 --> 00:00:58.940
Now they're telling us about
the bases and the heights

00:00:58.940 --> 00:01:00.840
that a lot of these
have the same base area,

00:01:00.840 --> 00:01:02.230
figure E's going to be different.

00:01:02.230 --> 00:01:04.540
And they also tell us they
all have the same height.

00:01:04.540 --> 00:01:06.450
So one way to think about volume

00:01:06.450 --> 00:01:08.980
is it's going to deal
with base and height.

00:01:08.980 --> 00:01:10.180
And so for figure A,

00:01:10.180 --> 00:01:11.330
it's pretty straight forward.

00:01:11.330 --> 00:01:14.120
If we call this area right over here,

00:01:14.120 --> 00:01:15.420
let's call that b,

00:01:15.420 --> 00:01:18.060
for the area of the base,

00:01:18.060 --> 00:01:22.410
and then it has some
height, h, right over here.

00:01:22.410 --> 00:01:25.170
We know that the base
area times the height

00:01:25.170 --> 00:01:26.950
is going to be the volume.

00:01:26.950 --> 00:01:29.830
So we can say that based on figure A,

00:01:29.830 --> 00:01:33.270
base times height is going to be equal

00:01:33.270 --> 00:01:36.993
to 28 cubic centimeters.

00:01:37.880 --> 00:01:38.840
Fair enough.

00:01:38.840 --> 00:01:41.650
Now what's going on
over here with figure B?

00:01:41.650 --> 00:01:43.000
Well, it's a cylinder.

00:01:43.000 --> 00:01:45.160
Now for a cylinder, what is
the volume of a cylinder?

00:01:45.160 --> 00:01:48.850
Well, it, too, is going
to be base times height.

00:01:48.850 --> 00:01:53.300
So it's going to be the area
of the base times the height.

00:01:53.300 --> 00:01:54.327
And if you're wondering,

00:01:54.327 --> 00:01:56.730
"How is that possible that it's the same

00:01:56.730 --> 00:01:58.840
as a volume of a rectangular
prism over here?"

00:01:58.840 --> 00:02:00.460
It's actually Cavalieri's principle.

00:02:00.460 --> 00:02:01.840
If they have the same height

00:02:01.840 --> 00:02:03.310
and if at any point on that height,

00:02:03.310 --> 00:02:05.280
they have the same cross-sectional area,

00:02:05.280 --> 00:02:07.250
then you're going to have the same volume.

00:02:07.250 --> 00:02:09.920
So this volume is also going
to be base times height.

00:02:09.920 --> 00:02:12.420
So let me just say this is figure A,

00:02:12.420 --> 00:02:14.690
figure B right over here,

00:02:14.690 --> 00:02:16.080
let me draw those dots a little better,

00:02:16.080 --> 00:02:17.510
these colons a little bit better.

00:02:17.510 --> 00:02:21.850
Figure B, the volume is also
going to be base times height,

00:02:21.850 --> 00:02:24.920
which is equal to 28 cubic centimeters.

00:02:24.920 --> 00:02:25.753
Let me make that clear.

00:02:25.753 --> 00:02:27.110
That's the volume's equals to that.

00:02:27.110 --> 00:02:28.930
Volume is equal to that.

00:02:28.930 --> 00:02:31.730
Now what about for figure C?

00:02:31.730 --> 00:02:33.050
What is the volume going to be,

00:02:33.050 --> 00:02:34.950
what is the formula for
the volume of a pyramid?

00:02:34.950 --> 00:02:36.020
And we've gotten the intuition

00:02:36.020 --> 00:02:38.800
and proven this to
ourselves in other videos.

00:02:38.800 --> 00:02:40.900
Well, we know that for pyramid,

00:02:40.900 --> 00:02:42.780
the volume is going to be equal

00:02:42.780 --> 00:02:47.780
to 1/3 times base times height.

00:02:48.040 --> 00:02:51.420
And we know that it has the same base area

00:02:51.420 --> 00:02:52.830
as these other characters here,

00:02:52.830 --> 00:02:54.310
it has the same height.

00:02:54.310 --> 00:02:56.353
And so we know what base times height is.

00:02:56.353 --> 00:02:59.130
It's 28 cubic centimeters.

00:02:59.130 --> 00:03:02.780
So this is going to be 1/3
times 28 cubic centimeters.

00:03:02.780 --> 00:03:03.613
So this is going to be equal

00:03:03.613 --> 00:03:08.613
to 1/3 times 28 cubic centimeters,

00:03:08.810 --> 00:03:13.810
which we could rewrite as 28
over three cubic centimeters.

00:03:13.910 --> 00:03:17.260
You could also write that as
nine and 1/3 cubic centimeters.

00:03:17.260 --> 00:03:19.020
So that's for figure C.

00:03:19.020 --> 00:03:20.803
Now let's think about figure D.

00:03:21.710 --> 00:03:23.720
I'll do that right over here.

00:03:23.720 --> 00:03:26.640
Well, for this oblique prism,

00:03:26.640 --> 00:03:27.710
I guess we could say,

00:03:27.710 --> 00:03:28.840
you're going to have the same idea

00:03:28.840 --> 00:03:30.860
that comes from Cavalieri's
principle again.

00:03:30.860 --> 00:03:32.230
It's going to have the same formula

00:03:32.230 --> 00:03:33.250
for volume as figure A.

00:03:33.250 --> 00:03:37.310
It's going to be the base times
the height right over here.

00:03:37.310 --> 00:03:39.740
So I could write volume is
going to be equal to base

00:03:39.740 --> 00:03:42.140
times height and we
already know what that is.

00:03:42.140 --> 00:03:42.973
They tell us.

00:03:42.973 --> 00:03:43.806
The base times the height

00:03:43.806 --> 00:03:45.510
is going to be the same as figure A.

00:03:45.510 --> 00:03:49.420
It's going to be 28 cubic centimeters.

00:03:49.420 --> 00:03:50.710
Now let's go to figure E.

00:03:50.710 --> 00:03:51.800
This is an interesting one

00:03:51.800 --> 00:03:55.160
'cause it has a different base area.

00:03:55.160 --> 00:03:59.590
What is gonna be the area right over here?

00:03:59.590 --> 00:04:01.430
Well, they tell us that the base

00:04:01.430 --> 00:04:03.680
of figure E is a dilation

00:04:03.680 --> 00:04:06.603
of the base of figure A
by a scale factor of 1.5.

00:04:08.079 --> 00:04:09.580
And these are both squares.

00:04:09.580 --> 00:04:13.480
So figure A, we'll say, x by x.

00:04:13.480 --> 00:04:18.480
This one over here is
going to be 1.5x by 1.5x.

00:04:18.580 --> 00:04:19.430
So let me write that down.

00:04:19.430 --> 00:04:24.430
1.5x times 1.5x.

00:04:24.470 --> 00:04:25.410
Or another way to think about it,

00:04:25.410 --> 00:04:26.243
let me do it over here

00:04:26.243 --> 00:04:27.290
where I have some free space.

00:04:27.290 --> 00:04:32.290
We know that b, which we
know is an area of figure A,

00:04:32.320 --> 00:04:35.680
that would be equal to x times x.

00:04:35.680 --> 00:04:39.500
Now what's the area of
the base of figure E?

00:04:39.500 --> 00:04:41.650
Well, it's going to be 1.5x times 1.5x

00:04:43.779 --> 00:04:46.590
or 1.5x squared, which is the same thing

00:04:46.590 --> 00:04:51.590
as 1.5 squared is 2.25x squared,

00:04:51.680 --> 00:04:54.570
and we know x squared or x times x

00:04:54.570 --> 00:04:56.050
that is equal to b.

00:04:56.050 --> 00:04:57.910
That is equal to our original base area

00:04:57.910 --> 00:04:59.460
in all of these other figures.

00:04:59.460 --> 00:05:01.630
So the area over here,

00:05:01.630 --> 00:05:02.980
this area right over here

00:05:02.980 --> 00:05:04.153
is going to be 2.25 times b.

00:05:06.726 --> 00:05:07.809
2.25 times b.

00:05:09.520 --> 00:05:11.500
Now that wasn't so easy to read.

00:05:11.500 --> 00:05:13.950
Let me write that a little bit clearer.

00:05:13.950 --> 00:05:18.950
So 2.25b is the base area right over here.

00:05:20.520 --> 00:05:22.570
And so what's the volume of this figure?

00:05:22.570 --> 00:05:25.940
The volume is going to
be the area of the base,

00:05:25.940 --> 00:05:30.100
which is 2.25 times the area

00:05:30.100 --> 00:05:32.250
of all these other figures' bases

00:05:32.250 --> 00:05:36.110
times the height, which
is the same, times h.

00:05:36.110 --> 00:05:38.340
Now we know what base times,

00:05:38.340 --> 00:05:40.070
we know what b times h is.

00:05:40.070 --> 00:05:42.120
Where b is the area of figure,

00:05:42.120 --> 00:05:43.860
the base area of figure A.

00:05:43.860 --> 00:05:47.010
We know that b times h
is 28 cubic centimeters.

00:05:47.010 --> 00:05:51.410
So the volume for figure
E is going to be 2.25

00:05:52.360 --> 00:05:54.890
times 28 cubic centimeters,

00:05:54.890 --> 00:05:58.370
times 28 cubic centimeters.

00:05:58.370 --> 00:06:00.370
And I don't have a calculator
here in front of me

00:06:00.370 --> 00:06:01.550
and I can do it by hand,

00:06:01.550 --> 00:06:02.840
but I think you get the general point.

00:06:02.840 --> 00:06:04.670
You just have to multiply 2.25

00:06:04.670 --> 00:06:07.280
times 28 to get the cubic,

00:06:07.280 --> 00:06:09.210
you get the volume of figure E.

00:06:09.210 --> 00:06:11.080
And that's because its
base has been scaled

00:06:11.080 --> 00:06:14.953
in each dimension by 1.5.


Volumes of cones intuition

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:01.370
- [Instructor] So I have two different

00:00:01.370 --> 00:00:02.750
three-dimensional figures here.

00:00:02.750 --> 00:00:04.690
I have a pyramid here on the left,

00:00:04.690 --> 00:00:07.820
and I have a cone here on the right.

00:00:07.820 --> 00:00:10.600
And we know a few things
about these two figures.

00:00:10.600 --> 00:00:13.190
First of all, they have
the exact same height.

00:00:13.190 --> 00:00:16.240
So this length right over here is h,

00:00:16.240 --> 00:00:18.290
and this length right over here,

00:00:18.290 --> 00:00:20.380
going from the peak to the
center of the base here,

00:00:20.380 --> 00:00:21.970
is h as well.

00:00:21.970 --> 00:00:26.690
We also know that the area
of the bases is the same.

00:00:26.690 --> 00:00:29.290
So for example, in this left pyramid,

00:00:29.290 --> 00:00:31.540
the area of the base would be x times,

00:00:31.540 --> 00:00:34.420
and let's just assume that it is a square,

00:00:34.420 --> 00:00:35.880
so x times x.

00:00:35.880 --> 00:00:39.200
So the area here is going
to be equal to x squared.

00:00:39.200 --> 00:00:40.880
And the area of the base,

00:00:40.880 --> 00:00:44.570
so that's area of this
base is equal to x squared.

00:00:44.570 --> 00:00:48.570
And the area of this base right
over here would be equal to

00:00:48.570 --> 00:00:52.950
area is equal to pi times r squared.

00:00:52.950 --> 00:00:54.780
And I'm saying that these
two things are the same.

00:00:54.780 --> 00:00:59.280
So we also know that x squared
is equal to pi r squared.

00:00:59.280 --> 00:01:03.070
Now, my question to you
is do these two figures

00:01:03.070 --> 00:01:05.520
have the same volume, or is it different?

00:01:05.520 --> 00:01:08.040
And if they are different,
which one has a larger volume?

00:01:08.040 --> 00:01:10.340
Pause this video, and
try to think about that.

00:01:11.610 --> 00:01:13.450
All right, now let's do this together.

00:01:13.450 --> 00:01:15.750
Now, given that we're
talking about two figures

00:01:15.750 --> 00:01:18.720
that have the same height
and at least the area

00:01:18.720 --> 00:01:22.060
of the base is the same,
you might be thinking

00:01:22.060 --> 00:01:24.230
that Cavalieri's
principle might be useful.

00:01:24.230 --> 00:01:25.760
And just a reminder of what that is,

00:01:25.760 --> 00:01:28.680
Cavalieri's principle tells us
that if you have two figures,

00:01:28.680 --> 00:01:29.960
and we're thinking in three dimension,

00:01:29.960 --> 00:01:32.170
three-dimensional version
of Cavalieri's principle,

00:01:32.170 --> 00:01:34.410
if you have two figures
that have the same height

00:01:34.410 --> 00:01:36.830
and at any point along that height

00:01:36.830 --> 00:01:40.530
the cross-sectional area is the same,

00:01:40.530 --> 00:01:43.340
then the figures have the same volume.

00:01:43.340 --> 00:01:45.830
So what we need to do
is figure out is it true

00:01:45.830 --> 00:01:47.420
that at any point in this height

00:01:47.420 --> 00:01:51.570
do these figures have the
same cross-sectional area?

00:01:51.570 --> 00:01:53.120
Well, to think about that,

00:01:53.120 --> 00:01:55.500
let's pick an arbitrary
point along this height.

00:01:55.500 --> 00:01:56.430
And just for simplicity,

00:01:56.430 --> 00:01:58.060
let's pick halfway along the height,

00:01:58.060 --> 00:01:59.140
although we could do this analysis

00:01:59.140 --> 00:02:00.780
at any point along the height.

00:02:00.780 --> 00:02:03.500
So halfway along the height there.

00:02:03.500 --> 00:02:05.230
Halfway along the height there.

00:02:05.230 --> 00:02:07.970
So this distance right over here,

00:02:07.970 --> 00:02:11.000
that would be h over two.

00:02:11.000 --> 00:02:15.370
This distance right over
here would be h over two.

00:02:15.370 --> 00:02:17.960
This whole thing is h.

00:02:17.960 --> 00:02:20.080
And what we can do is construct

00:02:20.080 --> 00:02:22.350
what look like similar triangles,

00:02:22.350 --> 00:02:24.290
and we can even prove it to ourselves

00:02:24.290 --> 00:02:27.220
that these are similar triangles.

00:02:27.220 --> 00:02:31.423
So let me construct them right over here.

00:02:33.020 --> 00:02:34.620
And the reason why we
know they're similar is

00:02:34.620 --> 00:02:38.170
that this line is going to
be parallel to this line

00:02:38.170 --> 00:02:42.240
and that this line is parallel
to that line, to that radius.

00:02:42.240 --> 00:02:43.410
And how do we know that?

00:02:43.410 --> 00:02:46.430
Well, we're taking cross-sectional areas

00:02:46.430 --> 00:02:48.710
that are parallel to the base,

00:02:48.710 --> 00:02:50.310
that are parallel to the surface

00:02:50.310 --> 00:02:52.820
on which it sits in this situation.

00:02:52.820 --> 00:02:54.253
So in either case,

00:02:55.324 --> 00:02:57.420
these cross-sections are
going to be parallel.

00:02:57.420 --> 00:03:00.020
So these lines, which sit
in these cross-sections

00:03:00.020 --> 00:03:01.820
or sit on the base and
sit in the cross-section,

00:03:01.820 --> 00:03:03.190
have to be parallel as well.

00:03:03.190 --> 00:03:05.250
Well, because these are parallel lines,

00:03:05.250 --> 00:03:08.370
this angle is congruent to that angle.

00:03:08.370 --> 00:03:11.330
This angle is congruent to this angle.

00:03:11.330 --> 00:03:13.730
'Cause these are transversals
across parallel lines,

00:03:13.730 --> 00:03:15.060
and these are just corresponding angles.

00:03:15.060 --> 00:03:17.450
And of course, they share
this angle in common.

00:03:17.450 --> 00:03:20.430
And here, you see very clearly
right angle, right angle.

00:03:20.430 --> 00:03:22.990
This angle is congruent to that angle,

00:03:22.990 --> 00:03:24.800
and then both triangles share that.

00:03:24.800 --> 00:03:28.480
And so this smaller triangle
in either case is similar

00:03:28.480 --> 00:03:30.640
to the larger triangle.

00:03:30.640 --> 00:03:33.040
And what that helps us
realize is that the ratio

00:03:33.040 --> 00:03:35.510
between corresponding sides
is going to be the same.

00:03:35.510 --> 00:03:38.280
So if this side is h over two

00:03:38.280 --> 00:03:39.790
and the entire height is h,

00:03:39.790 --> 00:03:41.630
so this is half of the entire height,

00:03:41.630 --> 00:03:44.580
that tells us that this side
is going to be half of r.

00:03:44.580 --> 00:03:47.930
So this right over here
is going to be r over two.

00:03:47.930 --> 00:03:50.270
And this side over here,
by the same argument,

00:03:50.270 --> 00:03:53.040
is going to be x over two.

00:03:53.040 --> 00:03:56.320
And so what's the
cross-sectional area here?

00:03:56.320 --> 00:03:58.590
Well, it's going to be x over two squared.

00:03:58.590 --> 00:04:01.460
So it's going to be x over two squared,

00:04:01.460 --> 00:04:04.750
which is equal to x squared over four,

00:04:04.750 --> 00:04:07.600
which is 1/4 of the base's area,

00:04:07.600 --> 00:04:11.090
which is equal to 1/4 of the base's area.

00:04:11.090 --> 00:04:12.440
And what about over here?

00:04:12.440 --> 00:04:17.220
Well, this cross-sectional
area is going to be pi

00:04:17.220 --> 00:04:20.160
times r over two squared,

00:04:20.160 --> 00:04:23.380
which is the same thing
as pi r squared over four,

00:04:23.380 --> 00:04:26.800
or we could say that is 1/4 pi r squared,

00:04:26.800 --> 00:04:30.030
which is the same thing as
1/4 of the area of the base.

00:04:30.030 --> 00:04:31.500
The area of the base is pi r squared.

00:04:31.500 --> 00:04:33.010
Now we're saying 1/4 pi r squared.

00:04:33.010 --> 00:04:36.270
So this is going to be
equal to 1/4 the area.

00:04:36.270 --> 00:04:38.490
And we already said that
these areas are the same,

00:04:38.490 --> 00:04:41.360
and so we've just seen that
the cross-sectional area

00:04:41.360 --> 00:04:42.560
at that point of the height

00:04:42.560 --> 00:04:44.240
of both of these figures is the same.

00:04:44.240 --> 00:04:46.480
And you could do that
1/4 along the height,

00:04:46.480 --> 00:04:47.580
3/4 along the height.

00:04:47.580 --> 00:04:49.650
You're going to get the
same exact analysis.

00:04:49.650 --> 00:04:51.260
You're gonna have two similar triangles,

00:04:51.260 --> 00:04:54.350
and you're going to see that
you have the same areas,

00:04:54.350 --> 00:04:57.490
same cross-sectional areas
at that point of the height.

00:04:57.490 --> 00:05:00.390
And so therefore, we see
by Cavalieri's principle

00:05:01.300 --> 00:05:03.510
in three dimensions that these two figures

00:05:03.510 --> 00:05:06.000
have the same volume.

00:05:06.000 --> 00:05:08.610
And what's interesting
about that is it allows us

00:05:08.610 --> 00:05:10.990
to take the formula, which we've proven

00:05:10.990 --> 00:05:12.930
and gotten the intuition
for in other videos,

00:05:12.930 --> 00:05:14.960
for the volume of a pyramid.

00:05:14.960 --> 00:05:17.860
We've learned that the volume
of a pyramid is equal to 1/3

00:05:17.860 --> 00:05:20.760
times base times height.

00:05:20.760 --> 00:05:24.970
And say, well, this one must
have the exact same volume.

00:05:24.970 --> 00:05:27.380
It must also be volume is equal to 1/3

00:05:27.380 --> 00:05:31.140
times the area of the
base times the height.

00:05:31.140 --> 00:05:33.420
Because in both of these cases,

00:05:33.420 --> 00:05:35.670
the area of the base is the same,

00:05:35.670 --> 00:05:37.300
and the height is the same.

00:05:37.300 --> 00:05:40.493
And we know that they
have the same volume.


Citing evidence in literary analysis

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:01.620
- [Instructor] Hello readers.

00:00:01.620 --> 00:00:04.630
The following video
contains explicit content.

00:00:04.630 --> 00:00:06.560
Uh, okay, not in the way you're thinking.

00:00:06.560 --> 00:00:09.940
It doesn't contain violence,
obscenity or profanity.

00:00:09.940 --> 00:00:12.750
Or even anything that wouldn't
appear in a G-rated movie,

00:00:12.750 --> 00:00:15.950
but it will contain explicit evidence.

00:00:15.950 --> 00:00:17.980
Yes, we're talking about citing evidence

00:00:17.980 --> 00:00:19.730
in literary analysis.

00:00:19.730 --> 00:00:20.840
When you're talking about a text

00:00:20.840 --> 00:00:22.610
and making arguments about it,

00:00:22.610 --> 00:00:25.040
in order to successfully
build that argument,

00:00:25.040 --> 00:00:27.900
you must make inferences
and draw conclusions.

00:00:27.900 --> 00:00:29.420
And those must be built

00:00:29.420 --> 00:00:31.260
on the back of evidence.

00:00:31.260 --> 00:00:33.960
Both explicit, that is stated in the text,

00:00:33.960 --> 00:00:38.060
or implicit or based on clues
or evidence in the text.

00:00:38.060 --> 00:00:39.680
So your responsibility is to tie

00:00:39.680 --> 00:00:42.760
those conclusions or
inferences back to explicit

00:00:42.760 --> 00:00:45.200
or implicit evidence in the text.

00:00:45.200 --> 00:00:46.277
It can't just be,

00:00:46.277 --> 00:00:47.750
"This is a feeling I have."

00:00:47.750 --> 00:00:51.060
It has to be, "My feeling
about this is backed up

00:00:51.060 --> 00:00:53.110
by this specific evidence."

00:00:53.110 --> 00:00:54.620
Say you're trying to be make an argument

00:00:54.620 --> 00:00:58.500
in a book that the captain
of this pirate ship,

00:00:58.500 --> 00:01:00.920
let's just say, I guess, he's a birdman.

00:01:00.920 --> 00:01:02.700
That's what I drew, he's a birdman.

00:01:02.700 --> 00:01:04.930
He's really unkind to the main character

00:01:04.930 --> 00:01:06.280
in the beginning of the book,

00:01:06.280 --> 00:01:08.200
but changes by the end of the book

00:01:08.200 --> 00:01:10.370
and treats everyone,
including the main character

00:01:10.370 --> 00:01:12.530
with respect and courtesy.

00:01:12.530 --> 00:01:13.680
So I'm gonna write that my argument

00:01:13.680 --> 00:01:15.530
is that the captain's behavior changes

00:01:15.530 --> 00:01:17.560
towards the MC, the main character.

00:01:17.560 --> 00:01:20.200
I have to back that
assertion up with evidence.

00:01:20.200 --> 00:01:22.530
So how do we find those details?

00:01:22.530 --> 00:01:24.230
First, you have to seek out parts

00:01:24.230 --> 00:01:25.510
of the book where the captain

00:01:25.510 --> 00:01:27.640
and the main character interact.

00:01:27.640 --> 00:01:31.130
Then look closely at
the pros and dialogue.

00:01:31.130 --> 00:01:33.520
What are the details
that prove your point?

00:01:33.520 --> 00:01:35.970
Which are the strongest,
most specific details

00:01:35.970 --> 00:01:37.980
that say, "Oh, yes, here is where

00:01:37.980 --> 00:01:39.320
the captain is being mean.

00:01:39.320 --> 00:01:42.210
Here is where the captain
is being respectful."

00:01:42.210 --> 00:01:45.540
If you can't find evidence
for your assertion,

00:01:45.540 --> 00:01:48.450
first, try searching in a
different part of the book.

00:01:48.450 --> 00:01:51.890
Or importantly,
acknowledge the possibility

00:01:51.890 --> 00:01:54.050
that you might have a weak argument.

00:01:54.050 --> 00:01:55.400
Maybe it's time to start over

00:01:55.400 --> 00:01:58.050
and find a new or
different argument to make

00:01:58.050 --> 00:01:59.880
and find support for.

00:01:59.880 --> 00:02:01.470
Once you've assembled your evidence

00:02:01.470 --> 00:02:03.387
work it into your analysis.

00:02:03.387 --> 00:02:05.160
"Captain Bigsby is dismissive

00:02:05.160 --> 00:02:07.540
and rude to Eniola when she first joins

00:02:07.540 --> 00:02:10.870
the crew of the pirate ship
Albatross," I might say.

00:02:10.870 --> 00:02:12.650
And then back it up with an explicit

00:02:12.650 --> 00:02:15.010
example of Captain Bigsby being dismissive

00:02:15.010 --> 00:02:18.640
and rude with a page
number citation like so.

00:02:18.640 --> 00:02:21.070
On page 34 of "To the Burbling Deep,"

00:02:21.070 --> 00:02:24.230
Bigsby says to Eniola, "You
there, what's her name!

00:02:24.230 --> 00:02:26.880
These portholes need to
be scrubbed yesterday.

00:02:26.880 --> 00:02:27.887
Get to work!."

00:02:27.887 --> 00:02:29.837
"Yes, sir, Captain," Eniola said.

00:02:29.837 --> 00:02:30.687
"But my name is-"

00:02:30.687 --> 00:02:34.630
"Does it look like I
care?" the captain snarled.

00:02:34.630 --> 00:02:35.780
But by the end of the story,

00:02:35.780 --> 00:02:37.680
when Eniola has proven her worth,

00:02:37.680 --> 00:02:39.720
saved the day, and humbled the captain,

00:02:39.720 --> 00:02:42.080
he treats her, and everyone around him,

00:02:42.080 --> 00:02:45.230
with much greater respect and deference.

00:02:45.230 --> 00:02:48.960
On page 225, after Bigsby
tends to Eniola's broken leg,

00:02:48.960 --> 00:02:51.450
he tells her, "Eniola, that was some

00:02:51.450 --> 00:02:53.630
mighty fine pirating you did.

00:02:53.630 --> 00:02:56.200
You showed no fear when
you punched that kraken,

00:02:56.200 --> 00:02:58.420
and more importantly, you showed no fear

00:02:58.420 --> 00:03:00.090
when you stood up to me.

00:03:00.090 --> 00:03:04.410
I was wrong, and I have been
a jerk, and I am sorry."

00:03:04.410 --> 00:03:06.630
Now both of those examples
use explicit evidence

00:03:06.630 --> 00:03:08.510
where Captain Bigsby is being a jerk

00:03:08.510 --> 00:03:11.540
and then when he is
apologizing for being a jerk.

00:03:11.540 --> 00:03:13.755
There's also implicit evidence too.

00:03:13.755 --> 00:03:16.300
Bigsby tends to Eniola's broken leg.

00:03:16.300 --> 00:03:17.307
So while he's not saying,

00:03:17.307 --> 00:03:19.960
"I will take care of you
little buddy," with his words,

00:03:19.960 --> 00:03:22.890
he is saying it with his actions.

00:03:22.890 --> 00:03:24.090
And in that first example

00:03:24.090 --> 00:03:26.300
when he demands that
Eniola clean the portholes,

00:03:26.300 --> 00:03:29.050
he's not literally
saying, "I'm impatient,"

00:03:29.050 --> 00:03:31.610
But by saying, "They need
to be scrubbed yesterday,"

00:03:31.610 --> 00:03:33.960
as an immediately because
you already messed up,

00:03:33.960 --> 00:03:36.920
he's implying that he's impatient.

00:03:36.920 --> 00:03:38.610
Now there is no such book that I know

00:03:38.610 --> 00:03:41.000
of about a kraken punching girl pirate.

00:03:41.000 --> 00:03:44.700
But if we're lucky, my
friend Jordan will write one.

00:03:44.700 --> 00:03:48.690
Remember to use strong
details to get good evidence.

00:03:48.690 --> 00:03:50.670
There might be a part of
"To The Burbling Deep"

00:03:50.670 --> 00:03:53.400
where Captain Bigsby huffs angrily

00:03:53.400 --> 00:03:55.500
at Eniola but doesn't say anything,

00:03:55.500 --> 00:03:57.780
and that's mean or, at least, impolite,

00:03:57.780 --> 00:04:01.170
but it's not as strong
as him yelling at her.

00:04:01.170 --> 00:04:03.290
So when you find the detail, ask yourself,

00:04:03.290 --> 00:04:05.320
how does that detail related back

00:04:05.320 --> 00:04:07.173
to your analysis or your argument.

00:04:08.530 --> 00:04:09.700
Is it repeated?

00:04:09.700 --> 00:04:12.600
Does that detail or detail similar to it

00:04:12.600 --> 00:04:14.650
appear elsewhere in the text?

00:04:14.650 --> 00:04:17.130
And if you see a lot of similar details,

00:04:17.130 --> 00:04:19.450
how do those details prove the argument

00:04:19.450 --> 00:04:20.730
that you're trying to make?

00:04:20.730 --> 00:04:22.600
If you have a sense of
what the central idea

00:04:22.600 --> 00:04:24.920
or ideas of the text is,

00:04:24.920 --> 00:04:26.790
try to connect those details

00:04:26.790 --> 00:04:28.730
back to that central idea and then connect

00:04:28.730 --> 00:04:32.090
that central idea to your own argument.

00:04:32.090 --> 00:04:33.780
I don't know what "To the Burbling Deep"

00:04:33.780 --> 00:04:36.680
is about 'cause I made
it up 20 minutes ago.

00:04:36.680 --> 00:04:38.380
But maybe one theme in it

00:04:38.380 --> 00:04:40.850
is that it's important to recognize

00:04:40.850 --> 00:04:42.720
the potential with another people.

00:04:42.720 --> 00:04:45.620
And that can be both true for Eniola

00:04:45.620 --> 00:04:48.870
who becomes a hero and punches
a giant octopus monster,

00:04:48.870 --> 00:04:51.360
but it can also be true of Captain Bigsby

00:04:51.360 --> 00:04:53.780
who occupies kind of antagonistic role

00:04:53.780 --> 00:04:55.810
and then changes through the story.

00:04:55.810 --> 00:04:58.970
And so we can build our
argument around that idea

00:04:58.970 --> 00:05:01.140
that character change is possible.

00:05:01.140 --> 00:05:02.450
Not just for the main character,

00:05:02.450 --> 00:05:03.730
but for everybody.

00:05:03.730 --> 00:05:05.300
So that's where I will leave you.

00:05:05.300 --> 00:05:07.100
Remember to choose the pieces of evidence

00:05:07.100 --> 00:05:08.810
that give you the strongest support

00:05:08.810 --> 00:05:10.310
for your idea and if the evidence

00:05:10.310 --> 00:05:11.930
doesn't match your idea,

00:05:11.930 --> 00:05:14.233
you might need to change the idea itself.

00:05:15.090 --> 00:05:17.523
You can learn anything.
David out.


Part-to-whole relationships in text structure

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:01.520
- [David] Hello readers.

00:00:01.520 --> 00:00:02.910
Today we're going to be talking

00:00:02.910 --> 00:00:05.950
about how smaller sections
of text work together

00:00:05.950 --> 00:00:07.700
to support the whole text.

00:00:07.700 --> 00:00:10.660
But first let us consider Voltron.

00:00:10.660 --> 00:00:13.950
It is a giant robot made
up of five smaller robots,

00:00:13.950 --> 00:00:16.290
each one piloted by a person.

00:00:16.290 --> 00:00:18.350
Five friends, each with control

00:00:18.350 --> 00:00:20.410
over a different part of the robot's body.

00:00:20.410 --> 00:00:23.100
You're up in the head, I'm
operating the right leg,

00:00:23.100 --> 00:00:25.960
our buddy's operating the left, and so on.

00:00:25.960 --> 00:00:28.240
In order to walk or pick up objects

00:00:28.240 --> 00:00:30.830
or fight space monsters
the size of battleships,

00:00:30.830 --> 00:00:34.930
all the disparate of the robot
have to function together.

00:00:34.930 --> 00:00:38.240
Well, texts work much the same way.

00:00:38.240 --> 00:00:39.570
Each portion of the text,

00:00:39.570 --> 00:00:42.390
from the section level down
to more granular divisions

00:00:42.390 --> 00:00:45.190
like individual paragraphs or sentences,

00:00:45.190 --> 00:00:49.380
is trying to serve the
broader point of the text.

00:00:49.380 --> 00:00:51.770
When we look at Voltron's foot in motion,

00:00:51.770 --> 00:00:53.430
we analyze what the foot is doing

00:00:53.430 --> 00:00:56.040
in the service of Voltron as a whole.

00:00:56.040 --> 00:00:57.930
If Voltron's foot kicks a ball,

00:00:57.930 --> 00:01:01.600
we have to zoom out to ask,
is Voltron playing soccer?

00:01:01.600 --> 00:01:05.260
This is a pretty abstract idea
without any text examples,

00:01:05.260 --> 00:01:08.000
so let's stop talking about space robots

00:01:08.000 --> 00:01:10.390
and start talking about the Armor

00:01:10.390 --> 00:01:13.170
of 10th-century Japanese horse archers.

00:01:13.170 --> 00:01:16.010
All right, so this whole
passage is about the armor

00:01:16.010 --> 00:01:19.950
worn by mounted archer
samurai of 10th-century Japan,

00:01:19.950 --> 00:01:22.450
o-yoroi, and how it's constructed.

00:01:22.450 --> 00:01:24.370
So this is a piece of o-yoroi armor.

00:01:24.370 --> 00:01:25.940
This is the whole assemblage.

00:01:25.940 --> 00:01:29.720
And that's how the
piece is divided, right?

00:01:29.720 --> 00:01:32.290
We've got the sections
here for each component.

00:01:32.290 --> 00:01:36.270
The cuirass, the o-sode
or shoulder guards,

00:01:36.270 --> 00:01:38.600
the kusazuri, or armored skirt,

00:01:38.600 --> 00:01:40.050
and the kabuto, the helmet.

00:01:40.050 --> 00:01:41.890
And each one of these
elements of the armor

00:01:41.890 --> 00:01:44.940
connects back to the
whole point of the piece,

00:01:44.940 --> 00:01:47.710
which is describing how
the armor does two things.

00:01:47.710 --> 00:01:50.530
One, it protects the wearer, and two,

00:01:50.530 --> 00:01:53.600
allows them to ride
horses and fire arrows.

00:01:53.600 --> 00:01:56.160
Now we see this in the
introductory paragraph.

00:01:56.160 --> 00:01:59.620
Their armor, called o-yoroi,
was designed specifically

00:01:59.620 --> 00:02:02.770
to withstand the demands
of mounted archer warfare.

00:02:02.770 --> 00:02:05.520
So let's dig into this first
section about the cuirass,

00:02:05.520 --> 00:02:08.130
the breastplate and backplate armor.

00:02:08.130 --> 00:02:09.877
I'll read the first paragraph.

00:02:09.877 --> 00:02:12.457
"A cuirass is usually a piece of armor

00:02:12.457 --> 00:02:14.997
"that consists of a
breastplate and a backplate

00:02:14.997 --> 00:02:16.517
"that are attached together.

00:02:16.517 --> 00:02:19.287
"The cuirass for the
o-yoroi was a bit different

00:02:19.287 --> 00:02:20.697
"than a typical cuirass,

00:02:20.697 --> 00:02:24.047
"as it was designed to be
more effective for an archer.

00:02:24.047 --> 00:02:27.967
"The o-yoroi cuirass had
three sections instead of two.

00:02:27.967 --> 00:02:29.627
"A section to protect the back,

00:02:29.627 --> 00:02:31.137
"a section to protect the chest,

00:02:31.137 --> 00:02:34.077
"and a section to protect
the left side body.

00:02:34.077 --> 00:02:35.517
"The right side was left open

00:02:35.517 --> 00:02:38.770
"so the warrior could best
utilize his bow and arrow."

00:02:38.770 --> 00:02:42.640
So this paragraph is
describing what a cuirass is

00:02:42.640 --> 00:02:44.010
and how it functions as armor,

00:02:44.010 --> 00:02:45.330
but then it goes into detailing

00:02:45.330 --> 00:02:47.520
how an o-yoroi cuirass is different

00:02:47.520 --> 00:02:49.980
because it's for someone
holding a bow, right?

00:02:49.980 --> 00:02:52.410
It has three sections instead of two.

00:02:52.410 --> 00:02:54.520
The right side was left
open so the warrior

00:02:54.520 --> 00:02:57.160
could best utilize his bow and arrow.

00:02:57.160 --> 00:02:59.500
This information down
to the sentence level

00:02:59.500 --> 00:03:01.560
all serves the same purpose.

00:03:01.560 --> 00:03:03.200
It serves to answer the question,

00:03:03.200 --> 00:03:07.000
how was o-yoroi armor
specifically designed

00:03:07.000 --> 00:03:09.420
to meet the needs of a mounted archer?

00:03:09.420 --> 00:03:11.440
Not every sentence or every paragraph

00:03:11.440 --> 00:03:14.080
will address every part
of that main question.

00:03:14.080 --> 00:03:15.930
Note that the cuirass
section doesn't cover

00:03:15.930 --> 00:03:19.120
anything about how the
archers are on horseback.

00:03:19.120 --> 00:03:20.110
To answer that question,

00:03:20.110 --> 00:03:22.060
we need to go down to the third section,

00:03:22.060 --> 00:03:24.360
the kusazuri, the battle skirt.

00:03:24.360 --> 00:03:25.760
I'm not gonna read the whole paragraph.

00:03:25.760 --> 00:03:28.597
Let's just zoom in on this one sentence.

00:03:28.597 --> 00:03:30.867
"The kusazuri were designed
so that when the warrior

00:03:30.867 --> 00:03:32.807
"was sitting in his horse's saddle,

00:03:32.807 --> 00:03:35.747
"they fit nicely over the
saddle in a skirtlike fashion

00:03:35.747 --> 00:03:38.800
"to protect his lower
body and upper legs."

00:03:38.800 --> 00:03:40.410
This particular sentence tells us

00:03:40.410 --> 00:03:43.160
how this part of the armor
protects the samurai's legs

00:03:43.160 --> 00:03:44.780
while they're on horseback.

00:03:44.780 --> 00:03:46.770
The paragraph and section it's part of

00:03:46.770 --> 00:03:49.570
detail the whole construction
of that part of the armor

00:03:49.570 --> 00:03:52.050
and how it relates to the other pieces.

00:03:52.050 --> 00:03:54.830
And all of those paragraphs
together form one text

00:03:54.830 --> 00:03:57.550
that explained the thing
it set out to explain.

00:03:57.550 --> 00:03:59.620
What is o-yoroi and how does it work

00:03:59.620 --> 00:04:02.560
for 10th-century horse archer samurai?

00:04:02.560 --> 00:04:05.250
Each section or paragraph is like a leg

00:04:05.250 --> 00:04:07.540
or an arm of Voltron.

00:04:07.540 --> 00:04:11.730
Each sentence is a muscle
or a finger or a robo-toe.

00:04:11.730 --> 00:04:14.280
Together, moving as one, they tell a story

00:04:14.280 --> 00:04:17.320
or make an argument, which
I guess is like Voltron

00:04:17.320 --> 00:04:19.130
winning a fight, maybe.

00:04:19.130 --> 00:04:21.630
I feel like I've carried this
metaphor as far as it'll go.

00:04:21.630 --> 00:04:23.520
The point is, if you're having trouble

00:04:23.520 --> 00:04:25.800
making sense of an informational text,

00:04:25.800 --> 00:04:28.233
think of it as a giant battle robot.

00:04:29.160 --> 00:04:31.000
What is each of component part of the text

00:04:31.000 --> 00:04:32.040
trying to accomplish?

00:04:32.040 --> 00:04:34.050
And when you put all those parts together,

00:04:34.050 --> 00:04:35.400
what are they trying to do?

00:04:36.320 --> 00:04:38.560
Well, my work here is done.

00:04:38.560 --> 00:04:40.050
You can learn anything.

00:04:40.050 --> 00:04:40.883
David out.


Cavalieri's principle in 3D

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:01.970
- [Instructor] So we
have two cylinders here.

00:00:01.970 --> 00:00:04.890
Let's say we know that they
have the exact same volume

00:00:04.890 --> 00:00:06.550
and that makes sense because it looks like

00:00:06.550 --> 00:00:09.750
they have the same area of their base

00:00:09.750 --> 00:00:12.040
and they have the same height.

00:00:12.040 --> 00:00:14.250
Now what I'm going to
do is start cutting up

00:00:14.250 --> 00:00:17.860
this left cylinder here
and shifting things around.

00:00:17.860 --> 00:00:21.390
So if I just cut it in two
and take that bottom cylinder,

00:00:21.390 --> 00:00:23.640
that bottom half and shift it a bit,

00:00:23.640 --> 00:00:25.423
have I changed its volume?

00:00:26.420 --> 00:00:28.890
Well, clearly I have
not changed its volume.

00:00:28.890 --> 00:00:30.190
I still have the same volume.

00:00:30.190 --> 00:00:33.730
The combined volume of both
of these half cylinders,

00:00:33.730 --> 00:00:37.720
I could say, are equal
to the original cylinder.

00:00:37.720 --> 00:00:40.290
Now what if I were to cut it up even more?

00:00:40.290 --> 00:00:43.610
So let me cut it up now into three.

00:00:43.610 --> 00:00:46.880
Well, once again I still haven't
changed my original volume.

00:00:46.880 --> 00:00:48.560
It's still the same volume as original

00:00:48.560 --> 00:00:50.250
and I just cut it up into thirds.

00:00:50.250 --> 00:00:51.950
And if I shift them around a little bit

00:00:51.950 --> 00:00:53.900
I'm not changing the volume.

00:00:53.900 --> 00:00:55.270
And I could keep doing that.

00:00:55.270 --> 00:00:57.210
I could cut it up into a bunch of them.

00:00:57.210 --> 00:01:01.530
Notice, this still has
the same original volume,

00:01:01.530 --> 00:01:04.420
I've just cut it up into
a bunch of sections.

00:01:04.420 --> 00:01:05.880
I've cut it horizontally

00:01:05.880 --> 00:01:08.010
and now I'm just shifting things around,

00:01:08.010 --> 00:01:10.080
but that doesn't change the volume.

00:01:10.080 --> 00:01:12.650
And I can do it a bunch of times.

00:01:12.650 --> 00:01:15.670
This looks like some type of
poker chips or gambling chips

00:01:15.670 --> 00:01:17.920
where I can have my original cylinder

00:01:17.920 --> 00:01:20.250
and now I've cut it horizontally

00:01:20.250 --> 00:01:22.370
into a bunch of these, I
guess you could say chips

00:01:22.370 --> 00:01:24.550
but clearly it has the
same combined volume.

00:01:24.550 --> 00:01:28.380
I can shift it around a bit
but it has the same volume.

00:01:28.380 --> 00:01:30.710
And this leads us to
an interesting question

00:01:30.710 --> 00:01:31.890
and it's actually a principle

00:01:31.890 --> 00:01:34.260
known as Cavalieri's principle,

00:01:34.260 --> 00:01:39.260
which is if I have two figures
that have the same height

00:01:39.290 --> 00:01:42.240
and at any point along that height,

00:01:42.240 --> 00:01:47.100
the cross-sectional area is the same,

00:01:47.100 --> 00:01:51.260
then the two figures have the same volume.

00:01:51.260 --> 00:01:53.470
Now how does what I just say
apply to what's going on here?

00:01:53.470 --> 00:01:57.860
Well, clearly both of these
figures have the same height

00:01:57.860 --> 00:02:01.317
and then at any point here,
wherever I did the cuts,

00:02:01.317 --> 00:02:04.200
at the same point on
this original cylinder,

00:02:04.200 --> 00:02:08.040
well, my cross-sectional
area is going to be the same

00:02:08.040 --> 00:02:09.650
because it's going to be the same area

00:02:09.650 --> 00:02:12.000
as the base in the case of this cylinder

00:02:12.000 --> 00:02:14.340
and so it meets Cavalieri's principle.

00:02:14.340 --> 00:02:16.380
But Cavalieri's
principle's nothing exotic.

00:02:16.380 --> 00:02:18.020
It comes straight out of common sense.

00:02:18.020 --> 00:02:20.510
I can just do more cuts like this

00:02:20.510 --> 00:02:22.080
and you can see that I have,

00:02:22.080 --> 00:02:25.920
you can see a more continuous
looking skewed cylinder

00:02:25.920 --> 00:02:30.100
but this will have the same
volume as our original cylinder.

00:02:30.100 --> 00:02:32.120
When I shift it around like this,

00:02:32.120 --> 00:02:33.910
it's not changing the volume.

00:02:33.910 --> 00:02:37.980
And that's not just true for cylinders.

00:02:37.980 --> 00:02:40.390
I could do the exact same argument

00:02:40.390 --> 00:02:43.010
with some form of a prism.

00:02:43.010 --> 00:02:44.730
Once again they have the same volume.

00:02:44.730 --> 00:02:46.800
I could shift, I could
cut the left one in half

00:02:46.800 --> 00:02:49.100
and shift it around,
doesn't change its volume.

00:02:49.100 --> 00:02:51.530
I could cut it more
and shift those around,

00:02:51.530 --> 00:02:53.480
still doesn't change the volume.

00:02:53.480 --> 00:02:56.150
So Cavalieri's principle
seems to make a lot

00:02:56.150 --> 00:02:57.550
of intuitive sense here.

00:02:57.550 --> 00:03:01.160
If I have two figures
that have the same height

00:03:01.160 --> 00:03:04.290
and at any point along that height,

00:03:04.290 --> 00:03:07.950
the cross-sectional area is the same,

00:03:07.950 --> 00:03:10.900
then the figures have the same volume.

00:03:10.900 --> 00:03:14.580
So these figures also
have the same volume.

00:03:14.580 --> 00:03:16.640
And I could do it with interesting things

00:03:16.640 --> 00:03:18.930
like, say, a pyramid.

00:03:18.930 --> 00:03:20.560
These two pyramids have the same volume

00:03:20.560 --> 00:03:22.470
and I were to cut the left pyramid

00:03:22.470 --> 00:03:26.570
halfway along its height and
shift the bottom like this,

00:03:26.570 --> 00:03:28.910
that doesn't change its volume.

00:03:28.910 --> 00:03:32.590
And I can keep doing that
with more and more cuts.

00:03:32.590 --> 00:03:35.100
And 'cause at any point here,

00:03:35.100 --> 00:03:37.360
these figures have the same height

00:03:37.360 --> 00:03:39.660
and at any point on that height,

00:03:39.660 --> 00:03:42.820
the cross-sectional area is the same,

00:03:42.820 --> 00:03:44.350
and so they have the same volume.

00:03:44.350 --> 00:03:47.010
But once again it is intuitive.

00:03:47.010 --> 00:03:49.910
And it goes all the way to
the case where you have,

00:03:49.910 --> 00:03:52.780
you could view it as a continuous
pyramid right over here

00:03:52.780 --> 00:03:54.220
that has been skewed.

00:03:54.220 --> 00:03:55.760
So no matter how much you skew it,

00:03:55.760 --> 00:03:58.250
it's gonna have the same
volume as our original pyramid

00:03:58.250 --> 00:03:59.460
'cause they have the same height.

00:03:59.460 --> 00:04:02.550
And the cross-sectional area
at any point in the height

00:04:02.550 --> 00:04:04.610
is going to be the same.

00:04:04.610 --> 00:04:06.610
We can actually do this with any figure.

00:04:07.810 --> 00:04:10.060
So these spheres have the same volume.

00:04:10.060 --> 00:04:11.790
I could cut the left one in half,

00:04:11.790 --> 00:04:14.550
halfway along its height
and shift it like this.

00:04:14.550 --> 00:04:16.750
Clearly, I'm not changing the volume.

00:04:16.750 --> 00:04:19.530
And I could make more cuts like that.

00:04:19.530 --> 00:04:22.270
And clearly it has still the same volume.

00:04:22.270 --> 00:04:24.270
And this meets Cavalieri's principle

00:04:24.270 --> 00:04:26.330
because they have the same height

00:04:26.330 --> 00:04:30.400
and the cross section at
any point along that height

00:04:30.400 --> 00:04:31.870
is going to be the same.

00:04:31.870 --> 00:04:35.930
So even though I can cut that
one up and I can shift it,

00:04:35.930 --> 00:04:38.870
it looks like a different type of object,

00:04:38.870 --> 00:04:40.280
a different type of thing,

00:04:40.280 --> 00:04:42.610
but they have the same
height and cross sections

00:04:42.610 --> 00:04:45.800
at any point are the same area,

00:04:45.800 --> 00:04:47.980
so we have the same volume,

00:04:47.980 --> 00:04:49.540
which is a useful thing to know

00:04:49.540 --> 00:04:51.100
not just to know the principle

00:04:51.100 --> 00:04:52.880
but hopefully this video helps you gain

00:04:52.880 --> 00:04:55.753
some of the intuition for
why it makes intuitive sense.


Dilating in 3D

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:01.860
- [Instructor] Let's say I
have some type of a surface,

00:00:01.860 --> 00:00:06.860
let's say that this right over
here is the top of your desk.

00:00:07.090 --> 00:00:12.090
And I were to draw a
triangle on that surface.

00:00:12.560 --> 00:00:15.573
So maybe the triangle looks like this,

00:00:16.890 --> 00:00:18.000
something like this,

00:00:18.000 --> 00:00:20.330
it doesn't have to be a right triangle.

00:00:20.330 --> 00:00:21.290
And so I'm not implying

00:00:21.290 --> 00:00:22.750
that this is necessarily a right triangle.

00:00:22.750 --> 00:00:24.670
Although it looks a little bit like one,

00:00:24.670 --> 00:00:29.670
and let's call it
triangle A, B, and then C.

00:00:30.160 --> 00:00:32.650
Now what I'm going to do
is something interesting.

00:00:32.650 --> 00:00:35.220
I'm gonna take a fourth point P

00:00:35.220 --> 00:00:37.720
that's not on the surface of this desk

00:00:37.720 --> 00:00:40.870
and it's going to be right above point B.

00:00:40.870 --> 00:00:42.620
So let me just take that point,

00:00:42.620 --> 00:00:44.250
go straight up,

00:00:44.250 --> 00:00:48.330
and I'm going to get to
point P right over here.

00:00:48.330 --> 00:00:51.080
Now, what I can do is construct a pyramid

00:00:51.080 --> 00:00:54.030
using point P as the peak of that pyramid.

00:00:54.030 --> 00:00:56.010
Now, what we're going
to start thinking about

00:00:56.010 --> 00:01:00.090
is what happens if I take
cross sections of this pyramid?

00:01:00.090 --> 00:01:00.950
So in this case,

00:01:00.950 --> 00:01:05.840
the length of segment PB is
the height of this pyramid.

00:01:05.840 --> 00:01:09.340
Now, if we were to go
halfway along that height,

00:01:09.340 --> 00:01:13.340
and if we were to take a
cross section of this pyramid

00:01:13.340 --> 00:01:16.600
that is parallel to the
surface of our original desk,

00:01:16.600 --> 00:01:18.430
what would that look like?

00:01:18.430 --> 00:01:22.963
Well, it would look something like this.

00:01:24.770 --> 00:01:28.650
Now you might be noticing
something really interesting.

00:01:28.650 --> 00:01:32.070
If you were to translate that
blue triangle straight down

00:01:32.070 --> 00:01:33.693
onto the surface of the table,

00:01:34.630 --> 00:01:38.080
it would look like this.

00:01:38.080 --> 00:01:39.500
And when you see it that way,

00:01:39.500 --> 00:01:43.470
it looks like it is a dilation
of our original triangle

00:01:43.470 --> 00:01:45.230
centered at point B.

00:01:45.230 --> 00:01:48.930
And in fact, it is a
dilation centered at point B

00:01:48.930 --> 00:01:52.500
with a scale factor of 0.5.

00:01:52.500 --> 00:01:54.630
And you can see it right over here,

00:01:54.630 --> 00:01:56.399
this length right over here,

00:01:56.399 --> 00:02:00.430
what BC was dilated down
to is half the length

00:02:00.430 --> 00:02:02.030
of the original BC.

00:02:02.030 --> 00:02:05.170
This is half the length
of the original AB,

00:02:05.170 --> 00:02:10.170
and then this is half the
length of the original AC.

00:02:10.190 --> 00:02:13.720
But you could do it at other
heights along this pyramid.

00:02:13.720 --> 00:02:18.720
What if we were to go 0.75
of the way between P and B.

00:02:19.820 --> 00:02:23.220
So if you were to go right over here.

00:02:23.220 --> 00:02:25.390
So it's closer to our original triangle,

00:02:25.390 --> 00:02:27.410
closer to our surface.

00:02:27.410 --> 00:02:29.360
So then the cross section

00:02:32.717 --> 00:02:34.467
would look like this.

00:02:35.940 --> 00:02:37.580
Now, if we were to translate that down

00:02:37.580 --> 00:02:38.946
onto our original surface,

00:02:38.946 --> 00:02:40.890
what would that look like?

00:02:40.890 --> 00:02:44.320
Well, it would look like this.

00:02:44.320 --> 00:02:45.930
It would look like a dilation

00:02:45.930 --> 00:02:49.960
of our original triangle
centered at point B.

00:02:49.960 --> 00:02:54.750
But this time with a scale factor of 0.75.

00:02:54.750 --> 00:02:56.760
And then what if you were to
go only a quarter of the way

00:02:56.760 --> 00:02:59.540
between point P and point B?

00:02:59.540 --> 00:03:02.330
Well, then you would
see something like this,

00:03:02.330 --> 00:03:03.690
a quarter of the way.

00:03:03.690 --> 00:03:05.070
If you were take the cross section

00:03:05.070 --> 00:03:07.670
parallel to our original surface,

00:03:07.670 --> 00:03:08.930
it would look like this.

00:03:08.930 --> 00:03:12.490
If you were to translate that
straight down onto our table,

00:03:12.490 --> 00:03:14.670
it would look something like this.

00:03:14.670 --> 00:03:16.760
And it looks like a dilation

00:03:16.760 --> 00:03:21.613
centered at point B with
a scale factor of 0.25.

00:03:23.160 --> 00:03:24.650
And the reason why all of these dilations

00:03:24.650 --> 00:03:28.500
look like dilations centered at point B

00:03:28.500 --> 00:03:33.500
is because point P is
directly above point B.

00:03:33.930 --> 00:03:36.700
But this is a way to
conceptualize dilations,

00:03:36.700 --> 00:03:40.080
or see the relationship
between cross sections

00:03:41.420 --> 00:03:43.100
of a three-dimensional shape,

00:03:43.100 --> 00:03:44.596
in this case like a pyramid,

00:03:44.596 --> 00:03:47.710
and how those cross sections relate

00:03:47.710 --> 00:03:50.000
to the base of the pyramid.

00:03:50.000 --> 00:03:51.675
Now, let me ask you an
interesting question.

00:03:51.675 --> 00:03:53.730
What if I were to try
to take a cross section

00:03:53.730 --> 00:03:55.840
right at point P.

00:03:55.840 --> 00:03:58.210
Well, then I would just get a point.

00:03:58.210 --> 00:04:00.490
I would not get an actual triangle,

00:04:00.490 --> 00:04:02.825
but you could view that as a dilation

00:04:02.825 --> 00:04:06.280
with a scale factor of zero.

00:04:06.280 --> 00:04:09.180
And what if I were to take
a cross section at the base?

00:04:09.180 --> 00:04:12.930
Well, then that would be my
original triangle, triangle ABC.

00:04:12.930 --> 00:04:16.166
And then you can view that as a dilation

00:04:16.166 --> 00:04:18.690
with a scale factor of one

00:04:18.690 --> 00:04:21.440
'cause you've gone all
the way down to the base.

00:04:21.440 --> 00:04:24.170
So hopefully this
connects some dots for you

00:04:24.170 --> 00:04:27.310
between cross sections of
a three-dimensional shape

00:04:27.310 --> 00:04:31.923
that is parallel to the base
and notions of dilation.


Calculating angle measures to verify congruence

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:02.080
- [Instructor] We have four
triangles depicted here,

00:00:02.080 --> 00:00:02.913
and they've told us

00:00:02.913 --> 00:00:05.550
that the triangles are not drawn to scale.

00:00:05.550 --> 00:00:10.550
And we are asked, "Which two
triangles must be congruent?"

00:00:10.830 --> 00:00:11.950
So pause this video

00:00:11.950 --> 00:00:13.610
and see if you can work
this out on your own

00:00:13.610 --> 00:00:15.763
before we work through this together.

00:00:17.010 --> 00:00:19.390
All right, now let's work
through this together.

00:00:19.390 --> 00:00:21.350
And it looks like for every one of these,

00:00:21.350 --> 00:00:22.890
or actually almost every one of these,

00:00:22.890 --> 00:00:26.120
they've given us two angles,
and they've given us a side.

00:00:26.120 --> 00:00:30.780
This triangle IJH, they've
only given us two angles.

00:00:30.780 --> 00:00:31.950
So what I'd like to do is

00:00:31.950 --> 00:00:33.900
if I know two angles of a triangle,

00:00:33.900 --> 00:00:35.670
I can figure out the third angle

00:00:35.670 --> 00:00:37.060
because the sum of the angles

00:00:37.060 --> 00:00:40.150
of a triangle have to
add up to 180 degrees.

00:00:40.150 --> 00:00:42.650
And then I can use that information,

00:00:42.650 --> 00:00:44.540
maybe with the sides that they give us,

00:00:44.540 --> 00:00:48.130
in order to judge which of
these triangles are congruent.

00:00:48.130 --> 00:00:50.070
So first of all, what is
going to be the measure

00:00:50.070 --> 00:00:53.700
of this angle right over here,
the measure of angle ACB?

00:00:53.700 --> 00:00:56.430
Pause the video and try
to think about that.

00:00:56.430 --> 00:00:57.450
Well, one way to think about it,

00:00:57.450 --> 00:00:59.890
if we call the measure of that angle X,

00:00:59.890 --> 00:01:03.000
we know that X plus 36

00:01:03.000 --> 00:01:07.130
plus 82 needs to be equals to 180.

00:01:07.130 --> 00:01:09.800
I'm just giving their
measures in degrees here.

00:01:09.800 --> 00:01:11.910
And so you could say x plus.

00:01:11.910 --> 00:01:16.890
Let's see, 36 plus 82 is 118.

00:01:16.890 --> 00:01:18.350
Did I do that right?

00:01:18.350 --> 00:01:22.600
Six plus two is eight, and
then three plus eight is 11.

00:01:22.600 --> 00:01:23.720
Yeah, that's right.

00:01:23.720 --> 00:01:25.470
So that's going to equal to 180.

00:01:25.470 --> 00:01:29.040
And then if I subtract
118 from both sides,

00:01:29.040 --> 00:01:30.950
I'm going to get x is equal

00:01:30.950 --> 00:01:35.430
to 180 minus 118 is 62.

00:01:35.430 --> 00:01:39.700
So this is x is equal to 62,
or this is a 62 degree angle,

00:01:39.700 --> 00:01:41.540
I guess is another way
of thinking about it.

00:01:41.540 --> 00:01:46.200
I could put everything in
terms of degrees if you like.

00:01:46.200 --> 00:01:47.360
All right, now let's do the same thing

00:01:47.360 --> 00:01:49.290
with this one right over here.

00:01:49.290 --> 00:01:51.440
Well, this one has an 82 degree angle

00:01:51.440 --> 00:01:55.010
and a 62 degree angle just
like this triangle over here.

00:01:55.010 --> 00:01:59.000
So we know that the third
angle needs to be 36 degrees.

00:01:59.000 --> 00:02:00.000
36 degrees.

00:02:00.000 --> 00:02:02.830
Because we know 82 and 62,

00:02:02.830 --> 00:02:05.580
if you need to get to
180, it has to be 36.

00:02:05.580 --> 00:02:09.300
We just figured that out from
this first triangle over here.

00:02:09.300 --> 00:02:13.400
Now, if we look over
here, 36 degrees and 59,

00:02:13.400 --> 00:02:15.270
this definitely looks like
it has different angles.

00:02:15.270 --> 00:02:18.710
But let's figure out what
this angle would have to be.

00:02:18.710 --> 00:02:22.490
So if we call that y degrees, we know.

00:02:22.490 --> 00:02:23.580
I'll do it over here.

00:02:23.580 --> 00:02:27.460
Y plus 36 plus 59

00:02:27.460 --> 00:02:29.770
is equal to 180.

00:02:29.770 --> 00:02:31.920
And I'm just thinking in
terms of degrees here.

00:02:31.920 --> 00:02:33.750
So y plus.

00:02:33.750 --> 00:02:36.870
This is going to be
equal to, what is this?

00:02:36.870 --> 00:02:40.880
This is going to be equal
to 95, is equal to 180.

00:02:40.880 --> 00:02:41.720
Did I do that right?

00:02:41.720 --> 00:02:44.330
Yeah, 80 plus 15, yeah, 95.

00:02:44.330 --> 00:02:47.773
And then if I subtract 95 from both sides,

00:02:48.750 --> 00:02:50.700
what am I left with?

00:02:50.700 --> 00:02:55.530
I'm left with y is equal to 85 degrees.

00:02:56.490 --> 00:03:00.280
And so this is going to
be equal to 85 degrees.

00:03:00.280 --> 00:03:02.470
And then this last
triangle right over here,

00:03:02.470 --> 00:03:06.050
I have an angle that has measured
36, another one that's 59.

00:03:06.050 --> 00:03:07.800
So by the same logic,

00:03:07.800 --> 00:03:10.960
this one over here has to be 85 degrees.

00:03:10.960 --> 00:03:12.250
So let's ask ourselves now

00:03:12.250 --> 00:03:13.690
that we've figured out a little bit more

00:03:13.690 --> 00:03:17.583
about these triangles, which
of these two must be congruent?

00:03:18.660 --> 00:03:19.980
So you might be tempted to look

00:03:19.980 --> 00:03:22.007
at these bottom two triangles and say,

00:03:22.007 --> 00:03:24.340
"Hey, look, all of their
angles are the same."

00:03:24.340 --> 00:03:27.040
You have angle, angle, angle
and angle, angle, angle.

00:03:27.040 --> 00:03:28.970
Well, they would be similar.

00:03:28.970 --> 00:03:31.480
If you have three angles
that are the same,

00:03:31.480 --> 00:03:33.680
you definitely have similar triangles.

00:03:33.680 --> 00:03:37.640
But we don't have any length
information for triangle IJH.

00:03:37.640 --> 00:03:39.700
You need to know at
least one of the lengths

00:03:39.700 --> 00:03:40.980
of one of the sides

00:03:40.980 --> 00:03:43.640
in order to even start to
think about congruence.

00:03:43.640 --> 00:03:48.180
And so we can't make any
conclusion that IJH and LMK,

00:03:49.110 --> 00:03:54.030
triangles IJL and triangles LMK
are congruent to each other.

00:03:54.030 --> 00:03:56.060
Now let's look at these
candidates up here.

00:03:56.060 --> 00:03:58.350
We know that their
angles are all the same.

00:03:58.350 --> 00:04:00.613
And so we could apply angle.

00:04:01.990 --> 00:04:03.340
I'll do this in a different color.

00:04:03.340 --> 00:04:06.250
Angle-side-angle.

00:04:06.250 --> 00:04:09.560
36 degrees, length six, 82 degrees.

00:04:09.560 --> 00:04:14.410
36 degrees, length six, 82 degrees.

00:04:14.410 --> 00:04:18.640
So by angle-side-angle,

00:04:18.640 --> 00:04:20.673
we know that triangle ABC

00:04:22.690 --> 00:04:25.923
is indeed congruent to triangle FDE.


Preview Get Ready for Grade Level

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:01.340
- [Instructor] Here's an example

00:00:01.340 --> 00:00:03.360
of a get ready for grade level course.

00:00:03.360 --> 00:00:05.853
In this case, it's get
ready for sixth grade.

00:00:05.853 --> 00:00:08.540
There's a couple of
interesting things here.

00:00:08.540 --> 00:00:10.500
First of all, you can see that the course

00:00:10.500 --> 00:00:12.320
is broken down into units like all

00:00:12.320 --> 00:00:13.750
of our courses are broken down,

00:00:13.750 --> 00:00:17.820
and these units are designed
to match fairly closely

00:00:17.820 --> 00:00:19.720
to the units in sixth grade.

00:00:19.720 --> 00:00:21.930
So, it can be used in two different ways.

00:00:21.930 --> 00:00:24.680
The get ready for sixth
grade course, for example,

00:00:24.680 --> 00:00:27.520
could be used at the
beginning of sixth grade,

00:00:27.520 --> 00:00:30.760
so that students can
fill in all of the gaps

00:00:30.760 --> 00:00:33.260
that they might have up to sixth grade,

00:00:33.260 --> 00:00:35.300
and then they could work
on the sixth grade course.

00:00:35.300 --> 00:00:37.240
Or, it could be done in parallel.

00:00:37.240 --> 00:00:39.270
For example, if in the sixth grade course,

00:00:39.270 --> 00:00:42.500
students are working
on the unit on ratios,

00:00:42.500 --> 00:00:45.000
rates, and percentages, then
this unit right over here

00:00:45.000 --> 00:00:47.350
in the get ready for
sixth grade can be used

00:00:47.350 --> 00:00:48.700
to make sure that students have all

00:00:48.700 --> 00:00:52.280
the prerequisites for
that sixth grade unit.

00:00:52.280 --> 00:00:55.790
Now, what we recommend, because
most sixth grade students

00:00:55.790 --> 00:00:58.020
will already have some of the mastery,

00:00:58.020 --> 00:00:59.750
will already know many of the skills

00:00:59.750 --> 00:01:01.920
in the get ready for sixth grade course,

00:01:01.920 --> 00:01:05.140
we recommend that they start
with the course challenge.

00:01:05.140 --> 00:01:06.920
By taking a course challenge,

00:01:06.920 --> 00:01:08.940
students are going to have an opportunity

00:01:08.940 --> 00:01:10.370
to show what they already know,

00:01:10.370 --> 00:01:13.340
and also identify what
their existing gaps are.

00:01:13.340 --> 00:01:15.200
If students are doing well
on the course challenge,

00:01:15.200 --> 00:01:16.290
they can keep taking it.

00:01:16.290 --> 00:01:17.930
They're going to get different exercises,

00:01:17.930 --> 00:01:19.300
different items every time,

00:01:19.300 --> 00:01:21.820
and they can accelerate
quite quickly through

00:01:21.820 --> 00:01:23.870
the get ready for grade level course.

00:01:23.870 --> 00:01:26.460
But of course, if they
need to fill in gaps,

00:01:26.460 --> 00:01:28.920
they can then go to those
corresponding units.

00:01:28.920 --> 00:01:31.920
So for example, I could go
to this unit right over here,

00:01:31.920 --> 00:01:34.550
and do more focused skill based practice

00:01:34.550 --> 00:01:37.000
on the things that they
haven't mastered yet.

00:01:37.000 --> 00:01:39.850
Even within a unit,
you can also use things

00:01:39.850 --> 00:01:44.170
like our unit tests, which
you can see right over here,

00:01:44.170 --> 00:01:46.430
to similar to a course challenge,

00:01:46.430 --> 00:01:48.380
accelerate through what you already know,

00:01:48.380 --> 00:01:50.130
but then also identify the gaps

00:01:50.130 --> 00:01:52.470
for what you don't already know.

00:01:52.470 --> 00:01:54.750
Hopefully, all of y'all enjoy using

00:01:54.750 --> 00:01:56.450
the get ready for grade level courses.

00:01:56.450 --> 00:01:58.000
I just used sixth grade as an example.

00:01:58.000 --> 00:02:00.480
It's a way to ensure that students enter

00:02:00.480 --> 00:02:02.590
this grade level and to
whatever grade level they are,

00:02:02.590 --> 00:02:04.010
whatever course they are in,

00:02:04.010 --> 00:02:05.710
without those Swiss cheese gaps.

00:02:05.710 --> 00:02:07.520
There's a way for them to fill them in,

00:02:07.520 --> 00:02:11.343
and then continue to proceed
at their own time and pace.


Reading within and across genres

WEBVTT
Kind: captions
Language: en

00:00:01.510 --> 00:00:02.600
- [David] Hello, readers.

00:00:02.600 --> 00:00:06.290
Let's talk about the
idea of genre in fiction.

00:00:06.290 --> 00:00:10.230
Genres are types of stories
that share similar themes,

00:00:10.230 --> 00:00:12.920
styles, or subject matter.

00:00:12.920 --> 00:00:16.770
So science fiction is a genre,
fairy tales are a genre,

00:00:16.770 --> 00:00:18.990
mysteries are a genre.

00:00:18.990 --> 00:00:21.630
Each one of these types of
stories has certain elements

00:00:21.630 --> 00:00:24.390
that you gradually come
to expect from them.

00:00:24.390 --> 00:00:28.150
Fantasy stories have magic
spells and imaginary creatures.

00:00:28.150 --> 00:00:30.410
Romance stories have lots of smooching.

00:00:30.410 --> 00:00:32.100
Mystery stories have a crime

00:00:32.100 --> 00:00:34.170
and a person who tries to solve it.

00:00:34.170 --> 00:00:37.100
You can call these tropes, you
can call these style elements

00:00:37.100 --> 00:00:40.740
but certain genres have
certain expectations

00:00:40.740 --> 00:00:42.120
embedded in them.

00:00:42.120 --> 00:00:43.770
The more you read of a genre,

00:00:43.770 --> 00:00:47.620
the more your expectations
are shaped for that genre.

00:00:47.620 --> 00:00:50.730
When a story begins with the
phrase, "Once upon a time,"

00:00:50.730 --> 00:00:53.800
and ends with "and they
lived happily ever after,"

00:00:53.800 --> 00:00:56.070
you know you're looking at a fairy tale.

00:00:56.070 --> 00:00:58.610
We all carry with us a unique collection

00:00:58.610 --> 00:01:01.030
of impressions and expectations.

00:01:01.030 --> 00:01:04.020
It's your background
knowledge, your schema.

00:01:04.020 --> 00:01:06.540
If you think of your brain as a closet,

00:01:06.540 --> 00:01:08.410
then schema is the hooks

00:01:08.410 --> 00:01:11.050
and clothes hangers inside that closet.

00:01:11.050 --> 00:01:13.410
You can put a new shirt
on a clothes hanger,

00:01:13.410 --> 00:01:17.580
you can hang pants or a
skirt on a clothes hanger,

00:01:17.580 --> 00:01:19.640
but if you wanted to hang up a dress,

00:01:19.640 --> 00:01:22.050
you might need one of those
fabric-covered hangers

00:01:22.050 --> 00:01:23.170
so it doesn't slip.

00:01:23.170 --> 00:01:24.890
And if you wanna store
shoes in your closet,

00:01:24.890 --> 00:01:26.120
you might need to get a shoe rack

00:01:26.120 --> 00:01:28.790
or one of those hanging shoe organizers.

00:01:28.790 --> 00:01:31.760
Just as different clothing
items require different,

00:01:31.760 --> 00:01:35.290
you know, closet infrastructure
like hangers or shoe racks,

00:01:35.290 --> 00:01:37.610
your schema, your background knowledge,

00:01:37.610 --> 00:01:41.510
informs the sort of literature
you know how to read.

00:01:41.510 --> 00:01:44.440
That's a weird sentiment to
express, I think, but it's true.

00:01:44.440 --> 00:01:47.220
The first time you read a
book in a particular genre,

00:01:47.220 --> 00:01:50.140
you're forming an
impression of that genre.

00:01:50.140 --> 00:01:53.250
And that impression
gets refined or revised

00:01:53.250 --> 00:01:56.170
with every similar book
you read after the first.

00:01:56.170 --> 00:01:59.070
Reading widely across
many genres of fiction

00:01:59.070 --> 00:02:02.562
expands your brain closet.

00:02:02.562 --> 00:02:03.720
But let's be clear here.

00:02:03.720 --> 00:02:06.480
Not every book is connected
with every other book.

00:02:06.480 --> 00:02:09.720
And when you try to apply
something you learned in one story

00:02:09.720 --> 00:02:12.240
to another story, it may not work.

00:02:12.240 --> 00:02:15.630
For example, in 20th
century detective fiction,

00:02:15.630 --> 00:02:20.270
there is a recurring theme or
trope that dates to the 1930s:

00:02:20.270 --> 00:02:22.450
the butler did it, which is to say that

00:02:22.450 --> 00:02:25.800
if there's a murder that takes
place at a fancy manor house,

00:02:25.800 --> 00:02:28.930
there's a good chance that
the butler is the murderer.

00:02:28.930 --> 00:02:32.430
But if you go into every mystery
set at a fancy manor house

00:02:32.430 --> 00:02:35.570
assuming that the butler
is the guilty party,

00:02:35.570 --> 00:02:38.220
you'll be wrong a lot.

00:02:38.220 --> 00:02:42.410
Famously, in Agatha Christie's
Murder on the Orient Express,

00:02:42.410 --> 00:02:43.623
everyone did it.

00:02:45.430 --> 00:02:48.950
Every suspect in the mystery
is responsible in some way

00:02:48.950 --> 00:02:50.780
for the murder in the title.

00:02:50.780 --> 00:02:52.820
It's an enormous conspiracy,

00:02:52.820 --> 00:02:54.990
and Christie plays with
the readers' assumptions

00:02:54.990 --> 00:02:57.750
as we go through the
story, knowing that you,

00:02:57.750 --> 00:03:00.470
as a person who has probably
read a mystery before,

00:03:00.470 --> 00:03:03.030
or who is at least familiar with the form,

00:03:03.030 --> 00:03:06.660
has an expectation that there
are only one or two culprits,

00:03:06.660 --> 00:03:09.330
only one or two people that did the murder

00:03:09.330 --> 00:03:11.280
to the guy on the train.

00:03:11.280 --> 00:03:16.020
And look, I apologize for
spoilers for a story from 1934,

00:03:16.020 --> 00:03:18.380
but you can't make an omelet
without breaking some eggs,

00:03:18.380 --> 00:03:19.450
you know what I mean?

00:03:19.450 --> 00:03:21.510
Anyway, it's neat to look at the way

00:03:21.510 --> 00:03:25.010
that an author can play with
the expectations of genre,

00:03:25.010 --> 00:03:27.750
how an author might
anticipate a reader's schema

00:03:27.750 --> 00:03:28.973
and play with that.

00:03:29.830 --> 00:03:32.370
Something that blew my
mind when I was in school

00:03:32.370 --> 00:03:35.400
was the idea that Star Wars was a western,

00:03:35.400 --> 00:03:37.500
or at least takes many of its cues

00:03:37.500 --> 00:03:40.210
from classic pulp western movies.

00:03:40.210 --> 00:03:41.960
Let's draw a little Venn diagram.

00:03:41.960 --> 00:03:44.380
All right, so over here,
we've got Star Wars

00:03:44.380 --> 00:03:47.640
and over here we've got
the western movie genre.

00:03:47.640 --> 00:03:49.450
Here are some things that Star Wars has

00:03:49.450 --> 00:03:51.130
that westerns don't have.

00:03:51.130 --> 00:03:53.283
Space wizards like the Jedi and Sith,

00:03:54.230 --> 00:03:58.290
magic like the force, or space ships.

00:03:58.290 --> 00:04:00.140
Here's some things that
western movies have

00:04:00.140 --> 00:04:02.853
that Star Wars movie's
don't, by and large, have.

00:04:04.130 --> 00:04:07.930
Western movies have horses,
they tend to have cowboys,

00:04:07.930 --> 00:04:10.300
and then tend to take place in settings

00:04:10.300 --> 00:04:13.990
like the western United States and Mexico.

00:04:13.990 --> 00:04:16.160
But here's our overlap, all right,

00:04:16.160 --> 00:04:20.370
so both Star Wars and western
movies have bar fights,

00:04:20.370 --> 00:04:23.840
both Star Wars and western
movies have bounty hunters,

00:04:23.840 --> 00:04:26.360
and both the Star Wars
films and western films

00:04:26.360 --> 00:04:28.570
tend to have a lot of desert settings.

00:04:28.570 --> 00:04:31.700
That could be the desert planet
of Tatooine from Star Wars,

00:04:31.700 --> 00:04:34.700
or Monument Valley in
the US state of Texas,

00:04:34.700 --> 00:04:38.140
or the Mexican state of
Durango in western films.

00:04:38.140 --> 00:04:40.410
Mind you, you could also
make a separate Venn diagram

00:04:40.410 --> 00:04:42.470
between Star Wars and samurai movies

00:04:42.470 --> 00:04:45.760
because Star Wars also
borrows liberally from those.

00:04:45.760 --> 00:04:48.070
This is a great activity for analysis.

00:04:48.070 --> 00:04:49.970
Take two stories that you love

00:04:49.970 --> 00:04:53.130
and compare their theme,
settings, and characters,

00:04:53.130 --> 00:04:54.780
and see if there's something you can find

00:04:54.780 --> 00:04:56.390
in common between them.

00:04:56.390 --> 00:04:59.200
You may discover connections
you didn't expect.

00:04:59.200 --> 00:05:02.380
As I've mentioned before,
good readers read widely.

00:05:02.380 --> 00:05:04.580
They read lots of books and they let

00:05:04.580 --> 00:05:07.710
what they know about one genre,
their schema of the genre,

00:05:07.710 --> 00:05:10.000
help them anticipate and make connections

00:05:10.000 --> 00:05:11.710
when they read a new book.

00:05:11.710 --> 00:05:14.500
The more you read, the
more schema you build,

00:05:14.500 --> 00:05:16.340
the easier and more interesting

00:05:16.340 --> 00:05:17.840
those connections will become.

00:05:18.690 --> 00:05:19.853
You can learn anything.

00:05:20.750 --> 00:05:21.583
David out.


Quantitative information in texts

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:01.810
- [Instructor] Hello, readers.

00:00:01.810 --> 00:00:02.740
Today we're going to talk

00:00:02.740 --> 00:00:05.090
about quantitative information in texts,

00:00:05.090 --> 00:00:07.570
but I want to start with a question.

00:00:07.570 --> 00:00:09.030
What's the best way to describe

00:00:09.030 --> 00:00:11.500
the way a horse looks as it runs?

00:00:11.500 --> 00:00:13.660
What's the most efficient way?

00:00:13.660 --> 00:00:16.010
I guess I could just use words, right?

00:00:16.010 --> 00:00:18.420
The horse pushes off with its back legs

00:00:18.420 --> 00:00:19.970
and then its front legs come up,

00:00:19.970 --> 00:00:22.470
and as the horse propels itself forward,

00:00:22.470 --> 00:00:24.960
both its back and front
hooves are off the ground

00:00:24.960 --> 00:00:27.160
at the same time as it gallops.

00:00:27.160 --> 00:00:30.180
It's like a series of jumps.

00:00:30.180 --> 00:00:31.740
Listening to myself say that,

00:00:31.740 --> 00:00:33.900
it doesn't feel very clear to me.

00:00:33.900 --> 00:00:35.590
At least it's certainly less clear

00:00:35.590 --> 00:00:39.060
than this moving image
of a galloping horse.

00:00:39.060 --> 00:00:41.580
Sometimes the most efficient
way to express information

00:00:41.580 --> 00:00:44.200
is to present it visually, to see it

00:00:44.200 --> 00:00:46.210
in order to best understand it.

00:00:46.210 --> 00:00:50.530
And so we're going to talk
today about graphs and charts,

00:00:50.530 --> 00:00:51.653
your friends and mine.

00:00:52.740 --> 00:00:54.680
This video is no longer about horses.

00:00:54.680 --> 00:00:56.630
I apologize if I gave that impression.

00:00:56.630 --> 00:00:58.560
I did give a pretty obvious title

00:00:58.560 --> 00:01:00.740
when I said quantitative
information in texts.

00:01:00.740 --> 00:01:04.270
But to those of you who
I've misled, I'm sorry.

00:01:04.270 --> 00:01:06.620
Now, graphs are one of many ways

00:01:06.620 --> 00:01:09.200
we can make visual
representations of data,

00:01:09.200 --> 00:01:11.943
information you can see, news you can use.

00:01:13.030 --> 00:01:16.060
And I guess a question that
follows from that is why do it?

00:01:16.060 --> 00:01:18.110
Why make a graph and a chart?

00:01:18.110 --> 00:01:19.660
What does that have to do with writing?

00:01:19.660 --> 00:01:23.843
Well, visualizing data can
make an argument stronger.

00:01:24.740 --> 00:01:27.220
Let's say my neighborhood
is having a bake sale

00:01:27.220 --> 00:01:30.270
to raise money for, I
don't know, a carnival.

00:01:30.270 --> 00:01:33.010
And some of my neighbors
have peanut allergies,

00:01:33.010 --> 00:01:35.000
so I want to write to
the neighborhood council

00:01:35.000 --> 00:01:37.490
to say let's make sure
that the baked goods

00:01:37.490 --> 00:01:40.060
don't have peanuts in them
so the whole neighborhood

00:01:40.060 --> 00:01:43.153
can participate without fear
of having an allergic attack.

00:01:44.010 --> 00:01:45.670
That's supposed to be a peanut.

00:01:45.670 --> 00:01:47.310
I realize it kinda looks like a dog treat,

00:01:47.310 --> 00:01:49.890
but let's pretend that's a peanut.

00:01:49.890 --> 00:01:52.810
Now, good readers know
that data is important

00:01:52.810 --> 00:01:55.230
to informational text, so as a writer,

00:01:55.230 --> 00:01:56.870
I wanna make sure that I'm backing up

00:01:56.870 --> 00:01:59.120
my claims with a chart.

00:01:59.120 --> 00:02:01.250
Here's the text of the letter I'd send

00:02:01.250 --> 00:02:02.640
to my neighborhood council:

00:02:02.640 --> 00:02:04.240
Dear Neighborhood Council,

00:02:04.240 --> 00:02:07.110
I'm writing to ask that
you officially ban peanuts

00:02:07.110 --> 00:02:09.560
and tree nuts from the
baked goods on offer

00:02:09.560 --> 00:02:12.060
during our upcoming carnival bake sale.

00:02:12.060 --> 00:02:14.090
Over a fifth of residents surveyed

00:02:14.090 --> 00:02:16.090
reported some kind of food allergy.

00:02:16.090 --> 00:02:18.040
Please see attached graph.

00:02:18.040 --> 00:02:20.180
If we want to have
their full participation

00:02:20.180 --> 00:02:22.380
in this fundraiser, we cannot permit foods

00:02:22.380 --> 00:02:24.280
that will send them to the hospital.

00:02:24.280 --> 00:02:26.210
All my best, David.

00:02:26.210 --> 00:02:27.940
Now, let's take a look at that graph.

00:02:27.940 --> 00:02:30.290
These are made-up numbers
in a made-up situation.

00:02:30.290 --> 00:02:33.390
But let's say I interviewed
50 of my neighbors,

00:02:33.390 --> 00:02:36.250
and 40 of them here report no allergies,

00:02:36.250 --> 00:02:37.730
but 10 of them do.

00:02:37.730 --> 00:02:38.930
So we can have this bar graph

00:02:38.930 --> 00:02:41.820
that breaks down those allergies by type.

00:02:41.820 --> 00:02:44.020
One person is allergic to strawberries.

00:02:44.020 --> 00:02:45.980
Two people are allergic to peanuts.

00:02:45.980 --> 00:02:47.670
Four people are allergic to tree nuts,

00:02:47.670 --> 00:02:49.050
like pecans or walnuts.

00:02:49.050 --> 00:02:52.000
And three people have
multiple food allergies.

00:02:52.000 --> 00:02:52.950
And then we have this bar here

00:02:52.950 --> 00:02:55.300
that shows all the folks that
don't have food allergies.

00:02:55.300 --> 00:02:56.670
That's the other 40.

00:02:56.670 --> 00:02:59.150
To put it another way, here's a pie chart

00:02:59.150 --> 00:03:02.020
of that same data set
showing just how many people

00:03:02.020 --> 00:03:04.830
in the neighborhood
have allergies in total.

00:03:04.830 --> 00:03:06.230
It's just a different
way of looking at it.

00:03:06.230 --> 00:03:07.630
This is all the same data,

00:03:07.630 --> 00:03:09.720
but we can see that the
percentage of people surveyed

00:03:09.720 --> 00:03:13.510
who have allergies adds
up to 20% of the total.

00:03:13.510 --> 00:03:15.610
Something that's in the
letter but not the graphs

00:03:15.610 --> 00:03:18.230
is the idea that these are
very serious allergies.

00:03:18.230 --> 00:03:19.560
If the wrong person ate a peanut,

00:03:19.560 --> 00:03:21.430
they could have a medical emergency.

00:03:21.430 --> 00:03:23.410
If the council were to look
at just the graph alone,

00:03:23.410 --> 00:03:25.697
they might say, "Eh, it's
not that many people.

00:03:25.697 --> 00:03:28.070
"Peanuts are great, let's include them."

00:03:28.070 --> 00:03:31.400
But combined with the text,
the stakes become more clear.

00:03:31.400 --> 00:03:33.640
And they may better
understand the consequences

00:03:33.640 --> 00:03:35.510
of having nuts at the bake sale.

00:03:35.510 --> 00:03:37.840
And something that's in the
graph that isn't in the letter

00:03:37.840 --> 00:03:40.330
is the breakdown of allergies by type.

00:03:40.330 --> 00:03:41.790
Maybe the council could decide

00:03:41.790 --> 00:03:44.630
that the one person with
a strawberry allergy

00:03:44.630 --> 00:03:47.760
is an acceptable risk and
that strawberry baked goods

00:03:47.760 --> 00:03:50.800
are kind of obvious-looking
and easy to avoid

00:03:50.800 --> 00:03:53.590
in a way that brownies with
walnuts in them aren't.

00:03:53.590 --> 00:03:56.090
This visual information allows
the neighborhood council

00:03:56.090 --> 00:03:59.930
to get my point more efficiently
than just the words alone.

00:03:59.930 --> 00:04:02.980
It helps me express a
sort of complicated idea

00:04:02.980 --> 00:04:05.610
that peanuts and other
allergens in the bake sale

00:04:05.610 --> 00:04:08.470
might constitute too much
of a risk to my neighbors

00:04:08.470 --> 00:04:10.380
and that they shouldn't be allowed.

00:04:10.380 --> 00:04:11.730
Now, there are other questions

00:04:11.730 --> 00:04:14.020
like is this a representative data set?

00:04:14.020 --> 00:04:16.070
But that's a question that
can be better answered

00:04:16.070 --> 00:04:18.050
by our statistics course.

00:04:18.050 --> 00:04:20.490
For now, think about it this way.

00:04:20.490 --> 00:04:22.770
As a reader, your job is to look

00:04:22.770 --> 00:04:26.380
at everything that is on
the page, not just the text.

00:04:26.380 --> 00:04:30.090
Then think about what both
the data and the words

00:04:30.090 --> 00:04:32.400
do for your understanding.

00:04:32.400 --> 00:04:35.110
Data and informational
text are two great tastes

00:04:35.110 --> 00:04:36.240
that go great together.

00:04:36.240 --> 00:04:39.030
One can support the other and vice versa,

00:04:39.030 --> 00:04:41.770
you know, like chocolate
and peanut butter.

00:04:41.770 --> 00:04:42.850
Oh, no, sorry, bad example,

00:04:42.850 --> 00:04:45.330
like chocolate and sunflower butter.

00:04:45.330 --> 00:04:46.483
You know what I mean.

00:04:47.390 --> 00:04:49.893
You can learn anything, David out.