Rectangle area as product of dimensions same as counting unit squares

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.590
I've got three rectangles
here, and I also

00:00:03.590 --> 00:00:05.640
have their dimensions.

00:00:05.640 --> 00:00:07.915
I have their height
and their width.

00:00:07.915 --> 00:00:10.540
And in fact, this one right here
has the same height and width,

00:00:10.540 --> 00:00:12.432
so this is actually a square.

00:00:12.432 --> 00:00:14.640
So let's think about how
much space they each take up

00:00:14.640 --> 00:00:15.294
on my screen.

00:00:15.294 --> 00:00:17.460
And since we're doing
everything in terms of meters,

00:00:17.460 --> 00:00:19.330
since all of the
dimensions are in meters,

00:00:19.330 --> 00:00:22.892
I'm going to measure the area
in terms of square meters.

00:00:22.892 --> 00:00:24.350
So let's see how
many square meters

00:00:24.350 --> 00:00:26.930
I can fit onto this
yellow rectangle

00:00:26.930 --> 00:00:28.760
without going outside
of its boundary

00:00:28.760 --> 00:00:30.680
and without overlapping.

00:00:30.680 --> 00:00:32.360
So I can fit 1 square meter.

00:00:32.360 --> 00:00:34.840
Remember, a square
meter is just a square

00:00:34.840 --> 00:00:38.510
where its length is 1 meter
and its width is 1 meter.

00:00:38.510 --> 00:00:53.980
So that's 1 square meter, 2, 3,
4, or 5, and 6 square meters.

00:00:53.980 --> 00:00:59.210
So we see here that the
area is 6 square meters.

00:00:59.210 --> 00:01:05.820
Area is equal to
6 square meters.

00:01:05.820 --> 00:01:07.800
But something might
be jumping out at you.

00:01:07.800 --> 00:01:13.040
Did I really have to sit
and count 1, 2, 3, 4, 5, 6?

00:01:13.040 --> 00:01:16.450
You might have recognized
that I could view this

00:01:16.450 --> 00:01:18.995
as really 2 groups of 3.

00:01:18.995 --> 00:01:21.380
And let me make that very clear.

00:01:21.380 --> 00:01:27.700
So, for example, I could
view this as one group of 3

00:01:27.700 --> 00:01:30.480
and then another group of 3.

00:01:30.480 --> 00:01:33.140
Now, how did I get groups of 3?

00:01:33.140 --> 00:01:35.680
Well, that's because
width here is 3 meters.

00:01:35.680 --> 00:01:40.630
So I could put 3 square
meters side by side.

00:01:40.630 --> 00:01:42.510
And how did I get the 2 groups?

00:01:42.510 --> 00:01:46.227
Well, this has a
length of 2 meters.

00:01:46.227 --> 00:01:48.060
So another way that I
could have essentially

00:01:48.060 --> 00:01:51.380
counted these six things
is I could have said,

00:01:51.380 --> 00:01:53.470
look, I have a
length of 2 meters.

00:01:53.470 --> 00:01:55.755
So I'm going to
have 2 groups of 3.

00:01:58.510 --> 00:02:05.990
So I could multiply 2 times
3, 2 of my groups of 3,

00:02:05.990 --> 00:02:07.255
and I would have gotten 6.

00:02:09.882 --> 00:02:11.090
And you might say, hey, wait.

00:02:11.090 --> 00:02:15.880
Is this just a coincidence
that if I took the length

00:02:15.880 --> 00:02:17.700
and I multiplied
it by the width,

00:02:17.700 --> 00:02:20.130
that I get the same
thing as its area?

00:02:20.130 --> 00:02:23.430
And no, it's not, because
when you took the length,

00:02:23.430 --> 00:02:26.220
you essentially said, well,
how many rows do I have?

00:02:26.220 --> 00:02:28.710
And then you say when you
multiply it by the width,

00:02:28.710 --> 00:02:32.850
you're saying, well,
how many of these square

00:02:32.850 --> 00:02:34.740
meters can I fit into a row?

00:02:34.740 --> 00:02:37.510
So this is really a
quick way of counting

00:02:37.510 --> 00:02:40.550
how many of these
square meters you have.

00:02:40.550 --> 00:02:47.430
So you could say that 2
meters multiplied by 3 meters

00:02:47.430 --> 00:02:49.310
is equal to 6 square meters.

00:02:55.059 --> 00:02:57.600
Now, you might say, hey, I'm
not sure if that always applies.

00:02:57.600 --> 00:03:00.340
Let's see if it applies
to these other rectangles

00:03:00.340 --> 00:03:01.910
right over here.

00:03:01.910 --> 00:03:04.020
So based on what
we just saw, let's

00:03:04.020 --> 00:03:10.710
take the length, 4 meters,
and multiply by the width,

00:03:10.710 --> 00:03:14.440
and multiply by 2 meters.

00:03:14.440 --> 00:03:17.670
Now, 4 times 2 is 8.

00:03:17.670 --> 00:03:23.550
So this should give
us 8 square meters.

00:03:23.550 --> 00:03:26.600
Let's see if this is
actually the case.

00:03:26.600 --> 00:03:36.440
So 1, 2, 3, 4, 5--
and you see it's

00:03:36.440 --> 00:03:44.540
going in the right
direction-- 6, 7, and 8.

00:03:44.540 --> 00:03:48.520
So the area of this rectangle
is, indeed, 8 square meters.

00:03:48.520 --> 00:03:51.100
And you could view
this as 4 groups of 2.

00:03:51.100 --> 00:03:54.010
So you could literally
view this as 4 groups of 2.

00:03:54.010 --> 00:03:56.610
That's where the 4
times 2 comes from.

00:03:56.610 --> 00:03:59.640
So you could view it as
4 groups of 2 like this.

00:03:59.640 --> 00:04:02.240
Or you could view
it as 2 groups of 4,

00:04:02.240 --> 00:04:05.130
So 1 group of 4 right over here.

00:04:05.130 --> 00:04:10.100
So you could view this is 2
times 4, and then 2 groups 4.

00:04:10.100 --> 00:04:12.100
I want to draw it a
little bit cleaner.

00:04:12.100 --> 00:04:13.900
Now, you could
probably figure out

00:04:13.900 --> 00:04:15.674
what the area of
this rectangle is.

00:04:15.674 --> 00:04:17.090
It's actually a
square, because it

00:04:17.090 --> 00:04:19.480
has the same length
and the same width.

00:04:19.480 --> 00:04:24.550
We multiply the length, 3
meters, times the width,

00:04:24.550 --> 00:04:36.330
so times 3 meters, to get 3
times 3 is 9-- 9 square meters.

00:04:38.920 --> 00:04:41.500
And let's verify it again just
to feel really good about this

00:04:41.500 --> 00:04:45.010
multiplying the dimensions
of these rectangles.

00:04:45.010 --> 00:05:02.850
So we have 1, 2, 3,
4, 5, 6, 7, 8, and 9.

00:05:02.850 --> 00:05:03.890
So it matches up.

00:05:03.890 --> 00:05:06.370
We figure out how many
square meters can we

00:05:06.370 --> 00:05:08.694
cover this thing with,
without overlapping,

00:05:08.694 --> 00:05:10.110
without going over
the boundaries.

00:05:10.110 --> 00:05:14.250
We get the exact same thing
as if we multiplied 3 times

00:05:14.250 --> 00:05:19.145
3, if we multiplied the length
times the width in meters.


Measuring the same rectangle with different unit squares

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.340
I have two identical
rectangles here,

00:00:03.340 --> 00:00:06.910
and I want to measure how
much space each of them

00:00:06.910 --> 00:00:09.360
take up on the
plane of my screen,

00:00:09.360 --> 00:00:12.010
the screen that you are
looking at right now.

00:00:12.010 --> 00:00:14.550
And I want to do it using
two different units.

00:00:14.550 --> 00:00:16.820
It's clear, since they're
two identical rectangles,

00:00:16.820 --> 00:00:19.550
that they take up the
exact same amount of space.

00:00:19.550 --> 00:00:21.887
They will have the same area.

00:00:21.887 --> 00:00:23.720
But what we could see
it that we can measure

00:00:23.720 --> 00:00:26.440
that area using different units.

00:00:26.440 --> 00:00:29.830
So first over here, let's
say that this figure

00:00:29.830 --> 00:00:31.860
is 1 foot in width.

00:00:38.040 --> 00:00:41.820
And it is also 1 foot in height.

00:00:41.820 --> 00:00:44.640
So this right over here
is equal to 1 square foot.

00:00:48.300 --> 00:00:49.300
It's clearly a square.

00:00:49.300 --> 00:00:51.690
It has the same width
and the same height.

00:00:51.690 --> 00:00:53.490
And each of these
dimensions are 1 foot,

00:00:53.490 --> 00:00:55.720
so we could call
it 1 square foot.

00:00:55.720 --> 00:00:57.250
So let's see how
many square feet

00:00:57.250 --> 00:00:59.540
we can get onto one
of these rectangles,

00:00:59.540 --> 00:01:01.870
and essentially we'll
be measuring its area

00:01:01.870 --> 00:01:03.970
in terms of square feet.

00:01:03.970 --> 00:01:05.830
And so we want to
cover the entire space

00:01:05.830 --> 00:01:08.440
without overlapping and without
going over the boundary.

00:01:08.440 --> 00:01:24.490
So that's 1, 2, 3, 4, 5, and 6.

00:01:24.490 --> 00:01:39.760
6 in that first row, and then
I have 7, 8, 9 10, 11, and 12.

00:01:39.760 --> 00:01:43.490
So that looks like this area,
which is the same as this area

00:01:43.490 --> 00:01:43.990
down here.

00:01:43.990 --> 00:01:47.770
If I were to measure
it in square feet,

00:01:47.770 --> 00:01:49.850
the area is equal to,
let me write this down,

00:01:49.850 --> 00:01:53.050
the area is equal to, we
have-- let me write it down.

00:01:53.050 --> 00:02:02.910
We have 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12 square feet.

00:02:02.910 --> 00:02:04.580
12 square feet.

00:02:08.372 --> 00:02:10.330
Now, I'm going to try to
measure that same area

00:02:10.330 --> 00:02:11.746
in a different
unit, and I'm going

00:02:11.746 --> 00:02:13.340
to just make up this unit.

00:02:13.340 --> 00:02:15.220
I'm going to call it a furgle.

00:02:15.220 --> 00:02:19.260
And a furgle in one dimension,
so a furgle in one dimension

00:02:19.260 --> 00:02:20.170
is twice a foot.

00:02:20.170 --> 00:02:21.980
So that distance
right over here,

00:02:21.980 --> 00:02:23.850
I'm going to call a furgle.

00:02:23.850 --> 00:02:25.829
That's one furgle.

00:02:25.829 --> 00:02:27.370
This is something
that I made up just

00:02:27.370 --> 00:02:28.900
for the purpose of this video.

00:02:28.900 --> 00:02:31.520
Most people will not
recognize what a furgle is.

00:02:31.520 --> 00:02:33.220
So its height is one furgle.

00:02:33.220 --> 00:02:37.020
Its width is a furgle.

00:02:37.020 --> 00:02:42.285
And so we could say that
this is 1 square furgle.

00:02:45.320 --> 00:02:48.290
So let's see how many
square furgles is

00:02:48.290 --> 00:02:52.520
this area, that same area
that is 12 square feet.

00:02:52.520 --> 00:02:54.863
So let me copy and paste this.

00:02:54.863 --> 00:03:01.130
So let me copy, and then paste.

00:03:01.130 --> 00:03:04.600
So let's see if we can get
one square furgle on there.

00:03:04.600 --> 00:03:07.380
We can get another
square furgle on there,

00:03:07.380 --> 00:03:11.320
and we can get a third
square furgle on there.

00:03:11.320 --> 00:03:15.760
So we get 1, 2,
3 square furgles.

00:03:15.760 --> 00:03:19.880
The area of this figure in
furgles, square furgles,

00:03:19.880 --> 00:03:23.030
I should say, area is
equal to 3 square furgles.

00:03:28.930 --> 00:03:31.010
So it's the same exact area.

00:03:31.010 --> 00:03:35.530
3 square furgles is equal
to 12 square feet, covering

00:03:35.530 --> 00:03:37.014
the exact same area.

00:03:37.014 --> 00:03:38.680
Now, what I encourage
you to think about

00:03:38.680 --> 00:03:42.944
is how many square feet
make a square furgle?


Introduction to area and unit squares

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.020
So we've got two
figures right over here,

00:00:03.020 --> 00:00:06.110
and I want to think about
how much space they take up

00:00:06.110 --> 00:00:07.700
on your screen.

00:00:07.700 --> 00:00:10.910
And this idea of how much
space something takes up

00:00:10.910 --> 00:00:15.190
on a surface, this idea is area.

00:00:15.190 --> 00:00:17.810
So right when you look at
it, it looks pretty clear

00:00:17.810 --> 00:00:20.560
that this purple figure
takes up more space

00:00:20.560 --> 00:00:23.350
on my screen than
this blue figure.

00:00:23.350 --> 00:00:25.190
But how do we
actually measure it?

00:00:25.190 --> 00:00:29.910
How do we actually know how much
more area this purple figure

00:00:29.910 --> 00:00:32.299
takes up than this blue one?

00:00:32.299 --> 00:00:34.950
Well, one way to do
it would be to define

00:00:34.950 --> 00:00:37.170
a unit amount of area.

00:00:37.170 --> 00:00:41.640
So, for example, I could create
a square right over here,

00:00:41.640 --> 00:00:45.000
and this square, whatever units
we're using, we could say it's

00:00:45.000 --> 00:00:46.050
a one unit.

00:00:46.050 --> 00:00:49.500
So if its width right
over here is one unit

00:00:49.500 --> 00:00:54.140
and its height right
over here is one unit,

00:00:54.140 --> 00:00:56.015
we could call this
a unit square.

00:00:59.670 --> 00:01:03.500
And so one way to measure
the area of these figures

00:01:03.500 --> 00:01:06.790
is to figure out how many
unit squares I could cover

00:01:06.790 --> 00:01:09.380
this thing with
without overlapping

00:01:09.380 --> 00:01:11.560
and while staying
in the boundaries.

00:01:11.560 --> 00:01:12.560
So let's try to do that.

00:01:12.560 --> 00:01:15.310
Let's try to cover each of
these with unit squares,

00:01:15.310 --> 00:01:18.010
and essentially we'll
have a measure of area.

00:01:18.010 --> 00:01:20.730
So I'll start with
this blue one.

00:01:20.730 --> 00:01:35.570
So we could put 1, 2, 3,
3, 4, 5, five unit squares.

00:01:35.570 --> 00:01:37.020
Let me write this down.

00:01:37.020 --> 00:01:43.331
So we got 1, 2, 3,
4, 5 unit squares,

00:01:43.331 --> 00:01:45.330
and I could draw the
boundary between those unit

00:01:45.330 --> 00:01:47.700
squares a little bit clearer.

00:01:47.700 --> 00:01:51.330
So we have 5 unit squares.

00:01:51.330 --> 00:01:54.090
And so we could say that
this figure right over here

00:01:54.090 --> 00:01:55.720
has an area.

00:01:55.720 --> 00:01:57.730
The area is 5.

00:01:57.730 --> 00:01:59.436
We could say 5 unit squares.

00:01:59.436 --> 00:02:00.810
The more typical
way of saying it

00:02:00.810 --> 00:02:07.290
is that you have 5 square units.

00:02:07.290 --> 00:02:08.570
That's the area over here.

00:02:08.570 --> 00:02:11.150
Now, let's do the same thing
with this purple figure.

00:02:11.150 --> 00:02:30.215
So with the purple figure, I
could put 1, 2, 3, 4, 5, 6, 7,

00:02:30.215 --> 00:02:38.210
8, 9, 10 of these unit squares.

00:02:38.210 --> 00:02:39.025
I can cover it.

00:02:39.025 --> 00:02:41.150
They're not overlapping,
or I'm trying pretty close

00:02:41.150 --> 00:02:42.191
to not make them overlap.

00:02:42.191 --> 00:02:43.710
You see, you can fit 10 of them.

00:02:43.710 --> 00:02:45.830
And let me draw the
boundary between them,

00:02:45.830 --> 00:02:48.770
so you can see a
little bit clearer.

00:02:48.770 --> 00:02:52.675
So that's the boundary
between my unit squares.

00:02:55.570 --> 00:02:57.760
So I think-- there you go.

00:02:57.760 --> 00:02:58.910
And we can count them.

00:02:58.910 --> 00:03:05.845
We have 1, 2, 3, 4,
5, 6, 7, 8, 9, 10.

00:03:05.845 --> 00:03:07.970
So we could say the area
here-- and let me actually

00:03:07.970 --> 00:03:09.720
divide these with the
black boundary, too.

00:03:09.720 --> 00:03:12.910
It makes it a little bit
clearer than that blue.

00:03:12.910 --> 00:03:16.760
So the area here for the
purple figure, we could say,

00:03:16.760 --> 00:03:21.060
so the area here is equal to 10.

00:03:21.060 --> 00:03:26.110
10 square, 10 square units.

00:03:26.110 --> 00:03:27.740
So what we have
here, we have an idea

00:03:27.740 --> 00:03:32.207
of how much space does
something take up on a surface.

00:03:32.207 --> 00:03:34.040
And you could eyeball
it, and say, hey, this

00:03:34.040 --> 00:03:34.873
takes up more space.

00:03:34.873 --> 00:03:37.230
But now we've come up with
a way of measuring it.

00:03:37.230 --> 00:03:39.500
We can define a unit square.

00:03:39.500 --> 00:03:41.710
Here it's a 1 unit by 1 unit.

00:03:41.710 --> 00:03:45.430
In the future we'll see that
it could be a unit centimeter.

00:03:45.430 --> 00:03:48.300
It could be a 1 centimeter
by 1 centimeter squared.

00:03:48.300 --> 00:03:50.420
It could be a 1 meter
by 1 meter squared.

00:03:50.420 --> 00:03:53.440
It could be a 1 foot
by 1 foot square,

00:03:53.440 --> 00:03:56.500
but then we can use
that to actually measure

00:03:56.500 --> 00:03:57.570
the area of things.

00:03:57.570 --> 00:04:00.110
This thing has an area
of 5 square units.

00:04:00.110 --> 00:04:02.710
This thing has an area
of 10 square units.

00:04:02.710 --> 00:04:06.350
So this one we can actually
say has twice the area.

00:04:06.350 --> 00:04:08.810
The purple figure
had twice the area--

00:04:08.810 --> 00:04:11.600
it's 10 square units--
as the blue figure.

00:04:11.600 --> 00:04:16.009
It takes up twice the amount
of space on the screen.


Compound events example with tree diagram

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:02.660
If you flip three
fair coins, what

00:00:02.660 --> 00:00:05.590
is the probability that
you'll get at least two tails?

00:00:05.590 --> 00:00:08.170
The tree diagram below shows
all the possible outcomes

00:00:08.170 --> 00:00:09.840
of flipping three coins.

00:00:09.840 --> 00:00:11.270
At the top of the
tree, this shows

00:00:11.270 --> 00:00:13.857
us the two outcomes
for the first coin,

00:00:13.857 --> 00:00:15.440
and then given each
of those outcomes,

00:00:15.440 --> 00:00:17.350
it shows us what's possible
for the second coins.

00:00:17.350 --> 00:00:18.895
If we got a heads
on the first coin,

00:00:18.895 --> 00:00:21.020
we could get a heads or a
tails on the second coin.

00:00:21.020 --> 00:00:22.170
Got a tails on the
first coin, well,

00:00:22.170 --> 00:00:24.560
we could get a heads or a
tails on the second coin.

00:00:24.560 --> 00:00:26.060
And then for each
of those outcomes,

00:00:26.060 --> 00:00:29.020
it shows us the different
outcomes for the third coins.

00:00:29.020 --> 00:00:31.530
So let's just think
about where on this--

00:00:31.530 --> 00:00:34.460
how do we represent
getting at least two tails?

00:00:34.460 --> 00:00:37.510
Or what are the total outcomes,
and which of those meet

00:00:37.510 --> 00:00:40.300
our constraints of getting
at least two tails?

00:00:40.300 --> 00:00:43.590
So this node right
over here, this

00:00:43.590 --> 00:00:45.550
is getting a heads
on the third coin,

00:00:45.550 --> 00:00:47.620
a heads on the
second coin, we just

00:00:47.620 --> 00:00:50.650
have to follow up the tree,
and a heads on the first coin.

00:00:50.650 --> 00:00:53.290
So this is getting three heads,
so this is definitely not

00:00:53.290 --> 00:00:55.450
going to meet our constraint.

00:00:55.450 --> 00:00:58.300
This node right over
here, we have a head,

00:00:58.300 --> 00:00:59.780
head-- this is
often called a leaf

00:00:59.780 --> 00:01:02.050
if we're talking
about a tree diagram--

00:01:02.050 --> 00:01:03.380
a head, head, and a tail.

00:01:03.380 --> 00:01:04.440
So that's one tail.

00:01:04.440 --> 00:01:07.340
That doesn't meet our constraint
of at least two tails.

00:01:07.340 --> 00:01:08.470
What about this one here?

00:01:08.470 --> 00:01:10.600
Heads, tails, heads.

00:01:10.600 --> 00:01:13.270
Once again, only one tail,
so that doesn't meet it.

00:01:13.270 --> 00:01:14.750
Heads, tails, tails.

00:01:14.750 --> 00:01:16.210
This one does.

00:01:16.210 --> 00:01:17.210
So let me color this in.

00:01:17.210 --> 00:01:19.480
Let me color all the ones
that meet our constraints.

00:01:19.480 --> 00:01:21.146
This is getting a
tail on the third one,

00:01:21.146 --> 00:01:23.570
a tail on the second one,
and a heads on the first.

00:01:23.570 --> 00:01:25.930
So that's at least two tails.

00:01:25.930 --> 00:01:28.640
Here we have tails,
heads, heads.

00:01:28.640 --> 00:01:29.900
That doesn't meet it.

00:01:29.900 --> 00:01:31.510
Tails, heads, tails.

00:01:31.510 --> 00:01:32.710
That does.

00:01:32.710 --> 00:01:35.230
So let's color this one in.

00:01:35.230 --> 00:01:37.060
And then tails, tails.

00:01:37.060 --> 00:01:39.390
Well, if you got a tail on
the first and the second,

00:01:39.390 --> 00:01:41.724
then either of these are
going to meet the constraints,

00:01:41.724 --> 00:01:43.140
because you already
got two tails.

00:01:43.140 --> 00:01:44.780
So that one meets it.

00:01:44.780 --> 00:01:48.110
That you've got tails, tails,
heads, and tails, tails, tails.

00:01:48.110 --> 00:01:49.860
Both of them.

00:01:49.860 --> 00:01:51.620
So, let's go back
to the question.

00:01:51.620 --> 00:01:54.310
What is the probability that
you'll get at least two tails?

00:01:54.310 --> 00:01:56.440
Well, how many equally
likely outcomes are there?

00:01:56.440 --> 00:01:58.020
We're assuming this
is a fair coin.

00:01:58.020 --> 00:02:03.077
We see that there are
1, 2, 3, 4, 5, 6, 7,

00:02:03.077 --> 00:02:06.080
8 equally likely outcomes.

00:02:06.080 --> 00:02:09.389
And how many of these
outcomes met our constraints?

00:02:09.389 --> 00:02:11.760
1, 2, 3, 4.

00:02:11.760 --> 00:02:14.340
4 out of the 8.

00:02:14.340 --> 00:02:18.150
4/8, which could also be
viewed is equivalent to 1/2.

00:02:18.150 --> 00:02:22.999
The probability that I'll get
at least two tails is 1/2.


Compound events example using diagram

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:04.180
If you roll a six-sided
die and a four-sided die,

00:00:04.180 --> 00:00:08.020
what is the probability
that neither die shows a 1?

00:00:08.020 --> 00:00:10.330
The grid below shows all
the possible outcomes

00:00:10.330 --> 00:00:13.306
of rolling a six-sided
die and a four-sided die.

00:00:13.306 --> 00:00:15.180
And we're fairly familiar
with six-sided die.

00:00:15.180 --> 00:00:18.884
Those are the cubes that we're
used to from the board games

00:00:18.884 --> 00:00:20.800
that we've been playing
our entire lives, that

00:00:20.800 --> 00:00:22.170
look something like that.

00:00:22.170 --> 00:00:24.670
And your roll is
whatever is the face up.

00:00:24.670 --> 00:00:27.120
You're probably less
familiar with four-sided die.

00:00:27.120 --> 00:00:30.450
These are more like these
triangular pyramids.

00:00:30.450 --> 00:00:32.130
They look something like this.

00:00:32.130 --> 00:00:35.300
So it has 1, 2, 3, and then
the one side on the bottom.

00:00:35.300 --> 00:00:37.880
And here your roll is
actually the face, the side,

00:00:37.880 --> 00:00:40.107
that is facing the bottom.

00:00:40.107 --> 00:00:41.690
And so what you have
here is a picture

00:00:41.690 --> 00:00:43.200
of all the possible rolls.

00:00:43.200 --> 00:00:46.070
They have the six-sided die, the
side that's facing on the top.

00:00:46.070 --> 00:00:47.700
And then on the
triangle, this shows

00:00:47.700 --> 00:00:51.140
what is the result of the
roll of the four-sided die.

00:00:51.140 --> 00:00:53.620
So this actually would be
the side that's facing down.

00:00:53.620 --> 00:00:55.000
Let's actually try to
answer their question,

00:00:55.000 --> 00:00:57.620
because they gave us all of the
possible outcomes of rolling

00:00:57.620 --> 00:01:00.800
a six-sided die and
a four-sided die.

00:01:00.800 --> 00:01:02.810
So we want to know,
what is the probability

00:01:02.810 --> 00:01:05.670
that neither dies shows a 1?

00:01:05.670 --> 00:01:07.440
So how many total
outcomes do we have?

00:01:07.440 --> 00:01:11.485
Let's see-- 1, 2, 3,
4 by 1, 2, 3, 4, 5, 6.

00:01:11.485 --> 00:01:13.190
4 times 6, that makes sense.

00:01:13.190 --> 00:01:15.990
There's six possible outcomes
for the six-sided die,

00:01:15.990 --> 00:01:18.900
four possible outcomes
for the four-sided die.

00:01:18.900 --> 00:01:23.556
4 times 6-- so there's 24
total possible outcomes.

00:01:23.556 --> 00:01:24.930
And then how many
of the outcomes

00:01:24.930 --> 00:01:29.050
meet our constraint that
neither died shows a 1?

00:01:29.050 --> 00:01:29.832
So this one.

00:01:29.832 --> 00:01:31.290
So anything in this
row isn't going

00:01:31.290 --> 00:01:34.070
to meet our constraint,
because these have a 1

00:01:34.070 --> 00:01:35.462
on the four-sided die.

00:01:35.462 --> 00:01:36.920
And nothing in this
column is going

00:01:36.920 --> 00:01:38.836
to meet our constraint,
because these have a 1

00:01:38.836 --> 00:01:40.290
on the six-sided die.

00:01:40.290 --> 00:01:43.710
But all of these,
everything left over,

00:01:43.710 --> 00:01:49.330
these are going to
meet our constraints.

00:01:49.330 --> 00:01:52.130
These are going to
meet our constraints.

00:01:52.130 --> 00:01:53.874
And how many of these
outcomes are there?

00:01:53.874 --> 00:01:54.540
Well, let's see.

00:01:54.540 --> 00:01:59.190
There's 3 by 4-- 3 times 4.

00:01:59.190 --> 00:02:03.290
So there's 12 possible
outcomes here.

00:02:03.290 --> 00:02:05.350
12/24 is the probability.

00:02:05.350 --> 00:02:07.490
You could say there's
a 12/24 probability.

00:02:07.490 --> 00:02:09.940
But if you wanted to write
it as an equivalent fraction

00:02:09.940 --> 00:02:13.030
that's using lower numbers, this
is the same thing as-- divide

00:02:13.030 --> 00:02:15.295
the numerator and
denominator by 12-- 1/2.

00:02:18.530 --> 00:02:24.100
So the probability that
neither die shows a 1 is 1/2.


Making sense of probabilities

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:05.600
Categorize each of the following
statements as true or false.

00:00:05.600 --> 00:00:07.850
If an event is
unlikely to occur,

00:00:07.850 --> 00:00:10.630
its probability
will be negative.

00:00:10.630 --> 00:00:11.210
No.

00:00:11.210 --> 00:00:13.870
Your probability can
never be negative.

00:00:13.870 --> 00:00:19.150
Even if something is impossible,
then the probability will be 0.

00:00:19.150 --> 00:00:22.270
If it's highly unlikely,
it'll be close to 0,

00:00:22.270 --> 00:00:24.940
but it'll be still positive.

00:00:24.940 --> 00:00:28.050
Probabilities are
always between 0 and 1.

00:00:28.050 --> 00:00:30.100
0 means it is impossible.

00:00:30.100 --> 00:00:31.330
It cannot happen.

00:00:31.330 --> 00:00:35.240
1 means that it's a certainty
that it will happen.

00:00:35.240 --> 00:00:38.060
And the values in between
say how likely or unlikely

00:00:38.060 --> 00:00:38.760
they are.

00:00:38.760 --> 00:00:40.780
Probabilities
greater than 0.5 say,

00:00:40.780 --> 00:00:42.890
hey, it's more likely than not.

00:00:42.890 --> 00:00:47.520
Probabilities less than 0.5,
it's more unlikely than not.

00:00:47.520 --> 00:00:49.810
And probabilities
of 0.5 say, hey,

00:00:49.810 --> 00:00:55.360
it's just as likely to
happen or as to not happen.

00:00:55.360 --> 00:00:58.180
An event with
probability 3/5 is less

00:00:58.180 --> 00:01:01.650
likely to occur than an
event with probability 3/4.

00:01:01.650 --> 00:01:05.645
So 3/5 is a smaller
number than 3/4.

00:01:05.645 --> 00:01:06.770
We have the same numerator.

00:01:06.770 --> 00:01:08.230
We have a larger denominator.

00:01:08.230 --> 00:01:11.560
3 is a smaller fraction
of 5 than it is of 4.

00:01:11.560 --> 00:01:14.310
So since 3/5 is a
smaller number than 3/4--

00:01:14.310 --> 00:01:17.210
and it's more obvious if you
see it as a decimal-- 3/5

00:01:17.210 --> 00:01:20.180
is 0.6 while 3/4 is 0.75.

00:01:20.180 --> 00:01:22.090
So this is less likely to occur.

00:01:22.090 --> 00:01:23.920
This is true.

00:01:23.920 --> 00:01:28.290
The probability of getting 1,000
tails in 1,000 tosses of coin

00:01:28.290 --> 00:01:30.310
is 0.

00:01:30.310 --> 00:01:32.804
So getting 1,000
tails in 1,000 tosses.

00:01:32.804 --> 00:01:35.470
In future videos, we'll actually
talk about how to calculate it.

00:01:35.470 --> 00:01:38.120
It's a very, very
low probability,

00:01:38.120 --> 00:01:40.040
but it still is possible.

00:01:40.040 --> 00:01:41.240
It still is possible.

00:01:41.240 --> 00:01:42.980
So the probability is not 0.

00:01:42.980 --> 00:01:46.200
It could be a very small
positive number close to 0,

00:01:46.200 --> 00:01:48.080
but it is not 0.

00:01:48.080 --> 00:01:49.480
It's actually possible.

00:01:49.480 --> 00:01:53.230
A probability of 0 means
that it is impossible.

00:01:53.230 --> 00:01:55.540
The probability of
winning the lottery is 0.

00:01:55.540 --> 00:01:57.820
Once again, you can
win the lottery.

00:01:57.820 --> 00:02:01.680
It's within the realm of
possibility, so it's not 0.

00:02:01.680 --> 00:02:03.030
It's a very small number.

00:02:03.030 --> 00:02:06.000
It could be close to
0, but it's not 0.

00:02:06.000 --> 00:02:09.250
So this is false.

00:02:09.250 --> 00:02:11.400
The probability
that the sun will

00:02:11.400 --> 00:02:16.020
rise in the west tomorrow
on earth is 0.25.

00:02:16.020 --> 00:02:18.190
So you probably already
know, or hopefully know,

00:02:18.190 --> 00:02:19.970
that the sun rises in the east.

00:02:19.970 --> 00:02:22.720
And assuming that the Earth
continues to rotate in the way

00:02:22.720 --> 00:02:26.570
that it rotates, it will
continue to rise in the east.

00:02:26.570 --> 00:02:30.670
Now, there is some ultra,
ultra small probability

00:02:30.670 --> 00:02:34.030
that an asteroid of the
right mass or a meteorite

00:02:34.030 --> 00:02:36.900
of the right mass hits
earth in the exact right way

00:02:36.900 --> 00:02:39.210
to somehow reverse
its spin to maybe make

00:02:39.210 --> 00:02:40.910
the sun rise in the
west, but that's

00:02:40.910 --> 00:02:44.110
a super, super, super,
duper small probability.

00:02:44.110 --> 00:02:48.460
That probability is very, very,
very, very, very close to 0.

00:02:48.460 --> 00:02:50.640
It's not going to be 0.25.

00:02:50.640 --> 00:02:53.980
0.25 is a fairly
large probability.

00:02:53.980 --> 00:02:57.430
So if this was
actually the case, then

00:02:57.430 --> 00:03:00.150
as we live our lives,
we would be seeing days

00:03:00.150 --> 00:03:01.860
where the sun rises on the west.

00:03:01.860 --> 00:03:06.030
And we clearly don't see
that, so this is false.

00:03:06.030 --> 00:03:09.256
The probability that the Obama
family's beloved dog, Bo,

00:03:09.256 --> 00:03:11.130
will gain access to the
President's computer,

00:03:11.130 --> 00:03:15.340
log on to dogancestors.com,
and leave the country in search

00:03:15.340 --> 00:03:19.660
of his long-lost relatives
in Portugal is 0.01.

00:03:19.660 --> 00:03:22.340
So this seems, if
not an impossibility,

00:03:22.340 --> 00:03:26.700
a near impossibility,
that the dog somehow

00:03:26.700 --> 00:03:28.480
will know how to
log onto a computer.

00:03:28.480 --> 00:03:30.700
This is probably
an impossibility.

00:03:30.700 --> 00:03:33.810
And then, actually, be
able to read something

00:03:33.810 --> 00:03:36.960
on a website, type on a website,
and then read on a website?

00:03:36.960 --> 00:03:39.240
This seems close to an
impossibility, if not

00:03:39.240 --> 00:03:40.450
an impossibility.

00:03:40.450 --> 00:03:42.930
And then, somehow getting a
ticket and leaving the country,

00:03:42.930 --> 00:03:44.870
and somehow the Secret
Service lets them?

00:03:44.870 --> 00:03:46.790
This seems pretty
close to impossible.

00:03:46.790 --> 00:03:49.940
So this probability is
pretty darn close to 0.

00:03:49.940 --> 00:03:51.620
If it's impossible,
it would be 0,

00:03:51.620 --> 00:03:55.710
but it'd be much, much
closer to 0 than 0.01.

00:03:55.710 --> 00:04:00.610
This is a small probability,
but this is not super unlikely.

00:04:00.610 --> 00:04:03.890
If this was the case, this means
that if we had 100 realities

00:04:03.890 --> 00:04:07.090
like we had today, 1 out
of those 100 realities,

00:04:07.090 --> 00:04:10.340
it would be reasonable to
expect this scenario to happen,

00:04:10.340 --> 00:04:12.830
and we know it's a much, much
lower probability than that.

00:04:12.830 --> 00:04:15.220
This probability is
very, very close to 0.

00:04:15.220 --> 00:04:19.550
If this said 0.000 and it had
like twenty zeroes and then

00:04:19.550 --> 00:04:22.500
a 1, I'd have to think
about a little bit more,

00:04:22.500 --> 00:04:26.100
but I would say that this is
still false, that even 0.01,

00:04:26.100 --> 00:04:31.910
as high as it is, is
still a low probability.

00:04:31.910 --> 00:04:34.320
So we only got one
true statement,

00:04:34.320 --> 00:04:37.880
and we got a bunch
of false ones.


Comparing probabilities example t2

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.720
The probability of winning
the Powerball lottery

00:00:02.720 --> 00:00:07.550
is 1 in 175,223,511.

00:00:07.550 --> 00:00:11.710
I'm assuming this is when you
buy a Powerball lottery ticket.

00:00:11.710 --> 00:00:12.686
This is a probability.

00:00:12.686 --> 00:00:14.560
Then they tell us the
probability of becoming

00:00:14.560 --> 00:00:20.560
a billionaire in the
United States is 0.000001.

00:00:20.560 --> 00:00:21.620
Which is more likely?

00:00:21.620 --> 00:00:23.200
And I assume this
is like saying,

00:00:23.200 --> 00:00:27.060
if you were to randomly
select a person at birth, what

00:00:27.060 --> 00:00:28.980
is the probability
that that person is

00:00:28.980 --> 00:00:30.530
going to become a billionaire?

00:00:30.530 --> 00:00:32.490
And that's this number
right over here.

00:00:32.490 --> 00:00:35.170
And just looking at this,
it's kind of hard to compare.

00:00:35.170 --> 00:00:37.550
So let me try to write the
second number, this decimal,

00:00:37.550 --> 00:00:38.134
as a fraction.

00:00:38.134 --> 00:00:39.508
And then we might
be able to make

00:00:39.508 --> 00:00:40.970
more of a direct comparison.

00:00:40.970 --> 00:00:44.140
So 0 point-- and let's
see, we have five 0's.

00:00:44.140 --> 00:00:47.650
1, 2, 3, 4, 5.

00:00:47.650 --> 00:00:49.300
And then we have a 1.

00:00:49.300 --> 00:00:52.110
And so let's just think about
what place that 1 is in.

00:00:52.110 --> 00:00:53.940
This is in the tenths place.

00:00:53.940 --> 00:00:57.710
That's the tenths place, that's
the hundredths place, that's

00:00:57.710 --> 00:01:03.220
the thousandths place, that's
the ten thousandths place,

00:01:03.220 --> 00:01:07.470
and that's the hundred
thousandths place.

00:01:07.470 --> 00:01:09.890
And this right over here
is the millionths place.

00:01:12.630 --> 00:01:16.300
So this 1 literally
represents 1/1,000,000.

00:01:16.300 --> 00:01:18.630
So this is the exact same thing.

00:01:18.630 --> 00:01:22.660
This is equivalent
to 1 over 1 million.

00:01:22.660 --> 00:01:25.100
This is 1/1,000,000.

00:01:25.100 --> 00:01:31.030
This is 1/775,223,511.

00:01:31.030 --> 00:01:34.860
So this whole thing,
this whole expression

00:01:34.860 --> 00:01:36.337
here is a much smaller value.

00:01:36.337 --> 00:01:37.420
We had the same numerator.

00:01:37.420 --> 00:01:39.200
We have a 1 in both cases.

00:01:39.200 --> 00:01:40.960
This denominator is much larger.

00:01:40.960 --> 00:01:45.640
So this entire fraction
is much, much smaller,

00:01:45.640 --> 00:01:49.620
actually, by roughly a
factor of 175-- much smaller.

00:01:49.620 --> 00:01:52.430
This is roughly 1
over 175 million.

00:01:52.430 --> 00:01:54.520
This is 1 over 1 million.

00:01:54.520 --> 00:01:56.210
So which is more likely?

00:01:56.210 --> 00:01:59.310
Which is more likely if you
were to pick a random person who

00:01:59.310 --> 00:02:02.230
bought a lottery ticket, and
the probability of them winning

00:02:02.230 --> 00:02:05.130
the Powerball lottery, or if
you were to pick a random person

00:02:05.130 --> 00:02:07.940
and you'd say the probability
of them becoming a billionaire?

00:02:07.940 --> 00:02:12.080
It's actually far more likely--
roughly 175 times more likely--

00:02:12.080 --> 00:02:14.590
that that person could
become a billionaire.

00:02:14.590 --> 00:02:17.472
So much more likely
becoming a billionaire.


Area and perimeter word problem: table dimensions

WEBVTT
Kind: captions
Language: en

00:00:00.980 --> 00:00:03.130
Charles built a
rectangular table

00:00:03.130 --> 00:00:09.160
that has a perimeter of 20 feet
and an area of 24 square feet.

00:00:09.160 --> 00:00:11.870
The table is longer
than it is wide.

00:00:11.870 --> 00:00:14.650
What are the length
and width of the table?

00:00:14.650 --> 00:00:17.190
The length and width
are whole numbers.

00:00:17.190 --> 00:00:18.790
So it's longer than it is wide.

00:00:18.790 --> 00:00:20.550
So let's draw this table here.

00:00:20.550 --> 00:00:23.470
So the table might
look something

00:00:23.470 --> 00:00:33.190
like this-- where this dimension
right over here is the length.

00:00:33.190 --> 00:00:36.849
So this distance right
over here is the length.

00:00:36.849 --> 00:00:38.390
We could also write
length here if we

00:00:38.390 --> 00:00:40.250
want to show that
this is the same,

00:00:40.250 --> 00:00:42.880
these two sides have
the same length.

00:00:42.880 --> 00:00:45.530
And then we could call this
dimension right over here--

00:00:45.530 --> 00:00:46.280
this is the width.

00:00:49.400 --> 00:00:52.600
And this is also, of
course, the width, as well.

00:00:52.600 --> 00:00:54.140
This is a rectangle.

00:00:54.140 --> 00:00:56.610
So these two sides are
going to be the same.

00:00:56.610 --> 00:00:58.950
Now, they tell us
that the perimeter

00:00:58.950 --> 00:01:01.670
is 20 feet, which is
another way of saying

00:01:01.670 --> 00:01:13.790
that the width plus the width
plus the length plus the length

00:01:13.790 --> 00:01:15.085
is equal to 20.

00:01:17.690 --> 00:01:20.100
And they tell us that the
area is 24 square feet.

00:01:20.100 --> 00:01:22.900
That's another way of saying
that the width times the length

00:01:22.900 --> 00:01:24.950
is going to be 24.

00:01:24.950 --> 00:01:26.540
So we could write that down.

00:01:26.540 --> 00:01:37.660
Width times the length is
going to be equal to 24.

00:01:37.660 --> 00:01:40.160
Now, there's a lot of ways
to solve this problem.

00:01:40.160 --> 00:01:42.130
And later on when you
learn more algebra,

00:01:42.130 --> 00:01:44.190
there's fancy algebraic
ways to do that.

00:01:44.190 --> 00:01:45.760
But we won't have
to resort to this.

00:01:45.760 --> 00:01:48.260
They tell us that the length
and width are whole numbers.

00:01:48.260 --> 00:01:50.634
So we really should just be
able to try out some numbers,

00:01:50.634 --> 00:01:53.760
because we know that the
width times the length is 24.

00:01:53.760 --> 00:01:55.760
So we just have try out
all of the whole numbers

00:01:55.760 --> 00:01:58.220
that when I were to take
their product, I get to 24.

00:01:58.220 --> 00:02:01.290
Essentially, the
factors of 24, and then

00:02:01.290 --> 00:02:03.750
figure out which of those
satisfy the perimeter up here.

00:02:03.750 --> 00:02:06.570
Where if I take the width plus
the width-- essentially 2 times

00:02:06.570 --> 00:02:08.660
the width plus 2
times the length--

00:02:08.660 --> 00:02:11.610
I am going to get to 20.

00:02:11.610 --> 00:02:13.290
So let's figure that out.

00:02:13.290 --> 00:02:15.450
So let me make two columns here.

00:02:15.450 --> 00:02:18.330
So one column, I'm going
to call it a width column.

00:02:18.330 --> 00:02:21.630
Another column I'm
going to call it length.

00:02:21.630 --> 00:02:25.650
And then I'm going to
write down the perimeter.

00:02:25.650 --> 00:02:27.080
I'll write perimeter.

00:02:27.080 --> 00:02:30.320
I'll just shorten it
with per-- maybe peri.

00:02:30.320 --> 00:02:32.960
I'll just write out the
whole word, perimeter.

00:02:32.960 --> 00:02:34.775
And then let's write out area.

00:02:38.400 --> 00:02:41.340
Actually, let's do all that-- I
could write it just like that.

00:02:41.340 --> 00:02:45.160
Let me try out--
make it a table here.

00:02:45.160 --> 00:02:47.170
So I have a table here.

00:02:47.170 --> 00:02:49.260
And then I can try things out.

00:02:49.260 --> 00:02:52.080
And what we can do is just make
sure that everything we try out

00:02:52.080 --> 00:02:55.250
has an area of 24 square feet.

00:02:55.250 --> 00:02:58.450
So let's just think
about the factors of 24.

00:02:58.450 --> 00:03:01.690
Well, it could be 1 and 24.

00:03:01.690 --> 00:03:06.790
So this literally could be 1, a
width of 1, and a length of 24.

00:03:06.790 --> 00:03:10.430
1 times 24 is 24.

00:03:10.430 --> 00:03:15.472
And they tell us that the
length is longer than the width,

00:03:15.472 --> 00:03:17.180
that the table is
longer than it is wide.

00:03:17.180 --> 00:03:19.620
So we want the larger
number under length.

00:03:19.620 --> 00:03:22.570
So let's see, 1 times 24 is 24.

00:03:22.570 --> 00:03:26.520
But what is 1 plus
1 plus 24 plus 24?

00:03:26.520 --> 00:03:30.830
Well, that's going to be
2 plus 48, which is 50.

00:03:30.830 --> 00:03:32.360
So this doesn't
meet our condition

00:03:32.360 --> 00:03:34.100
that the perimeter is 20.

00:03:34.100 --> 00:03:35.450
So let's cross that out.

00:03:35.450 --> 00:03:40.020
So this one right over
here does not work out.

00:03:40.020 --> 00:03:41.900
Let's try the other
factors of 24.

00:03:41.900 --> 00:03:45.040
It could be 2 and 12.

00:03:45.040 --> 00:03:48.380
Once again, 2 times 12 is 24.

00:03:48.380 --> 00:03:49.950
But what's 2 plus 2?

00:03:49.950 --> 00:03:52.560
It's 4, plus 12 plus 12.

00:03:52.560 --> 00:03:54.250
So it's 4 plus 24.

00:03:54.250 --> 00:03:55.640
That's going to be 28.

00:03:55.640 --> 00:03:58.550
Well, that doesn't meet
our perimeter constraint.

00:03:58.550 --> 00:04:01.590
So we can-- that's
not going to be right.

00:04:01.590 --> 00:04:07.860
Well, what about-- let's see,
3 times 8 is also equal to 24.

00:04:07.860 --> 00:04:12.040
And what's 3 plus 3-- is
6, plus 8 plus 8 is 16.

00:04:12.040 --> 00:04:14.102
6 plus 16 is 22.

00:04:14.102 --> 00:04:15.810
Well, we're getting
close, but it's still

00:04:15.810 --> 00:04:17.769
not a perimeter of 20.

00:04:17.769 --> 00:04:19.779
So that's not going to be right.

00:04:19.779 --> 00:04:23.760
Now, what about 4 and 6?

00:04:23.760 --> 00:04:27.560
Once again, 4 times 6 is 24.

00:04:27.560 --> 00:04:30.720
And what's 4 plus
4 plus 6 plus 6?

00:04:30.720 --> 00:04:36.170
Well, that's 8 plus 12,
which is indeed equal to 20.

00:04:36.170 --> 00:04:37.650
So that works out.

00:04:37.650 --> 00:04:41.900
Our width is going to be
4 feet, and our length

00:04:41.900 --> 00:04:45.510
is going to be 6 feet.


Area and perimeter word problem: width of a dog pen

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:05.210
Mike built a rectangular
dog pen that is 21 feet long

00:00:05.210 --> 00:00:08.850
and has a perimeter of 78 feet.

00:00:08.850 --> 00:00:12.050
What is the width
of Mike's dog pen?

00:00:12.050 --> 00:00:14.670
So let's draw what
it might look like.

00:00:14.670 --> 00:00:16.930
So it might look
something like this.

00:00:16.930 --> 00:00:22.160
I'll do my best to draw it, a
rectangle or potential dog pen.

00:00:22.160 --> 00:00:25.330
So let's say that this is
the dog pen right over here.

00:00:25.330 --> 00:00:26.900
And they give us its length.

00:00:26.900 --> 00:00:28.540
Its length is 21 feet.

00:00:31.290 --> 00:00:34.720
So this distance right
over here is 21 feet.

00:00:34.720 --> 00:00:37.010
And of course, if that
distance is 21 feet,

00:00:37.010 --> 00:00:43.080
this distance is also
going to be 21 feet.

00:00:43.080 --> 00:00:46.210
Now, they also tell us that
the perimeter is 78 feet.

00:00:46.210 --> 00:00:50.230
So the perimeter is 78 feet.

00:00:50.230 --> 00:00:52.120
So we need to figure
out the width.

00:00:52.120 --> 00:00:56.240
And the width would be this
length-- or this width,

00:00:56.240 --> 00:00:58.570
I should say-- this
width, which is

00:00:58.570 --> 00:01:02.390
going to be the same thing as
that width right over here.

00:01:02.390 --> 00:01:03.480
So let's think about.

00:01:03.480 --> 00:01:05.379
And we know how we
figure out the perimeter.

00:01:05.379 --> 00:01:07.870
If we added this
distance, 21 feet,

00:01:07.870 --> 00:01:12.070
to this distance, the width,
plus 21 feet to the width

00:01:12.070 --> 00:01:15.050
again, we're going
to get to 78 feet.

00:01:15.050 --> 00:01:18.426
So one way to back into
what the width is is to say,

00:01:18.426 --> 00:01:20.300
well, look if we sum up
all of these lengths,

00:01:20.300 --> 00:01:21.424
we're going to get 28 feet.

00:01:21.424 --> 00:01:25.490
So let's at least subtract these
two orange lengths, the two 21

00:01:25.490 --> 00:01:27.630
feet, from the 78 feet.

00:01:27.630 --> 00:01:30.250
And then we figure out how
much length essentially

00:01:30.250 --> 00:01:33.020
these two widths
have to make up.

00:01:33.020 --> 00:01:34.730
So let's subtract.

00:01:34.730 --> 00:01:39.890
Let's do 78 minus 21 minus 21.

00:01:39.890 --> 00:01:41.580
And what's that
going to be equal to?

00:01:41.580 --> 00:01:45.540
So that's 78 minus 42.

00:01:45.540 --> 00:01:49.500
And 78 minus 42-- I don't want
to make a careless mistake-- 78

00:01:49.500 --> 00:01:53.440
minus 42 is going
to get me to 36.

00:01:53.440 --> 00:01:55.460
So this is equal to 36.

00:01:55.460 --> 00:02:00.300
And let's remind
ourselves what 36 feet is.

00:02:00.300 --> 00:02:03.540
That's the total length-- or
I should say the total width--

00:02:03.540 --> 00:02:06.140
if we were to take the
width down here and add it

00:02:06.140 --> 00:02:07.604
to the width up here.

00:02:07.604 --> 00:02:08.770
Now, we know something else.

00:02:08.770 --> 00:02:11.990
These are the exact same
width, that this distance

00:02:11.990 --> 00:02:13.850
is the same as this distance.

00:02:13.850 --> 00:02:17.990
And if we add them
together, we get to 36.

00:02:17.990 --> 00:02:19.750
So what are two
distances, or what

00:02:19.750 --> 00:02:24.480
are two numbers that I can
add together to get to 36?

00:02:24.480 --> 00:02:34.680
Well, we know that 18
plus 18 is equal to 36.

00:02:34.680 --> 00:02:37.230
So this one down
here must be 18.

00:02:37.230 --> 00:02:40.790
And this one up here
must also be 18.

00:02:40.790 --> 00:02:42.070
And we can verify that.

00:02:42.070 --> 00:02:44.290
Try adding all of
these up together.

00:02:44.290 --> 00:02:45.890
You add the two widths.

00:02:45.890 --> 00:02:48.070
You get 18 plus 18.

00:02:48.070 --> 00:02:53.240
And you add the two
lengths, plus 21 plus 21.

00:02:53.240 --> 00:02:58.290
And you are left with 36, which
is exactly what we got here.

00:02:58.290 --> 00:03:01.715
This is what we needed the two
widths to add up to-- plus 42.

00:03:05.220 --> 00:03:08.580
Which is, of course,
equal to-- 36

00:03:08.580 --> 00:03:12.874
plus 42 is equal to 78,
which is the perimeter.


Area and perimeter word problem: comparing areas of rectangles

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:05.950
Mary's rectangular poster
is 36 inches by 20 inches.

00:00:05.950 --> 00:00:10.900
Susan's rectangular poster
is 26 inches by 30 inches.

00:00:10.900 --> 00:00:15.325
Which poster has a larger area
and by how many square inches?

00:00:15.325 --> 00:00:16.450
So let's think about these.

00:00:16.450 --> 00:00:19.120
So this is Mary's poster.

00:00:19.120 --> 00:00:23.050
Mary's poster is 36
inches by 20 inches.

00:00:23.050 --> 00:00:27.010
So it's 36 inches by 20 inches.

00:00:31.830 --> 00:00:33.930
So it might look
something like that.

00:00:33.930 --> 00:00:40.500
So the area is going to be
36 times 20 square inches.

00:00:40.500 --> 00:00:43.430
36 times 2 is 72.

00:00:43.430 --> 00:00:47.950
So 36 times 20 is going
to be 720 square inches.

00:00:50.910 --> 00:00:53.280
Now let's think about
Susan's situation.

00:00:53.280 --> 00:00:55.710
So let's draw Susan's poster.

00:00:55.710 --> 00:01:00.350
Susan's poster is 26
inches by 30 inches,

00:01:00.350 --> 00:01:05.860
so 26 inches by 30 inches.

00:01:05.860 --> 00:01:09.516
So Susan's poster might
look something like that.

00:01:09.516 --> 00:01:13.280
That's Susan's poster, my best
attempt to draw a rectangle.

00:01:13.280 --> 00:01:15.000
What's the area here?

00:01:15.000 --> 00:01:20.210
The area is 26 times
30 square inches,

00:01:20.210 --> 00:01:21.640
which is equal
to-- let's actually

00:01:21.640 --> 00:01:25.360
multiply this one
out-- 26 times 30.

00:01:28.470 --> 00:01:31.500
We could do 26 times 3 and
essentially add a 0 there.

00:01:31.500 --> 00:01:35.650
So 3 times 6 is 18.

00:01:35.650 --> 00:01:39.546
3 times 2 is 6, plus 1 is 78.

00:01:39.546 --> 00:01:41.920
And actually, I could have
probably done that in my head.

00:01:41.920 --> 00:01:47.060
3 times 20 is 60, plus 3
times 6 is 18, gets us 78.

00:01:47.060 --> 00:01:48.630
But this isn't 3 times 26.

00:01:48.630 --> 00:01:50.820
3 times 26 would be 78.

00:01:50.820 --> 00:01:53.620
30 times 26 is 780.

00:01:53.620 --> 00:01:58.510
So it's 780 square inches.

00:01:58.510 --> 00:02:02.340
So whose poster, which
poster has a larger area?

00:02:02.340 --> 00:02:03.650
Susan's.

00:02:03.650 --> 00:02:05.810
Susan's poster
has a larger area.

00:02:05.810 --> 00:02:07.060
And by how many square inches?

00:02:07.060 --> 00:02:09.889
Well, hers is 780 square
inches while Mary's is

00:02:09.889 --> 00:02:12.200
720 square inches.

00:02:12.200 --> 00:02:16.070
So it's by 60 square inches.

00:02:16.070 --> 00:02:19.725
780 minus 720 would be 60.


Choose pounds or ounces to measure weight

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.000
Identify the most
reasonable units

00:00:03.000 --> 00:00:05.970
to measure the weight
of the choices below.

00:00:05.970 --> 00:00:09.100
So we have an average
adult, a slice of bread,

00:00:09.100 --> 00:00:13.090
a kitchen table,
or a pack of gum.

00:00:13.090 --> 00:00:16.200
So I am an average
adult, and I know

00:00:16.200 --> 00:00:19.020
that I weigh about 160 pounds.

00:00:19.020 --> 00:00:21.240
So I know that pounds are
a pretty good measurement

00:00:21.240 --> 00:00:23.710
for about something
the weight of an adult.

00:00:23.710 --> 00:00:27.430
So I'll put adults in that
category right over there.

00:00:27.430 --> 00:00:30.820
Now we have a single
slice of bread.

00:00:30.820 --> 00:00:34.610
So even a full loaf of
bread doesn't weigh a lot.

00:00:34.610 --> 00:00:38.010
Maybe that gets close to a
pound, but a slice of bread

00:00:38.010 --> 00:00:39.680
is going to be
less than a pound.

00:00:39.680 --> 00:00:44.000
So I would do this in ounces.

00:00:44.000 --> 00:00:44.880
A kitchen table.

00:00:44.880 --> 00:00:48.160
Well, a kitchen table
kind of weighs the same--

00:00:48.160 --> 00:00:50.460
it could, if it's a very
large kitchen table,

00:00:50.460 --> 00:00:52.350
could weigh the same
or more than an adult.

00:00:52.350 --> 00:00:55.780
So I would do this in pounds.

00:00:55.780 --> 00:00:56.950
And then a pack of gum.

00:00:56.950 --> 00:01:00.190
So once again, a pack
of gum is fairly light.

00:01:00.190 --> 00:01:02.050
If you've ever been
to the gym and held

00:01:02.050 --> 00:01:04.290
even a 1-pound or
2-pound weight,

00:01:04.290 --> 00:01:06.100
it weighs more
than a pack of gum.

00:01:06.100 --> 00:01:08.960
So a pack of gum is going
to be less than a pound,

00:01:08.960 --> 00:01:12.880
so ounces seem to
make sense here.


Unit measurement word problem: distance (US customary)

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.450
On Judy's way home from
school yesterday, she

00:00:03.450 --> 00:00:06.430
ran for half the distance.

00:00:06.430 --> 00:00:11.700
Then she skipped for 1/4
of the remaining distance

00:00:11.700 --> 00:00:15.260
and walked the rest
of the way home.

00:00:15.260 --> 00:00:18.840
She walked for a total 360 feet.

00:00:18.840 --> 00:00:22.070
What is the distance between
school and Judy's home?

00:00:22.070 --> 00:00:25.149
And they say write the
answer in the yards.

00:00:25.149 --> 00:00:26.940
So this is a little
tricky right over here.

00:00:26.940 --> 00:00:30.250
They say, 360 feet, write
the answer in yards.

00:00:30.250 --> 00:00:34.000
And we have to remind
ourselves that one yard is

00:00:34.000 --> 00:00:36.230
equal to 3 feet.

00:00:36.230 --> 00:00:38.730
So just so we make sure we don't
make that careless mistake,

00:00:38.730 --> 00:00:42.220
let's put all of our
information in yards right now.

00:00:42.220 --> 00:00:45.180
So 360 feet is how many yards?

00:00:45.180 --> 00:00:47.840
Well there's 3 feet per yard.

00:00:47.840 --> 00:00:51.510
So we're just going to
divide 360 feet by 3.

00:00:51.510 --> 00:00:57.990
And so 360 divided
by 3 is 120 yards.

00:00:57.990 --> 00:01:02.500
And now we can do the whole
problem in terms of yards.

00:01:02.500 --> 00:01:05.710
So let's think about it.

00:01:05.710 --> 00:01:08.340
So let me say that this
is her entire distance.

00:01:08.340 --> 00:01:10.150
This is on her way
home from school.

00:01:12.720 --> 00:01:16.110
And then let me draw a line that
shows the distance all the way

00:01:16.110 --> 00:01:17.130
home.

00:01:17.130 --> 00:01:19.610
So I want to draw this
as large as possible,

00:01:19.610 --> 00:01:22.670
so we have room to
write inside of it.

00:01:22.670 --> 00:01:24.060
So that's school.

00:01:24.060 --> 00:01:26.770
And that is home.

00:01:26.770 --> 00:01:29.370
And they first tells us that
she ran for half the distance.

00:01:29.370 --> 00:01:31.660
So let's split this in two.

00:01:31.660 --> 00:01:33.110
Let's split that in two.

00:01:33.110 --> 00:01:35.410
And so she ran for
half the distance.

00:01:35.410 --> 00:01:37.720
This is the distance
that she ran.

00:01:37.720 --> 00:01:39.150
It's half the total distance.

00:01:42.370 --> 00:01:45.380
Then they say, then
she skipped for one

00:01:45.380 --> 00:01:48.460
fourth of the
remaining distance.

00:01:48.460 --> 00:01:50.180
So let's be clear,
the remaining distance

00:01:50.180 --> 00:01:53.110
is what I haven't
filled in orange yet.

00:01:53.110 --> 00:01:56.090
So let's mark that off into
fourths of the remaining

00:01:56.090 --> 00:01:56.590
distance.

00:01:56.590 --> 00:01:58.214
So that is half the
remaining distance.

00:02:01.530 --> 00:02:04.466
That's just 1/4, 2/4, 3/4.

00:02:04.466 --> 00:02:05.840
So we took the
remaining distance

00:02:05.840 --> 00:02:07.910
and we divided it into fourths.

00:02:07.910 --> 00:02:10.270
And they say, then
she skipped for 1/4

00:02:10.270 --> 00:02:12.200
of the remaining distance.

00:02:12.200 --> 00:02:14.300
So this is the remaining
distance after running.

00:02:14.300 --> 00:02:19.110
She skipped for 1/4 of it.

00:02:19.110 --> 00:02:21.980
And then she walked the
rest of the way home.

00:02:21.980 --> 00:02:24.060
So she walked to the
rest of the way home,

00:02:24.060 --> 00:02:27.990
this is the distance
that she walked.

00:02:27.990 --> 00:02:33.900
And this distance right
over here is 120 yards.

00:02:33.900 --> 00:02:37.420
What is the distance between
school and Judy's home?

00:02:37.420 --> 00:02:40.369
So there's a bunch of ways that
you could think about this.

00:02:40.369 --> 00:02:42.160
When we talked about
the remaining distance

00:02:42.160 --> 00:02:44.770
after running, so this
was the remaining distance

00:02:44.770 --> 00:02:47.790
after she ran, after this
orange part right over here.

00:02:47.790 --> 00:02:50.140
You see that this
blue section is

00:02:50.140 --> 00:02:52.990
3/4 of that remaining distance.

00:02:52.990 --> 00:02:57.220
120 yards is 3/4 of
this remaining distance.

00:02:57.220 --> 00:03:02.280
So this is equal to 3/4
of the remaining distance.

00:03:07.170 --> 00:03:09.690
So how much is 1/4 of each
of those remaining distance?

00:03:09.690 --> 00:03:14.140
Well since this is 3/4, 1/4
is going to be 1/3 of this.

00:03:14.140 --> 00:03:18.840
1/3 of 120 is 40.

00:03:18.840 --> 00:03:21.000
So this distance
right over here is 40.

00:03:21.000 --> 00:03:22.280
This is 40.

00:03:22.280 --> 00:03:23.090
This is 40.

00:03:23.090 --> 00:03:26.030
Notice 40 plus 40
plus 40 is 120.

00:03:26.030 --> 00:03:30.510
And this right over here
is also going to be 40.

00:03:30.510 --> 00:03:32.690
These are all in terms of yards.

00:03:32.690 --> 00:03:38.050
So the remaining distance is
going to be 40 times 4 or 160

00:03:38.050 --> 00:03:38.780
yards.

00:03:38.780 --> 00:03:43.320
So this is 160 yards.

00:03:43.320 --> 00:03:45.290
Now, what is the total distance?

00:03:45.290 --> 00:03:48.130
Well this 160 yards,
the remaining distance

00:03:48.130 --> 00:03:51.620
after she ran, that's half
of the total distance.

00:03:51.620 --> 00:03:54.630
So this is 160
yards, then this is

00:03:54.630 --> 00:04:03.290
going to be 160 yards as well.

00:04:03.290 --> 00:04:06.120
So 160 yards plus 160 yards?

00:04:06.120 --> 00:04:13.061
Gives us a total distance from
school to home of 320 yards.


Units of measurement word problem: volume (metric)

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.880
Mary made 15,000 milliliters
of tea for a party,

00:00:04.880 --> 00:00:08.420
and she served the tea
divided equally in 8 pitchers.

00:00:08.420 --> 00:00:11.370
Her guests drank
6 pitchers of tea.

00:00:11.370 --> 00:00:13.760
How much tea did
Mary have leftover?

00:00:13.760 --> 00:00:15.600
Write the answer as a
whole number of liters

00:00:15.600 --> 00:00:17.910
and a whole number
of milliliters.

00:00:17.910 --> 00:00:19.640
So let's think about
this a little bit.

00:00:19.640 --> 00:00:21.260
She's got 8 pitchers.

00:00:21.260 --> 00:00:23.710
So let's visualize
these pitchers.

00:00:23.710 --> 00:00:28.780
So let me see if I can, so
let me draw a pitcher here.

00:00:28.780 --> 00:00:30.840
So this is one pitcher.

00:00:30.840 --> 00:00:33.990
I'll do my best to
draw a nice pitcher.

00:00:33.990 --> 00:00:35.900
So this is one pitcher.

00:00:35.900 --> 00:00:39.790
And she's going to put the same
amount of tea in every pitcher.

00:00:39.790 --> 00:00:42.340
So the same amount of
tea in every pitcher.

00:00:42.340 --> 00:00:47.864
Let me copy and paste
this, so copy and paste.

00:00:47.864 --> 00:00:49.280
So she's going to
have 8 of these.

00:00:49.280 --> 00:01:02.890
So that's 2, 3,
4, 5, 6, 7, and 8.

00:01:02.890 --> 00:01:05.680
So she wants to put the
exact same amount of fluid

00:01:05.680 --> 00:01:06.960
in each of these 8.

00:01:06.960 --> 00:01:10.480
So she's going to divide
evenly the 15,000 milliliters

00:01:10.480 --> 00:01:12.230
into 8 pitchers.

00:01:12.230 --> 00:01:13.850
So that's straight up division.

00:01:13.850 --> 00:01:18.890
She's going to start
with 15,000 milliliters,

00:01:18.890 --> 00:01:20.660
And she's going
to divide it by 8.

00:01:20.660 --> 00:01:24.840
She's going to divide into
8 equal groupings or 8

00:01:24.840 --> 00:01:27.920
equal pitchers with the same
amount in each of the pitchers.

00:01:27.920 --> 00:01:31.441
So 8 goes into, it doesn't go
into 1, it goes into 15 one

00:01:31.441 --> 00:01:31.940
time.

00:01:31.940 --> 00:01:37.150
1 times 8 is 8, subtract
15 minus 8 is 7.

00:01:37.150 --> 00:01:38.960
So bring down a 0.

00:01:38.960 --> 00:01:41.440
8 goes into 70, 8 times.

00:01:41.440 --> 00:01:43.390
8 times 8 is 64.

00:01:43.390 --> 00:01:46.820
Subtract, we get a 6,
bring down another zero.

00:01:46.820 --> 00:01:50.130
8 goes into 60 seven times.

00:01:50.130 --> 00:01:52.620
7 times 8 is 56.

00:01:52.620 --> 00:01:56.700
Subtract again, we get a 4 and
then bring down a zero again.

00:01:56.700 --> 00:01:59.130
8 goes into 40 exactly 5 times.

00:01:59.130 --> 00:02:00.880
5 times 8 is 40.

00:02:00.880 --> 00:02:02.860
And we're not left
with a remainder.

00:02:02.860 --> 00:02:05.200
So if we divide
15,000 milliliters

00:02:05.200 --> 00:02:07.710
into 8 equal
sections, each pitcher

00:02:07.710 --> 00:02:11.750
is going to have exactly
1,875 milliliters.

00:02:11.750 --> 00:02:16.690
So that's 1,875 milliliters.

00:02:16.690 --> 00:02:18.480
But that's not what
they're asking us for.

00:02:18.480 --> 00:02:21.530
They tell us that her guests
drank 6 pitchers of tea,

00:02:21.530 --> 00:02:24.440
and how much tea did
Mary have leftover.

00:02:24.440 --> 00:02:32.400
So the guests drank 1, 2, 3, 4,
5, 6, leaving 2 pitchers left.

00:02:32.400 --> 00:02:34.230
How much total tea is
going to be in that?

00:02:34.230 --> 00:02:44.440
It's going to be 2 times 1,875,
or 1,875 milliliters times 2.

00:02:44.440 --> 00:02:45.660
Figure out what that is.

00:02:45.660 --> 00:02:51.720
2 times 5 is 10, 2 times
7 is 14 plus 1 is 15.

00:02:51.720 --> 00:02:55.575
2 times 8 is 16 plus 1 is 17.

00:02:55.575 --> 00:02:58.670
2 times 1 is 2 plus 1 is 3.

00:02:58.670 --> 00:03:00.720
So what she's left
with, what she's

00:03:00.720 --> 00:03:06.536
leftover with is
3,750 milliliters.

00:03:06.536 --> 00:03:07.910
Now they want our
answer in terms

00:03:07.910 --> 00:03:11.940
of a whole number of liters and
a whole number of milliliters.

00:03:11.940 --> 00:03:13.890
And we just have
to remind ourselves

00:03:13.890 --> 00:03:20.460
that 1,000 milliliters
is equal to 1 liter.

00:03:20.460 --> 00:03:21.755
So you could rewrite this.

00:03:21.755 --> 00:03:23.380
This is the total
number of milliliters

00:03:23.380 --> 00:03:24.620
that she's left with.

00:03:24.620 --> 00:03:29.470
We could rewrite this
as 3,000 milliliters

00:03:29.470 --> 00:03:32.744
plus 750 milliliters.

00:03:32.744 --> 00:03:34.160
Now the reason I
wrote it this way

00:03:34.160 --> 00:03:36.285
is because if 3,000
milliliters-- this is literally

00:03:36.285 --> 00:03:39.730
3,000 one-thousandths
of a milliliter.

00:03:39.730 --> 00:03:41.580
We already saw that
1,000 milliliters

00:03:41.580 --> 00:03:44.830
is equal to 1 liter, so this
piece right over here, this

00:03:44.830 --> 00:03:46.870
is equal to 3 liters.

00:03:46.870 --> 00:03:48.939
This is equal to 3 liters.

00:03:48.939 --> 00:03:51.230
So if we wanted to write it
as a whole number of liters

00:03:51.230 --> 00:03:52.646
and a whole number
of milliliters,

00:03:52.646 --> 00:04:04.160
this would be 3 liters
and 750 milliliters.


Time word problem: How long is Susan's break?

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:05.060
Susan has 96 minutes to
complete a test at school.

00:00:05.060 --> 00:00:09.340
The test begins at 1:59 PM.

00:00:09.340 --> 00:00:13.940
And Susan has volleyball
practice at 4:00 PM.

00:00:13.940 --> 00:00:16.740
How much time will Susan have
between the end of the test

00:00:16.740 --> 00:00:18.750
and the beginning of
volleyball practice?

00:00:18.750 --> 00:00:20.640
Let's think about
this a little bit.

00:00:20.640 --> 00:00:22.410
So the test begins at 1:59.

00:00:22.410 --> 00:00:25.124
So it's a very strange
time to begin a test,

00:00:25.124 --> 00:00:26.290
but let's just go with that.

00:00:26.290 --> 00:00:28.630
So let me draw a
little time line here.

00:00:28.630 --> 00:00:35.030
So let's say that
this is 2 o'clock PM.

00:00:35.030 --> 00:00:37.650
Then we're going
to have 3:00 PM.

00:00:42.060 --> 00:00:47.990
Then we're going to have 4
o'clock PM, which is also

00:00:47.990 --> 00:00:49.960
when volleyball practice begins.

00:00:49.960 --> 00:00:51.440
So this is volleyball practice.

00:00:51.440 --> 00:00:54.760
It's going to start then
and then go on from 4:00 PM.

00:00:54.760 --> 00:00:57.000
And then the test
begins at 1:59.

00:00:57.000 --> 00:00:58.850
So let me continue
the number line.

00:00:58.850 --> 00:01:02.530
So this right over
here is 1:00 PM.

00:01:05.880 --> 00:01:10.900
So the test is going to begin at
1:59, so 1 minute before 2 PM.

00:01:10.900 --> 00:01:12.950
We know that there are
60 minutes in an hour.

00:01:12.950 --> 00:01:16.460
So 1:59 is 1 minute
before 2:00 PM.

00:01:16.460 --> 00:01:18.210
So I'm not drawing it
completely it scale,

00:01:18.210 --> 00:01:24.640
but this right over
here is 1:59 PM.

00:01:24.640 --> 00:01:29.220
So we use 1 minute
to get to 2:00 PM.

00:01:29.220 --> 00:01:32.440
And then, let's see, we have
to get up to 96 minutes.

00:01:32.440 --> 00:01:36.430
So then you get another 60
minutes to go to 3:00 PM.

00:01:36.430 --> 00:01:39.580
So, so far, we've
used a cumulative 61.

00:01:39.580 --> 00:01:40.654
So this is 1 minute.

00:01:40.654 --> 00:01:42.570
So this is how long the
test has been going on

00:01:42.570 --> 00:01:43.403
since the beginning.

00:01:43.403 --> 00:01:48.360
So 1 minute by 2:00 PM,
61 minutes by 3:00 PM.

00:01:48.360 --> 00:01:52.225
If we go all the way to 4:00 PM,
that's going to be another 60.

00:01:52.225 --> 00:01:56.110
That'd be 121 minutes, which
is longer than the actual test.

00:01:56.110 --> 00:01:58.952
So we know that the
entire test is 96 minutes.

00:01:58.952 --> 00:02:00.660
So we're going to get
to some point right

00:02:00.660 --> 00:02:03.140
over here, some point like this.

00:02:03.140 --> 00:02:06.030
That's 96 minutes.

00:02:06.030 --> 00:02:09.590
And what we need to figure
out is, is what is left over?

00:02:09.590 --> 00:02:12.355
What is this distance
right over here?

00:02:12.355 --> 00:02:14.490
Or what is this
time, this difference

00:02:14.490 --> 00:02:15.690
in time right over here?

00:02:15.690 --> 00:02:17.176
And so to figure
that out, we had

00:02:17.176 --> 00:02:19.950
to figure out the time at
which 96 minutes have passed

00:02:19.950 --> 00:02:21.950
since the beginning of the exam.

00:02:21.950 --> 00:02:25.090
So at 3 o'clock, 61
minutes have passed.

00:02:25.090 --> 00:02:27.500
And so if we, if we say
well how many more minutes

00:02:27.500 --> 00:02:29.430
have to go on for
the end of the test?

00:02:29.430 --> 00:02:32.420
It's going to be 96 minus 61.

00:02:32.420 --> 00:02:36.600
So the amount of time that
elapses past 3 minutes

00:02:36.600 --> 00:02:40.930
is 96 minus 61,
which is 35 minutes.

00:02:40.930 --> 00:02:48.717
So this right over here
is going to be 3:35.

00:02:48.717 --> 00:02:50.800
And so now our question
is, how much time is there

00:02:50.800 --> 00:02:55.800
between 3:35 and 4:00 PM?

00:02:55.800 --> 00:02:58.950
Well, once again, 60
minutes in an hour.

00:02:58.950 --> 00:03:02.020
4:00 PM is essentially
the 60th minute.

00:03:02.020 --> 00:03:09.360
So if we take 60
minus 35, we get 2:25.

00:03:09.360 --> 00:03:15.300
So she has a total of 25
minutes remaining after the test

00:03:15.300 --> 00:03:17.580
before she has to get
to volleyball practice.

00:03:17.580 --> 00:03:18.560
Just as a review.

00:03:18.560 --> 00:03:21.660
One minute between 1:59
and 2:00, then another 60.

00:03:21.660 --> 00:03:23.870
That gets us to
61 total minutes.

00:03:23.870 --> 00:03:30.160
Then another 35 minutes gets
us to 96 minutes have passed.

00:03:30.160 --> 00:03:32.810
And that also gets
us to 3:35, which

00:03:32.810 --> 00:03:37.255
means we have 25
minutes until 4:00.


Unit conversion: centimeters to meters

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:04.700
Convert 37
centimeters to meters.

00:00:04.700 --> 00:00:05.850
Let me write this over.

00:00:05.850 --> 00:00:09.910
37, and I'll write centi
in a different color

00:00:09.910 --> 00:00:11.610
to emphasize the prefix.

00:00:11.610 --> 00:00:19.600
37 centimeters, and we want
to convert it to meters.

00:00:19.600 --> 00:00:21.350
So we really just have
to remind ourselves

00:00:21.350 --> 00:00:24.420
what the prefix centi means.

00:00:24.420 --> 00:00:32.920
Centi literally means 1/100th,
or one hundredth of a meter.

00:00:32.920 --> 00:00:37.030
So this is really saying
37 hundredth meters,

00:00:37.030 --> 00:00:38.660
so let's write it that way.

00:00:38.660 --> 00:00:42.900
This is literally
37/100 of a meter.

00:00:42.900 --> 00:00:46.010
So 1/100 meters.

00:00:46.010 --> 00:00:48.300
These are equivalent statements.

00:00:48.300 --> 00:00:51.980
So what are 37/100 of a meter?

00:00:51.980 --> 00:01:03.620
Well, that's going to be
37/100 of a meter, which

00:01:03.620 --> 00:01:05.900
can be rewritten
as, if you wanted

00:01:05.900 --> 00:01:09.520
to write it as a
decimal, that's 0.

00:01:09.520 --> 00:01:15.260
you could view it as 3/10
and 7/100, or 37/100.

00:01:15.260 --> 00:01:21.850
So 0.37, 0.37 meters.

00:01:21.850 --> 00:01:24.580
Another way that you could have
thought about this is look,

00:01:24.580 --> 00:01:27.570
I'm going to go from
centimeters to meters.

00:01:27.570 --> 00:01:28.840
I have 100.

00:01:28.840 --> 00:01:32.370
I need 100 centimeters
to get to one meter,

00:01:32.370 --> 00:01:35.270
so I'm going to have to
divide by 100 in order

00:01:35.270 --> 00:01:37.590
to figure out how
many meters I have.

00:01:37.590 --> 00:01:39.670
And you should always
do a reality check.

00:01:39.670 --> 00:01:41.500
If I convert
centimeters to meters,

00:01:41.500 --> 00:01:44.080
should I get a larger
number or a smaller number?

00:01:44.080 --> 00:01:46.190
Well, however many
centimeters I have,

00:01:46.190 --> 00:01:48.720
I'm going to have a
fewer number of meters.

00:01:48.720 --> 00:01:50.850
Meters is a larger unit.

00:01:50.850 --> 00:01:53.280
So you should have a
smaller value here,

00:01:53.280 --> 00:01:56.035
and it should be a smaller
value by a factor of 100.

00:01:56.035 --> 00:01:57.660
So you literally
could have started off

00:01:57.660 --> 00:01:58.650
with 37 centimeters.

00:02:01.982 --> 00:02:03.690
Let me write it this
way, 37 centimeters.

00:02:07.884 --> 00:02:10.050
And actually, let's go
through a couple of the units

00:02:10.050 --> 00:02:11.300
right over here.

00:02:11.300 --> 00:02:14.110
Now, if you wanted to turn it
to decimeters, so this is 1/100.

00:02:14.110 --> 00:02:16.580
Decimeters, one over 1/10.

00:02:16.580 --> 00:02:18.000
So you would divide.

00:02:18.000 --> 00:02:22.445
This would be 3.7 decimeters.

00:02:22.445 --> 00:02:24.953
Let me write this
as 3.7 decimeters.

00:02:31.160 --> 00:02:33.850
And a decimeter is
1/10 of a meter,

00:02:33.850 --> 00:02:35.760
so you would divide by 10 again.

00:02:35.760 --> 00:02:44.030
So this is 0.37 meters.

00:02:44.030 --> 00:02:46.730
So one way to think about
it is to go from centimeters

00:02:46.730 --> 00:02:49.610
to meters, you're
going to divide by 100.

00:02:49.610 --> 00:02:53.590
Dividing by 100, you would
move the decimal space over

00:02:53.590 --> 00:02:55.140
to the left two times.

00:02:55.140 --> 00:02:57.130
Doing it once divides by 10.

00:02:57.130 --> 00:02:59.230
Doing it twice divides by 100.

00:02:59.230 --> 00:03:01.290
So you get to 0.37 meters.


Unit conversion: ordering metric distances

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.360
We're asked to arrange the
following measurements in order

00:00:03.360 --> 00:00:04.904
from smallest to largest.

00:00:04.904 --> 00:00:07.320
And we have the measurements,
and they're different units.

00:00:07.320 --> 00:00:10.060
This is in decameters,
then we have meters,

00:00:10.060 --> 00:00:13.580
then we have millimeters,
then we have centimeters.

00:00:13.580 --> 00:00:15.070
So the way I would
tackle this is

00:00:15.070 --> 00:00:18.320
I would try to convert them
all to the same units, maybe

00:00:18.320 --> 00:00:21.940
meters, and then compare
them all in terms of meters.

00:00:21.940 --> 00:00:23.620
So let's do that.

00:00:23.620 --> 00:00:27.220
So this is just a screenshot of
that exact same screen we just

00:00:27.220 --> 00:00:28.110
saw there.

00:00:28.110 --> 00:00:32.020
And so let's just convert
each of these into meters.

00:00:32.020 --> 00:00:37.710
So the first one we have right
over here is one decameter.

00:00:37.710 --> 00:00:42.260
So we have to remind ourselves
what deca even means.

00:00:42.260 --> 00:00:46.456
Well, deca is
equal to 10 meters.

00:00:46.456 --> 00:00:48.080
And actually, let me
add more of these,

00:00:48.080 --> 00:00:50.810
just so that we know all
the different prefixes

00:00:50.810 --> 00:00:51.410
we could have.

00:00:51.410 --> 00:00:52.690
So you have deca.

00:00:52.690 --> 00:00:56.150
You have hecto,
which would be 100.

00:00:56.150 --> 00:01:00.070
You have kilo, which
would be 1,000.

00:01:00.070 --> 00:01:02.370
And then, of course,
you just have meters,

00:01:02.370 --> 00:01:04.230
and you'd have no prefix there.

00:01:04.230 --> 00:01:06.040
That would just
be equal 1 if you

00:01:06.040 --> 00:01:08.260
have no prefix in
front of the meters.

00:01:08.260 --> 00:01:11.190
And then if you have
a tenth of a meter,

00:01:11.190 --> 00:01:15.570
that is decimeter,
so this is 1/10.

00:01:15.570 --> 00:01:20.550
Then you have centi,
which is 1/100.

00:01:20.550 --> 00:01:28.256
And then you have milli,
which is equal to 1/1,000.

00:01:28.256 --> 00:01:30.130
So let's use this
information right over here

00:01:30.130 --> 00:01:32.640
to figure out how many
meters each of these are.

00:01:32.640 --> 00:01:40.730
So one decameter-- we
just saw deca means 10.

00:01:40.730 --> 00:01:43.870
That's 1 times 10 meters.

00:01:43.870 --> 00:01:47.900
So this is going to
be equal to 10 meters.

00:01:47.900 --> 00:01:51.340
So this right over
here is 10 meters.

00:01:51.340 --> 00:01:52.860
This is already
written in meters.

00:01:52.860 --> 00:01:54.980
This is 13 meters.

00:01:54.980 --> 00:02:06.710
Then we have 15,000 millimeters,
but milli means 1/1,000,

00:02:06.710 --> 00:02:12.710
So it's 15,000 times
1/1,000 of a meter.

00:02:12.710 --> 00:02:15.099
So each millimeter is
1/1,000 of a meter.

00:02:15.099 --> 00:02:17.390
You could view this as instead
of writing a milli here,

00:02:17.390 --> 00:02:19.030
I wrote 1/1,000.

00:02:19.030 --> 00:02:24.190
So 15,000 times 1/1,000, that's
just going to give me 15.

00:02:24.190 --> 00:02:27.610
So this is going
to be 15 meters.

00:02:27.610 --> 00:02:29.950
So this is 15 meters.

00:02:29.950 --> 00:02:33.590
Another way of thinking
about it is, look,

00:02:33.590 --> 00:02:36.230
1,000 millimeters
is equal to a meter.

00:02:36.230 --> 00:02:38.480
So let's divide this
into groups of 1,000.

00:02:38.480 --> 00:02:41.000
Well, this is literally
15 groups of 1,000.

00:02:41.000 --> 00:02:44.120
This is 15 groups of
1,000 millimeters,

00:02:44.120 --> 00:02:47.350
so that's going to be 15 meters.

00:02:47.350 --> 00:02:51.150
And then, finally, we
have 1,900 centimeters.

00:02:51.150 --> 00:02:55.820
So 1,900, instead
of writing centi,

00:02:55.820 --> 00:03:01.940
I'm going to write
1/100 of a meter.

00:03:01.940 --> 00:03:04.705
Well, 1,900 times
1 over 100 is 19,

00:03:04.705 --> 00:03:09.190
so this is equal to 19 meters.

00:03:09.190 --> 00:03:11.150
Another way of
thinking about it is,

00:03:11.150 --> 00:03:14.620
this is 19 groups
of 100 centimeters,

00:03:14.620 --> 00:03:18.395
and 100 centimeters
is equal to a meter.

00:03:18.395 --> 00:03:22.870
This is 19 meters
right over here.

00:03:22.870 --> 00:03:26.390
So let's go back to
the actual problem.

00:03:26.390 --> 00:03:28.150
So let's remind ourselves.

00:03:28.150 --> 00:03:30.690
This right over here,
this is 10 meters.

00:03:33.230 --> 00:03:38.520
This is 13 meters, so 10, 13.

00:03:38.520 --> 00:03:41.040
This right over
here was 15 meters,

00:03:41.040 --> 00:03:44.709
and this right over
here was 19 meters.

00:03:44.709 --> 00:03:47.250
So actually, the order that they
already gave it to us in, we

00:03:47.250 --> 00:03:49.870
could obviously swap things
around if we wanted to.

00:03:49.870 --> 00:03:51.620
But the order that
they already gave it to

00:03:51.620 --> 00:03:54.220
was the order from
smallest to largest.

00:03:54.220 --> 00:04:00.330
This is 10 meters, 13 meters,
15 meters, and 19 meters.


Unit conversion: minutes to hours

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:04.580
How many hours are
in 549 minutes?

00:00:04.580 --> 00:00:07.120
And we can write it as
a decimal or a fraction.

00:00:07.120 --> 00:00:09.860
So essentially, we're
going to take 549 minutes

00:00:09.860 --> 00:00:12.210
and divide them
into groups of 60.

00:00:12.210 --> 00:00:13.370
Why 60?

00:00:13.370 --> 00:00:21.030
Because we know that one
hour is equal to 60 minutes.

00:00:21.030 --> 00:00:24.020
So it's essentially saying, how
many groups of 60 minutes can

00:00:24.020 --> 00:00:27.220
we divide 549 into.

00:00:27.220 --> 00:00:28.780
Or another way of
thinking about that

00:00:28.780 --> 00:00:34.590
is, well, what is 549
divided into groups of 60.

00:00:34.590 --> 00:00:37.090
This is how many hours
we're going to have.

00:00:37.090 --> 00:00:38.300
So let's do that.

00:00:38.300 --> 00:00:46.290
Let's take 549 and
divide it by 60.

00:00:46.290 --> 00:00:49.670
So let's see, 6 goes
into 54 9 times.

00:00:49.670 --> 00:00:52.359
So 60 is going to
go into 540 9 times.

00:00:52.359 --> 00:00:54.150
We're going to have a
little bit left over.

00:00:54.150 --> 00:00:58.080
So we have 9 times 60 is 540.

00:00:58.080 --> 00:00:59.270
We subtract.

00:00:59.270 --> 00:01:00.695
We have 9 left over.

00:01:05.204 --> 00:01:07.120
And now let's see, we
have a little left over,

00:01:07.120 --> 00:01:08.453
so we're going to get a decimal.

00:01:08.453 --> 00:01:11.060
So let's put a decimal
place right over here

00:01:11.060 --> 00:01:13.580
and let's throw
some 0's over there.

00:01:13.580 --> 00:01:15.220
Let's bring down a 0.

00:01:15.220 --> 00:01:16.400
So we bring down a 0.

00:01:16.400 --> 00:01:19.000
60 goes into 90 1 time.

00:01:19.000 --> 00:01:21.390
1 times 60 is 60.

00:01:21.390 --> 00:01:22.250
And we subtract.

00:01:22.250 --> 00:01:23.510
We get 30.

00:01:23.510 --> 00:01:26.130
Let's bring down another 0.

00:01:26.130 --> 00:01:27.822
And so we get to 300.

00:01:27.822 --> 00:01:30.690
60 goes into 300 5 times.

00:01:30.690 --> 00:01:36.660
5 times 60, 6 times 6 is
30, so 5 times 60 is 300.

00:01:36.660 --> 00:01:39.390
Subtract and we are done.

00:01:39.390 --> 00:01:42.110
So you divide 549
into groups of 60.

00:01:42.110 --> 00:01:45.960
You can divide it into
9.15 groups of 60 minutes.

00:01:45.960 --> 00:01:47.720
A group of 60
minutes is an hour.

00:01:47.720 --> 00:01:52.650
So this 549 minutes
is 9.15 hours.

00:01:52.650 --> 00:01:58.340
I want to make sure that we
can visualize that properly.

00:01:58.340 --> 00:02:04.490
So let's actually construct
what 9.15 hours looks like.

00:02:04.490 --> 00:02:08.240
So let me draw a little
line here, and on the top

00:02:08.240 --> 00:02:10.660
I'll label Hours, and on the
bottom I'll label Minutes.

00:02:10.660 --> 00:02:13.820
So this is 0 hours, 0 minutes.

00:02:13.820 --> 00:02:17.520
And now we have 1 hour,
which is 60 minutes.

00:02:17.520 --> 00:02:21.860
Now we have 2 hours,
which is 120 minutes.

00:02:21.860 --> 00:02:25.880
Then you have 3 hours,
which is 180 minutes.

00:02:25.880 --> 00:02:31.170
Then you have 4 hours,
which is 240 minutes.

00:02:31.170 --> 00:02:34.810
5 hours is 300 minutes.

00:02:34.810 --> 00:02:39.770
6 hours is 360 minutes.

00:02:39.770 --> 00:02:43.111
7 hours, I might be running out
of space, is for 420 minutes.

00:02:43.111 --> 00:02:44.860
Let me copy and paste
this someplace where

00:02:44.860 --> 00:02:46.900
I have more real estate.

00:02:46.900 --> 00:02:49.580
So let me clear that.

00:02:49.580 --> 00:02:51.090
And then let me
paste that someplace

00:02:51.090 --> 00:02:54.400
where I don't run into
my other math that I did.

00:02:54.400 --> 00:02:55.530
All right.

00:02:55.530 --> 00:03:00.980
So then you have 8
hours is 480 minutes.

00:03:00.980 --> 00:03:03.070
And then you have 9 hours.

00:03:03.070 --> 00:03:05.570
Notice, I'm just adding
60 minutes every time.

00:03:05.570 --> 00:03:08.099
9 hours is 540 minutes.

00:03:08.099 --> 00:03:10.140
Or another way you could
think about it is, well,

00:03:10.140 --> 00:03:13.700
if each hour is 60,
9 times 60 is 540.

00:03:13.700 --> 00:03:15.710
And we don't want
to go to just 540.

00:03:15.710 --> 00:03:17.260
We have another 9 left.

00:03:17.260 --> 00:03:25.320
So then we have to go another
9 minutes to go to 549.

00:03:25.320 --> 00:03:28.120
So you have 9 minutes left over.

00:03:28.120 --> 00:03:34.550
So another way of thinking about
this, is that 549 is 9 hours.

00:03:34.550 --> 00:03:37.450
And then you have 9
minutes left over.

00:03:37.450 --> 00:03:41.160
And 9 minutes is what
fraction of an hour?

00:03:41.160 --> 00:03:43.900
Well, 60 minutes
is a whole hour.

00:03:43.900 --> 00:03:49.289
So 9 minutes is 9/60 of an hour.

00:03:49.289 --> 00:03:50.580
So you could write it this way.

00:03:50.580 --> 00:03:53.150
It's 9 and 9/60 hours.

00:03:53.150 --> 00:03:55.720
Or we could write this as
an equivalent fraction.

00:03:55.720 --> 00:03:58.200
9/60 is the same
thing if we divide

00:03:58.200 --> 00:04:00.070
the numerator and
the denominator by 3,

00:04:00.070 --> 00:04:02.060
is the same thing as 3/20.

00:04:02.060 --> 00:04:05.230
So we could write
this as 9 and 3/20.

00:04:08.530 --> 00:04:12.510
And 3/20, well we could figure
out what that is going to be.

00:04:12.510 --> 00:04:15.292
Let's see, 20 divided by 3.

00:04:15.292 --> 00:04:16.750
It's definitely
going to be smaller

00:04:16.750 --> 00:04:19.390
than 1, because 3
is smaller than 20.

00:04:19.390 --> 00:04:21.450
So let's throw some 0's on here.

00:04:21.450 --> 00:04:26.300
20 doesn't go into 3, but
it does go into 30 1 time.

00:04:26.300 --> 00:04:28.200
1 times 20 is 20.

00:04:28.200 --> 00:04:30.620
Subtract, we get a 10.

00:04:30.620 --> 00:04:31.650
Bring down a 0.

00:04:31.650 --> 00:04:34.270
20 goes into 100 5 times.

00:04:34.270 --> 00:04:36.650
5 times 20 is 100.

00:04:36.650 --> 00:04:38.730
And we are done.

00:04:38.730 --> 00:04:41.780
So notice, 3 over 20 is
the exact same thing.

00:04:41.780 --> 00:04:49.840
So 9 and 3/20 is the exact
same thing as 9 and 15/100.

00:04:49.840 --> 00:04:52.605
These are all
equivalent answers.


Quadrilaterals: find the type exercise

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.600
What is the type of
this quadrilateral?

00:00:02.600 --> 00:00:05.950
Be as specific as possible
with the given data.

00:00:05.950 --> 00:00:07.680
So it clearly is
a quadrilateral.

00:00:07.680 --> 00:00:09.620
We have four sides here.

00:00:09.620 --> 00:00:13.370
And we see that we have two
pairs of parallel sides.

00:00:13.370 --> 00:00:16.400
Or we could also say there are
two pairs of congruent sides

00:00:16.400 --> 00:00:17.520
here as well.

00:00:17.520 --> 00:00:20.480
This side is parallel and
congruent to this side.

00:00:20.480 --> 00:00:23.110
This side is parallel and
congruent to that side.

00:00:23.110 --> 00:00:26.350
So we're dealing
with a parallelogram.

00:00:26.350 --> 00:00:29.250
Let's do more of these.

00:00:29.250 --> 00:00:31.200
So here it looks like
a same type of scenario

00:00:31.200 --> 00:00:32.366
we just saw in the last one.

00:00:32.366 --> 00:00:35.430
We have two pairs of
parallel and congruent sides,

00:00:35.430 --> 00:00:37.330
but all the sides aren't
equal to each other.

00:00:37.330 --> 00:00:38.420
If they're all
equal to each other,

00:00:38.420 --> 00:00:39.940
we'd be dealing with a rhombus.

00:00:39.940 --> 00:00:42.250
But here, they're not
all equal to each other.

00:00:42.250 --> 00:00:44.307
This side is congruent
to the side opposite.

00:00:44.307 --> 00:00:46.140
This side is congruent
to the side opposite.

00:00:46.140 --> 00:00:47.348
That's another parallelogram.

00:00:50.840 --> 00:00:52.150
Now this is interesting.

00:00:52.150 --> 00:00:55.340
We have two pairs of sides that
are parallel to each other,

00:00:55.340 --> 00:00:58.600
but now all the sides
have an equal length.

00:00:58.600 --> 00:01:00.640
So this would be
a parallelogram.

00:01:00.640 --> 00:01:02.470
And it is a
parallelogram, but they're

00:01:02.470 --> 00:01:05.630
saying to be as specific as
possible with the given data.

00:01:05.630 --> 00:01:07.090
So saying it's a
rhombus would be

00:01:07.090 --> 00:01:09.650
more specific than saying
it's a parallelogram.

00:01:09.650 --> 00:01:11.080
This does satisfy
the constraints

00:01:11.080 --> 00:01:13.930
for being a parallelogram,
but saying it's a rhombus

00:01:13.930 --> 00:01:15.290
tells us even more.

00:01:15.290 --> 00:01:17.490
Not every parallelogram
is a rhombus,

00:01:17.490 --> 00:01:19.730
but every rhombus
is a parallelogram.

00:01:19.730 --> 00:01:23.350
Here, they have the sides are
parallel to the side opposite

00:01:23.350 --> 00:01:26.330
and all of the sides are equal.

00:01:26.330 --> 00:01:28.906
Let's do a few more of these.

00:01:28.906 --> 00:01:30.530
What is the type of
this quadrilateral?

00:01:30.530 --> 00:01:34.150
Be as specific as possible
with the given data .

00:01:34.150 --> 00:01:37.340
So we have two pairs of
sides that are parallel,

00:01:37.340 --> 00:01:38.920
or I should say one pair.

00:01:38.920 --> 00:01:41.650
We have a pair of sides
that are parallel.

00:01:41.650 --> 00:01:46.580
And then we have another
pair of sides that are not.

00:01:46.580 --> 00:01:48.351
So this is a trapezoid.

00:01:48.351 --> 00:01:49.850
But then they have
two choices here.

00:01:49.850 --> 00:01:53.280
They have trapezoid and
isosceles trapezoid.

00:01:53.280 --> 00:01:57.010
Now an isosceles
trapezoid is a trapezoid

00:01:57.010 --> 00:02:00.770
where the two non-parallel
sides have the same length, just

00:02:00.770 --> 00:02:02.300
like an isosceles
triangle, you have

00:02:02.300 --> 00:02:04.230
two sides have the same length.

00:02:04.230 --> 00:02:07.110
Well we could see these two
non-parallel sides do not

00:02:07.110 --> 00:02:08.400
have the same length.

00:02:08.400 --> 00:02:10.637
So this is not an
isosceles trapezoid.

00:02:10.637 --> 00:02:12.220
If they did have the
same length, then

00:02:12.220 --> 00:02:13.761
we would pick that
because that would

00:02:13.761 --> 00:02:15.400
be more specific
than just trapezoid.

00:02:15.400 --> 00:02:19.070
But this case right over here,
this is just a trapezoid.

00:02:19.070 --> 00:02:21.150
Let's do one more of these.

00:02:21.150 --> 00:02:23.610
What is the type of
this quadrilateral?

00:02:23.610 --> 00:02:25.537
Well we could say
it's a parallelogram

00:02:25.537 --> 00:02:27.120
because all of the
sides are parallel.

00:02:27.120 --> 00:02:28.740
But if we wanted to
be more specific,

00:02:28.740 --> 00:02:31.290
you could also see that
all the sides are the same.

00:02:31.290 --> 00:02:33.090
So you could say it's
a rhombus, but you

00:02:33.090 --> 00:02:34.720
could get even more
specific than that.

00:02:34.720 --> 00:02:36.250
You notice that
all the sides are

00:02:36.250 --> 00:02:38.280
intersecting at right angles.

00:02:38.280 --> 00:02:42.450
So this is-- if we wanted to be
as specific as possible-- this

00:02:42.450 --> 00:02:45.370
is a square.

00:02:45.370 --> 00:02:46.990
Let me check the answer.

00:02:46.990 --> 00:02:48.780
Got it right.


Quadrilaterals: kites as a geometric shape

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:02.960
In everyday language, we
know what a kite means.

00:00:02.960 --> 00:00:05.050
It's these flimsy things
that we take to the beach

00:00:05.050 --> 00:00:07.950
to fly in the wind
with our families.

00:00:07.950 --> 00:00:09.450
But you could imagine
mathematicians

00:00:09.450 --> 00:00:11.450
have looked at the general
shape of these kites,

00:00:11.450 --> 00:00:13.690
or at least the way
they're drawn in cartoons,

00:00:13.690 --> 00:00:15.439
and say, well, that's
an interesting shape

00:00:15.439 --> 00:00:16.160
in its own right.

00:00:16.160 --> 00:00:17.990
Let's also make this
a mathematical term.

00:00:17.990 --> 00:00:20.540
This is a shape
like a parallelogram

00:00:20.540 --> 00:00:21.960
or like a rhombus.

00:00:21.960 --> 00:00:25.270
It's just another
type of quadrilateral.

00:00:25.270 --> 00:00:28.470
But in order for it to be used
in mathematics in a useful way,

00:00:28.470 --> 00:00:31.007
we have to define it a
little bit more precisely.

00:00:31.007 --> 00:00:32.840
So let's see if we can
come up with a couple

00:00:32.840 --> 00:00:35.120
of interesting definitions
of what a kite could

00:00:35.120 --> 00:00:38.420
be or a couple of interesting
ways to construct a kite.

00:00:38.420 --> 00:00:40.620
Well, one way that you
could think about a kite

00:00:40.620 --> 00:00:43.145
is it looks like it
has two pairs of sides

00:00:43.145 --> 00:00:44.520
that are congruent
to each other.

00:00:44.520 --> 00:00:48.440
So, for example, it looks
like this side and this side

00:00:48.440 --> 00:00:50.780
need to be congruent
to each other.

00:00:50.780 --> 00:00:52.900
So let's make that a constraint.

00:00:52.900 --> 00:00:54.210
And they touch each other.

00:00:54.210 --> 00:00:55.910
They have a shared
common endpoint.

00:00:55.910 --> 00:00:57.820
So you have one pair
of congruent sides

00:00:57.820 --> 00:00:59.070
that's adjacent to each other.

00:00:59.070 --> 00:01:01.060
They have a common endpoint.

00:01:01.060 --> 00:01:06.230
And then you have
another pair of sides

00:01:06.230 --> 00:01:08.120
that are congruent
to each other.

00:01:10.650 --> 00:01:11.820
And they are adjacent.

00:01:11.820 --> 00:01:13.610
They share a common endpoint.

00:01:13.610 --> 00:01:16.690
So one definition that
you could make for a kite

00:01:16.690 --> 00:01:27.880
is that you have two
pairs of congruent sides,

00:01:27.880 --> 00:01:31.630
where the congruent
sides are adjacent.

00:01:38.541 --> 00:01:40.790
And you might say, well,
what's the other alternative?

00:01:40.790 --> 00:01:42.610
If the congruent
sides aren't adjacent,

00:01:42.610 --> 00:01:43.610
what else could they be?

00:01:43.610 --> 00:01:45.960
Well, the congruent sides
could be opposite each other.

00:01:45.960 --> 00:01:47.660
And what happens if
you were to do that?

00:01:47.660 --> 00:01:49.400
So if these two
sides are congruent,

00:01:49.400 --> 00:01:51.202
but they didn't have
a common endpoint,

00:01:51.202 --> 00:01:52.910
we're still dealing
with a quadrilateral.

00:01:52.910 --> 00:01:54.300
What would it look like?

00:01:54.300 --> 00:01:57.230
Well, you would have
one congruent site here,

00:01:57.230 --> 00:02:00.475
and that would be congruent
to this side right over here.

00:02:04.110 --> 00:02:07.350
And then you would have a
congruent side right over here

00:02:07.350 --> 00:02:08.759
that is congruent to this side.

00:02:08.759 --> 00:02:10.217
This would be a
situation where you

00:02:10.217 --> 00:02:11.759
have two pairs of
congruent sides,

00:02:11.759 --> 00:02:12.800
but they're not adjacent.

00:02:12.800 --> 00:02:16.260
They don't have any common
endpoints with each other.

00:02:16.260 --> 00:02:19.430
Each side in the
congruent side pair,

00:02:19.430 --> 00:02:21.130
they're opposite to each other.

00:02:21.130 --> 00:02:23.070
So here, once again,
we get a quadrilateral.

00:02:23.070 --> 00:02:24.140
We still get four sides.

00:02:24.140 --> 00:02:25.290
A kite is a quadrilateral.

00:02:25.290 --> 00:02:26.620
This is a quadrilateral.

00:02:26.620 --> 00:02:27.850
But this isn't a kite.

00:02:27.850 --> 00:02:32.310
This right over here
is a parallelogram,

00:02:32.310 --> 00:02:34.760
and we've seen that
multiple times before.

00:02:34.760 --> 00:02:36.230
But kites can also
be constructed

00:02:36.230 --> 00:02:37.830
in other interesting ways.

00:02:37.830 --> 00:02:42.300
You might see that what looks
right here, that these two

00:02:42.300 --> 00:02:44.357
diagonals of this kite
are perpendicular.

00:02:44.357 --> 00:02:46.190
And that indeed-- and
I'm not going to prove

00:02:46.190 --> 00:02:48.565
it here-- is a
property of a kite.

00:02:51.990 --> 00:02:54.910
These two lines,
these two diagonals,

00:02:54.910 --> 00:02:58.272
intersect at a 90-degree angle.

00:02:58.272 --> 00:02:59.730
The other thing we
know about kites

00:02:59.730 --> 00:03:03.100
is that one of these lines is
bisecting the other of the two.

00:03:03.100 --> 00:03:05.570
So you could actually
construct a kite that way.

00:03:05.570 --> 00:03:10.150
You could start with
a line, and then you

00:03:10.150 --> 00:03:12.280
could construct a
perpendicular bisector

00:03:12.280 --> 00:03:14.260
of that line,
another segment that

00:03:14.260 --> 00:03:16.760
bisects it at a 90-degree angle.

00:03:16.760 --> 00:03:18.550
So here, there you go.

00:03:18.550 --> 00:03:20.200
So that bisects
it, so that means

00:03:20.200 --> 00:03:22.630
that this segment is
equal to this segment.

00:03:22.630 --> 00:03:24.050
We split it in two.

00:03:24.050 --> 00:03:26.820
And then if you connect the
endpoints of the segments,

00:03:26.820 --> 00:03:28.190
you should get a kite.

00:03:28.190 --> 00:03:30.080
And you will indeed get a kite.

00:03:30.080 --> 00:03:36.030
So it would look
something like this.

00:03:36.030 --> 00:03:38.380
And once again, this
segment is congruent

00:03:38.380 --> 00:03:41.170
to this adjacent
segment, and this segment

00:03:41.170 --> 00:03:43.710
is congruent to this
adjacent segment.

00:03:43.710 --> 00:03:46.450
But what would happen if
these two diagonals are

00:03:46.450 --> 00:03:50.020
both perpendicular
bisectors of each other?

00:03:50.020 --> 00:03:52.100
So what would happen
in this scenario,

00:03:52.100 --> 00:03:56.779
where-- let me draw one segment.

00:03:56.779 --> 00:03:58.570
And then I'm going to
make another segment,

00:03:58.570 --> 00:04:00.111
but they're going
to be perpendicular

00:04:00.111 --> 00:04:02.130
bisectors of each other.

00:04:02.130 --> 00:04:03.010
So let's do that.

00:04:03.010 --> 00:04:05.427
So now they're both
perpendicular bisectors

00:04:05.427 --> 00:04:06.010
of each other.

00:04:06.010 --> 00:04:08.640
So this segment is
equal to this segment,

00:04:08.640 --> 00:04:11.740
and this segment is
equal to this segment.

00:04:11.740 --> 00:04:16.110
Well, now, once again,
you still have a kite,

00:04:16.110 --> 00:04:18.350
but now you're also
satisfying the constraint

00:04:18.350 --> 00:04:21.750
for another type of
quadrilateral that we've seen.

00:04:21.750 --> 00:04:25.390
So now you're satisfying
the constraint.

00:04:25.390 --> 00:04:27.230
All your sides are equal.

00:04:27.230 --> 00:04:29.190
All of your sides are parallel.

00:04:29.190 --> 00:04:33.640
You're now dealing with
a rhombus, which is also

00:04:33.640 --> 00:04:36.050
a special type of parallelogram.

00:04:36.050 --> 00:04:37.810
And then if you were
to go even further,

00:04:37.810 --> 00:04:40.350
where these two diagonals
have the exact same length

00:04:40.350 --> 00:04:43.520
and they're both perpendicular
bisectors of each other,

00:04:43.520 --> 00:04:46.380
so you have both the
exact same length.

00:04:46.380 --> 00:04:48.470
I'll try to draw it
as cleanly as I can.

00:04:48.470 --> 00:04:50.810
So they're both the
exact same length,

00:04:50.810 --> 00:04:53.830
and they're both perpendicular
bisectors of each other.

00:04:53.830 --> 00:04:56.580
So each of these halves would
be the same length as well.

00:04:56.580 --> 00:04:59.570
Then you have a subset of--
I guess I could say-- rhombi,

00:04:59.570 --> 00:05:01.510
and you get to a square.

00:05:08.060 --> 00:05:12.010
So one way of thinking
about it is any square

00:05:12.010 --> 00:05:14.096
is also a rhombus.

00:05:16.950 --> 00:05:20.850
And any rhombus is also going
to satisfy your constraints

00:05:20.850 --> 00:05:21.890
for being a kite.

00:05:21.890 --> 00:05:23.950
But there's a
bunch of types that

00:05:23.950 --> 00:05:28.060
don't satisfy your constraints
of being a rhombus or a square.

00:05:28.060 --> 00:05:30.780
A kite is just two
pairs of congruent sides

00:05:30.780 --> 00:05:33.620
that are adjacent to each
other, and they're usually

00:05:33.620 --> 00:05:37.395
pretty easy to spot out
because they look like kites.