Regrouping with decimals example 1

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:02.050
We're asked to
fill in the blanks

00:00:02.050 --> 00:00:04.460
to complete the equation.

00:00:04.460 --> 00:00:07.660
So they have 86.93
is equal to 5 tens

00:00:07.660 --> 00:00:13.490
plus blank ones plus 9
tenths plus 3 hundredths.

00:00:13.490 --> 00:00:16.200
So let me expand out
86.93 and just think

00:00:16.200 --> 00:00:18.920
about how I could regroup the
value in the different places

00:00:18.920 --> 00:00:20.515
so that I get
something more close

00:00:20.515 --> 00:00:23.520
or exactly what we have
on the right-hand side.

00:00:23.520 --> 00:00:27.440
So the 8 in 86.93,
that's in the tens place.

00:00:27.440 --> 00:00:29.940
So it literally
represents 8 tens.

00:00:29.940 --> 00:00:34.000
So I can write that down as 80.

00:00:34.000 --> 00:00:38.160
The 6 in 86 represents 6 ones.

00:00:38.160 --> 00:00:40.520
So I can write that down as 6.

00:00:40.520 --> 00:00:46.100
Then the 9 in 86.93,
that's 9 tenths.

00:00:46.100 --> 00:00:48.270
So I could write that down.

00:00:48.270 --> 00:00:50.420
I could write this 9.

00:00:50.420 --> 00:00:52.610
That's right of
the decimal place.

00:00:52.610 --> 00:00:56.190
This is plus 9 tenths.

00:00:56.190 --> 00:01:01.760
And then the 3 in 86.93, that
is in the hundredths place.

00:01:01.760 --> 00:01:04.510
It represents 3 hundredths.

00:01:04.510 --> 00:01:08.780
So this is plus 3 over 100.

00:01:08.780 --> 00:01:12.060
So I've expanded out the
number that I have on the left.

00:01:12.060 --> 00:01:14.840
Now let me write out what
they have on the right again.

00:01:14.840 --> 00:01:19.520
So this is going to be equal
to-- they have 5 tens here.

00:01:19.520 --> 00:01:23.510
So I'll write all the tens
places in this red color.

00:01:23.510 --> 00:01:25.470
Then they have blank ones.

00:01:25.470 --> 00:01:27.680
So I'm just going to
literally write out

00:01:27.680 --> 00:01:32.630
a blank plus 9 tenths.

00:01:32.630 --> 00:01:38.620
So the 9 tenths is still 9
tenths plus 3 hundredths.

00:01:38.620 --> 00:01:40.820
So let me make sure
that we color code this.

00:01:40.820 --> 00:01:42.610
So that's the 9 tenths.

00:01:42.610 --> 00:01:44.250
This is the 5 tens.

00:01:44.250 --> 00:01:45.710
That's the 50.

00:01:45.710 --> 00:01:47.770
We have the blank ones.

00:01:47.770 --> 00:01:49.990
That's this right
over here, 9 tenths.

00:01:49.990 --> 00:01:56.780
And then finally we have the 3
hundredths, so plus 3 over 100.

00:01:56.780 --> 00:01:58.680
So how do we get
the right-hand side

00:01:58.680 --> 00:02:01.780
to be the exact same thing
as the left-hand side

00:02:01.780 --> 00:02:04.320
by adjusting this blank?

00:02:04.320 --> 00:02:07.300
Well, both sides
have a 3/100 on it,

00:02:07.300 --> 00:02:10.020
both sides have a 9/10 on it.

00:02:10.020 --> 00:02:13.000
And so what's really different
about the two sides is

00:02:13.000 --> 00:02:14.775
on the left-hand
side-- let me do

00:02:14.775 --> 00:02:16.970
this in a color we
can actually see.

00:02:16.970 --> 00:02:19.710
On the left-hand side
you have an 80 plus a 6.

00:02:19.710 --> 00:02:24.740
On the right-hand side you
have a 50 plus a blank.

00:02:24.740 --> 00:02:29.910
So 80 plus 6 is clearly 86.

00:02:29.910 --> 00:02:33.566
Well, what has to be
added to 50 to get to 86?

00:02:33.566 --> 00:02:34.607
Well, you have to add 36.

00:02:39.050 --> 00:02:41.370
So now both sides add
up to the same thing.

00:02:41.370 --> 00:02:42.980
80 plus 6 is 86.

00:02:42.980 --> 00:02:45.680
50 plus 36 is 86.

00:02:45.680 --> 00:02:50.520
Then you have 9/10,
9/10, 3/100, 3/100.

00:02:50.520 --> 00:02:53.110
Now, you might be saying, hey
wait, you wrote a 36 here.

00:02:53.110 --> 00:02:55.330
Does this literally
means 36 ones?

00:02:55.330 --> 00:03:00.730
And I will say yes, it
literally represents 36 ones.

00:03:00.730 --> 00:03:04.600
This is the same
thing as 36 times 1.

00:03:04.600 --> 00:03:08.090
So we can fill in
the blank here.

00:03:08.090 --> 00:03:13.040
86.93 is equal to
5 tens plus 36 ones

00:03:13.040 --> 00:03:15.272
plus 9 tenths plus 3 hundredths.

00:03:15.272 --> 00:03:16.730
And I encourage
you to add this up.

00:03:16.730 --> 00:03:19.390
You will get 86.93.

00:03:19.390 --> 00:03:22.040
Now, what have we
essentially done here?

00:03:22.040 --> 00:03:27.520
What we essentially did, if you
compare the 8 and the 5 right

00:03:27.520 --> 00:03:33.980
over here, you see that we
took 3 from the tens place.

00:03:33.980 --> 00:03:36.530
But taking 3 from the
tens place is essentially

00:03:36.530 --> 00:03:42.200
taking 30 from the value
that it represents.

00:03:42.200 --> 00:03:45.070
So we took 30 from
the tens place,

00:03:45.070 --> 00:03:47.660
and we gave that 30
to the ones place.

00:03:50.680 --> 00:03:51.660
So you had 86.

00:03:51.660 --> 00:03:54.490
You take 30 away, you
have 50 in the tens place,

00:03:54.490 --> 00:03:56.150
represented by this 5.

00:03:56.150 --> 00:03:59.060
And then you give that 30 to
the ones place, you have 36.

00:03:59.060 --> 00:04:01.799
You have 36 ones.


Regrouping with decimals

WEBVTT
Kind: captions
Language: en

00:00:00.472 --> 00:00:02.430
I want to think about
all of the different ways

00:00:02.430 --> 00:00:10.230
we can represent value
in the number 21.3.

00:00:10.230 --> 00:00:12.750
So one is to just look straight
up at the place values.

00:00:12.750 --> 00:00:16.960
This 2 is in the tens place, so
it literally represents 2 tens.

00:00:16.960 --> 00:00:20.680
So this is equal
to 20, 2 times 10.

00:00:20.680 --> 00:00:22.500
This 1 is literally equal to 1.

00:00:22.500 --> 00:00:23.600
It's 1 one.

00:00:23.600 --> 00:00:28.010
And then this 3 is
3/10, so plus 3/10.

00:00:28.010 --> 00:00:30.320
But now I want to
rearrange or regroup

00:00:30.320 --> 00:00:32.400
the value in these places.

00:00:32.400 --> 00:00:37.000
So, for example, I could
take 1 from the ones place

00:00:37.000 --> 00:00:39.070
and give it to the tenths place.

00:00:39.070 --> 00:00:40.990
So let's see how
that would work.

00:00:40.990 --> 00:00:44.490
So we're going to take 1
away from the ones place,

00:00:44.490 --> 00:00:46.280
and so it's going to become a 0.

00:00:46.280 --> 00:00:48.400
And we're going to give
it to the tenths place.

00:00:48.400 --> 00:00:50.149
And what we're going
to see is that that's

00:00:50.149 --> 00:00:55.250
going to make the
tenths place into 13/10.

00:00:55.250 --> 00:00:58.870
Now, does that actually make
sense that I took 1 from here

00:00:58.870 --> 00:01:02.000
and it essentially added
10 to the tenths place?

00:01:02.000 --> 00:01:04.470
Well, let's rewrite
what this represents.

00:01:04.470 --> 00:01:06.670
So we still have 2 tens.

00:01:06.670 --> 00:01:09.310
So this is still
going to be 2 tens.

00:01:09.310 --> 00:01:13.440
Now we have plus 0 ones.

00:01:13.440 --> 00:01:17.150
And we essentially wanted to
write that 1 that we took away

00:01:17.150 --> 00:01:20.050
from the ones place
in terms of tenths.

00:01:20.050 --> 00:01:22.190
So if we were to write
this in terms of tenths,

00:01:22.190 --> 00:01:28.060
it would be 10/10 plus the
3/10 that were already there.

00:01:28.060 --> 00:01:30.872
And so this is going
to be equal to 13/10.

00:01:30.872 --> 00:01:31.830
Let me write that down.

00:01:31.830 --> 00:01:34.460
So this is equal to 20.

00:01:34.460 --> 00:01:35.760
That's the color you can't see.

00:01:35.760 --> 00:01:41.480
This is equal to 20 plus 0
ones, so 2 tens plus 0 ones

00:01:41.480 --> 00:01:43.255
plus 13 tens.

00:01:47.240 --> 00:01:49.790
Let's do another example
with this exact same number.

00:01:49.790 --> 00:01:57.520
So once again, 21.3.

00:01:57.520 --> 00:01:59.150
And I'll write it out again.

00:01:59.150 --> 00:02:01.785
This is equal to 20 plus 1.

00:02:04.530 --> 00:02:07.090
We'll do that in
the purple color.

00:02:07.090 --> 00:02:13.270
Plus 1 plus 3/10,
plus 3 over 10.

00:02:13.270 --> 00:02:16.210
Now, I could take 1
from the tens place

00:02:16.210 --> 00:02:19.390
so that this becomes just 1.

00:02:19.390 --> 00:02:22.040
Now what do I do with that 10?

00:02:22.040 --> 00:02:29.010
Well, let's say with that 10 I
give 9 of it to the ones place.

00:02:29.010 --> 00:02:34.010
So I give 9 of it to the ones
place so that this becomes 10.

00:02:34.010 --> 00:02:36.730
And I still have 1 left
over, and I give it

00:02:36.730 --> 00:02:39.653
to the tenths place, so
that's going to become 13/10.

00:02:42.420 --> 00:02:44.320
So what did I just do?

00:02:44.320 --> 00:02:45.900
Well, I could rewrite this.

00:02:45.900 --> 00:02:47.180
Let me be clear what I did.

00:02:47.180 --> 00:02:55.170
This is the same
thing as 1 plus 9.

00:02:55.170 --> 00:02:57.870
Actually, let me
write it this way.

00:02:57.870 --> 00:03:00.010
1 plus 9 plus 1.

00:03:00.010 --> 00:03:03.280
That's obviously
the same thing-- 10

00:03:03.280 --> 00:03:08.470
plus 9 plus 1 is the
same thing as 20.

00:03:08.470 --> 00:03:11.100
And of course, we have what
we have in our ones place,

00:03:11.100 --> 00:03:15.340
plus 1 plus 3/10.

00:03:15.340 --> 00:03:19.690
And what I want to do is I want
to take this 9, the 9 that I

00:03:19.690 --> 00:03:23.660
took from the tens place
and give to the ones place.

00:03:23.660 --> 00:03:27.260
And I'm going to take this 1
that I took from the tens place

00:03:27.260 --> 00:03:29.850
and give it to the tenths place.

00:03:29.850 --> 00:03:34.130
So 1 is the same thing as 10/10.

00:03:34.130 --> 00:03:37.320
And so when you
regroup this value,

00:03:37.320 --> 00:03:43.830
you get this as being equal
to 10 plus-- 9 plus 1 is 10,

00:03:43.830 --> 00:03:49.130
and then 10/10
plus 3/10 is 13/10.

00:03:49.130 --> 00:03:50.610
So that's all that
happened here.

00:03:50.610 --> 00:03:52.580
I changed the value
in the places.

00:03:52.580 --> 00:03:54.840
I took 1 ten away.

00:03:54.840 --> 00:03:55.750
I had 2 tens.

00:03:55.750 --> 00:03:57.630
Now I'm only left with 1 ten.

00:03:57.630 --> 00:04:00.510
And that extra 10 of
value, I regrouped it.

00:04:00.510 --> 00:04:02.540
I gave 9 to the ones place.

00:04:02.540 --> 00:04:04.390
So 1 plus 9 is 10.

00:04:04.390 --> 00:04:08.470
And then I gave 1
to the tenths place.

00:04:08.470 --> 00:04:13.390
So 1 plus 3/10 is the
same thing as 13/10.


Regrouping decimals example 2

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:02.850
Fill in the table
with whole numbers

00:00:02.850 --> 00:00:07.120
to write 0.25 in different ways.

00:00:07.120 --> 00:00:10.060
So one way to look at it
is just this 2 in 0.25

00:00:10.060 --> 00:00:14.640
is in the tenths place, so it
literally represents 2 tenths.

00:00:14.640 --> 00:00:16.870
While the 5 is in
the hundredths place,

00:00:16.870 --> 00:00:19.350
so it literally
represents 5 hundredths.

00:00:19.350 --> 00:00:21.240
So that's one way
to represent it.

00:00:21.240 --> 00:00:23.220
But then we can start
to rearrange value

00:00:23.220 --> 00:00:25.670
amongst these two place values.

00:00:25.670 --> 00:00:29.190
For example, what happens
if we only have 1 tenth,

00:00:29.190 --> 00:00:31.930
then how many hundredths
would we have to have?

00:00:31.930 --> 00:00:35.650
Well, 1 tenth is 10 hundredths.

00:00:35.650 --> 00:00:38.500
So if we take the 10 hundredths
that we took away from here,

00:00:38.500 --> 00:00:40.510
and give it to the
hundredths place,

00:00:40.510 --> 00:00:44.070
you're going to
have 15 hundredths.

00:00:44.070 --> 00:00:47.980
And you could verify that
1 tenth plus 15 hundredths

00:00:47.980 --> 00:00:50.710
is equal to 25 hundredths.

00:00:50.710 --> 00:00:53.490
Or another way you could say
it is 10 hundredths plus 15

00:00:53.490 --> 00:00:55.950
hundredths is equal
to 25 hundredths.

00:00:55.950 --> 00:00:58.860
Now another way you could do
it, you could have no tenths

00:00:58.860 --> 00:01:03.570
and do it as all hundredths,
because 2 tenths is

00:01:03.570 --> 00:01:05.810
the same thing as 20 hundredths.

00:01:05.810 --> 00:01:11.010
So 20 hundredths plus 5
hundredths is 25 hundredths.

00:01:11.010 --> 00:01:12.370
Let's do a few more of these.

00:01:12.370 --> 00:01:14.051
These are a lot of fun.

00:01:14.051 --> 00:01:15.550
Fill in the table
with whole numbers

00:01:15.550 --> 00:01:18.630
to write 20.1 in different ways.

00:01:18.630 --> 00:01:21.290
So here, the table, the two
columns are tens and tenths.

00:01:21.290 --> 00:01:23.040
So I guess the assumption
is, is we're not

00:01:23.040 --> 00:01:25.610
going to give anything
to the ones place here.

00:01:25.610 --> 00:01:27.210
And in the original
example, we didn't

00:01:27.210 --> 00:01:29.340
have anything in the
ones place either.

00:01:29.340 --> 00:01:34.420
So we could write the 2 in the
tens place is literally 2 tens.

00:01:34.420 --> 00:01:37.760
And the 1 in the tenths
place is literally 1 tenth.

00:01:37.760 --> 00:01:40.110
But let's think about what
happens as we regroup value

00:01:40.110 --> 00:01:43.450
from the tens place
to the tenths place.

00:01:43.450 --> 00:01:45.890
So if we only have
1 ten, then we

00:01:45.890 --> 00:01:47.510
have another 10 to
play with that we

00:01:47.510 --> 00:01:49.205
have to express in
terms of tenths.

00:01:53.460 --> 00:01:56.710
1 ten is the same
thing as 10 ones, which

00:01:56.710 --> 00:01:59.700
is the same thing as 100 tenths.

00:01:59.700 --> 00:02:02.180
So we're essentially
giving 100 tenths.

00:02:02.180 --> 00:02:06.700
Notice, we have an extra 10
to play with, 10 of value.

00:02:06.700 --> 00:02:11.370
That is the same thing as
100 over 10, or 100 tenths.

00:02:11.370 --> 00:02:13.220
And we already had 1 tenth.

00:02:13.220 --> 00:02:17.572
So 100 tenths plus 1 tenth
is going to be 101 tenths.

00:02:17.572 --> 00:02:18.780
I know it's really confusing.

00:02:18.780 --> 00:02:21.210
We have tens and tenths.

00:02:21.210 --> 00:02:22.720
So now let's go again.

00:02:22.720 --> 00:02:26.550
What happens if we take all
the value in the tens place?

00:02:26.550 --> 00:02:30.000
So now we have 2 tens,
or 20, to play with.

00:02:30.000 --> 00:02:32.800
Well, 20 is how many tenths?

00:02:32.800 --> 00:02:35.740
Well, you have 10 tenths per 1.

00:02:35.740 --> 00:02:39.080
So 20 is 200 tenths.

00:02:39.080 --> 00:02:41.230
And so you take the 200
tenths that you took out

00:02:41.230 --> 00:02:42.620
of the tens place.

00:02:42.620 --> 00:02:44.040
Add it to the 1 tenth.

00:02:44.040 --> 00:02:48.350
And you have 201 tenths.

00:02:48.350 --> 00:02:50.820
Let's do one more example.

00:02:50.820 --> 00:02:52.840
So let's write 2.1 in
three different ways.

00:02:52.840 --> 00:02:55.130
So here it's between the
one and the tenths place.

00:02:55.130 --> 00:02:57.040
So we could write it as 2 ones.

00:02:57.040 --> 00:02:58.790
And 1 tenth.

00:02:58.790 --> 00:03:00.670
We could also write it as 1 one.

00:03:00.670 --> 00:03:02.180
And now we've seen
this game before.

00:03:02.180 --> 00:03:03.880
We have one extra
1 to play with.

00:03:03.880 --> 00:03:05.230
So that's going to be 10 tenths.

00:03:05.230 --> 00:03:07.230
Plus the 1 tenth that
was already there,

00:03:07.230 --> 00:03:09.020
gives us 11 tenths.

00:03:09.020 --> 00:03:15.330
Or if we have no ones, well, 2.1
is the same thing as 21 tenths.

00:03:15.330 --> 00:03:18.980
The 2 represents,
literally, 20 tenths.

00:03:18.980 --> 00:03:20.730
And then you have the 1 tenth.

00:03:20.730 --> 00:03:26.410
So 20 tenths plus 1
tenth is 21 tenths.


Absolute value of a complex number

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.330
I have the complex
number 3 minus 4i.

00:00:03.330 --> 00:00:05.140
I've plotted it on
the complex plane.

00:00:05.140 --> 00:00:08.450
We see that the real part
is 3, so we've gone 3

00:00:08.450 --> 00:00:11.100
along the horizontal
axis, or the real axis.

00:00:11.100 --> 00:00:12.740
And the imaginary
part is negative 4,

00:00:12.740 --> 00:00:16.120
so we've gone down 4
along the vertical axis.

00:00:16.120 --> 00:00:19.595
So this right here is
the point 3 minus 4i.

00:00:19.595 --> 00:00:20.970
Now what I want
to think about is

00:00:20.970 --> 00:00:24.760
what the absolute
value of 3 minus 4i is.

00:00:24.760 --> 00:00:26.760
And just as a reminder,
absolute value literally

00:00:26.760 --> 00:00:28.926
means-- whether we're talking
about a complex number

00:00:28.926 --> 00:00:31.620
or a real number, it literally
just means distance away

00:00:31.620 --> 00:00:33.190
from 0.

00:00:33.190 --> 00:00:35.420
So the absolute
value of 3 minus 4i

00:00:35.420 --> 00:00:37.520
is going to be the
distance between 0,

00:00:37.520 --> 00:00:43.310
between the origin on the
complex plane, and that point,

00:00:43.310 --> 00:00:44.652
and the point 3 minus 4i.

00:00:44.652 --> 00:00:46.110
So this distance
right over here is

00:00:46.110 --> 00:00:51.030
going to be the absolute
value of 3 minus 4i.

00:00:51.030 --> 00:00:52.900
So how can we think about that?

00:00:52.900 --> 00:00:57.050
Well, we could literally
just set up a right triangle

00:00:57.050 --> 00:00:59.480
and then use the
Pythagorean theorem.

00:00:59.480 --> 00:01:00.540
So let's think about it.

00:01:00.540 --> 00:01:09.160
If we wanted to set up a right
triangle, the height here,

00:01:09.160 --> 00:01:11.330
the distance between 0
and negative 4, well,

00:01:11.330 --> 00:01:13.360
that distance is going to be 4.

00:01:13.360 --> 00:01:16.240
And then the base
of this triangle,

00:01:16.240 --> 00:01:20.760
the distance between 0 and
3, is just going to be 3.

00:01:20.760 --> 00:01:22.460
And this is definitely
a right angle.

00:01:22.460 --> 00:01:23.740
This is a horizontal line.

00:01:23.740 --> 00:01:25.060
This is a vertical line.

00:01:25.060 --> 00:01:27.720
We can now use the Pythagorean
theorem to figure out

00:01:27.720 --> 00:01:29.720
the absolute value
of 3 minus 4i.

00:01:29.720 --> 00:01:32.440
The distance between
this point and 0--

00:01:32.440 --> 00:01:34.825
it's the hypotenuse of
this right triangle.

00:01:34.825 --> 00:01:36.450
So we just use the
Pythagorean theorem.

00:01:36.450 --> 00:01:41.280
This side squared, 3 squared,
plus this side squared,

00:01:41.280 --> 00:01:47.670
plus 4 squared, is going to
be equal to the absolute value

00:01:47.670 --> 00:01:56.750
of 3 minus 4i squared, the
absolute value squared.

00:01:56.750 --> 00:02:03.740
So 3 squared plus 4 squared,
that's 9 plus 16, which is 25.

00:02:03.740 --> 00:02:08.683
So you get 25 is equal to the
absolute value of 3 minus 4i

00:02:08.683 --> 00:02:09.183
squared.

00:02:13.157 --> 00:02:15.490
And we know if you take the
absolute value of something,

00:02:15.490 --> 00:02:16.520
this is just a distance.

00:02:16.520 --> 00:02:17.657
It's going to be positive.

00:02:17.657 --> 00:02:19.490
So we want to take the
positive square root,

00:02:19.490 --> 00:02:21.980
the principal square root,
of both sides of this.

00:02:21.980 --> 00:02:23.730
And so we're going to
be left with-- well,

00:02:23.730 --> 00:02:26.320
the principal square root, the
positive square root of 25,

00:02:26.320 --> 00:02:31.510
is 5, is equal to the
absolute value of 3 minus 4i.

00:02:35.200 --> 00:02:37.840
So another way of saying it,
this thing right over here

00:02:37.840 --> 00:02:40.250
is going to be equal to 5.

00:02:40.250 --> 00:02:43.440
This distance right
over here is equal to 5.

00:02:43.440 --> 00:02:45.070
Now, without having
to draw it, one way

00:02:45.070 --> 00:02:47.486
you could just think about
this is I'll have my real part.

00:02:47.486 --> 00:02:49.170
I have my imaginary part.

00:02:49.170 --> 00:02:51.390
I could literally take
each of those parts,

00:02:51.390 --> 00:02:53.962
square them, take the sum,
and take the square root.

00:02:53.962 --> 00:02:55.420
So another way of
taking it, if you

00:02:55.420 --> 00:02:57.253
didn't want to visualize
all this-- but this

00:02:57.253 --> 00:02:58.466
is really what we're doing.

00:02:58.466 --> 00:02:59.840
You could say,
well, this is just

00:02:59.840 --> 00:03:03.085
going to be equal to take
the real part squared.

00:03:05.910 --> 00:03:08.870
Take the imaginary part
squared-- so let me write this.

00:03:11.990 --> 00:03:17.480
Add them together, and
then take the square root,

00:03:17.480 --> 00:03:19.370
or the principal root.

00:03:19.370 --> 00:03:21.670
The principal root's just
the positive square root.

00:03:21.670 --> 00:03:25.390
So that's going to be
the square root of 9

00:03:25.390 --> 00:03:31.868
plus 16, which, once
again, is equal to 5.


Using factors and multiples to figure out days of the week

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:04.560
Let's say that today-- and
we'll call today Day 1--

00:00:04.560 --> 00:00:08.000
Day 1 is a Monday.

00:00:08.000 --> 00:00:09.830
What I want to
figure out is what

00:00:09.830 --> 00:00:13.600
is the Day 300 going to be?

00:00:13.600 --> 00:00:15.857
What day of the week
will Day 300 be?

00:00:15.857 --> 00:00:17.440
And I encourage you
to pause the video

00:00:17.440 --> 00:00:19.800
and think about
that a little bit.

00:00:19.800 --> 00:00:22.100
So let's just write out
the days of the week.

00:00:22.100 --> 00:00:26.090
You have Monday-- I'll do this
in a different color-- you have

00:00:26.090 --> 00:00:37.140
Monday, Tuesday, Wednesday,
Thursday, Friday, Saturday

00:00:37.140 --> 00:00:39.226
and Sunday.

00:00:39.226 --> 00:00:40.600
So if we had a
lower number here,

00:00:40.600 --> 00:00:43.230
we could just fill this out.

00:00:43.230 --> 00:00:44.380
Monday is Day 1.

00:00:44.380 --> 00:00:45.490
Tuesday's Day 2.

00:00:45.490 --> 00:00:46.450
Wednesday, Day 3.

00:00:46.450 --> 00:00:48.794
4, 5, 6, 7.

00:00:48.794 --> 00:00:49.460
I'll keep going.

00:00:49.460 --> 00:00:52.030
Day 8, well, that's going
to be a Monday again.

00:00:52.030 --> 00:00:53.190
9 10.

00:00:53.190 --> 00:00:54.920
I'm almost writing
a calendar out.

00:00:54.920 --> 00:01:01.270
11, 12, 13, 14, 15, 16.

00:01:01.270 --> 00:01:02.510
So this is kind of useful.

00:01:02.510 --> 00:01:04.051
I could just write
it out if I wanted

00:01:04.051 --> 00:01:06.520
to figure out something
Day 16 or Day 20.

00:01:06.520 --> 00:01:08.220
I could just write that out.

00:01:08.220 --> 00:01:10.806
But this isn't that helpful if
I want to figure out Day 300

00:01:10.806 --> 00:01:12.180
or especially
wouldn't be helpful

00:01:12.180 --> 00:01:14.920
if I wanted to
figure out Day 3,000.

00:01:14.920 --> 00:01:17.360
So can I come up with
some mathematical way

00:01:17.360 --> 00:01:20.750
of thinking about what
Day 300 is going to be?

00:01:20.750 --> 00:01:24.130
Well, as you see when I
started drawing this grid here,

00:01:24.130 --> 00:01:25.540
this grid has rows.

00:01:25.540 --> 00:01:27.895
And each row, you
have seven days in it.

00:01:27.895 --> 00:01:28.770
And that makes sense.

00:01:28.770 --> 00:01:31.307
There are seven
days of the week.

00:01:31.307 --> 00:01:33.390
So is there a way that if
someone just gave you 16

00:01:33.390 --> 00:01:34.960
without drawing
this grid, then you

00:01:34.960 --> 00:01:38.060
would know that 16 is a Tuesday?

00:01:38.060 --> 00:01:40.800
Well, one way to think about
it that might jump out at you

00:01:40.800 --> 00:01:43.180
is you could divide 16 by 7.

00:01:43.180 --> 00:01:47.680
That will tell you how many of
these rows will come before 16.

00:01:52.510 --> 00:01:55.110
So 16 divided by 7 is 2.

00:01:55.110 --> 00:01:58.440
You have 2 rows before
16 right over here.

00:01:58.440 --> 00:02:01.340
You could get 7 into 16 2 times.

00:02:01.340 --> 00:02:04.720
And then you have a remainder.

00:02:04.720 --> 00:02:08.020
What is the remainder
when you divide 16 by 7?

00:02:08.020 --> 00:02:12.690
16 divided by 7
is going to be 2.

00:02:12.690 --> 00:02:14.400
2 times 7 is 14.

00:02:14.400 --> 00:02:16.610
You're going to have
a remainder of 2.

00:02:16.610 --> 00:02:18.870
So when we divide,
we've historically

00:02:18.870 --> 00:02:21.230
cared more about this 2.

00:02:21.230 --> 00:02:23.010
We normally care
more about well,

00:02:23.010 --> 00:02:24.430
how many times
does it go into it.

00:02:24.430 --> 00:02:27.590
But now, the remainder
is actually interesting.

00:02:27.590 --> 00:02:30.050
The remainder is really
interesting here.

00:02:30.050 --> 00:02:32.810
Because the remainder
tells you-- the first two

00:02:32.810 --> 00:02:35.200
just tells you 7
goes into 16 2 times.

00:02:35.200 --> 00:02:38.020
That's how many rows you have
before getting to the 16.

00:02:38.020 --> 00:02:42.410
But then the remainder tells
you in that row where is the 16?

00:02:42.410 --> 00:02:44.630
So the 16 is remainder 2.

00:02:44.630 --> 00:02:46.250
So the 16 is not the first.

00:02:46.250 --> 00:02:49.940
It's the second entry
in the third row.

00:02:49.940 --> 00:02:52.300
And so it's going
to be a Tuesday.

00:02:52.300 --> 00:02:54.257
Tuesday is the second day.

00:02:54.257 --> 00:02:55.340
I know what you're saying.

00:02:55.340 --> 00:02:56.430
Does that always work?

00:02:56.430 --> 00:02:59.090
Well, let's try it out
with some other examples.

00:02:59.090 --> 00:03:03.490
Let's imagine Day 25.

00:03:03.490 --> 00:03:06.240
So let's just divide 25 by 7.

00:03:06.240 --> 00:03:08.330
So I'll do it right
over-- I could do it.

00:03:08.330 --> 00:03:10.140
I'm going to make sure
I have enough space.

00:03:10.140 --> 00:03:15.940
So if I have 7 goes into
25, it goes 3 times.

00:03:15.940 --> 00:03:17.770
3 times 7 is 21.

00:03:17.770 --> 00:03:20.080
You have a remainder of 4.

00:03:20.080 --> 00:03:21.340
So let's see.

00:03:21.340 --> 00:03:25.240
So based on that-- so let
me rewrite this-- so 25

00:03:25.240 --> 00:03:29.785
divided by 7 is equal
to 3, remainder 4.

00:03:33.970 --> 00:03:36.340
So based on this, if we
were to write out the grid,

00:03:36.340 --> 00:03:39.740
we should have three rows of
7 before we get to the 25.

00:03:39.740 --> 00:03:43.680
And then 25 should sit
in the fourth column.

00:03:43.680 --> 00:03:46.310
So if it's sitting
in the fourth column,

00:03:46.310 --> 00:03:47.920
it should be a Thursday.

00:03:47.920 --> 00:03:52.120
So Day 25, based on this
little math we just did,

00:03:52.120 --> 00:03:53.300
should be a Thursday.

00:03:53.300 --> 00:03:55.810
Let's see if that
actually works out.

00:03:55.810 --> 00:04:06.086
So let's go to 17, 18, 19,
20, 21, 22, 23, 24 and 25.

00:04:06.086 --> 00:04:08.390
It is indeed a Thursday.

00:04:08.390 --> 00:04:09.970
But it makes complete sense.

00:04:09.970 --> 00:04:14.390
You could get seven rows in it
before the row that gets to 25.

00:04:14.390 --> 00:04:18.180
And then in that row, it's
going to be the fourth entry,

00:04:18.180 --> 00:04:19.940
because you have
a remainder of 4.

00:04:19.940 --> 00:04:21.360
1, 2, 3, 4.

00:04:21.360 --> 00:04:22.900
It's going to be a Thursday.

00:04:22.900 --> 00:04:24.950
So now we're ready to
answer the question.

00:04:24.950 --> 00:04:27.570
What is Day 300 going to be?

00:04:27.570 --> 00:04:34.560
So let's just divide 300
by 7 and see where we get.

00:04:34.560 --> 00:04:38.510
7 goes into 30 4 times.

00:04:38.510 --> 00:04:40.900
4 times 7 is 28.

00:04:40.900 --> 00:04:42.240
Subtract, you get 2.

00:04:42.240 --> 00:04:43.760
Bring down a 0.

00:04:43.760 --> 00:04:46.530
7 goes into 20 2 times.

00:04:46.530 --> 00:04:49.450
2 times 7 is 14.

00:04:49.450 --> 00:04:51.730
And then we get our remainder.

00:04:51.730 --> 00:04:53.860
And now we care much
more about the remainder.

00:04:53.860 --> 00:04:56.970
20 minus 14 is 6.

00:04:56.970 --> 00:04:59.870
So our remainder is 6.

00:04:59.870 --> 00:05:03.080
So if we think about what day
of the week it is, in its row

00:05:03.080 --> 00:05:04.684
it's going to be
the sixth entry.

00:05:04.684 --> 00:05:06.100
It's going to be
the sixth column.

00:05:06.100 --> 00:05:08.290
There's going to be
42 rows above it.

00:05:08.290 --> 00:05:11.180
But we care about which
entry it is in its row.

00:05:11.180 --> 00:05:15.510
So Day 300 is going to be
the sixth day of the week,

00:05:15.510 --> 00:05:16.810
the way we've written it out.

00:05:16.810 --> 00:05:21.398
It is going to be a Saturday.


New Khan Academy learning flow

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:01.790
Hello.

00:00:01.790 --> 00:00:04.860
I just wanted to make a
quick video to introduce you

00:00:04.860 --> 00:00:07.590
to what is probably or what
is definitely the biggest

00:00:07.590 --> 00:00:10.510
change that we've ever made
in the history of the Khan

00:00:10.510 --> 00:00:11.420
Academy.

00:00:11.420 --> 00:00:13.420
We've introduced a
new learning flow

00:00:13.420 --> 00:00:16.110
that tries to
engage people more.

00:00:16.110 --> 00:00:18.210
We've heard from a lot
of users that there's

00:00:18.210 --> 00:00:19.630
a lot of content
on Khan Academy.

00:00:19.630 --> 00:00:21.040
They didn't know where to start.

00:00:21.040 --> 00:00:23.730
They didn't know what level
was appropriate for them.

00:00:23.730 --> 00:00:26.480
And so we've spent a lot of
time thinking about this,

00:00:26.480 --> 00:00:29.404
and what you're about to see
is our new learning flow.

00:00:29.404 --> 00:00:30.820
So I'm a new user,
and it gives us

00:00:30.820 --> 00:00:32.580
a nice little welcome message.

00:00:32.580 --> 00:00:33.720
And I say, cool.

00:00:33.720 --> 00:00:35.190
Let's explore.

00:00:35.190 --> 00:00:38.950
So when you log in,
you'll see this mission--

00:00:38.950 --> 00:00:39.950
The World of Math.

00:00:39.950 --> 00:00:42.610
And right now as we launch
that is the only mission

00:00:42.610 --> 00:00:43.620
that we have.

00:00:43.620 --> 00:00:46.140
And it will essentially
walk through all of the math

00:00:46.140 --> 00:00:48.197
exercises that we
have on Khan Academy.

00:00:48.197 --> 00:00:49.780
Over the next few
months, you're going

00:00:49.780 --> 00:00:51.490
to see other missions--
missions aligned

00:00:51.490 --> 00:00:53.400
to grade levels,
missions aligned

00:00:53.400 --> 00:00:57.400
to preparing for a test,
missions aligned to being

00:00:57.400 --> 00:00:59.860
able to do some
type of a project.

00:00:59.860 --> 00:01:02.970
You'll also start to see
missions in other subjects.

00:01:02.970 --> 00:01:05.630
Now these squares that you
see-- This is, for those of you

00:01:05.630 --> 00:01:07.380
who are familiar with
our Knowledge Map,

00:01:07.380 --> 00:01:09.500
you can kind of view it
as a condensed Knowledge

00:01:09.500 --> 00:01:12.650
Map for the particular mission
that you're thinking about.

00:01:12.650 --> 00:01:15.250
So here there are 482 boxes.

00:01:15.250 --> 00:01:17.810
Each of these
represents a skill.

00:01:17.810 --> 00:01:21.454
It represents a type
of math problem.

00:01:21.454 --> 00:01:23.120
So what we're going
to start right now--

00:01:23.120 --> 00:01:24.730
and I highly recommend
you do this as well,

00:01:24.730 --> 00:01:25.983
it's actually kind
of fun-- is we're

00:01:25.983 --> 00:01:27.310
going to start with a pretest.

00:01:27.310 --> 00:01:29.100
This is to address
the issue of where

00:01:29.100 --> 00:01:30.480
do I start on Khan Academy.

00:01:30.480 --> 00:01:33.842
What are the exercises that
are most appropriate for me?

00:01:33.842 --> 00:01:36.050
So I'm going to actually
take this pretest right now.

00:01:36.050 --> 00:01:38.960
So let's start the pretest.

00:01:38.960 --> 00:01:41.450
So if I see anything that is
completely daunting to me,

00:01:41.450 --> 00:01:45.210
I should just say, "I
haven't learned this yet."

00:01:45.210 --> 00:01:46.880
"What is the area
of the rectangle?"

00:01:46.880 --> 00:01:49.790
Well that's just going
to be 3 times 9, which

00:01:49.790 --> 00:01:53.990
is 27 square inches.

00:01:53.990 --> 00:01:55.960
Well, let's just say for
the sake of argument,

00:01:55.960 --> 00:01:58.460
let's pretend that we haven't
learned algebra or geometry

00:01:58.460 --> 00:02:00.725
yet, so I haven't
learned this yet.

00:02:00.725 --> 00:02:01.850
I haven't learned this yet.

00:02:01.850 --> 00:02:03.266
And this is an
adaptive test here.

00:02:03.266 --> 00:02:04.690
If I answer
questions right, it's

00:02:04.690 --> 00:02:07.320
going to get harder or go
to more advanced topics.

00:02:07.320 --> 00:02:08.930
If I get questions
wrong, it's going

00:02:08.930 --> 00:02:11.600
to give us simpler
or more basic topics.

00:02:11.600 --> 00:02:14.050
Move the orange dot to 4/3.

00:02:14.050 --> 00:02:18.920
So 4/3-- that's 1/3,
2/3, 3/3, and 4/3.

00:02:18.920 --> 00:02:20.740
Submit my answer.

00:02:20.740 --> 00:02:22.270
And I have to do
five more problems.

00:02:22.270 --> 00:02:24.690
So this isn't too
long of a thing here.

00:02:24.690 --> 00:02:26.130
So similar triangles--
well I said

00:02:26.130 --> 00:02:28.460
I'm going to pretend like I
haven't taken geometry yet,

00:02:28.460 --> 00:02:30.251
so I'm going to say I
haven't learned this.

00:02:30.251 --> 00:02:32.130
Now they gave us a
proportion, 9 over 2

00:02:32.130 --> 00:02:33.950
is equal to what over 18.

00:02:33.950 --> 00:02:35.840
Well, we multiply
the denominator by 9,

00:02:35.840 --> 00:02:38.180
so we have to multiply
the numerator by 9.

00:02:38.180 --> 00:02:42.900
So it's going to 9 times
9 is 81, and there we go.

00:02:42.900 --> 00:02:44.586
I've done my pretest.

00:02:44.586 --> 00:02:46.210
And now the system
is actually building

00:02:46.210 --> 00:02:50.640
a model for what I likely know
and what I likely don't know.

00:02:50.640 --> 00:02:52.760
And we'll look at
that in a second.

00:02:52.760 --> 00:02:56.510
So now you see that some of
these tiles changed colors.

00:02:56.510 --> 00:03:00.130
Gray means that I haven't
even practiced this thing yet,

00:03:00.130 --> 00:03:04.377
while blue means that I've shown
that the system thinks that I

00:03:04.377 --> 00:03:05.960
might know that at
a reasonable level.

00:03:05.960 --> 00:03:09.670
And the deeper the
darkness of the blue,

00:03:09.670 --> 00:03:11.640
the better that I
actually know that.

00:03:11.640 --> 00:03:13.990
So let's actually
start looking at these.

00:03:13.990 --> 00:03:18.500
So you can see here that I can
go on each of the concepts,

00:03:18.500 --> 00:03:23.790
and it says I'm Practiced
for congruent segments.

00:03:23.790 --> 00:03:25.400
It says right here
Need Practice,

00:03:25.400 --> 00:03:27.759
or it hasn't started for
points, lines, and planes.

00:03:27.759 --> 00:03:29.300
And if you click on
any one of these,

00:03:29.300 --> 00:03:31.590
you can go directly
to that exercise.

00:03:31.590 --> 00:03:33.160
So based on how I
did the pretest,

00:03:33.160 --> 00:03:35.920
it immediately figured out
what I probably know well,

00:03:35.920 --> 00:03:38.790
what I might not know that well.

00:03:38.790 --> 00:03:39.870
And now this is key.

00:03:39.870 --> 00:03:41.600
Instead of having
the old Khan Academy

00:03:41.600 --> 00:03:43.570
where you just had this huge
Knowledge Map, and you said,

00:03:43.570 --> 00:03:46.020
oh, I guess I'll work on
something-- one of these 500

00:03:46.020 --> 00:03:47.630
things that I see
in front of me.

00:03:47.630 --> 00:03:51.870
This now will recommend what
the next best thing is for me.

00:03:51.870 --> 00:03:52.610
So let me see.

00:03:52.610 --> 00:03:56.800
So Reading tables 2, so I
can start working on that.

00:03:56.800 --> 00:03:59.620
The table below shows solar
panel installations by state

00:03:59.620 --> 00:04:02.170
during the last fiscal year,
except one entry is missing.

00:04:02.170 --> 00:04:03.530
So this entry is missing.

00:04:03.530 --> 00:04:06.450
And if I'm completely confused
by this-- I've never seen it--

00:04:06.450 --> 00:04:08.800
we have a video right
here that should hopefully

00:04:08.800 --> 00:04:11.616
help explain that concept.

00:04:11.616 --> 00:04:14.040
Let me go over here.

00:04:14.040 --> 00:04:16.738
And it says for this one I
have to get five in a row.

00:04:16.738 --> 00:04:18.279
And if I get five
in a row, I'm going

00:04:18.279 --> 00:04:20.019
to get to that Practiced state.

00:04:20.019 --> 00:04:24.184
So, I will just
continue touring.

00:04:24.184 --> 00:04:25.850
But I don't have to
do exactly this one.

00:04:25.850 --> 00:04:27.266
This is what the
system recommends

00:04:27.266 --> 00:04:29.069
the most, is the
next best thing.

00:04:29.069 --> 00:04:30.610
But then there's
other things that it

00:04:30.610 --> 00:04:33.620
thinks I should also be
working on right now.

00:04:33.620 --> 00:04:36.350
So say finding absolute values.

00:04:36.350 --> 00:04:38.500
What is the absolute
value of negative 6.5?

00:04:38.500 --> 00:04:39.810
Well that's 6.5.

00:04:39.810 --> 00:04:42.770
Likewise, we still
have videos there,

00:04:42.770 --> 00:04:46.440
and I got one out
of my five correct.

00:04:46.440 --> 00:04:49.950
And if you have a coach or
if you are someone's coach,

00:04:49.950 --> 00:04:52.500
the coach can also make
their own recommendations.

00:04:52.500 --> 00:04:54.120
Obviously we have a
lot of coach tools

00:04:54.120 --> 00:04:56.300
so that a coach can
keep track of what

00:04:56.300 --> 00:04:57.430
a student is working on.

00:04:57.430 --> 00:04:58.804
And then based on
that, the coach

00:04:58.804 --> 00:05:02.220
can recommend particular
exercises to be focused on

00:05:02.220 --> 00:05:04.210
or to be prioritized.

00:05:04.210 --> 00:05:07.140
Now once you've spent
some time on the system,

00:05:07.140 --> 00:05:10.040
your Learning Dashboard
might look a little bit more

00:05:10.040 --> 00:05:10.632
like this.

00:05:10.632 --> 00:05:12.340
And to get closer and
closer to Mastery--

00:05:12.340 --> 00:05:14.900
to get deeper and deeper
shades of blue there--

00:05:14.900 --> 00:05:17.250
not only do you need to
practice it-- get five in a row

00:05:17.250 --> 00:05:20.330
and that sometimes might change
depending on the exercise--

00:05:20.330 --> 00:05:22.460
but later on you need
to show that you still

00:05:22.460 --> 00:05:24.390
know how to do
that thing in a mix

00:05:24.390 --> 00:05:26.900
of many, many, many
different types of problems.

00:05:26.900 --> 00:05:29.190
And the place that happens
is right over here,

00:05:29.190 --> 00:05:31.490
which we call our
Mastery Challenge.

00:05:31.490 --> 00:05:34.670
This is the tool that you want
to use repeatedly in order

00:05:34.670 --> 00:05:37.630
to get those squares
as dark as possible.

00:05:37.630 --> 00:05:39.400
So let me take a
Mastery Challenge.

00:05:39.400 --> 00:05:43.322
Negative 7 minus 3i.

00:05:43.322 --> 00:05:44.690
It'd be the absolute value of 5.

00:05:44.690 --> 00:05:45.523
Well, that's just 5.

00:05:48.400 --> 00:05:52.130
Is this triangle equilateral,
isosceles, or scalene?

00:05:52.130 --> 00:05:55.390
So it is definitely isosceles.

00:05:55.390 --> 00:05:57.609
If it was equilateral,
these would be 60 degrees.

00:05:57.609 --> 00:05:58.400
So it is isosceles.

00:05:58.400 --> 00:06:01.060
You have two common base angles.

00:06:01.060 --> 00:06:02.509
So you see it's
going through all

00:06:02.509 --> 00:06:04.050
of the different
types of mathematics

00:06:04.050 --> 00:06:06.050
to make sure that not
only have I remembered how

00:06:06.050 --> 00:06:07.760
to do these, not
only have I retained

00:06:07.760 --> 00:06:09.820
the understanding
of these concepts,

00:06:09.820 --> 00:06:12.082
but I also can do it
in a mixed environment

00:06:12.082 --> 00:06:13.790
where I have to also
recognize, hey, what

00:06:13.790 --> 00:06:17.230
is a tool I need
to use right now.

00:06:17.230 --> 00:06:19.420
What are the factors of 18?

00:06:19.420 --> 00:06:20.120
So let's see.

00:06:20.120 --> 00:06:21.230
We could have 1.

00:06:21.230 --> 00:06:23.190
We could have 2, 3.

00:06:23.190 --> 00:06:24.660
Let's see, 4 isn't divisible.

00:06:24.660 --> 00:06:25.740
5 isn't divisible.

00:06:25.740 --> 00:06:27.350
6 is.

00:06:27.350 --> 00:06:30.617
9 is, and 18.

00:06:30.617 --> 00:06:32.200
And I could have
used this little tool

00:06:32.200 --> 00:06:34.340
to help me with that, but
if you know the factors

00:06:34.340 --> 00:06:35.430
you can just type them in.

00:06:35.430 --> 00:06:37.130
So let's check our answer.

00:06:37.130 --> 00:06:40.060
Plot 10 comma 6 and
select the quadrant

00:06:40.060 --> 00:06:42.650
in which the point lies.

00:06:42.650 --> 00:06:47.016
10 comma 6, and that's
sitting in the first quadrant.

00:06:51.420 --> 00:06:55.140
Move the orange dot to
negative 3 minus 2i.

00:06:55.140 --> 00:06:57.960
Negative 3 minus 2i.

00:06:57.960 --> 00:07:00.370
And now I'm completing
my Mastery Challenge,

00:07:00.370 --> 00:07:02.810
and I'm going to get essentially
a report for what did I

00:07:02.810 --> 00:07:05.180
achieve by completing
this Mastery Challenge.

00:07:09.060 --> 00:07:11.490
And it's showing us the
things that have gotten darker

00:07:11.490 --> 00:07:13.600
based on the fact
that these are things

00:07:13.600 --> 00:07:15.320
that I had practiced
in the past.

00:07:15.320 --> 00:07:17.330
But in this last
eight problems, I

00:07:17.330 --> 00:07:19.330
showed that I still
remembered them.

00:07:19.330 --> 00:07:21.810
And I could do it
in a situation where

00:07:21.810 --> 00:07:24.980
I'm getting a whole
mix of problems.

00:07:24.980 --> 00:07:28.690
And so it's our hope that you
enjoy this as much as I just

00:07:28.690 --> 00:07:31.490
did, that this
learning flow really

00:07:31.490 --> 00:07:33.900
makes sure that you
know where to start,

00:07:33.900 --> 00:07:36.480
that you're starting on
things that are at your level.

00:07:36.480 --> 00:07:38.330
And not only are you
learning new things,

00:07:38.330 --> 00:07:40.734
but you have an opportunity
to review those new things,

00:07:40.734 --> 00:07:42.400
and you have an
opportunity to make sure

00:07:42.400 --> 00:07:45.360
that you know when to use
those things when you're

00:07:45.360 --> 00:07:47.480
faced with a whole
mix of problems.

00:07:47.480 --> 00:07:52.090
So I look forward to seeing how
you can fill out your Knowledge

00:07:52.090 --> 00:07:54.000
Map here, fill out
all of these skills,

00:07:54.000 --> 00:07:56.890
and fulfill your mission
in the world of math,

00:07:56.890 --> 00:07:58.410
and in the future,
other missions.

00:07:58.410 --> 00:08:00.280
And we really want
to hear from you

00:08:00.280 --> 00:08:03.910
to figure out how we can make
this entire experience better.

00:08:03.910 --> 00:08:06.480
So I encourage you go down
to the bottom of these pages

00:08:06.480 --> 00:08:08.540
and say Give Feedback.

00:08:08.540 --> 00:08:09.860
And tell us what you think.

00:08:09.860 --> 00:08:12.320
Tell us what's working for you,
what's not working for you.

00:08:12.320 --> 00:08:15.320
And if you have ideas for things
that might make it even better,

00:08:15.320 --> 00:08:17.260
let us know.


Constructing an angle bisector using a compass and straightedge

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.481
We're asked to construct an
angle bisector for the given

00:00:03.481 --> 00:00:03.980
angle.

00:00:03.980 --> 00:00:05.841
So this is the angle
they're talking about.

00:00:05.841 --> 00:00:07.340
And they want us
to make a line that

00:00:07.340 --> 00:00:09.460
goes right in between
that angle, that

00:00:09.460 --> 00:00:12.450
divides that angle
into two angles that

00:00:12.450 --> 00:00:13.930
have equal measure,
that have half

00:00:13.930 --> 00:00:15.890
the measure of the first angle.

00:00:15.890 --> 00:00:18.610
So let's first find
two points that

00:00:18.610 --> 00:00:20.980
are equidistant from this
point right over here

00:00:20.980 --> 00:00:22.880
on each of these rays.

00:00:22.880 --> 00:00:31.120
So to do that, let's
draw one circle here.

00:00:31.120 --> 00:00:32.920
And I can make
this of any radius.

00:00:32.920 --> 00:00:34.947
Wherever this intersects
with the rays,

00:00:34.947 --> 00:00:36.530
that's where I'm
going to put a point.

00:00:36.530 --> 00:00:38.460
So let's say, here and here.

00:00:38.460 --> 00:00:40.300
Notice both of these
points, since they're

00:00:40.300 --> 00:00:42.180
both on this circle,
are going to be

00:00:42.180 --> 00:00:43.780
equidistant from
this point, which

00:00:43.780 --> 00:00:45.870
is the center of the circle.

00:00:45.870 --> 00:00:48.770
Now, what I want
to do is construct

00:00:48.770 --> 00:00:54.770
a line that is equidistant
from both of these points.

00:00:54.770 --> 00:00:57.600
And we've done that already
when we looked at perpendicular

00:00:57.600 --> 00:01:01.260
bisectors for lines in
this construction module.

00:01:01.260 --> 00:01:02.800
So let's do that.

00:01:02.800 --> 00:01:05.319
So let's add their compass.

00:01:05.319 --> 00:01:07.985
And so what I want
to do, this circle

00:01:07.985 --> 00:01:09.790
is centered at this point.

00:01:09.790 --> 00:01:11.800
And it has a radius
equal to the distance

00:01:11.800 --> 00:01:13.860
between this point
and that point.

00:01:13.860 --> 00:01:16.100
And then I do that again.

00:01:16.100 --> 00:01:18.690
So this circle is
centered at this point

00:01:18.690 --> 00:01:22.160
and has a radius
equal to the distance

00:01:22.160 --> 00:01:24.070
between that point
and that point.

00:01:24.070 --> 00:01:26.170
And then the two places
where they intersect

00:01:26.170 --> 00:01:28.940
are equidistant to
both of these points.

00:01:32.690 --> 00:01:36.820
And so we can now draw our
angle bisector, just like that.

00:01:36.820 --> 00:01:40.360
And you might say,
well, how do we really

00:01:40.360 --> 00:01:43.540
know that this angle
is equal to this angle?

00:01:43.540 --> 00:01:45.260
Well, there's a couple
ways we can tell.

00:01:45.260 --> 00:01:47.580
We know this distance
right over here

00:01:47.580 --> 00:01:50.650
is equal to this distance
right over there.

00:01:50.650 --> 00:01:52.950
We know that this
distance over here

00:01:52.950 --> 00:01:55.110
is equal to this
distance over here.

00:01:55.110 --> 00:01:58.060
And both of these
triangles share this line.

00:01:58.060 --> 00:02:00.450
So essentially, if you look
at this point, this point,

00:02:00.450 --> 00:02:02.080
and this point, that
forms a triangle.

00:02:02.080 --> 00:02:04.020
And if you look at this point,
this point, and this point,

00:02:04.020 --> 00:02:05.240
that forms a triangle.

00:02:05.240 --> 00:02:07.480
We know those two
triangles are congruent,

00:02:07.480 --> 00:02:09.970
so this angle must be
equal to this angle.

00:02:09.970 --> 00:02:11.730
These are the
corresponding angles.

00:02:11.730 --> 00:02:13.300
So they're going
to be congruent.

00:02:13.300 --> 00:02:16.170
This is an angle bisector.


Constructing a perpendicular line using a compass and straightedge

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:04.590
Construct a line perpendicular
to the given line.

00:00:04.590 --> 00:00:08.210
So if I can pick two
arbitrary points on this line,

00:00:08.210 --> 00:00:10.650
and if I can make a
line that is always

00:00:10.650 --> 00:00:13.010
equidistant from
those two points,

00:00:13.010 --> 00:00:14.629
then that line will
be perpendicular.

00:00:14.629 --> 00:00:16.670
And actually, it will be
a perpendicular bisector

00:00:16.670 --> 00:00:19.140
of the segment formed
by those two points.

00:00:19.140 --> 00:00:21.404
Now, they don't care whether
we're bisecting anything.

00:00:21.404 --> 00:00:23.320
But they do care about
it being perpendicular.

00:00:23.320 --> 00:00:24.250
So let's do this.

00:00:24.250 --> 00:00:27.340
So I'm going to draw a
circle with my compass.

00:00:27.340 --> 00:00:30.690
And so let's just pick that
point right over there.

00:00:30.690 --> 00:00:32.890
I could adjust the
radius if I like.

00:00:32.890 --> 00:00:35.920
But I might as well-- well, I'll
just leave it right over there.

00:00:35.920 --> 00:00:37.230
Now let me draw another circle.

00:00:37.230 --> 00:00:38.646
And this time, I'm
going to center

00:00:38.646 --> 00:00:43.530
it where the first circle
intersects with my line.

00:00:43.530 --> 00:00:46.180
And then I'm going to
adjust the radius to overlap

00:00:46.180 --> 00:00:49.480
with the first dot.

00:00:49.480 --> 00:00:52.550
And now, where these
two circles intersect,

00:00:52.550 --> 00:00:56.320
those are points that
are equidistant from both

00:00:56.320 --> 00:00:59.500
of these centers that
I just constructed.

00:00:59.500 --> 00:01:02.620
So let me draw a line
that connects those two.

00:01:02.620 --> 00:01:06.940
And that line is going
to be perpendicular

00:01:06.940 --> 00:01:10.000
to our original line.


Constructing a perpendicular bisector using a compass and straightedge

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:02.770
We're asked to construct
a perpendicular

00:00:02.770 --> 00:00:06.092
bisector of the line segment AB.

00:00:06.092 --> 00:00:07.550
So the fact that
it's perpendicular

00:00:07.550 --> 00:00:10.100
means that this line will
make a 90-degree angle

00:00:10.100 --> 00:00:11.555
where it intersects with AB.

00:00:11.555 --> 00:00:13.120
And it's going to
bisect it, so it's

00:00:13.120 --> 00:00:15.310
going to go halfway in between.

00:00:15.310 --> 00:00:18.580
And I have at my disposal
some tools I can put out.

00:00:18.580 --> 00:00:19.990
I can draw things
with a compass,

00:00:19.990 --> 00:00:21.770
and I can add a straight edge.

00:00:21.770 --> 00:00:22.730
So let's try this out.

00:00:22.730 --> 00:00:24.260
So let me add a compass.

00:00:24.260 --> 00:00:25.780
And so this is a
virtual compass.

00:00:25.780 --> 00:00:27.946
So in a real compass, it's
one of those little metal

00:00:27.946 --> 00:00:29.880
things where you can
pivot it on one point,

00:00:29.880 --> 00:00:31.900
and you can draw a
circle of any radius.

00:00:31.900 --> 00:00:36.620
And so here I'm going to draw--
I'm going to center it at A.

00:00:36.620 --> 00:00:41.340
And I'm going to make the radius
equal to the length of AB.

00:00:41.340 --> 00:00:45.260
Now I'm going to add another
circle with my compass.

00:00:45.260 --> 00:00:47.290
And now, I'm going
to center it at B

00:00:47.290 --> 00:00:52.000
and make the radius equal to AB.

00:00:52.000 --> 00:00:53.950
And now, this
gives me two points

00:00:53.950 --> 00:00:57.640
that I can actually use to
draw my perpendicular bisector.

00:00:57.640 --> 00:00:59.840
If I connected this
point and this point,

00:00:59.840 --> 00:01:03.360
it is going to bisect
AB, and it's also

00:01:03.360 --> 00:01:04.590
going to be perpendicular.

00:01:04.590 --> 00:01:06.600
So let's add a
straight edge here,

00:01:06.600 --> 00:01:08.310
so this is to draw a line.

00:01:08.310 --> 00:01:10.580
So I'm going to draw a
line between that point

00:01:10.580 --> 00:01:12.724
and that point right over there.

00:01:12.724 --> 00:01:14.140
And let me scroll
down, so you can

00:01:14.140 --> 00:01:15.630
look at it a little bit clearer.

00:01:15.630 --> 00:01:16.450
So there you go.

00:01:16.450 --> 00:01:17.740
That's my construction.

00:01:17.740 --> 00:01:22.395
I've made a perpendicular
bisector for segment AB.

00:01:22.395 --> 00:01:23.650
Check my answer.

00:01:23.650 --> 00:01:25.670
We got it right.


Radius and center for a circle equation in standard form

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.400
The equation of a
circle C is x plus 3

00:00:03.400 --> 00:00:06.340
squared plus y minus 4
squared is equal to 49.

00:00:06.340 --> 00:00:09.470
What are its center
h, k and its radius r?

00:00:09.470 --> 00:00:12.210
So let's just remind
ourselves what a circle is.

00:00:12.210 --> 00:00:15.910
You have some point,
let's call that h, k.

00:00:15.910 --> 00:00:18.560
The circle is the set
of all points that

00:00:18.560 --> 00:00:21.490
are equidistant from that point.

00:00:21.490 --> 00:00:23.550
So let's take the set
of all points that

00:00:23.550 --> 00:00:26.380
are, say, r away from h, k.

00:00:26.380 --> 00:00:30.780
So let's say that this
distance right over here

00:00:30.780 --> 00:00:34.940
is r, and so we want all of the
set of points that are exactly

00:00:34.940 --> 00:00:36.060
r away.

00:00:36.060 --> 00:00:39.771
So all the points x comma
y that are exactly r way.

00:00:39.771 --> 00:00:41.770
And so you could imagine
you could rotate around

00:00:41.770 --> 00:00:44.300
and all of these points are
going to be exactly r away.

00:00:44.300 --> 00:00:48.470
And I'm going to try my
best to draw at least a

00:00:48.470 --> 00:00:50.760
somewhat perfect looking circle.

00:00:50.760 --> 00:00:53.590
I won't be able to do a perfect
job of it, but you get a sense.

00:00:53.590 --> 00:00:57.830
All of these are
exactly r away, at least

00:00:57.830 --> 00:00:59.510
if I were to draw it properly.

00:00:59.510 --> 00:01:01.170
They are r away.

00:01:01.170 --> 00:01:05.319
So how do we find an equation
in terms of r and h, k, and x

00:01:05.319 --> 00:01:08.540
and y that describes
all these points?

00:01:08.540 --> 00:01:10.740
Well, we know how to find
the distance between two

00:01:10.740 --> 00:01:12.610
points on a coordinate plane.

00:01:12.610 --> 00:01:15.820
In fact, it comes straight out
of the Pythagorean theorem.

00:01:15.820 --> 00:01:18.800
If we were to draw a vertical
line right over here, that

00:01:18.800 --> 00:01:22.320
essentially is the change in the
vertical axis between these two

00:01:22.320 --> 00:01:27.720
points, up here, we're
at y, here we're k,

00:01:27.720 --> 00:01:31.200
so this distance is
going to be y minus k.

00:01:31.200 --> 00:01:33.625
We can do the exact same
thing on the horizontal axis.

00:01:39.030 --> 00:01:43.840
This x-coordinate is x while
this x-coordinate is h.

00:01:43.840 --> 00:01:50.060
So this is going to be x
minus h is this distance.

00:01:50.060 --> 00:01:52.610
And this is a right triangle,
because by definition, we're

00:01:52.610 --> 00:01:55.780
saying, hey, we're measuring
vertical distance here.

00:01:55.780 --> 00:01:57.640
We're measuring
horizontal distance here,

00:01:57.640 --> 00:01:59.500
so these two things
are perpendicular.

00:01:59.500 --> 00:02:01.510
And so from the
Pythagorean theorem,

00:02:01.510 --> 00:02:04.860
we know that this squared
plus this squared must

00:02:04.860 --> 00:02:07.040
be equal to our
distance squared,

00:02:07.040 --> 00:02:09.340
and this is where the
distance formula comes from.

00:02:09.340 --> 00:02:18.530
So we know that x minus h
squared plus y minus k squared

00:02:18.530 --> 00:02:21.430
must be equal to r squared.

00:02:21.430 --> 00:02:23.450
This is the equation
for the set--

00:02:23.450 --> 00:02:26.120
this describes any x
and y that satisfies

00:02:26.120 --> 00:02:29.607
this equation will
sit on this circle.

00:02:29.607 --> 00:02:32.190
Now, with that out of the way,
let's go answer their question.

00:02:32.190 --> 00:02:34.530
The equation of the
circle is this thing.

00:02:34.530 --> 00:02:37.075
And this looks awfully
close to what we just wrote,

00:02:37.075 --> 00:02:38.450
we just have to
make sure that we

00:02:38.450 --> 00:02:40.033
don't get confused
with the negatives.

00:02:40.033 --> 00:02:43.620
Remember, it has to be in the
form x minus h, y minus k.

00:02:43.620 --> 00:02:45.510
So let's write it a
little bit differently.

00:02:45.510 --> 00:02:47.680
Instead of x plus
3 squared, we can

00:02:47.680 --> 00:02:53.840
write that as x minus
negative 3 squared.

00:02:53.840 --> 00:02:55.340
And then plus--
well this is already

00:02:55.340 --> 00:03:00.490
in the form-- plus
y minus 4 squared

00:03:00.490 --> 00:03:05.310
is equal to, instead of 49, we
can just call that 7 squared.

00:03:05.310 --> 00:03:08.740
And so now it
becomes pretty clear

00:03:08.740 --> 00:03:11.990
that our h is
negative 3-- I want

00:03:11.990 --> 00:03:20.250
to do that in the red color--
that our h is negative 3,

00:03:20.250 --> 00:03:26.500
and that our k is positive
4, and that our r is 7.

00:03:26.500 --> 00:03:31.460
So we could say h comma k
is equal to negative 3 comma

00:03:31.460 --> 00:03:32.350
positive 4.

00:03:32.350 --> 00:03:34.100
Make sure to get-- you
know you might say,

00:03:34.100 --> 00:03:35.558
hey, there's a
negative 4 here, no.

00:03:35.558 --> 00:03:37.540
But look, it's minus k, minus 4.

00:03:37.540 --> 00:03:38.850
So k is 4.

00:03:38.850 --> 00:03:40.917
Likewise, it's minus h.

00:03:40.917 --> 00:03:42.750
You might say, hey,
maybe h is a positive 3,

00:03:42.750 --> 00:03:44.580
but no you're subtracting the h.

00:03:44.580 --> 00:03:50.328
So you'd say minus negative 3,
and similarly, the radius is 7.


Plotting complex numbers on the complex plane

WEBVTT
Kind: captions
Language: en

00:00:00.940 --> 00:00:04.270
Move the orange dot
to negative 2 plus 2i.

00:00:04.270 --> 00:00:05.690
So we have a
complex number here.

00:00:05.690 --> 00:00:07.380
It has a real part, negative 2.

00:00:07.380 --> 00:00:10.520
It has an imaginary
part, you have 2 times i.

00:00:10.520 --> 00:00:12.020
And what you see
here is we're going

00:00:12.020 --> 00:00:14.650
to plot it on this
two-dimensional grid,

00:00:14.650 --> 00:00:16.600
but it's not our
traditional coordinate axes.

00:00:16.600 --> 00:00:18.058
In our traditional
coordinate axis,

00:00:18.058 --> 00:00:22.440
you're plotting a real x value
versus a real y-coordinate.

00:00:22.440 --> 00:00:24.320
Here on the horizontal
axis, that's

00:00:24.320 --> 00:00:26.570
going to be the real part
of our complex number.

00:00:26.570 --> 00:00:29.460
And our vertical axis is going
to be the imaginary part.

00:00:29.460 --> 00:00:31.240
So in this example,
this complex number,

00:00:31.240 --> 00:00:33.900
our real part is the
negative 2 and then

00:00:33.900 --> 00:00:36.460
our imaginary part
is a positive 2.

00:00:36.460 --> 00:00:38.720
And so that right over
there in the complex plane

00:00:38.720 --> 00:00:41.940
is the point negative 2 plus 2i.

00:00:41.940 --> 00:00:44.160
Let's do a few more of these.

00:00:44.160 --> 00:00:46.100
So 5 plus 2i.

00:00:46.100 --> 00:00:50.420
Once again, real part is
5, imaginary part is 2,

00:00:50.420 --> 00:00:52.020
and we're done.

00:00:52.020 --> 00:00:53.630
Let's do two more of these.

00:00:53.630 --> 00:00:55.310
1 plus 5i.

00:00:55.310 --> 00:01:00.632
1-- that's the real part--
plus 5i right over that Im.

00:01:00.632 --> 00:01:03.360
All right, let's do
one more of these.

00:01:03.360 --> 00:01:05.099
4 minus 4i.

00:01:05.099 --> 00:01:08.690
Real part is 4, imaginary
part is negative 4.

00:01:11.330 --> 00:01:13.420
And we're done.


Rotation of polygons example

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:03.480
What is the image of the polygon
below after the rotation?

00:00:03.480 --> 00:00:08.879
So this is a rotational
transformation of 270 degrees.

00:00:08.879 --> 00:00:10.420
So what we-- and
when they talk about

00:00:10.420 --> 00:00:11.850
a rotational
transformation, at least

00:00:11.850 --> 00:00:13.433
in the context of
this exercise, we're

00:00:13.433 --> 00:00:16.140
talking about a rotation
around the origin,

00:00:16.140 --> 00:00:17.552
so around this point.

00:00:17.552 --> 00:00:19.010
So the way I like
to think about it

00:00:19.010 --> 00:00:21.540
is pick one of these points,
one of these orange points,

00:00:21.540 --> 00:00:25.570
and, essentially, rotate it
around the origin 270 degrees.

00:00:25.570 --> 00:00:26.620
So let's do that.

00:00:26.620 --> 00:00:28.500
So this would be rotating it.

00:00:28.500 --> 00:00:30.730
And when we talk
about positive angles,

00:00:30.730 --> 00:00:33.520
we're talking about going in
the counterclockwise directions.

00:00:33.520 --> 00:00:36.070
The same way that we,
typically, measure angles

00:00:36.070 --> 00:00:39.120
when we're doing trigonometry
with the unit circle and things

00:00:39.120 --> 00:00:39.930
like that.

00:00:39.930 --> 00:00:44.720
So here, so we started here,
and let's rotate 90 degrees.

00:00:44.720 --> 00:00:47.200
Then we could keep rotating,
rotate another 90 degrees

00:00:47.200 --> 00:00:49.640
so we've rotated a
total of 180 degrees.

00:00:49.640 --> 00:00:54.790
And then we rotate 90 more
degrees to get to 270.

00:00:54.790 --> 00:00:58.300
Let's do a couple more of these.

00:00:58.300 --> 00:01:03.010
What is the transformation that
rotates the blue solid shape

00:01:03.010 --> 00:01:05.260
to the orange dashed shape?

00:01:05.260 --> 00:01:06.810
Now, this one's
a little bit-- we

00:01:06.810 --> 00:01:09.500
don't have the manipulative
here to help us

00:01:09.500 --> 00:01:11.210
answer that question
so this is really

00:01:11.210 --> 00:01:15.240
going to challenge our
powers of visualization.

00:01:15.240 --> 00:01:18.240
So the main thing is to pick
a point on the blue shape

00:01:18.240 --> 00:01:21.910
and see where does it end up
on the orange dashed shape.

00:01:21.910 --> 00:01:23.680
And I'll pick this
point right over here

00:01:23.680 --> 00:01:27.430
because you can see
it's pointing straight

00:01:27.430 --> 00:01:28.830
to the top left.

00:01:28.830 --> 00:01:31.510
And then when you go onto
the orange dashed shape,

00:01:31.510 --> 00:01:33.450
it's pointing to
the bottom right.

00:01:33.450 --> 00:01:35.891
So this one went
all the way around.

00:01:35.891 --> 00:01:37.640
So you could say it
went-- it was pointing

00:01:37.640 --> 00:01:39.520
from this direction
and went all the way

00:01:39.520 --> 00:01:42.350
to going in the exact
opposite direction.

00:01:42.350 --> 00:01:45.870
So this was a 180-degree
transformation,

00:01:45.870 --> 00:01:49.980
a 180-degree rotation.

00:01:49.980 --> 00:01:52.360
Let's do a few more.

00:01:52.360 --> 00:01:53.940
So what's the image
of the polygon?

00:01:53.940 --> 00:01:56.580
So they want us to do
another 270-degree rotation.

00:01:56.580 --> 00:01:58.160
So like before.

00:01:58.160 --> 00:02:03.130
So let's rotate this
3/4 of the way around.

00:02:03.130 --> 00:02:04.840
So let's see how we can do that.

00:02:04.840 --> 00:02:07.896
So right now-- so this is going
to be a little bit trickier.

00:02:07.896 --> 00:02:10.479
So if we-- actually, I'm going
to pick this point because it's

00:02:10.479 --> 00:02:12.900
easier to visualize
the 270 degrees.

00:02:12.900 --> 00:02:15.910
So now we've done 90 degrees.

00:02:15.910 --> 00:02:18.470
Now we can do 180 degrees.

00:02:18.470 --> 00:02:24.580
And now we've done 270 degrees.


Determining a translation for a shape

WEBVTT
Kind: captions
Language: en

00:00:01.037 --> 00:00:03.735
PROBLEM: What was the translation applied to move

00:00:03.735 --> 00:00:07.353
the blue solid shape to the orange dashed shape?

00:00:07.353 --> 00:00:11.853
So if you were to start with this blue solid shape,

00:00:11.853 --> 00:00:13.567
and you wanted to shift it

00:00:13.567 --> 00:00:15.486
to where the orange dotted shape is,

00:00:15.486 --> 00:00:16.751
how would you do that?

00:00:16.751 --> 00:00:17.938
And the way that I like to think about it

00:00:17.938 --> 00:00:19.291
is to just pick a point,

00:00:19.291 --> 00:00:22.681
and see what had to happen to that point.

00:00:22.681 --> 00:00:24.794
So this point at the top of this – I guess –

00:00:24.794 --> 00:00:26.325
this quadrilateral-looking thing,

00:00:26.325 --> 00:00:27.659
it moved down here.

00:00:27.659 --> 00:00:29.601
So it moved one to the left.

00:00:29.601 --> 00:00:31.951
And it moved one down.

00:00:31.951 --> 00:00:33.600
So we can denote that with a transformation.

00:00:33.600 --> 00:00:35.788
Moving one to the left is negative one (-1)

00:00:35.788 --> 00:00:37.186
in the horizontal direction.

00:00:37.186 --> 00:00:37.946
If we were moving to the right,

00:00:37.946 --> 00:00:39.456
it would have been a positive one (+1).

00:00:39.456 --> 00:00:42.497
And we moved one down, in the vertical direction.

00:00:42.497 --> 00:00:45.050
And that is another negative one (-1).


Translations of polygons

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:04.350
What is the image of the polygon
below after the translation is

00:00:04.350 --> 00:00:07.930
capital T with the
subscript 5 comma 2?

00:00:07.930 --> 00:00:10.950
So that means let's translate
5 in the horizontal direction

00:00:10.950 --> 00:00:12.340
and 2 in the vertical direction.

00:00:12.340 --> 00:00:13.048
So let's do that.

00:00:13.048 --> 00:00:15.290
So I'm going to focus on
this point right over here.

00:00:15.290 --> 00:00:18.320
So that point is at
the point 3 comma 3.

00:00:18.320 --> 00:00:21.530
If I want to translate five
in the horizontal direction,

00:00:21.530 --> 00:00:23.790
the x value is
going to get to 8.

00:00:23.790 --> 00:00:29.320
So the x value is
going to 1, 2, 3, 4, 5.

00:00:29.320 --> 00:00:32.150
And then we want to
increase the y value by 2.

00:00:32.150 --> 00:00:34.140
1, 2.

00:00:34.140 --> 00:00:35.470
Let's check our answer.

00:00:35.470 --> 00:00:37.150
Let's do a couple more of these.

00:00:37.150 --> 00:00:38.640
That was lot of fun.

00:00:38.640 --> 00:00:41.000
So here we need to
shift it by negative 8

00:00:41.000 --> 00:00:42.830
in the horizontal direction.

00:00:42.830 --> 00:00:45.024
So this one right over
here, it's x value is 0,

00:00:45.024 --> 00:00:47.190
so we need to go essentially
negative 8, negative 1,

00:00:47.190 --> 00:00:56.720
negative 2, or we move to the
left 1, 2, 3, 4, 5, 6, 7, 8.

00:00:56.720 --> 00:00:59.300
And then we have to
move it down by 2.

00:00:59.300 --> 00:01:02.690
So we have to move
down 1, move it down 2.

00:01:02.690 --> 00:01:04.910
And we're done.


Completing the square to write equation in standard form of a circle

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:02.140
We're asked to graph the circle.

00:00:02.140 --> 00:00:04.819
And they give us this somewhat
crazy looking equation.

00:00:04.819 --> 00:00:06.610
And then we could graph
it right over here.

00:00:06.610 --> 00:00:09.109
And to graph a circle, you have
to know where its center is,

00:00:09.109 --> 00:00:12.080
and you have to know
what its radius is.

00:00:12.080 --> 00:00:13.590
So let me see if
I can change that.

00:00:13.590 --> 00:00:15.400
And you have to know
what its radius is.

00:00:15.400 --> 00:00:17.480
So what we need to do
is put this in some form

00:00:17.480 --> 00:00:20.330
where we can pick out its
center and its radius.

00:00:20.330 --> 00:00:23.510
Let me get my little scratch pad
out and see if we can do that.

00:00:23.510 --> 00:00:24.995
So this is that same equation.

00:00:24.995 --> 00:00:26.370
And what I
essentially want to do

00:00:26.370 --> 00:00:28.840
is I want to complete
the square in terms of x,

00:00:28.840 --> 00:00:31.930
and complete the square in terms
of y, to put it into a form

00:00:31.930 --> 00:00:33.730
that we can recognize.

00:00:33.730 --> 00:00:35.865
So first let's take
all of the x terms.

00:00:38.650 --> 00:00:42.630
So you have x squared and
4x on the left-hand side.

00:00:42.630 --> 00:00:46.357
So I could rewrite this
as x squared plus 4x.

00:00:46.357 --> 00:00:48.440
And I'm going to put some
parentheses around here,

00:00:48.440 --> 00:00:50.370
because I'm going to
complete the square.

00:00:50.370 --> 00:00:51.690
And then I have my y terms.

00:00:51.690 --> 00:00:53.280
I'll circle those
in-- well, the red

00:00:53.280 --> 00:00:54.571
looks too much like the purple.

00:00:54.571 --> 00:00:56.010
I'll circle those in blue.

00:00:56.010 --> 00:00:58.860
y squared and negative 4y.

00:00:58.860 --> 00:01:05.420
So we have plus y
squared minus 4y.

00:01:05.420 --> 00:01:08.972
And then we have a minus 17.

00:01:08.972 --> 00:01:10.680
And I'll just do that
in a neutral color.

00:01:10.680 --> 00:01:16.820
So minus 17 is equal to 0.

00:01:16.820 --> 00:01:18.920
Now, what I want
to do is make each

00:01:18.920 --> 00:01:22.050
of these purple expressions
perfect squares.

00:01:22.050 --> 00:01:23.260
So how could I do that here?

00:01:23.260 --> 00:01:27.070
Well, this would be a perfect
square if I took half of this 4

00:01:27.070 --> 00:01:28.000
and I squared it.

00:01:28.000 --> 00:01:31.280
So if I made this plus 4,
then this entire expression

00:01:31.280 --> 00:01:34.920
would be x plus 2 squared.

00:01:34.920 --> 00:01:37.576
And you can verify
that if you like.

00:01:37.576 --> 00:01:39.640
If you need to review on
completing the square,

00:01:39.640 --> 00:01:41.990
there's plenty of videos
on Khan Academy on that.

00:01:41.990 --> 00:01:46.500
All we did is we took
half of this coefficient

00:01:46.500 --> 00:01:48.690
and then squared it to get 4.

00:01:48.690 --> 00:01:51.530
Half of 4 is 2,
square it to get 4.

00:01:51.530 --> 00:01:54.140
And that comes straight out of
the idea if you take x plus 2

00:01:54.140 --> 00:01:57.380
and square it, it's going
to be x squared plus twice

00:01:57.380 --> 00:02:01.450
the product of 2 and
x, plus 2 squared.

00:02:01.450 --> 00:02:03.832
Now, we can't just
willy-nilly add a 4 here.

00:02:03.832 --> 00:02:05.790
We had an equality before,
and just adding a 4,

00:02:05.790 --> 00:02:07.140
it wouldn't be equal anymore.

00:02:07.140 --> 00:02:09.009
So if we want to
maintain the equality,

00:02:09.009 --> 00:02:13.110
we have to add 4 on the
right-hand side as well.

00:02:13.110 --> 00:02:15.260
Now, let's do the same
thing for the y's.

00:02:15.260 --> 00:02:19.020
Half of this coefficient right
over here is a negative 2.

00:02:19.020 --> 00:02:21.837
If we square negative 2,
it becomes a positive 4.

00:02:21.837 --> 00:02:23.670
We can't just do that
on the left-hand side.

00:02:23.670 --> 00:02:26.530
We have to do that on the
right-hand side as well.

00:02:26.530 --> 00:02:31.750
Now, what we have in blue
becomes y minus 2 squared.

00:02:31.750 --> 00:02:34.640
And of course, we
have the minus 17.

00:02:34.640 --> 00:02:37.150
But why don't we
add 17 to both sides

00:02:37.150 --> 00:02:39.320
as well to get rid of
this minus 17 here?

00:02:39.320 --> 00:02:44.300
So let's add 17 on the left
and add 17 on the right.

00:02:44.300 --> 00:02:47.100
So on the left, we're just left
with these two expressions.

00:02:47.100 --> 00:02:50.190
And on the right, we
have 4 plus 4 plus 17.

00:02:50.190 --> 00:02:55.190
Well, that's 8 plus 17,
which is equal to 25.

00:02:55.190 --> 00:02:57.710
Now, this is a form
that we recognize.

00:02:57.710 --> 00:03:04.230
If you have the form x minus a
squared plus y minus b squared

00:03:04.230 --> 00:03:10.190
is equal to r squared, we know
that the center is at the point

00:03:10.190 --> 00:03:12.740
a, b, essentially,
the point that

00:03:12.740 --> 00:03:15.830
makes both of these equal to 0.

00:03:15.830 --> 00:03:20.420
And that the radius
is going to be r.

00:03:20.420 --> 00:03:22.310
So if we look over
here, what is our a?

00:03:22.310 --> 00:03:23.670
We have to be careful here.

00:03:23.670 --> 00:03:24.700
Our a isn't 2.

00:03:24.700 --> 00:03:29.550
Our a is negative 2. x minus
negative 2 is equal to 2.

00:03:29.550 --> 00:03:31.740
So the x-coordinate
of our center

00:03:31.740 --> 00:03:34.912
is going to be negative 2, and
the y-coordinate of our center

00:03:34.912 --> 00:03:35.620
is going to be 2.

00:03:35.620 --> 00:03:37.965
Remember, we care about the
x value that makes this 0,

00:03:37.965 --> 00:03:40.870
and the y value
that makes this 0.

00:03:40.870 --> 00:03:43.890
So the center is negative 2, 2.

00:03:43.890 --> 00:03:45.750
And this is the radius squared.

00:03:45.750 --> 00:03:49.560
So the radius is equal to 5.

00:03:49.560 --> 00:03:52.790
So let's go back to the
exercise and actually plot this.

00:03:52.790 --> 00:03:55.230
So it's negative 2, 2.

00:03:55.230 --> 00:03:58.546
So our center is negative 2, 2.

00:03:58.546 --> 00:03:59.670
So that's right over there.

00:03:59.670 --> 00:04:02.320
X is negative 2,
y is positive 2.

00:04:02.320 --> 00:04:04.040
And the radius is 5.

00:04:04.040 --> 00:04:07.350
So let's see, this
would be 1, 2, 3, 4, 5.

00:04:07.350 --> 00:04:10.000
So you have to go a little
bit wider than this.

00:04:10.000 --> 00:04:12.450
My pen is having trouble.

00:04:12.450 --> 00:04:13.190
There you go.

00:04:13.190 --> 00:04:15.870
1, 2, 3, 4, 5.

00:04:15.870 --> 00:04:17.589
Let's check our answer.

00:04:17.589 --> 00:04:19.540
We got it right.


Axis of symmetry

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:02.500
For each of these
diagrams, I want

00:00:02.500 --> 00:00:04.950
to think about whether
this blue line represents

00:00:04.950 --> 00:00:07.370
an axis of symmetry.

00:00:07.370 --> 00:00:10.740
And the way we can tell is if
on both sides of the blue line

00:00:10.740 --> 00:00:13.172
we essentially
have mirror images.

00:00:13.172 --> 00:00:13.880
So let's imagine.

00:00:13.880 --> 00:00:18.010
Let's take this top part of
this polygon, the part that

00:00:18.010 --> 00:00:20.596
is above this blue line
here, and let's reflect it

00:00:20.596 --> 00:00:22.220
across the blue line--
you could almost

00:00:22.220 --> 00:00:25.060
imagine that it's a reflection
over some type of a lake

00:00:25.060 --> 00:00:28.590
or something-- and see if we
get exactly what we have below.

00:00:28.590 --> 00:00:30.890
Then this would be
an axis of symmetry.

00:00:30.890 --> 00:00:33.770
So this point right over here,
this distance to the blue line,

00:00:33.770 --> 00:00:35.830
let's go-- the same
amount on the other side

00:00:35.830 --> 00:00:36.965
would get you right there.

00:00:36.965 --> 00:00:38.840
And so you immediately
see we start ending up

00:00:38.840 --> 00:00:43.090
with a point that is off
what's actually here in black,

00:00:43.090 --> 00:00:45.354
the actual bottom
part of the polygon.

00:00:45.354 --> 00:00:47.270
So this is a pretty good
clue that this is not

00:00:47.270 --> 00:00:48.103
an axis of symmetry.

00:00:48.103 --> 00:00:51.190
But let's just continue it,
just to go through the exercise.

00:00:51.190 --> 00:00:53.790
So this point, if you reflected
it across this blue line,

00:00:53.790 --> 00:00:55.240
would get you here.

00:00:55.240 --> 00:00:57.600
This point-- I'll do it
in a different color.

00:00:57.600 --> 00:01:00.010
This point, if you
were to reflect it

00:01:00.010 --> 00:01:02.440
across this blue
line, it would get

00:01:02.440 --> 00:01:04.940
you-- let me make sure I can
do that relatively straight.

00:01:04.940 --> 00:01:07.210
I can do a straighter
job than that.

00:01:07.210 --> 00:01:09.690
So if you go about
that distance about it,

00:01:09.690 --> 00:01:12.430
and I want to go straight
down into the blue line,

00:01:12.430 --> 00:01:15.390
and I'm going to go the same
distance on the other side,

00:01:15.390 --> 00:01:18.640
it gets me to
right around there.

00:01:18.640 --> 00:01:22.900
And then this point
over here, if I

00:01:22.900 --> 00:01:26.880
were to drop it
straight down, then

00:01:26.880 --> 00:01:29.270
if I were to go the same
distance on the other side,

00:01:29.270 --> 00:01:31.310
it gets me right around there.

00:01:31.310 --> 00:01:34.550
And then finally, this point
gets me right around there.

00:01:34.550 --> 00:01:36.260
So its mirror image
of this top part

00:01:36.260 --> 00:01:37.650
would look something like this.

00:01:42.830 --> 00:01:45.710
My best attempt to draw
it would look something

00:01:45.710 --> 00:01:48.780
like this, which is very
different than the part

00:01:48.780 --> 00:01:50.780
of the polygon that's
actually on the other side

00:01:50.780 --> 00:01:52.110
of this blue line.

00:01:52.110 --> 00:01:56.180
So in this case, the blue line
is not an axis of symmetry.

00:01:56.180 --> 00:01:57.822
So this is no.

00:01:57.822 --> 00:02:00.210
No, this blue line is
not an axis of symmetry.

00:02:00.210 --> 00:02:01.920
Now let's look at it over here.

00:02:01.920 --> 00:02:04.110
And our eyes pick this
out very naturally.

00:02:04.110 --> 00:02:06.700
Here you can see that it looks
like this blue line really

00:02:06.700 --> 00:02:08.330
divides this polygon in half.

00:02:08.330 --> 00:02:10.340
It really does look
like mirror images.

00:02:10.340 --> 00:02:12.200
It really does look
like, if you imagine

00:02:12.200 --> 00:02:16.430
that this is some type
of a lake, a still lake,

00:02:16.430 --> 00:02:17.900
so I shouldn't
actually draw waves,

00:02:17.900 --> 00:02:20.540
but this is some type of a lake,
that this is the reflection.

00:02:20.540 --> 00:02:22.760
And we can even go
point by point here.

00:02:22.760 --> 00:02:27.430
So this point right over
here is the same distance

00:02:27.430 --> 00:02:30.270
from, if we dropped a
perpendicular to this point

00:02:30.270 --> 00:02:33.720
as this one right over
here; this one over here,

00:02:33.720 --> 00:02:37.770
same distance, same distance
as this point right over here;

00:02:37.770 --> 00:02:39.940
and we could do that
for all of the points.

00:02:39.940 --> 00:02:42.110
So in this case,
the blue line does

00:02:42.110 --> 00:02:45.475
represent an axis of symmetry.


Shifting & reflecting functions

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.660
So this red curve is
the graph of f of x.

00:00:03.660 --> 00:00:06.910
And this blue curve is
the graph of g of x.

00:00:06.910 --> 00:00:10.275
And I want to try to express
g of x in terms of f of x.

00:00:10.275 --> 00:00:12.280
And so let's see
how they're related.

00:00:12.280 --> 00:00:13.750
So we pick any x.

00:00:13.750 --> 00:00:17.320
And we could start right
here at the vertex of f of x.

00:00:17.320 --> 00:00:19.520
And we see that, at least
at that point, g of x

00:00:19.520 --> 00:00:22.260
is exactly 1 higher than that.

00:00:22.260 --> 00:00:30.200
So g of 2-- I could
write this down-- g of 2

00:00:30.200 --> 00:00:36.130
is equal to f of 2 plus 1.

00:00:36.130 --> 00:00:38.149
Let's see if that's
true for any x.

00:00:38.149 --> 00:00:39.690
So then we can just
sample over here.

00:00:39.690 --> 00:00:43.310
Let's see, f of 4
is right over here.

00:00:43.310 --> 00:00:46.360
g of 4 is one more than that.

00:00:46.360 --> 00:00:49.840
f of 6 is right here.

00:00:49.840 --> 00:00:52.451
g of 6 is 1 more than that.

00:00:52.451 --> 00:00:54.700
So it looks like if we pick
any point over here-- even

00:00:54.700 --> 00:00:56.100
though there's a little bit
of an optical illusion--

00:00:56.100 --> 00:00:57.725
it looks like they
get closer together.

00:00:57.725 --> 00:00:59.659
They do if you look
try to find the closest

00:00:59.659 --> 00:01:00.700
distance between the two.

00:01:00.700 --> 00:01:02.199
But if you look at
vertical distance

00:01:02.199 --> 00:01:05.750
you see that it
stays a constant 1.

00:01:05.750 --> 00:01:07.545
So we can actually
generalize this.

00:01:07.545 --> 00:01:11.100
This is true for
any x. g of x is

00:01:11.100 --> 00:01:16.850
equal to f of x is
equal to f of x plus 1.

00:01:16.850 --> 00:01:19.160
Let's do a few more
examples of this.

00:01:19.160 --> 00:01:22.920
So right over here, here
is f of x in red again,

00:01:22.920 --> 00:01:25.060
and here is g of x.

00:01:25.060 --> 00:01:28.400
And so let's say we picked
x equals negative 4.

00:01:28.400 --> 00:01:30.340
This is f of negative 4.

00:01:30.340 --> 00:01:34.890
And we see g of negative
4 is 2 less than that.

00:01:34.890 --> 00:01:38.630
And we see whatever f of
x is, g of x-- no matter

00:01:38.630 --> 00:01:43.270
what x we pick-- g of x
seems to be exactly 2 less.

00:01:43.270 --> 00:01:48.000
g of x is exactly 2 less.

00:01:48.000 --> 00:01:50.820
So in this case, very
similar to the other one,

00:01:50.820 --> 00:01:54.310
g of x is going to
be equal to f of x.

00:01:54.310 --> 00:01:55.730
But instead of
adding, we're going

00:01:55.730 --> 00:02:00.110
to subtract 2 from f
of x. f of x minus 2.

00:02:00.110 --> 00:02:03.020
Let's do a few more examples.

00:02:03.020 --> 00:02:06.020
So here we have f
of x in red again.

00:02:06.020 --> 00:02:07.460
I'll label it.

00:02:07.460 --> 00:02:08.520
f of x.

00:02:08.520 --> 00:02:12.009
And here is g of x.

00:02:12.009 --> 00:02:13.550
So let's think about
it a little bit.

00:02:13.550 --> 00:02:16.750
Let's pick an
arbitrary point here.

00:02:16.750 --> 00:02:21.760
Let's say we have in red here,
this point right over there

00:02:21.760 --> 00:02:27.120
is the value of f of negative 3.

00:02:27.120 --> 00:02:28.570
This is negative 3.

00:02:28.570 --> 00:02:31.095
This is the point
negative 3, f of 3.

00:02:34.820 --> 00:02:40.675
Now g hits that same value
when x is equal to negative 1.

00:02:44.340 --> 00:02:45.890
So let's think about this.

00:02:45.890 --> 00:02:51.925
g of negative 1 is equal
to f of negative 3.

00:02:57.857 --> 00:02:59.690
And we could do that
with a bunch of points.

00:02:59.690 --> 00:03:07.632
We could see that g of 0, which
is right there-- let me do it

00:03:07.632 --> 00:03:10.490
in a color you can
see-- g of 0 is

00:03:10.490 --> 00:03:15.130
equivalent to f of negative 2.

00:03:15.130 --> 00:03:16.800
So let me write that down.

00:03:16.800 --> 00:03:23.990
g of 0 is equal to
f of negative 2.

00:03:23.990 --> 00:03:25.100
We could keep doing that.

00:03:25.100 --> 00:03:31.980
We could say g of 1,
which is right over here.

00:03:31.980 --> 00:03:32.980
This is 1.

00:03:32.980 --> 00:03:37.790
g of 1 is equal to
f of negative 1.

00:03:37.790 --> 00:03:44.710
g of 1 is equal to
f of negative 1.

00:03:44.710 --> 00:03:46.930
So I think you see
the pattern here.

00:03:46.930 --> 00:03:55.350
g of whatever is equal to the
function evaluated at 2 less

00:03:55.350 --> 00:03:56.630
than whatever is here.

00:03:56.630 --> 00:04:04.095
So we could say that g of
x is equal to f of-- well

00:04:04.095 --> 00:04:05.930
it's going to be 2 less than x.

00:04:05.930 --> 00:04:08.954
So f of x minus 2.

00:04:08.954 --> 00:04:10.120
So this is the relationship.

00:04:10.120 --> 00:04:12.830
g of x is equal
to f of x minus 2.

00:04:12.830 --> 00:04:14.600
And it's important
to realize here.

00:04:14.600 --> 00:04:19.250
When I get f of x minus 2 here--
and remember the function is

00:04:19.250 --> 00:04:23.500
being evaluated, this is the
input. x minus 2 is the input.

00:04:23.500 --> 00:04:26.340
When I subtract the 2, this
is shifting the function

00:04:26.340 --> 00:04:29.200
to the right, which is a
little bit counter-intuitive

00:04:29.200 --> 00:04:31.540
unless you go through this
exercise right over here.

00:04:31.540 --> 00:04:34.550
So g of x is equal
to f of x minus 2.

00:04:34.550 --> 00:04:37.480
If it was f of x plus 2 we
would have actually shifted

00:04:37.480 --> 00:04:39.779
f to the left.

00:04:39.779 --> 00:04:41.070
Now let's think about this one.

00:04:41.070 --> 00:04:43.350
This one seems kind of wacky.

00:04:43.350 --> 00:04:46.647
So first of all,
g of x, it almost

00:04:46.647 --> 00:04:48.230
looks like a mirror
image but it looks

00:04:48.230 --> 00:04:49.970
like it's been flattened out.

00:04:49.970 --> 00:04:52.010
So let's think of it this way.

00:04:52.010 --> 00:04:54.390
Let's take the mirror
image of what g of x is.

00:04:54.390 --> 00:04:56.900
So I'm going to try my best to
take the mirror image of it.

00:04:56.900 --> 00:04:57.524
So let's see...

00:04:57.524 --> 00:04:59.470
It gets to about
2 there, then it

00:04:59.470 --> 00:05:02.310
gets pretty close to
1 right over there.

00:05:02.310 --> 00:05:05.100
And then it gets about
right over there.

00:05:05.100 --> 00:05:06.750
So if I were to take
its mirror image,

00:05:06.750 --> 00:05:10.530
it looks something like this.

00:05:10.530 --> 00:05:15.500
Its mirror image if I were to
reflect it across the x-axis.

00:05:15.500 --> 00:05:16.815
It looks something like this.

00:05:20.820 --> 00:05:24.840
So this right over
here we would call--

00:05:24.840 --> 00:05:30.900
so if this is g of x,
when we flip it that way,

00:05:30.900 --> 00:05:34.950
this is the negative g of x.

00:05:34.950 --> 00:05:38.400
When x equals 4, g of
x looks like it's about

00:05:38.400 --> 00:05:39.660
negative 3 and 1/2.

00:05:39.660 --> 00:05:42.139
You take the negative of
that, you get positive.

00:05:42.139 --> 00:05:43.680
I guess it should
be closer to here--

00:05:43.680 --> 00:05:45.771
You get positive
3 and 1/2 if you

00:05:45.771 --> 00:05:47.270
were to take the
exact mirror image.

00:05:47.270 --> 00:05:48.690
So that's negative g of x.

00:05:48.690 --> 00:05:50.160
But that still doesn't get us.

00:05:50.160 --> 00:05:53.030
It looks like we
actually have to triple

00:05:53.030 --> 00:05:54.710
this value for any point.

00:05:54.710 --> 00:05:55.680
And you see it here.

00:05:55.680 --> 00:05:59.400
This gets to 2, but
we need to get to 6.

00:05:59.400 --> 00:06:02.220
This gets to 1, but
we need to get to 3.

00:06:02.220 --> 00:06:04.590
So it looks like this
red graph right over here

00:06:04.590 --> 00:06:06.840
is 3 times this graph.

00:06:06.840 --> 00:06:11.260
So this is 3 times
negative g of x,

00:06:11.260 --> 00:06:16.860
which is equal to
negative 3 g of x.

00:06:16.860 --> 00:06:22.660
So here we have f of x is equal
to negative 3 times g of x.

00:06:22.660 --> 00:06:25.330
And if we wanted to solve for
g of x, right-- g of x in terms

00:06:25.330 --> 00:06:28.940
of f of x-- we would
write, dividing both sides

00:06:28.940 --> 00:06:35.980
by negative 3, g of x is
equal to negative 1/3 f of x.


Area of a sector given a central angle

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.890
A circle with area
81 pi has a sector

00:00:04.890 --> 00:00:07.221
with a 350-degree central angle.

00:00:07.221 --> 00:00:09.470
So this whole sector right
over here that's shaded in,

00:00:09.470 --> 00:00:13.800
this pale orange-yellowish
color, that

00:00:13.800 --> 00:00:16.450
has a 350-degree central angle.

00:00:16.450 --> 00:00:19.850
So you see the central angle,
it's a very large angle.

00:00:19.850 --> 00:00:23.710
It's going all the
way around like that.

00:00:23.710 --> 00:00:27.090
And they ask us, what is
the area of the sector?

00:00:27.090 --> 00:00:30.730
So we just need to realize
that the ratio between the area

00:00:30.730 --> 00:00:41.570
of the sector and the
total area of the circle.

00:00:41.570 --> 00:00:43.390
And they tell us what
the total area is.

00:00:43.390 --> 00:00:44.990
It's 81 pi.

00:00:44.990 --> 00:00:49.440
And 81 pi is going to
be equal to the ratio

00:00:49.440 --> 00:00:54.310
of its central angle,
which is 350 degrees,

00:00:54.310 --> 00:01:00.800
over the total number of
degrees in a circle-- over 360.

00:01:00.800 --> 00:01:03.270
So the area of the sector
over the total area

00:01:03.270 --> 00:01:05.630
is equal to the degrees
in the central angle

00:01:05.630 --> 00:01:08.020
over the total
degrees in a circle.

00:01:08.020 --> 00:01:10.090
And then we just can
solve for area of a sector

00:01:10.090 --> 00:01:13.220
by multiplying both
sides by 81 pi.

00:01:13.220 --> 00:01:19.450
81 pi, 81 pi-- so
these cancel out.

00:01:19.450 --> 00:01:23.500
350 divided by 360 is 35/36.

00:01:23.500 --> 00:01:31.376
And so our area, our sector
area, is equal to-- let's see,

00:01:31.376 --> 00:01:37.630
in the numerator, we have
35 times-- instead of 81,

00:01:37.630 --> 00:01:42.310
let's see, that's going
to be 9 times 9 pi.

00:01:42.310 --> 00:01:44.680
And in the
denominator, I have 36.

00:01:44.680 --> 00:01:48.000
Well, that's the same
thing as 9 times 4.

00:01:48.000 --> 00:01:50.250
And so we can divide the
numerator and the denominator

00:01:50.250 --> 00:01:57.400
both by 9, and so we are
left with 35 times 9.

00:01:57.400 --> 00:01:59.140
And neither of these
are divisible by 4,

00:01:59.140 --> 00:02:01.220
so that's about as
simplified as we can get it.

00:02:01.220 --> 00:02:03.740
So let's think about
what 35 times 9 is.

00:02:03.740 --> 00:02:10.130
35 times 9, it's going to be 350
minus 35, which would be 315,

00:02:10.130 --> 00:02:12.810
I guess.

00:02:12.810 --> 00:02:13.910
Did I do that right?

00:02:13.910 --> 00:02:19.690
Yeah, it's going to be 270 plus
45, which is 315 pi over 4.

00:02:19.690 --> 00:02:24.856
315 pi over 4 is the
area of the sector.


Finding central angle measure given arc length

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:04.210
A circle has a
circumference of 20 pi.

00:00:04.210 --> 00:00:11.060
It has an arc
length of 221/18 pi.

00:00:11.060 --> 00:00:15.479
What is the central angle
of the arc in degrees?

00:00:15.479 --> 00:00:16.770
So they're asking for this one.

00:00:16.770 --> 00:00:18.686
So this is the arc that
they're talking about,

00:00:18.686 --> 00:00:21.760
that's 221/18 pi long.

00:00:21.760 --> 00:00:24.720
And they want to know this
angle that it subtends,

00:00:24.720 --> 00:00:27.012
this central angle
right over here.

00:00:27.012 --> 00:00:28.470
So we just have to
remind ourselves

00:00:28.470 --> 00:00:30.630
that the ratio of
this arc length

00:00:30.630 --> 00:00:32.479
to the entire
circumference-- let

00:00:32.479 --> 00:00:35.320
me write that down-- the
ratio of this arc length,

00:00:35.320 --> 00:00:41.460
which is 221/18 pi, to the
entire circumference, which

00:00:41.460 --> 00:00:46.840
is 20 pi, is going to
be equal to the ratio

00:00:46.840 --> 00:00:50.430
of this central angle,
which we can call theta,

00:00:50.430 --> 00:00:54.142
the ratio of theta
to 360 degrees

00:00:54.142 --> 00:00:56.100
if we were to go all the
way around the circle.

00:00:59.790 --> 00:01:02.390
This will give us
our theta in degrees.

00:01:02.390 --> 00:01:04.920
If we wanted it in radians,
we would think of it

00:01:04.920 --> 00:01:07.500
in terms of 2 pi radians
around the circle,

00:01:07.500 --> 00:01:10.250
but it was 360 degrees
since we're in degrees.

00:01:10.250 --> 00:01:11.802
Now we just have to simplify.

00:01:11.802 --> 00:01:13.510
Now the easiest thing
is just to multiply

00:01:13.510 --> 00:01:16.200
both sides times 360 degrees.

00:01:16.200 --> 00:01:17.770
So let's do that.

00:01:17.770 --> 00:01:22.290
So if we multiply the
left-hand side by 360 degrees,

00:01:22.290 --> 00:01:33.190
we get 360 times 221
times pi over-- let's see,

00:01:33.190 --> 00:01:39.280
we have 18 times 20 times pi.

00:01:41.870 --> 00:01:45.850
And on the right-hand side,
if we multiply it by 360,

00:01:45.850 --> 00:01:49.160
we are just left with theta.

00:01:49.160 --> 00:01:51.820
So we really just have
to simplify this now.

00:01:51.820 --> 00:01:54.920
Pi divided by pi
is going to be 1.

00:01:54.920 --> 00:01:58.580
360 divided by 20,
well, it's going

00:01:58.580 --> 00:02:04.464
to be the same thing as 36/2,
which is the same thing as 18.

00:02:04.464 --> 00:02:07.330
And 18 divided by 18 is 1.

00:02:07.330 --> 00:02:11.690
So this all simplified
to 221 degrees.

00:02:11.690 --> 00:02:17.007
Theta is 221 degrees.


Length of an arc that subtends a central angle

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:04.600
I have a circle here whose
circumference is 18 pi.

00:00:04.600 --> 00:00:07.270
So if we were to measure all
the way around the circle,

00:00:07.270 --> 00:00:09.560
we would get 18 pi.

00:00:09.560 --> 00:00:11.780
And we also have a
central angle here.

00:00:11.780 --> 00:00:14.130
So this is the
center of the circle.

00:00:14.130 --> 00:00:18.040
And this central angle
that I'm about to draw

00:00:18.040 --> 00:00:22.470
has a measure of 10 degrees.

00:00:22.470 --> 00:00:25.090
So this angle right
over here is 10 degrees.

00:00:25.090 --> 00:00:27.560
And what I'm curious
about is the length

00:00:27.560 --> 00:00:30.940
of the arc that subtends
that central angle.

00:00:30.940 --> 00:00:34.720
So what is the length of
what I just did in magenta?

00:00:34.720 --> 00:00:37.280
And one way to think about
it, or actually maybe the way

00:00:37.280 --> 00:00:40.210
to think about it, is
that the ratio of this arc

00:00:40.210 --> 00:00:42.140
length to the entire
circumference--

00:00:42.140 --> 00:00:52.030
let me write this
down-- should be

00:00:52.030 --> 00:00:59.130
the same as the ratio
of the central angle

00:00:59.130 --> 00:01:01.220
to the total number
of angles if you were

00:01:01.220 --> 00:01:07.899
to go all the way around the
circle-- so to 360 degrees.

00:01:07.899 --> 00:01:09.190
So let's just think about that.

00:01:09.190 --> 00:01:10.990
We know the
circumference is 18 pi.

00:01:14.580 --> 00:01:16.380
We're looking for
the arc length.

00:01:16.380 --> 00:01:19.070
I'm just going to call
that a. a for arc length.

00:01:19.070 --> 00:01:20.910
That's what we're going
to try to solve for.

00:01:20.910 --> 00:01:23.955
We know that the central
angle is 10 degrees.

00:01:23.955 --> 00:01:27.025
So you have 10 degrees
over 360 degrees.

00:01:32.530 --> 00:01:35.620
So we could simplify this
by multiplying both sides

00:01:35.620 --> 00:01:36.570
by 18 pi.

00:01:41.570 --> 00:01:46.090
And we get that our arc
length is equal to-- well,

00:01:46.090 --> 00:01:49.310
10/360 is the same
thing as 1/36.

00:01:49.310 --> 00:01:53.690
So it's equal to
1/36 times 18 pi,

00:01:53.690 --> 00:02:01.830
so it's 18 pi over 36, which
is the same thing as pi/2.

00:02:01.830 --> 00:02:03.630
So this arc right
over here is going

00:02:03.630 --> 00:02:08.250
to be pi/2, whatever units
we're talking about, long.

00:02:08.250 --> 00:02:10.440
Now let's think about
another scenario.

00:02:10.440 --> 00:02:13.010
Let's imagine the same circle.

00:02:13.010 --> 00:02:14.880
So it's the same circle here.

00:02:14.880 --> 00:02:17.430
Our circumference
is still 18 pi.

00:02:19.771 --> 00:02:21.270
There are people
having a conference

00:02:21.270 --> 00:02:22.228
behind me or something.

00:02:22.228 --> 00:02:24.540
That's why you might hear
those mumbling voices.

00:02:24.540 --> 00:02:29.990
But this circumference
is also 18 pi.

00:02:29.990 --> 00:02:36.210
But now I'm going to make the
central angle an obtuse angle.

00:02:36.210 --> 00:02:38.560
So let's say we were to
start right over here.

00:02:38.560 --> 00:02:40.970
This is one side of the angle.

00:02:40.970 --> 00:02:45.010
I'm going to go and
make a 350 degree angle.

00:02:45.010 --> 00:02:47.060
So I'm going to go all
the way around like that.

00:02:47.060 --> 00:02:51.820
So this right over here
is a 350 degree angle.

00:02:51.820 --> 00:02:54.850
And now I'm curious
about this arc that

00:02:54.850 --> 00:02:57.920
subtends this really huge angle.

00:02:57.920 --> 00:03:02.420
So now I want to figure out this
arc length-- so all of this.

00:03:02.420 --> 00:03:05.540
I want to figure out this
arc length, the arc that

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subtends this really obtuse
angle right over here.

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Well, same exact logic-- the
ratio between our arc length,

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a, and the circumference of
the entire circle, 18 pi,

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should be the same as the
ratio between our central angle

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that the arc subtends, so
350, over the total number

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of degrees in a
circle, over 360.

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So multiply both sides by 18 pi.

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We get a is equal to-- this
is 35 times 18 over 36 pi.

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350 divided by 360 is 35/36.

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So this is 35 times
18 times pi over 36.

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Well both 36 and 18
are divisible by 18,

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so let's divide them both by 18.

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And so we are left with 35/2 pi.

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Let me just write it
that way-- 35 pi over 2.

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Or, if you wanted to
write it as a decimal,

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this would be 17.5 pi.

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Now does this makes sense?

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This right over here,
this other arc length,

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when our central
angle was 10 degrees,

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this had an arc
length of 0.5 pi.

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So when you add these two
together, this arc length

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and this arc length,
0.5 plus 17.5,

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you get to 18 pi, which was
the circumference, which

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makes complete sense because
if you add these angles, 10

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degrees and 350 degrees, you
get 360 degrees in a circle.