Slope and intercept in tables

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:02.170
- [Instructor] Flynn's
sister loaned him some money,

00:00:02.170 --> 00:00:04.240
and he paid her back over time.

00:00:04.240 --> 00:00:06.830
Flynn graphed the relationship
between how much time

00:00:06.830 --> 00:00:09.505
had passed, in weeks, since
the loan and how much money

00:00:09.505 --> 00:00:11.740
he still owed his sister.

00:00:11.740 --> 00:00:13.710
What feature of the graph represents

00:00:13.710 --> 00:00:17.260
how long it took Flynn
to pay back the loan?

00:00:17.260 --> 00:00:19.970
Pause this video, see if
you can figure that out.

00:00:19.970 --> 00:00:22.960
All right, now let's go
through each of these choices.

00:00:22.960 --> 00:00:25.030
So the first one is the slope.

00:00:25.030 --> 00:00:26.170
So the slope tells us

00:00:26.170 --> 00:00:28.150
how much do we change in
the vertical direction

00:00:28.150 --> 00:00:30.710
for any given change in
the horizontal direction.

00:00:30.710 --> 00:00:32.890
So for example, if we start from here,

00:00:32.890 --> 00:00:35.770
we can see that if one week goes by,

00:00:35.770 --> 00:00:37.650
we go from one week to two weeks,

00:00:37.650 --> 00:00:39.620
we can see that our loan went down.

00:00:39.620 --> 00:00:41.930
It went from $15 to $10.

00:00:41.930 --> 00:00:44.550
So when we had plus one weeks in time,

00:00:44.550 --> 00:00:48.070
our loan went down by $5.

00:00:48.070 --> 00:00:50.760
And then that happens over the next week.

00:00:50.760 --> 00:00:53.730
And that rate of how
quickly the loan is paid,

00:00:53.730 --> 00:00:55.410
that's what the slope tells us.

00:00:55.410 --> 00:00:57.580
And it would be constant over the course

00:00:57.580 --> 00:01:00.510
of the entire time period
is what makes this a line,

00:01:00.510 --> 00:01:02.820
is that slope is always
going to be constant.

00:01:02.820 --> 00:01:04.600
So the slope is useful for the rate

00:01:04.600 --> 00:01:06.360
at which the loan is being paid back,

00:01:06.360 --> 00:01:08.321
but it's not the clearest
way to figure out

00:01:08.321 --> 00:01:11.040
how long it took Flynn
to pay back the loan.

00:01:11.040 --> 00:01:12.720
So I would rule that one out.

00:01:12.720 --> 00:01:15.410
The x-intercept, that's
where the graph intersects

00:01:15.410 --> 00:01:16.690
the horizontal axis,

00:01:16.690 --> 00:01:19.072
which is often referred to as the x-axis.

00:01:19.072 --> 00:01:21.270
Or another way to think about it,

00:01:21.270 --> 00:01:24.297
that tells us, that
says, what is the x-value

00:01:24.297 --> 00:01:29.150
when our vertical value, our
y-value, is equal to zero?

00:01:29.150 --> 00:01:30.680
And our y-value is the money owed.

00:01:30.680 --> 00:01:33.540
So it says, hey how much
time has passed when we owe,

00:01:33.540 --> 00:01:34.810
when we don't owe any more money

00:01:34.810 --> 00:01:37.740
or when Flynn doesn't owe
any more money to his sister?

00:01:37.740 --> 00:01:39.140
Well that's exactly
what they're asking for.

00:01:39.140 --> 00:01:41.170
How long did it take Flynn
to pay back the loan?

00:01:41.170 --> 00:01:42.870
And that's what that x-intercept tells us.

00:01:42.870 --> 00:01:44.230
It took him four weeks.

00:01:44.230 --> 00:01:45.868
After four weeks, he
didn't owe any more money

00:01:45.868 --> 00:01:47.460
to his sister.

00:01:47.460 --> 00:01:49.640
So I like this choice,
but let's just review

00:01:49.640 --> 00:01:50.473
the other ones.

00:01:50.473 --> 00:01:52.570
The y-intercept, well
that's where the graph

00:01:52.570 --> 00:01:55.460
intersects the y-axis right over there.

00:01:55.460 --> 00:01:58.670
This tells us what's going
on at time t equals zero.

00:01:58.670 --> 00:02:01.690
At time t equal zero or time equal zero,

00:02:01.690 --> 00:02:04.410
when the horizontal
variable is equal to zero,

00:02:04.410 --> 00:02:07.020
we see that Flynn owes $20.

00:02:07.020 --> 00:02:09.840
So this one isn't valuable
for figuring out when he,

00:02:09.840 --> 00:02:11.280
how long it takes to pay it back.

00:02:11.280 --> 00:02:12.896
This one is useful for figuring out

00:02:12.896 --> 00:02:15.650
how much did he owe initially.

00:02:15.650 --> 00:02:16.650
So we'll rule that one out.

00:02:16.650 --> 00:02:17.890
And we already picked one choice,

00:02:17.890 --> 00:02:19.740
so we can rule out none of the above.


Calculating slope from tables

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:00.980
- [Instructor] We are asked,

00:00:00.980 --> 00:00:05.070
what is the slope of the line
that contains these points?

00:00:05.070 --> 00:00:07.440
So pause this video and see
if you can work through this

00:00:07.440 --> 00:00:10.060
on your own before we do it together.

00:00:10.060 --> 00:00:11.760
Alright, now let's do it together,

00:00:11.760 --> 00:00:14.270
and let's just remind
ourselves what slope is.

00:00:14.270 --> 00:00:19.060
Slope is equal to change in y,

00:00:19.060 --> 00:00:21.090
this is the Greek letter delta,

00:00:21.090 --> 00:00:21.980
look likes a triangle,

00:00:21.980 --> 00:00:24.410
but it's shorthand for change in y

00:00:24.410 --> 00:00:27.370
over change in x.

00:00:27.370 --> 00:00:31.523
Sometimes you would see
it written as y2 minus y1

00:00:31.523 --> 00:00:34.810
over x2 minus x1

00:00:34.810 --> 00:00:37.610
where you could kind of view
x1 y1 as the starting point

00:00:37.610 --> 00:00:40.862
and x2 y2 as the ending point.

00:00:40.862 --> 00:00:44.790
So let's just pick two xy pairs here,

00:00:44.790 --> 00:00:46.140
and we can actually pick any two

00:00:46.140 --> 00:00:49.120
if we can assume that this is
actually describing a line.

00:00:49.120 --> 00:00:51.100
So we might as well
just pick the first two.

00:00:51.100 --> 00:00:52.890
So let's say that's our starting point

00:00:52.890 --> 00:00:54.332
and that's our finishing point.

00:00:54.332 --> 00:00:56.686
So what is our change in x here?

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So we're going from two to three,

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so our change in x is
equal to three minus two

00:01:03.090 --> 00:01:03.950
which is equal to one,

00:01:03.950 --> 00:01:05.550
and you can see that
to go from two to three

00:01:05.550 --> 00:01:06.740
you're just adding one.

00:01:06.740 --> 00:01:08.182
And what's our change in y?

00:01:08.182 --> 00:01:10.911
Our change in y is our finishing y one

00:01:10.911 --> 00:01:15.000
minus our starting y four, which
is equal to negative three.

00:01:15.000 --> 00:01:16.970
And you could of, you didn't
even have to do this math,

00:01:16.970 --> 00:01:17.803
you would have been able to see

00:01:17.803 --> 00:01:19.610
to go from two to three you added one,

00:01:19.610 --> 00:01:22.602
and to go from four to one,
you have to subtract three.

00:01:22.602 --> 00:01:25.050
For there we have all
the information we need.

00:01:25.050 --> 00:01:27.802
What is change in y over change in x?

00:01:27.802 --> 00:01:29.405
Well, it's going to be,

00:01:29.405 --> 00:01:31.560
our change in y is negative three

00:01:31.560 --> 00:01:33.080
and our change in x is one.

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So our slope is negative
three divided by one

00:01:34.999 --> 00:01:37.007
is negative three.

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Let's do another example.

00:01:39.930 --> 00:01:42.743
Here we are asked, what is the slope

00:01:42.743 --> 00:01:45.320
of the line that contains these points?

00:01:45.320 --> 00:01:47.410
So pause this video and see
if you can figure it out

00:01:47.410 --> 00:01:50.720
or pause the video again and
see if you can figure it out.

00:01:50.720 --> 00:01:55.720
Alright, so remember, slope
is equal to change in y

00:01:55.876 --> 00:01:57.610
over change in x.

00:01:57.610 --> 00:02:00.229
And we should be able to
pick any two of these pairs

00:02:00.229 --> 00:02:01.990
in order to figure that out if we assume

00:02:01.990 --> 00:02:03.511
that this is indeed a line.

00:02:03.511 --> 00:02:07.713
Well, just for variety, let's
pick these middle two pairs.

00:02:07.713 --> 00:02:09.730
So what's our change in x?

00:02:09.730 --> 00:02:12.260
To go from one to five, we added four.

00:02:12.260 --> 00:02:13.730
And what's our change in y?

00:02:13.730 --> 00:02:16.920
To go from seven to 13, we added six.

00:02:16.920 --> 00:02:21.920
So our change in y is six
when our change in x is four.

00:02:22.770 --> 00:02:25.970
And I got the signs right,
in both case it's a positive.

00:02:25.970 --> 00:02:28.290
When x increases, y increased as well.

00:02:28.290 --> 00:02:29.560
So our slope is six fourths,

00:02:29.560 --> 00:02:31.250
and we could rewrite that if we like.

00:02:31.250 --> 00:02:33.040
Both six and four are divisible by two,

00:02:33.040 --> 00:02:35.280
so let be divide both the
numerator and the denominator

00:02:35.280 --> 00:02:38.143
by two and we get three
halves, and we're done.


Graphing a line given point and slope

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:01.970
- [Instructor] We are told graph a line

00:00:01.970 --> 00:00:03.490
with the slope of negative two,

00:00:03.490 --> 00:00:06.150
that contains the point
four comma negative three.

00:00:06.150 --> 00:00:09.310
And we have our little Khan
Academy graphing widget

00:00:09.310 --> 00:00:11.400
right over here, where
we just have to find

00:00:11.400 --> 00:00:12.630
two points on that line,

00:00:12.630 --> 00:00:15.220
and then that will graph the line for us.

00:00:15.220 --> 00:00:17.350
So pause this video and even
if you don't have access

00:00:17.350 --> 00:00:18.183
to the widget right now,

00:00:18.183 --> 00:00:20.360
although it's all
available on Khan Academy,

00:00:21.410 --> 00:00:23.050
at least think about how
you would approach this.

00:00:23.050 --> 00:00:24.900
And if you have paper and pencil handy,

00:00:24.900 --> 00:00:27.530
I encourage you to try to
graph this line on your own,

00:00:27.530 --> 00:00:30.070
before I work through it
with this little widget.

00:00:30.070 --> 00:00:31.730
All right, now let's do it together.

00:00:31.730 --> 00:00:33.630
So we do know that it contains

00:00:33.630 --> 00:00:35.490
point four comma negative three.

00:00:35.490 --> 00:00:37.610
So that's I guess you
could say the easy part,

00:00:37.610 --> 00:00:38.561
we just have to find the point

00:00:38.561 --> 00:00:41.320
x is four y is negative three.

00:00:41.320 --> 00:00:44.750
So it's from the origin four
to the right, three down.

00:00:44.750 --> 00:00:46.740
But then we have to figure out
where could another point be?

00:00:46.740 --> 00:00:47.980
Because if we can figure
out another point,

00:00:47.980 --> 00:00:49.720
then we would have graphed the line.

00:00:49.720 --> 00:00:53.000
And the clue here is that they
say a slope of negative two.

00:00:53.000 --> 00:00:56.150
So one way to think about it
is, we can start at the point

00:00:56.150 --> 00:00:59.577
that we know is on the line,
and a slope of negative two

00:00:59.577 --> 00:01:04.577
tells us that as x increases
by one, y goes down by two.

00:01:05.850 --> 00:01:07.760
The change in why would be negative two.

00:01:07.760 --> 00:01:10.730
And so this could be
another point on that line.

00:01:10.730 --> 00:01:14.010
So I could graph it like
this is x goes up by one,

00:01:14.010 --> 00:01:17.070
as x goes from four to five, y will go,

00:01:17.070 --> 00:01:18.870
or y will change by negative two.

00:01:18.870 --> 00:01:21.860
So why we'll go from negative
three to negative five.

00:01:21.860 --> 00:01:25.310
So this will be done, we
have just graphed that line.

00:01:25.310 --> 00:01:27.420
Now another way that you could do it,

00:01:27.420 --> 00:01:30.400
because sometimes you might
not have space on the paper,

00:01:30.400 --> 00:01:32.810
or on the widget to be
able to go to the right

00:01:32.810 --> 00:01:35.500
for x to increase, is to go the other way.

00:01:35.500 --> 00:01:37.270
If you have a slope of negative two,

00:01:37.270 --> 00:01:42.270
another way to think about
it is, if x goes down by one,

00:01:42.400 --> 00:01:45.960
if x goes down by one,
then y goes up by two.

00:01:45.960 --> 00:01:48.810
'Cause remember, slope is
change of y over change in x.

00:01:48.810 --> 00:01:50.532
So you could either say
you have a positive change

00:01:50.532 --> 00:01:54.590
in y of two when x has
a negative one change,

00:01:54.590 --> 00:01:57.400
or you could think of it when
x is a positive one change,

00:01:57.400 --> 00:01:58.900
y has a negative two change.

00:01:58.900 --> 00:02:01.090
But either way notice,
you got the same line.

00:02:01.090 --> 00:02:03.160
Notice this line is the same thing,

00:02:03.160 --> 00:02:06.370
as if we did the first way
is we had x going up by one

00:02:06.370 --> 00:02:08.200
and y going down by two,

00:02:08.200 --> 00:02:09.743
it's the exact same line.


Dividing whole numbers by decimals examples

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:01.320
- [Instructor] Let's
say we wanna figure out

00:00:01.320 --> 00:00:04.060
what eight divided by 4/10 is.

00:00:04.060 --> 00:00:06.250
Pause this video and try to
figure it out on your own

00:00:06.250 --> 00:00:08.240
before we do it together.

00:00:08.240 --> 00:00:11.150
All right, now one way to approach this is

00:00:11.150 --> 00:00:13.280
to think about everything
in terms of tenths.

00:00:13.280 --> 00:00:15.040
And why tenths, you ask?

00:00:15.040 --> 00:00:19.110
Well, I just described
the second number as 4/10.

00:00:19.110 --> 00:00:21.660
So what would eight be in terms of tenths?

00:00:21.660 --> 00:00:24.510
Well, we know that one whole

00:00:24.510 --> 00:00:27.163
is equal to 10/10,

00:00:28.070 --> 00:00:32.063
so eight wholes is going
to be equal to 80/10.

00:00:33.370 --> 00:00:34.680
So I could write that down over here.

00:00:34.680 --> 00:00:38.400
Instead of eight, I can
write that as 80/10.

00:00:38.400 --> 00:00:41.580
And I'm taking 80/10 and
I'm dividing that by,

00:00:41.580 --> 00:00:43.520
divided by 4/10.

00:00:43.520 --> 00:00:46.770
I'm just writing out, 4/10.

00:00:46.770 --> 00:00:48.860
So if I have 80 of something,

00:00:48.860 --> 00:00:52.370
and if I'm diving it into groups
of four of that something,

00:00:52.370 --> 00:00:54.660
how many equal groups am I going to have?

00:00:54.660 --> 00:00:57.430
Well, I'm gonna have 80
divided by four groups,

00:00:57.430 --> 00:01:00.390
or 80 divided by four,
eight divided by four.

00:01:00.390 --> 00:01:01.540
Actually, let me just write that down.

00:01:01.540 --> 00:01:05.150
This is going to be equal to 80 over four

00:01:05.150 --> 00:01:06.890
or 80 divided by four.

00:01:06.890 --> 00:01:08.800
And eight divided by four is two,

00:01:08.800 --> 00:01:12.500
so 80 divided by four is
going to be equal to 20.

00:01:12.500 --> 00:01:16.510
So eight divided by 4/10 is equal to 20.

00:01:16.510 --> 00:01:18.630
Another way that you
could think about that is

00:01:18.630 --> 00:01:20.900
to rewrite eight divided by 4/10

00:01:20.900 --> 00:01:25.900
as eight over 4/10.

00:01:26.290 --> 00:01:28.320
Over 4/10.

00:01:28.320 --> 00:01:30.480
And then if you wanted to rewrite this,

00:01:30.480 --> 00:01:32.867
you could say, "Hey, can
I multiply the numerator

00:01:32.867 --> 00:01:35.727
"and the denominator by some
quantity that'll get rid

00:01:35.727 --> 00:01:37.840
"of the decimal in the denominator?"

00:01:37.840 --> 00:01:40.970
Well, I could multiply
the denominator by 10.

00:01:40.970 --> 00:01:42.700
And if I don't wanna change
the value of the fraction,

00:01:42.700 --> 00:01:45.240
I have to multiply the
numerator by that same amount.

00:01:45.240 --> 00:01:47.180
Well, what does that get us?

00:01:47.180 --> 00:01:49.380
Well, this is going to be equal to,

00:01:49.380 --> 00:01:51.200
in the denominator right over here,

00:01:51.200 --> 00:01:53.180
I'm just going to have a four.

00:01:53.180 --> 00:01:55.190
So we got rid of the decimal.

00:01:55.190 --> 00:01:58.730
In the numerator, I am going to get 80.

00:01:58.730 --> 00:02:02.530
So once again, we get 80 over
four, which is equal to 20.

00:02:02.530 --> 00:02:03.610
Let's do another example,

00:02:03.610 --> 00:02:06.460
but in this case we'll
deal with hundredths.

00:02:06.460 --> 00:02:09.170
So let's say we want to figure out

00:02:09.170 --> 00:02:12.790
what is 48

00:02:12.790 --> 00:02:16.710
divided by 0.24

00:02:17.870 --> 00:02:20.200
or 24/100?

00:02:20.200 --> 00:02:22.400
Pause this video and
try to figure that out.

00:02:23.410 --> 00:02:26.410
All right, well, we can
do something very similar.

00:02:26.410 --> 00:02:28.250
This time we're dealing with hundredths,

00:02:28.250 --> 00:02:29.940
so we could try to express everything

00:02:29.940 --> 00:02:31.630
in terms of hundredths.

00:02:31.630 --> 00:02:35.320
So 48 is equal to how many hundredths?

00:02:35.320 --> 00:02:39.733
Well, we know that one
is equal to 100/100,

00:02:42.507 --> 00:02:46.450
so 48 is going to be equal

00:02:46.450 --> 00:02:51.450
to 48 times 100/100.

00:02:51.670 --> 00:02:53.600
Maybe I'll just write it like that.

00:02:53.600 --> 00:02:55.417
Or you might be able to
say, "Hey, that's 4,800

00:02:55.417 --> 00:02:57.980
"or 4,800/100."

00:02:57.980 --> 00:02:59.750
So let's just, actually,
I'll write it that way.

00:02:59.750 --> 00:03:04.750
This is going to be 4,800/100

00:03:07.120 --> 00:03:11.450
divided by, this is 24/100.

00:03:11.450 --> 00:03:13.843
24/100.

00:03:15.500 --> 00:03:17.360
And so what's that going to be?

00:03:17.360 --> 00:03:19.290
Well, same logic as we used up here.

00:03:19.290 --> 00:03:22.400
If I have 4,800 of something
or 4,800 of something

00:03:22.400 --> 00:03:24.540
and if I'm dividing it into equal groups

00:03:24.540 --> 00:03:27.100
of 24 of that something,

00:03:27.100 --> 00:03:31.430
well, I'm going to have 4,800
divided by 24 equal groups.

00:03:31.430 --> 00:03:34.210
So this is going to be equal to 4,800

00:03:36.029 --> 00:03:38.160
over 24.

00:03:38.160 --> 00:03:42.970
We could express 4,800 as, of
course, 48 times a hundred.

00:03:42.970 --> 00:03:44.810
So this is equal to 48

00:03:44.810 --> 00:03:48.730
times 100 over 24.

00:03:48.730 --> 00:03:51.450
And of course, you could
view this as the same thing

00:03:51.450 --> 00:03:55.103
as this is equal to 48 over 24.

00:03:56.070 --> 00:03:59.580
48 over 24, and many of you
might have already done it

00:03:59.580 --> 00:04:02.193
in your head, times 100.

00:04:03.130 --> 00:04:05.820
What's 48 divided by 24?

00:04:05.820 --> 00:04:08.290
Well, that is going to be equal to two.

00:04:08.290 --> 00:04:11.340
So two times a hundred is equal to 200.

00:04:11.340 --> 00:04:14.460
So this right over here is equal to 200.

00:04:14.460 --> 00:04:16.300
You can also do it in this type

00:04:16.300 --> 00:04:18.630
or do it similar to the way we did this

00:04:18.630 --> 00:04:22.710
where instead of
expressing it out as words,

00:04:22.710 --> 00:04:26.000
you could express it as 48 over 24/100.

00:04:26.000 --> 00:04:27.510
And instead of multiplying the numerator

00:04:27.510 --> 00:04:28.500
and denominator by 10,

00:04:28.500 --> 00:04:30.540
maybe you wanna multiply
it by something else

00:04:30.540 --> 00:04:33.640
to get whole numbers in both
the numerator and denominator.

00:04:33.640 --> 00:04:35.030
Think about that after this video,

00:04:35.030 --> 00:04:37.053
and try to work it out that way as well.


Pattern when dividing by tenths and hundredths

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:01.620
- [Instructor] Let's
see if we can figure out

00:00:01.620 --> 00:00:06.620
what two divided by 0.1 or one tenth is.

00:00:08.540 --> 00:00:12.070
Pause this video and see
if you can figure that out.

00:00:12.070 --> 00:00:13.730
All right, now let's work
through this together.

00:00:13.730 --> 00:00:16.010
And there's a couple of ways
that we can approach it.

00:00:16.010 --> 00:00:20.420
One way is to think about
everything in terms of tenths.

00:00:20.420 --> 00:00:24.060
So two wholes is how many tenths?

00:00:24.060 --> 00:00:26.280
Well, a whole is equal to 10 tenths,

00:00:26.280 --> 00:00:29.520
so two wholes is equal to 20 tenths.

00:00:29.520 --> 00:00:34.403
So I could write the two,
rewrite the two, as 20 tenths.

00:00:35.270 --> 00:00:38.543
And so that's going to
be 20 tenths divided by,

00:00:39.780 --> 00:00:41.330
and instead of writing
it this way, I could,

00:00:41.330 --> 00:00:43.160
instead of just writing it as 0.1,

00:00:43.160 --> 00:00:45.170
we know this is the
same thing as one tenth.

00:00:45.170 --> 00:00:47.623
So divided by one tenth.

00:00:48.590 --> 00:00:50.400
And so if I have 20 of something

00:00:50.400 --> 00:00:53.310
and if I divide it by, into
groups of one of that something,

00:00:53.310 --> 00:00:55.170
how many groups am I going to have?

00:00:55.170 --> 00:00:59.610
Well, I'm going to have
20 equal groups of one.

00:00:59.610 --> 00:01:03.830
So 20 tenths divided by
one tenth is equal to 20.

00:01:03.830 --> 00:01:06.060
Another way that you
could approach that is

00:01:06.060 --> 00:01:10.360
you could rewrite the 0.1,
the one tenth, as a fraction.

00:01:10.360 --> 00:01:15.360
We could rewrite this as
two divided by, divided by,

00:01:16.629 --> 00:01:18.028
instead of writing one tenth like this,

00:01:18.028 --> 00:01:21.430
I could write it as a
fraction, divided by 1/10.

00:01:21.430 --> 00:01:24.560
Well, we know that dividing by a tenth is

00:01:24.560 --> 00:01:28.300
the same thing as multiplying by 10.

00:01:28.300 --> 00:01:33.300
So this is going to be equal
to two times, two times 10.

00:01:36.690 --> 00:01:39.470
And that gets us to the same
place that we had before.

00:01:39.470 --> 00:01:40.840
What is two times 10?

00:01:40.840 --> 00:01:44.840
Well, that is going to be equal to 20.

00:01:44.840 --> 00:01:47.160
That's good, that we
got to the same answer,

00:01:47.160 --> 00:01:50.460
otherwise these would not
be equivalent methods.

00:01:50.460 --> 00:01:52.310
Let's do another example.

00:01:52.310 --> 00:01:54.020
So let's say we wanted to figure out

00:01:54.020 --> 00:01:59.020
what six divided by one tenth is.

00:01:59.550 --> 00:02:02.150
Pause this video and see
if you can figure that out.

00:02:03.150 --> 00:02:06.370
All right, well, you
could do the same idea.

00:02:06.370 --> 00:02:08.500
Six wholes is equal to how many tenths?

00:02:08.500 --> 00:02:10.680
It's equal to 60 tenths.

00:02:10.680 --> 00:02:15.680
So I'll rewrite this as 60
tenths, divided by one tenth,

00:02:19.550 --> 00:02:23.530
is equal to how many, is equal to what?

00:02:23.530 --> 00:02:25.500
Well, if I have 60 of
something and if I divide it

00:02:25.500 --> 00:02:27.170
into groups of one of that something,

00:02:27.170 --> 00:02:29.550
I'm gonna have 60 equal groups of one.

00:02:29.550 --> 00:02:32.510
So this is going to be equal to 60.

00:02:32.510 --> 00:02:33.840
So you might see a pattern.

00:02:33.840 --> 00:02:37.720
When we divide it by a tenth,
we end up multiplying by 10.

00:02:37.720 --> 00:02:41.320
When we divide by a tenth,
we are multiplying by 10.

00:02:41.320 --> 00:02:43.530
And you could do the same
thing as we saw up here.

00:02:43.530 --> 00:02:46.880
You could take, you could
say six divided by one tenth

00:02:46.880 --> 00:02:51.880
is the same thing as
six divided by one tenth

00:02:52.150 --> 00:02:56.080
written as a fraction, which
is going to be equal to

00:02:56.080 --> 00:02:58.820
six times, you could say six times 10

00:02:58.820 --> 00:03:01.370
or six times 10 over one, either way.

00:03:01.370 --> 00:03:05.750
Six times 10 over one.

00:03:05.750 --> 00:03:07.240
Or, which is the same thing as 10,

00:03:07.240 --> 00:03:10.100
which is once again
going to be equal to 60.

00:03:10.100 --> 00:03:11.410
So I think you see the general pattern.

00:03:11.410 --> 00:03:15.060
Divide by a tenth, same
thing as multiplying by 10.

00:03:15.060 --> 00:03:17.920
Now, what about if we
dealt with hundredths?

00:03:17.920 --> 00:03:19.610
So let's say we wanna figure out

00:03:19.610 --> 00:03:24.610
what seven divided by a hundredth is.

00:03:26.400 --> 00:03:27.233
What would this be?

00:03:27.233 --> 00:03:29.040
Pause the video and try to figure it out.

00:03:30.360 --> 00:03:32.520
All right, well, we can do the same drill.

00:03:32.520 --> 00:03:36.050
So seven wholes, one whole is
equal to a hundred hundredths,

00:03:36.050 --> 00:03:39.520
so seven wholes is
equal to 700 hundredths.

00:03:39.520 --> 00:03:42.398
So this is equal to,
I'll write it like this,

00:03:42.398 --> 00:03:47.398
700 hundredths, hundredths, divided by,

00:03:49.220 --> 00:03:54.220
divided by one hundredth, one hundredth.

00:03:55.150 --> 00:03:57.240
If I have 700 of something
and if I'm dividing it

00:03:57.240 --> 00:03:59.290
into equal groups of
one of that something,

00:03:59.290 --> 00:04:01.760
I'm gonna have 700 equal groups.

00:04:01.760 --> 00:04:04.800
So this is going to be equal to 700.

00:04:04.800 --> 00:04:06.200
So you divide by a hundredth,

00:04:06.200 --> 00:04:09.430
is the same thing as
multiplying by a hundred.

00:04:09.430 --> 00:04:12.480
Now, you could also rewrite
this as fraction if you like,

00:04:12.480 --> 00:04:15.540
so that some of the principles
we saw up here still apply.

00:04:15.540 --> 00:04:20.540
So we could rewrite this
as seven divided by 1/100,

00:04:21.821 --> 00:04:24.220
and so this is going
to be the same thing as

00:04:24.220 --> 00:04:28.430
seven times a hundred over
one, or seven times a hundred.

00:04:28.430 --> 00:04:33.430
Seven times 100, which, once
again, will get you to 700.

00:04:35.460 --> 00:04:36.540
So I think you see a pattern.

00:04:36.540 --> 00:04:39.920
Divide by a tenth, same
thing as multiplying by 10.

00:04:39.920 --> 00:04:42.150
Divide by a hundredth, same thing as

00:04:42.150 --> 00:04:44.403
multiplying by a hundred.


WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:01.620
- [Instructor] Let's
see if we can figure out

00:00:01.620 --> 00:00:06.620
what two divided by 0.1 or one tenth is.

00:00:08.540 --> 00:00:12.070
Pause this video and see
if you can figure that out.

00:00:12.070 --> 00:00:13.730
Alright, now let's work
through it together

00:00:13.730 --> 00:00:16.010
and there's a couple of ways
that we can approach it.

00:00:16.010 --> 00:00:20.420
One way is to think about
everything in terms of tenths.

00:00:20.420 --> 00:00:24.060
So two wholes is how many tenths?

00:00:24.060 --> 00:00:26.280
Well a whole is equal to 10 tenths,

00:00:26.280 --> 00:00:29.520
so two wholes is equal to 20 tenths.

00:00:29.520 --> 00:00:34.403
So I could rewrite the two as 20 tenths.

00:00:35.270 --> 00:00:38.543
And so that's gonna be
20 tenths divided by,

00:00:39.830 --> 00:00:41.013
instead of writing it this way,

00:00:41.013 --> 00:00:43.160
instead of just writing it as 0.1,

00:00:43.160 --> 00:00:45.170
we know this is the
same thing as one tenth.

00:00:45.170 --> 00:00:47.623
So divided by one tenth.

00:00:48.590 --> 00:00:50.400
And so if I have 20 of something

00:00:50.400 --> 00:00:53.310
and if I divide it into groups
of one of that something,

00:00:53.310 --> 00:00:55.170
how many groups am I going to have?

00:00:55.170 --> 00:00:59.610
Well I'm going to have
20 equal groups of one.

00:00:59.610 --> 00:01:03.830
So 20 tenths divided by
one tenth is equal to 20.

00:01:03.830 --> 00:01:06.470
Another way that you could
approach that is you could

00:01:06.470 --> 00:01:10.360
rewrite the 0.1, the one
tenth, as a fraction.

00:01:10.360 --> 00:01:15.251
We could rewrite this as two divided by.

00:01:15.251 --> 00:01:16.410
Divided by.

00:01:16.410 --> 00:01:17.960
Instead of writing one tenth like this,

00:01:17.960 --> 00:01:19.410
I could write it as a fraction.

00:01:19.410 --> 00:01:21.340
Divided by one tenth.

00:01:21.340 --> 00:01:25.270
Well, we know that dividing
by a tenth is the same thing

00:01:25.270 --> 00:01:28.300
as multiplying by 10.

00:01:28.300 --> 00:01:32.783
So this is going to be equal to two times.

00:01:34.000 --> 00:01:36.690
Two times 10.

00:01:36.690 --> 00:01:39.470
And that gets us to the same
place that we had before.

00:01:39.470 --> 00:01:40.840
What is two times 10?

00:01:40.840 --> 00:01:44.650
Well that is going to be equal to 20.

00:01:44.650 --> 00:01:47.160
And that's good that we
got to the same answer.

00:01:47.160 --> 00:01:50.460
Otherwise, these would
not be equivalent methods.

00:01:50.460 --> 00:01:52.310
Let's do another example.

00:01:52.310 --> 00:01:56.410
So let's say we wanted to
figure out what six divided

00:01:56.410 --> 00:01:59.550
by one tenth is.

00:01:59.550 --> 00:02:02.150
Pause this video and see
if you can figure that out.

00:02:03.150 --> 00:02:04.290
Alright.

00:02:04.290 --> 00:02:06.370
Well, you could do the same idea.

00:02:06.370 --> 00:02:08.500
Six wholes is equal to how many tenths?

00:02:08.500 --> 00:02:10.680
It's equal to 60 tenths.

00:02:10.680 --> 00:02:15.680
So I'll rewrite this as 60
tenths divided by one tenth

00:02:19.550 --> 00:02:23.530
is equal to how many is equal to what?

00:02:23.530 --> 00:02:24.760
Well if I have 60 of something

00:02:24.760 --> 00:02:27.170
and if I divide it into groups
of one of that something,

00:02:27.170 --> 00:02:29.550
I'm gonna have 60 equal groups of one.

00:02:29.550 --> 00:02:32.510
So this is going to be equal to 60.

00:02:32.510 --> 00:02:33.840
So you might see a pattern.

00:02:33.840 --> 00:02:37.720
When we divided by a tenth,
we ended up multiplying by 10.

00:02:37.720 --> 00:02:41.320
When we divide by a tenth,
we are multiplying by 10.

00:02:41.320 --> 00:02:43.530
And you could do the same
thing as we saw up here.

00:02:43.530 --> 00:02:47.890
You could say six divided by
one tenth is the same thing

00:02:47.890 --> 00:02:52.890
as six divided by one
tenth written as a fraction

00:02:53.860 --> 00:02:57.570
which is going to be equal to six times.

00:02:57.570 --> 00:03:00.390
You can say six times 10
or six times 10 over one.

00:03:00.390 --> 00:03:01.370
Either way.

00:03:01.370 --> 00:03:06.370
Six times 10 over one or
which is the same thing as 10

00:03:07.240 --> 00:03:10.100
which is once again
going to be equal to 60.

00:03:10.100 --> 00:03:11.410
So I think you see the general pattern.

00:03:11.410 --> 00:03:15.060
Divide by a tenth same
thing as multiplying by 10.

00:03:15.060 --> 00:03:17.920
Now what about if we
dealt with hundredths?

00:03:17.920 --> 00:03:22.080
So let's say we wanna figure
out what seven divided

00:03:22.080 --> 00:03:26.400
by a hundredth is?

00:03:26.400 --> 00:03:27.233
What would this be?

00:03:27.233 --> 00:03:29.040
Pause the video and try to figure it out.

00:03:30.360 --> 00:03:32.520
Alright, well we can do the same drill.

00:03:32.520 --> 00:03:34.250
So seven wholes.

00:03:34.250 --> 00:03:36.050
One whole is equal to a hundred hundredths

00:03:36.050 --> 00:03:39.520
so seven wholes is
equal to 700 hundredths.

00:03:39.520 --> 00:03:40.800
So this is equal to.

00:03:40.800 --> 00:03:41.850
I'll write it like this.

00:03:41.850 --> 00:03:46.390
700 hundredths divided by.

00:03:49.220 --> 00:03:51.830
Divided by one hundredth.

00:03:51.830 --> 00:03:55.150
One hundredth.

00:03:55.150 --> 00:03:57.890
If I have 700 of something and
if I'm dividing it into equal

00:03:57.890 --> 00:03:59.290
groups of one of that something,

00:03:59.290 --> 00:04:01.760
I'm gonna have 700 equal groups.

00:04:01.760 --> 00:04:04.800
So this is going to be equal to 700.

00:04:04.800 --> 00:04:06.200
So you divide by a hundredth

00:04:06.200 --> 00:04:09.430
is the same thing as multiplying by a 100.

00:04:09.430 --> 00:04:12.480
Now you could also rewrite
this as a fraction if you like

00:04:12.480 --> 00:04:15.540
so that some of the principles
we saw up here still apply.

00:04:15.540 --> 00:04:20.540
So we could rewrite this as
seven divided by one hundredth.

00:04:21.680 --> 00:04:25.170
And so this is going to be
the same thing as seven times

00:04:25.170 --> 00:04:26.110
100 over one.

00:04:26.110 --> 00:04:28.430
Or seven times 100.

00:04:28.430 --> 00:04:33.430
Seven times 100 which once
again, will get you to 700.

00:04:35.460 --> 00:04:36.540
So I think you see a pattern.

00:04:36.540 --> 00:04:39.920
Divide by a tenth, same
thing as multiplying by 10.

00:04:39.920 --> 00:04:43.523
Divide by a hundredth, same
thing as multiplying by a 100.


Dividing a whole number by a decimal on a number line

WEBVTT
Kind: captions
Language: en

00:00:00.040 --> 00:00:02.090
- [Instructor] What we
want to do in this video

00:00:02.090 --> 00:00:06.230
is figure out what two divided by 0.4 is,

00:00:06.230 --> 00:00:09.100
or two divided by 4/10.

00:00:09.100 --> 00:00:10.660
So why don't you pause this video

00:00:10.660 --> 00:00:12.010
and try to figure out what it is.

00:00:12.010 --> 00:00:13.420
And as a little bit of a hint,

00:00:13.420 --> 00:00:15.800
think about two on the number line

00:00:15.800 --> 00:00:18.664
and think about how many jumps of 4/10

00:00:18.664 --> 00:00:22.600
do you have to make starting
at zero to get to two,

00:00:22.600 --> 00:00:26.750
to figure out what two divided by 4/10 is.

00:00:26.750 --> 00:00:28.700
All right, now let's do it together.

00:00:28.700 --> 00:00:31.110
And as I mentioned in my hint,

00:00:31.110 --> 00:00:32.740
let's just think about how many jumps

00:00:32.740 --> 00:00:36.330
of 4/10 we have to make
to go from zero to two.

00:00:36.330 --> 00:00:39.390
And on this number line,
each of these, this is a 10th

00:00:39.390 --> 00:00:41.600
right over here because we can
see there's 10 equal spaces

00:00:41.600 --> 00:00:43.790
between zero and one.

00:00:43.790 --> 00:00:46.430
So let's take some jumps of 4/10.

00:00:46.430 --> 00:00:48.780
So if we start at zero, we can make,

00:00:48.780 --> 00:00:52.530
let's see there's 1/10, 2/10, 3/10, 4/10.

00:00:52.530 --> 00:00:55.140
That's one jump.

00:00:55.140 --> 00:00:56.270
I'll number that.

00:00:56.270 --> 00:01:01.120
And then we can make our next
jump of one, two, three, 4/10.

00:01:01.120 --> 00:01:03.320
Guess that's right over there.

00:01:03.320 --> 00:01:04.610
That's two jumps.

00:01:04.610 --> 00:01:09.070
Then our next jump will
take us right over there.

00:01:09.070 --> 00:01:12.450
So once again I'm just
making equal jumps of 4/10.

00:01:12.450 --> 00:01:13.300
Then my next jump

00:01:13.300 --> 00:01:16.880
will take us one, two,
three, 4/10 further.

00:01:16.880 --> 00:01:19.190
So that gets us right over there.

00:01:19.190 --> 00:01:20.710
We've taken four jumps.

00:01:20.710 --> 00:01:23.530
And then it looks like our
fifth jump gets us there.

00:01:23.530 --> 00:01:25.610
One, two, three, 4/10.

00:01:25.610 --> 00:01:26.443
Yup.

00:01:26.443 --> 00:01:30.070
Our fifth jump gets us exactly to two.

00:01:30.070 --> 00:01:35.070
So we've taken five jumps of
4/10 to go from zero to two

00:01:35.820 --> 00:01:37.080
or another way to think about it is

00:01:37.080 --> 00:01:41.440
if you were to divide two
into equal chunks of 4/10,

00:01:41.440 --> 00:01:45.460
you could divide into
five equal chunks of 4/10.

00:01:45.460 --> 00:01:47.030
Or you could make five equal jumps

00:01:47.030 --> 00:01:49.520
to go from five equal jumps of 4/10

00:01:49.520 --> 00:01:52.873
to go from zero to two.


Dividing a decimal by a whole number on the number line

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:01.990
- [Instructor] In this
video, we're going to try

00:00:01.990 --> 00:00:06.990
to figure out what 0.6 or
6/10 divided by two is.

00:00:07.060 --> 00:00:09.200
And I'll give you a little bit of a hint.

00:00:09.200 --> 00:00:12.760
We have 6/10 plotted on the
number line right over here.

00:00:12.760 --> 00:00:14.550
And so one way to think about it is,

00:00:14.550 --> 00:00:19.540
if I wanted to go from
zero to 6/10 in two jumps,

00:00:19.540 --> 00:00:22.610
how big would each of
those jumps have to be?

00:00:22.610 --> 00:00:24.990
Pause this video and try to
figure that out on your own

00:00:24.990 --> 00:00:26.740
before we work through it together.

00:00:27.720 --> 00:00:29.760
All right, so as I mentioned,

00:00:29.760 --> 00:00:30.840
one way to think about it is

00:00:30.840 --> 00:00:35.230
how can I go from zero
to 6/10 in two jumps?

00:00:35.230 --> 00:00:38.100
Two equal jumps, I should say.

00:00:38.100 --> 00:00:40.830
Well, if we just eyeball
it right over here,

00:00:40.830 --> 00:00:42.730
I could do one jump like that

00:00:42.730 --> 00:00:45.330
and then another jump like that.

00:00:45.330 --> 00:00:48.440
So I did one jump and then two jumps.

00:00:48.440 --> 00:00:50.310
And so one way to think about it is

00:00:50.310 --> 00:00:54.000
we've divided 6/10 into two jumps,

00:00:54.000 --> 00:00:56.480
and each of those jumps are how far?

00:00:56.480 --> 00:01:00.780
Well, each of those
jumps are one, two, 3/10.

00:01:00.780 --> 00:01:04.520
So this is 3/10 or I could
write it like this, 3/10.

00:01:04.520 --> 00:01:07.640
We have another one, two, 3/10.

00:01:07.640 --> 00:01:11.190
So this is going to be equal to 3/10.

00:01:11.190 --> 00:01:12.670
And we've thought about it in other ways.

00:01:12.670 --> 00:01:15.830
If you have six of something
and you divide it by two,

00:01:15.830 --> 00:01:18.580
you're going to have two groups
of three of that something.

00:01:18.580 --> 00:01:20.890
So if you have 6/10, and you wanna

00:01:20.890 --> 00:01:22.730
put it into two equal groups, well,

00:01:22.730 --> 00:01:25.783
each of those groups are going to be 3/10.


How interest rates affect interest rates, financial flows, and exchange rates

WEBVTT
Kind: captions
Language: en

00:00:00.180 --> 00:00:01.520
- [Instructor] What we're
gonna do in this video

00:00:01.520 --> 00:00:03.930
is try to think of the chain of events

00:00:03.930 --> 00:00:08.390
that would happen if the
supply of loanable funds

00:00:08.390 --> 00:00:10.400
were to increase in the United States.

00:00:10.400 --> 00:00:12.110
And the way that that could happen is

00:00:12.110 --> 00:00:15.310
let's say the Federal Reserve were to

00:00:15.310 --> 00:00:18.090
so to speak print money,
and then use that money

00:00:18.090 --> 00:00:20.490
to buy treasuries in the US.

00:00:20.490 --> 00:00:23.210
So it's inserting those
Federal Reserve notes

00:00:23.210 --> 00:00:26.560
into the quantity of loanable funds.

00:00:26.560 --> 00:00:28.660
Well what would happen is that the supply

00:00:28.660 --> 00:00:31.740
of loanable funds would
shift to the right.

00:00:31.740 --> 00:00:36.200
So our new supply would look like this,

00:00:36.200 --> 00:00:37.550
would look like this.

00:00:37.550 --> 00:00:38.990
I'll call that S prime.

00:00:38.990 --> 00:00:43.100
And then we'd have a new
equilibrium price of those funds,

00:00:43.100 --> 00:00:45.580
which we would call
our real interest rate.

00:00:45.580 --> 00:00:48.920
So then we get to r prime.

00:00:48.920 --> 00:00:51.310
So our real interest rates have gone down,

00:00:51.310 --> 00:00:56.210
and we have a higher quantity of money

00:00:56.210 --> 00:00:58.940
that is being loaned, Q prime.

00:00:58.940 --> 00:01:01.090
But what would be the effect of that?

00:01:01.090 --> 00:01:02.420
What would be the effect of that

00:01:02.420 --> 00:01:04.250
relative to other countries?

00:01:04.250 --> 00:01:06.640
And I'm just picking Japan
here as another country,

00:01:06.640 --> 00:01:09.190
but this could be the case
with many other countries

00:01:09.190 --> 00:01:11.120
where they have relatively free flows

00:01:11.120 --> 00:01:14.750
of goods and financial capital.

00:01:14.750 --> 00:01:16.060
And to help us think through that,

00:01:16.060 --> 00:01:18.500
I drew the balance of
payments for each country.

00:01:18.500 --> 00:01:19.960
And the balance of payments is made up of

00:01:19.960 --> 00:01:23.180
the current account, which
is talking about the flows

00:01:23.180 --> 00:01:25.810
of goods and services, and then you have

00:01:25.810 --> 00:01:27.600
what's sometimes called a capital account,

00:01:27.600 --> 00:01:30.280
sometimes the capital
and financial account,

00:01:30.280 --> 00:01:33.350
which is talking about the flow of,

00:01:33.350 --> 00:01:36.010
it oftentimes, you could think
of it as financial investment

00:01:36.010 --> 00:01:40.050
or investments of some
kind or the flow of funds.

00:01:40.050 --> 00:01:43.563
So pause this video and think
about what would happen.

00:01:45.140 --> 00:01:50.140
Well, if the real interest rate
go down in the United States

00:01:50.190 --> 00:01:52.130
and we're assuming that all else equal

00:01:52.130 --> 00:01:55.160
in every other country, well
then you have a situation

00:01:55.160 --> 00:01:59.190
where in Japan, the
relative real interest rates

00:01:59.190 --> 00:02:04.190
are now higher, so relative,
relative real interest rates,

00:02:08.490 --> 00:02:13.343
interest rates, are higher.

00:02:14.530 --> 00:02:17.700
Now in general, people
might want to say hey,

00:02:17.700 --> 00:02:20.730
if I can get a higher
or a higher than before

00:02:20.730 --> 00:02:22.750
relative real interest
rate, it might not be

00:02:22.750 --> 00:02:25.640
absolutely higher, but it's
higher than it was before

00:02:25.640 --> 00:02:29.380
relative to the United States,
well that might increase

00:02:29.380 --> 00:02:33.630
the financial flows from
the United States to Japan.

00:02:33.630 --> 00:02:36.170
And so you might have some more people,

00:02:36.170 --> 00:02:38.750
not everyone, but some
more people than before,

00:02:38.750 --> 00:02:42.797
who want to take their
dollars, convert it into yen,

00:02:44.290 --> 00:02:48.560
and buy financial assets in Japan

00:02:48.560 --> 00:02:51.010
where they can get that relatively now

00:02:51.010 --> 00:02:53.440
higher real interest rate.

00:02:53.440 --> 00:02:57.080
Well if these folks are
converting from dollars to yen,

00:02:57.080 --> 00:02:59.310
what's the immediate effect of that?

00:02:59.310 --> 00:03:02.750
Well, it will increase demand for the yen

00:03:02.750 --> 00:03:06.830
and so the price of the yen
in dollar terms will go up.

00:03:06.830 --> 00:03:08.720
Or another way to think about it is

00:03:08.720 --> 00:03:13.330
this is going to cause
the dollar to depreciate,

00:03:13.330 --> 00:03:18.330
depreciate, relative, relative to the yen.

00:03:20.270 --> 00:03:22.450
So we're just gonna put
that aside right over here,

00:03:22.450 --> 00:03:24.730
'cause this is gonna
have other implications.

00:03:24.730 --> 00:03:27.510
But that dollar, it's used to buy yen,

00:03:27.510 --> 00:03:32.291
and then those investors will
maybe buy Japanese bonds.

00:03:32.291 --> 00:03:34.260
And so what's happening?

00:03:34.260 --> 00:03:37.710
Well we're talking about the
transfer of financial assets.

00:03:37.710 --> 00:03:39.450
And so in the United States,

00:03:39.450 --> 00:03:44.330
the capital and financial
account, that will go down.

00:03:44.330 --> 00:03:45.780
This will go down.

00:03:45.780 --> 00:03:48.793
You could think of it as being debited.

00:03:49.720 --> 00:03:51.960
And in Japan, they're getting,

00:03:51.960 --> 00:03:54.190
they're getting an increase
in financial assets.

00:03:54.190 --> 00:03:56.210
People are investing
more in Japanese bonds.

00:03:56.210 --> 00:03:57.510
However you want to think about it.

00:03:57.510 --> 00:04:00.650
And so that increase
in funds, that goes up,

00:04:00.650 --> 00:04:03.950
and so this is getting credited.

00:04:03.950 --> 00:04:07.350
Now, in other videos, we've
talked about how over time,

00:04:07.350 --> 00:04:09.200
the balance of payments
tend to balance out.

00:04:09.200 --> 00:04:10.610
If one side is getting debited,

00:04:10.610 --> 00:04:13.750
the other side is getting
credited, or vice versa.

00:04:13.750 --> 00:04:17.110
So how is that going to
work out in this situation?

00:04:17.110 --> 00:04:19.800
Well that all goes back to
the fact that the dollar

00:04:19.800 --> 00:04:23.100
has depreciated relative to the yen.

00:04:23.100 --> 00:04:26.200
If the dollar depreciates
relative to the yen,

00:04:26.200 --> 00:04:27.810
what is that going to do?

00:04:27.810 --> 00:04:32.503
Well now American goods, American goods,

00:04:33.750 --> 00:04:38.040
are relatively cheaper,
relatively cheaper,

00:04:38.040 --> 00:04:43.040
or cheaper than they were
before, cheaper in Japan.

00:04:44.780 --> 00:04:46.100
That's hard to read.

00:04:46.100 --> 00:04:50.833
And Japanese goods more expensive,

00:04:52.760 --> 00:04:57.760
more expensive, in the US.

00:04:59.230 --> 00:05:01.310
And so what is going to happen?

00:05:01.310 --> 00:05:03.970
Well in that situation,
that means that the US

00:05:03.970 --> 00:05:06.700
is going to export more to Japan.

00:05:06.700 --> 00:05:09.290
Their goods are now relative,
are now cheaper in Japan,

00:05:09.290 --> 00:05:12.580
Japanese goods are now
more expensive in the US.

00:05:12.580 --> 00:05:17.580
So they're going to buy fewer
Japanese, Japanese goods

00:05:23.010 --> 00:05:28.010
and sell, and export, export more

00:05:28.510 --> 00:05:32.960
American goods, American goods.

00:05:32.960 --> 00:05:35.480
And so what's going to happen
on our current account?

00:05:35.480 --> 00:05:37.070
Well if your exporting more,

00:05:37.070 --> 00:05:39.190
that means your current account goes up.

00:05:39.190 --> 00:05:40.913
It is going to be credited.

00:05:41.760 --> 00:05:44.280
It is going to be credited.

00:05:44.280 --> 00:05:46.780
And then the opposite's
going to happen to Japan.

00:05:46.780 --> 00:05:49.210
Relative to the US, it's going to export

00:05:49.210 --> 00:05:52.030
less and import more.

00:05:52.030 --> 00:05:55.480
So its current account is going to be,

00:05:55.480 --> 00:05:58.870
is going to be debited.

00:05:58.870 --> 00:06:02.240
Now economies are complex things,

00:06:02.240 --> 00:06:05.050
and what I've just done is a
little bit of a simplification.

00:06:05.050 --> 00:06:07.770
But these are the general
trends that you would expect.

00:06:07.770 --> 00:06:10.400
Other things that you
might expect is well,

00:06:10.400 --> 00:06:15.400
if you have this flow of
financial capital into Japan,

00:06:17.030 --> 00:06:19.940
well that might increase
their loanable funds.

00:06:19.940 --> 00:06:23.100
And so their real interest
rate might eventually go down

00:06:23.100 --> 00:06:25.290
and all of these cycles would keep going

00:06:25.290 --> 00:06:28.890
and reverberating back
and forth over time.

00:06:28.890 --> 00:06:30.760
But this is the general chain of events

00:06:30.760 --> 00:06:35.160
that you might expect, that
the interest rates in Japan

00:06:35.160 --> 00:06:37.890
will become relatively higher.

00:06:37.890 --> 00:06:41.940
And so you have a flow of financial funds

00:06:41.940 --> 00:06:43.550
going from the US to Japan.

00:06:43.550 --> 00:06:46.410
In the process, when they
convert from dollar to yen,

00:06:46.410 --> 00:06:47.980
the dollar is going to get cheaper,

00:06:47.980 --> 00:06:50.190
the yen's going to get more expensive.

00:06:50.190 --> 00:06:53.760
The American capital and
financial account goes down

00:06:53.760 --> 00:06:56.600
'cause you have this net
outflow of financial funds.

00:06:56.600 --> 00:06:59.100
But because of the
depreciation of the dollar,

00:06:59.100 --> 00:07:03.491
the US is now importing
less and exporting more.

00:07:03.491 --> 00:07:07.340
Now a question is is this good
or bad for either country?

00:07:07.340 --> 00:07:09.590
Well, it depends what
the country's goals are.

00:07:09.590 --> 00:07:12.060
This might be good for
the US if their goal

00:07:12.060 --> 00:07:15.100
was to export more American goods.

00:07:15.100 --> 00:07:17.860
Or it might be bad for
the US if they said hey,

00:07:17.860 --> 00:07:19.620
now Japanese goods are more expensive,

00:07:19.620 --> 00:07:22.355
and maybe they're dependent
on some type of Japanese goods

00:07:22.355 --> 00:07:24.380
in some way, shape, or form.

00:07:24.380 --> 00:07:25.750
That might not be the case with the US,

00:07:25.750 --> 00:07:27.660
but in another country,
let's say they're dependent

00:07:27.660 --> 00:07:30.680
on oil from other countries
or they're dependent

00:07:30.680 --> 00:07:32.930
on military hardware from other countries.

00:07:32.930 --> 00:07:35.990
And if those things become
more expensive, or food,

00:07:35.990 --> 00:07:38.010
that might make it a lot
harder for their citizens.

00:07:38.010 --> 00:07:40.000
So it's an interesting
thing to think about,

00:07:40.000 --> 00:07:41.730
whether this is good or bad and how

00:07:41.730 --> 00:07:43.380
all of these things fit together.


Failure of Reconstruction

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:03.360
- [Kim] How do you define freedom?

00:00:03.360 --> 00:00:07.100
Stop for a minute and picture
what it means to be free.

00:00:07.100 --> 00:00:08.560
What comes into your mind?

00:00:08.560 --> 00:00:11.040
Traveling wherever you please,

00:00:11.040 --> 00:00:13.150
having enough money to do what you want,

00:00:13.150 --> 00:00:16.030
or is freedom better
defined by what it's not?

00:00:16.030 --> 00:00:18.430
Not having anyone telling you what to do,

00:00:18.430 --> 00:00:20.040
not being in prison?

00:00:20.040 --> 00:00:24.380
Freedom is a core aspect
of US national identity,

00:00:24.380 --> 00:00:28.540
but if someone gave you box
labeled Contents: Freedom,

00:00:28.540 --> 00:00:31.030
what would you expect to find inside?

00:00:31.030 --> 00:00:33.260
This was the question that
the United States faced

00:00:33.260 --> 00:00:36.780
during Reconstruction, the
period following the Civil War,

00:00:36.780 --> 00:00:39.750
when the US government,
Southern state governments,

00:00:39.750 --> 00:00:42.570
and African Americans
attempted to negotiate

00:00:42.570 --> 00:00:45.970
a new social and political
order for the South.

00:00:45.970 --> 00:00:49.080
But what African Americans
expected to find in the box

00:00:49.080 --> 00:00:51.180
labeled Freedom was very different

00:00:51.180 --> 00:00:54.230
from what their former
enslavers wanted to put there.

00:00:54.230 --> 00:00:57.230
Was freedom just the absence of slavery,

00:00:57.230 --> 00:00:59.330
as most white Southerners believed,

00:00:59.330 --> 00:01:02.910
or did it imply citizenship,
political power,

00:01:02.910 --> 00:01:04.950
and economic self-sufficiency?

00:01:04.950 --> 00:01:07.330
Try to solve this dilemma,

00:01:07.330 --> 00:01:09.930
Congress passed, and the states ratified,

00:01:09.930 --> 00:01:12.420
three new Constitutional amendments

00:01:12.420 --> 00:01:15.300
during the Reconstruction
era, the 13th Amendment,

00:01:15.300 --> 00:01:18.790
which ended the system of slavery in 1865,

00:01:18.790 --> 00:01:22.010
the 14th Amendment, which
extended citizenship

00:01:22.010 --> 00:01:25.882
to all persons born or
naturalized in the United States

00:01:25.882 --> 00:01:29.210
in 1868, and the 15th Amendment,

00:01:29.210 --> 00:01:32.950
which gave black men the
right to vote in 1870.

00:01:32.950 --> 00:01:35.690
So in just five years,

00:01:35.690 --> 00:01:39.330
African Americans in the South
went from personal property

00:01:39.330 --> 00:01:43.170
to full civic participants,
at least in theory.

00:01:43.170 --> 00:01:46.339
In reality, how different were definitions

00:01:46.339 --> 00:01:50.290
of freedom, citizenship, and democracy

00:01:50.290 --> 00:01:52.530
before and after Reconstruction?

00:01:52.530 --> 00:01:54.250
To really answer this question,

00:01:54.250 --> 00:01:57.429
we need to examine continuity and change

00:01:57.429 --> 00:01:59.470
in the Reconstruction era.

00:01:59.470 --> 00:02:02.220
What stayed the same and what changed

00:02:02.220 --> 00:02:04.200
in each of these three areas

00:02:04.200 --> 00:02:07.210
following the passage of the
Reconstruction Amendments?

00:02:07.210 --> 00:02:10.920
Okay, first let's look at
continuities and changes

00:02:10.920 --> 00:02:13.090
in the definition of freedom.

00:02:13.090 --> 00:02:14.790
Before the end of slavery,

00:02:14.790 --> 00:02:18.940
African Americans had neither
economic nor physical freedom.

00:02:18.940 --> 00:02:20.930
They didn't have control of their bodies

00:02:20.930 --> 00:02:22.370
or of their labor.

00:02:22.370 --> 00:02:25.040
The pass system kept
them from moving freely,

00:02:25.040 --> 00:02:27.840
and slavery itself meant
that they couldn't choose

00:02:27.840 --> 00:02:31.110
where to work or earn
money from their own work.

00:02:31.110 --> 00:02:34.570
So how much did their
physical and economic freedom

00:02:34.570 --> 00:02:38.110
change after the 13th
Amendment outlawed slavery?

00:02:38.110 --> 00:02:40.440
Well, their economic self-sufficiency

00:02:40.440 --> 00:02:42.380
went through some ups and downs.

00:02:42.380 --> 00:02:44.480
Most African Americans believed

00:02:44.480 --> 00:02:47.650
that their years of
unpaid toil entitled them

00:02:47.650 --> 00:02:48.940
to land of their own.

00:02:48.940 --> 00:02:51.520
US Army general William Tecumseh Sherman

00:02:51.520 --> 00:02:53.860
redistributed Confederate territory

00:02:53.860 --> 00:02:56.680
on the coasts of Georgia
and South Carolina

00:02:56.680 --> 00:03:00.260
to black families, who
farmed there for a few years

00:03:00.260 --> 00:03:03.420
until Lincoln's successor, Andrew Johnson,

00:03:03.420 --> 00:03:07.060
gave all confiscated land
back to its former owners.

00:03:07.060 --> 00:03:10.630
Instead, most black farmers
became sharecroppers,

00:03:10.630 --> 00:03:13.710
renting a portion of a
white landowner's farm

00:03:13.710 --> 00:03:16.200
in exchange for part of the crop yield.

00:03:16.200 --> 00:03:18.510
This gave black farmers a lot more freedom

00:03:18.510 --> 00:03:19.720
over their own work,

00:03:19.720 --> 00:03:22.060
since they didn't have to
work under an overseer.

00:03:22.060 --> 00:03:25.680
But economically, sharecropping
kept black farmers,

00:03:25.680 --> 00:03:27.319
as well as small white farmers,

00:03:27.319 --> 00:03:30.690
in an endless cycle of debt and poverty.

00:03:30.690 --> 00:03:33.960
After the 13th Amendment, most
Southern state governments

00:03:33.960 --> 00:03:36.370
attempted to limit the physical freedom

00:03:36.370 --> 00:03:38.020
of African Americans as well,

00:03:38.020 --> 00:03:40.720
with statutes known as the Black Codes.

00:03:40.720 --> 00:03:42.950
Many of these codes defined anyone

00:03:42.950 --> 00:03:46.481
who wasn't under a labor
contract as a vagrant

00:03:46.481 --> 00:03:49.490
who could be arrested and
have their labor sold.

00:03:49.490 --> 00:03:52.500
Later, segregation limited
the physical freedom

00:03:52.500 --> 00:03:55.150
of where Southern African
Americans could go

00:03:55.150 --> 00:03:56.480
and what they could do.

00:03:56.480 --> 00:03:57.900
Laws like the Black Codes,

00:03:57.900 --> 00:04:01.500
which so obviously attempted
to institute slavery

00:04:01.500 --> 00:04:05.320
by another name, led Congress
to pass the 14th Amendment,

00:04:05.320 --> 00:04:07.140
which defined a US citizen

00:04:07.140 --> 00:04:10.910
as anyone born or naturalized
in the United States

00:04:10.910 --> 00:04:12.930
and specifically prevented states

00:04:12.930 --> 00:04:15.830
from infringing upon
the rights of citizens.

00:04:15.830 --> 00:04:19.870
Before the Civil War,
citizenship was exclusively

00:04:19.870 --> 00:04:22.090
the privilege of white Americans.

00:04:22.090 --> 00:04:25.286
Non-white immigrants weren't
eligible to become US citizens,

00:04:25.286 --> 00:04:29.679
and the 1857 Supreme Court
decision in Dred Scott

00:04:29.679 --> 00:04:33.860
declared that no African Americans
could be citizens at all.

00:04:33.860 --> 00:04:36.810
The 14th Amendment, ratified in 1868,

00:04:36.810 --> 00:04:40.720
led to a huge increase in
the number of US citizens

00:04:40.720 --> 00:04:44.410
and it decoupled
citizenship from whiteness.

00:04:44.410 --> 00:04:47.720
Even the American-born
children of Asian immigrants

00:04:47.720 --> 00:04:48.830
were citizens.

00:04:48.830 --> 00:04:51.910
But the Supreme Court
defined the 14th Amendment

00:04:51.910 --> 00:04:56.220
very narrowly in the late 19th
century, permitting many laws

00:04:56.220 --> 00:04:58.570
that discriminated on the basis of race.

00:04:58.570 --> 00:05:01.880
Only in the 20th century
would the 14th Amendment

00:05:01.880 --> 00:05:05.680
become an important tool
for civil rights activists

00:05:05.680 --> 00:05:07.490
to break down segregation.

00:05:07.490 --> 00:05:11.170
Lastly, the 15th Amendment,
ratified in 1870,

00:05:11.170 --> 00:05:13.890
extended the right to vote to black men.

00:05:13.890 --> 00:05:16.160
In the years leading up to the Civil War,

00:05:16.160 --> 00:05:20.150
with few exceptions, only white
man had the right to vote.

00:05:20.150 --> 00:05:23.490
The 15th Amendment radically
redefined the terms

00:05:23.490 --> 00:05:24.770
of American democracy.

00:05:24.770 --> 00:05:28.770
During Reconstruction, more
than 2,000 African Americans

00:05:28.770 --> 00:05:32.230
held public office,
including two US senators.

00:05:32.230 --> 00:05:33.210
But there were limits

00:05:33.210 --> 00:05:35.710
to this new broader
definition of democracy.

00:05:35.710 --> 00:05:37.640
First, it didn't include women,

00:05:37.640 --> 00:05:40.470
much to the frustration of
the women's suffrage movement.

00:05:40.470 --> 00:05:44.302
Then, as the federal
government ceased to intervene

00:05:44.302 --> 00:05:48.600
to protect black citizens in
the South in the late 1870s,

00:05:48.600 --> 00:05:51.393
Southern state governments imposed a range

00:05:51.393 --> 00:05:53.640
of voter suppression tactics

00:05:53.640 --> 00:05:56.610
to effectively bar African
Americans from voting,

00:05:56.610 --> 00:05:58.480
which then reduced the likelihood

00:05:58.480 --> 00:06:00.900
of black politicians winning office.

00:06:00.900 --> 00:06:03.590
Not until the 1960s

00:06:03.590 --> 00:06:06.050
would African American voter registration

00:06:06.050 --> 00:06:09.330
once again reach
Reconstruction-era levels.

00:06:09.330 --> 00:06:12.420
So how much did the
Reconstruction Amendments

00:06:12.420 --> 00:06:17.100
change definitions of freedom,
citizenship, and democracy?

00:06:17.100 --> 00:06:19.030
Well, after the amendments,

00:06:19.030 --> 00:06:22.093
African Americans were free
to own their own bodies

00:06:22.093 --> 00:06:24.580
and labor, but that was about it.

00:06:24.580 --> 00:06:28.496
The 14th and 15th Amendments
led to short-lived revolutions

00:06:28.496 --> 00:06:32.350
in the concept of citizenship
and in voting rights,

00:06:32.350 --> 00:06:34.930
but those rights had all but evaporated

00:06:34.930 --> 00:06:36.280
by the end of the century.

00:06:36.280 --> 00:06:39.854
Nevertheless, although they
didn't have much of an impact

00:06:39.854 --> 00:06:44.140
in the short term, these
amendments would lay the foundation

00:06:44.140 --> 00:06:48.123
for the civil rights movement
of the 1950s and '60s.


The Mexican-American War

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:01.470
- [Kim] This is a painting

00:00:01.470 --> 00:00:04.180
of US general Winfield Scott

00:00:04.180 --> 00:00:08.080
entering Mexico City on
September 15th, 1847.

00:00:08.080 --> 00:00:11.950
Scott landed with a US naval
fleet several weeks beforehand.

00:00:11.950 --> 00:00:15.070
He bombarded the coastal
stronghold of Veracruz

00:00:15.070 --> 00:00:18.020
and then fought his way
inland toward the capital.

00:00:18.020 --> 00:00:20.020
Scott actually followed the same route

00:00:20.020 --> 00:00:23.340
that Spanish conquistador
Hernan Cortes took

00:00:23.340 --> 00:00:25.560
more than 300 years earlier.

00:00:25.560 --> 00:00:28.330
Winfield Scott's campaign to Mexico City

00:00:28.330 --> 00:00:30.500
was just one of three fronts

00:00:30.500 --> 00:00:34.340
in the two-year-long,
continent-spanning effort

00:00:34.340 --> 00:00:37.870
of the United States to take
Mexican territory by force.

00:00:37.870 --> 00:00:41.200
The other two fronts were in
California and New Mexico.

00:00:41.200 --> 00:00:43.200
After the two nations made peace

00:00:43.200 --> 00:00:47.610
by signing the Treaty of
Guadalupe Hidalgo in mid 1848,

00:00:47.610 --> 00:00:51.250
the United States gained
over a million square miles

00:00:51.250 --> 00:00:52.350
of new territory,

00:00:52.350 --> 00:00:55.220
a landmass larger than
the Louisiana Purchase.

00:00:55.220 --> 00:00:59.170
For Mexico, this war was
a catastrophic defeat,

00:00:59.170 --> 00:01:03.330
which resulted in the loss of
about 1/3 of its total area.

00:01:03.330 --> 00:01:06.440
The Mexican-American War
doesn't really loom large

00:01:06.440 --> 00:01:09.880
in American memory, compared
to the Revolutionary War

00:01:09.880 --> 00:01:13.580
or the Civil War, but it
was a transformative event

00:01:13.580 --> 00:01:16.570
in the history of the United
States and North America.

00:01:16.570 --> 00:01:19.120
On the scale of national politics,

00:01:19.120 --> 00:01:22.130
the war led to political realignment,

00:01:22.130 --> 00:01:24.040
and eventually, the Civil War.

00:01:24.040 --> 00:01:26.700
But on a human scale, it
led to transformations

00:01:26.700 --> 00:01:28.990
in the lives of people
who lived in the West

00:01:28.990 --> 00:01:31.410
who went to bed one day in Mexico

00:01:31.410 --> 00:01:34.170
and woke up the next day
in the United States.

00:01:34.170 --> 00:01:36.830
National boundaries
shifted under their feet.

00:01:36.830 --> 00:01:40.630
For those people, the outcome
of the war meant new laws,

00:01:40.630 --> 00:01:43.380
customs, new friends and enemies,

00:01:43.380 --> 00:01:46.680
and even the loss of
rights and privileges.

00:01:46.680 --> 00:01:50.810
So let's dive a little deeper
into the causes and effects

00:01:50.810 --> 00:01:52.440
of the Mexican-American War.

00:01:52.440 --> 00:01:56.330
The war began in April of 1846.

00:01:56.330 --> 00:02:00.100
A Mexican cavalry brigade
attacked US forces

00:02:00.100 --> 00:02:02.750
who were under the command
of General Zachary Taylor

00:02:02.750 --> 00:02:04.560
across the Rio Grande River

00:02:04.560 --> 00:02:07.110
from the town of Matamoros, Mexico.

00:02:07.110 --> 00:02:10.340
After this attack, President James K. Polk

00:02:10.340 --> 00:02:12.600
sent a war message to Congress.

00:02:12.600 --> 00:02:16.390
He fumed that the Mexican
troops had invaded our territory

00:02:16.390 --> 00:02:19.690
and shed American blood on American soil.

00:02:19.690 --> 00:02:20.750
Now back up a minute.

00:02:20.750 --> 00:02:21.840
You may be wondering,

00:02:21.840 --> 00:02:24.630
as many keen observers did at the time,

00:02:24.630 --> 00:02:27.572
what exactly were US forces doing there

00:02:27.572 --> 00:02:30.290
near Rio Grande River in the first place?

00:02:30.290 --> 00:02:33.900
And the answer to that
reveals the two major causes

00:02:33.900 --> 00:02:37.640
of the war, Texas annexation
and Manifest Destiny.

00:02:37.640 --> 00:02:40.960
Let's start by talking
about Texas annexation.

00:02:40.960 --> 00:02:44.420
American settlers, many
of whom were slave owners,

00:02:44.420 --> 00:02:46.950
had been moving to Texas since the 1820s,

00:02:46.950 --> 00:02:49.300
when the region was still
controlled by Spain.

00:02:49.300 --> 00:02:53.780
After Mexican independence,
the country outlawed slavery.

00:02:53.780 --> 00:02:56.250
But the American settlers resisted

00:02:56.250 --> 00:02:58.180
the Mexican government's authority.

00:02:58.180 --> 00:03:02.400
In 1836, they rebelled and
won independence for Texas.

00:03:02.400 --> 00:03:05.410
They requested the United
States annex the new nation

00:03:05.410 --> 00:03:08.870
shortly thereafter, but
adding another slave state

00:03:08.870 --> 00:03:11.770
to the Union was politically dangerous

00:03:11.770 --> 00:03:13.310
for the administration at that time.

00:03:13.310 --> 00:03:17.780
So Texas remained an
independent nation until 1845.

00:03:17.780 --> 00:03:22.780
In 1845, Democratic president
James K. Polk took office.

00:03:23.260 --> 00:03:25.720
Now Polk was an ardent expansionist.

00:03:25.720 --> 00:03:28.070
He was a believer in Manifest Destiny,

00:03:28.070 --> 00:03:30.680
this idea that God
wanted the United States

00:03:30.680 --> 00:03:33.800
to expand across the
North American continent.

00:03:33.800 --> 00:03:35.770
Polk wanted to annex Texas,

00:03:35.770 --> 00:03:38.420
which his administration
undertook immediately.

00:03:38.420 --> 00:03:41.270
He also desperately wanted California,

00:03:41.270 --> 00:03:44.560
which was a hub of commerce
on the Pacific Ocean.

00:03:44.560 --> 00:03:47.110
This is actually before
gold was discovered there.

00:03:47.110 --> 00:03:50.600
So Polk sent a representative
to the Mexican government

00:03:50.600 --> 00:03:52.860
offering to buy California.

00:03:52.860 --> 00:03:55.990
But Mexico said California
was not for sale.

00:03:55.990 --> 00:03:58.910
Now Polk was determined
to get this territory

00:03:58.910 --> 00:04:00.380
with blood or money.

00:04:00.380 --> 00:04:03.120
So he came up with an alternate plan.

00:04:03.120 --> 00:04:07.250
The border between Mexico
and Texas was under dispute.

00:04:07.250 --> 00:04:10.420
So Polk directed General Zachary Taylor

00:04:10.420 --> 00:04:13.100
to go down into this disputed territory

00:04:13.100 --> 00:04:15.060
and provoke hostilities.

00:04:15.060 --> 00:04:16.550
And that's exactly what happened

00:04:16.550 --> 00:04:20.110
when the Mexican cavalry
attacked Taylor's forces.

00:04:20.110 --> 00:04:22.250
As far as Mexico was concerned,

00:04:22.250 --> 00:04:24.920
Taylor's troops were
invading their country,

00:04:24.920 --> 00:04:27.080
and they had no choice but to defend it.

00:04:27.080 --> 00:04:29.830
Despite Polk's war message saying that

00:04:29.830 --> 00:04:32.760
American blood had been
shed on American soil,

00:04:32.760 --> 00:04:36.080
many US politicians were
also skeptical about

00:04:36.080 --> 00:04:38.310
who started the war and where.

00:04:38.310 --> 00:04:42.000
A young Whig congressman from
Illinois named Abraham Lincoln

00:04:42.000 --> 00:04:45.021
demanded that Polk show him the exact spot

00:04:45.021 --> 00:04:47.440
where American blood had been shed.

00:04:47.440 --> 00:04:52.155
The war that ensued was
longer, costlier, and deadlier

00:04:52.155 --> 00:04:54.570
than the US government had estimated,

00:04:54.570 --> 00:04:56.410
which is often the case with wars.

00:04:56.410 --> 00:05:00.380
At its conclusion, Polk
had achieved his vision

00:05:00.380 --> 00:05:01.660
for Manifest Destiny.

00:05:01.660 --> 00:05:05.310
In the Treaty of Guadalupe
Hidalgo, which ended the war,

00:05:05.310 --> 00:05:08.770
the United States agreed
to pay Mexico $15 million,

00:05:08.770 --> 00:05:12.740
and in exchange, Mexico
ceded Texas, California,

00:05:12.740 --> 00:05:16.950
and most of the modern-day
Southwest to the United States.

00:05:16.950 --> 00:05:19.260
So what were the effects of this war?

00:05:19.260 --> 00:05:22.940
Well, the addition of this
Mexican cession territory

00:05:22.940 --> 00:05:25.890
had far-reaching consequences
for both the United States

00:05:25.890 --> 00:05:28.070
and the residents of the West.

00:05:28.070 --> 00:05:30.610
The existing resident of the territory,

00:05:30.610 --> 00:05:33.339
including Mexicans, Native Americans,

00:05:33.339 --> 00:05:36.890
and the descendants of Spanish colonists,

00:05:36.890 --> 00:05:39.800
found that life under the
rule of the United States

00:05:39.800 --> 00:05:43.190
could be very different than
under the rule of Mexico.

00:05:43.190 --> 00:05:46.500
Where Mexican law had abolished slavery

00:05:46.500 --> 00:05:48.870
and proscribed equality under the law

00:05:48.870 --> 00:05:50.800
for people regardless of color,

00:05:50.800 --> 00:05:53.800
the Texas constitution permitted slavery

00:05:53.800 --> 00:05:57.220
and denied civil rights
to non-white residents.

00:05:57.220 --> 00:05:59.330
For other residents of the territory,

00:05:59.330 --> 00:06:00.890
life didn't change much at all.

00:06:00.890 --> 00:06:03.310
Huge swathes of the West
were actually controlled

00:06:03.310 --> 00:06:06.570
by Native American nations,
like the Comanche Empire,

00:06:06.570 --> 00:06:09.900
which didn't care whether
the distant government

00:06:09.900 --> 00:06:12.210
who claimed their territory on paper

00:06:12.210 --> 00:06:15.740
was located in Mexico
City or in Washington DC.

00:06:15.740 --> 00:06:17.670
For the United States government,

00:06:17.670 --> 00:06:21.850
the addition of this new territory
was political kryptonite.

00:06:21.850 --> 00:06:25.120
Both Northerners and
Southerners were convinced

00:06:25.120 --> 00:06:27.742
that the opposite region was conspiring

00:06:27.742 --> 00:06:31.360
to limit their economic
opportunities in the West.

00:06:31.360 --> 00:06:35.060
During the war, Congressman
David Wilmot of Pennsylvania

00:06:35.060 --> 00:06:37.640
introduced a resolution in the House

00:06:37.640 --> 00:06:39.350
that would prohibit slavery

00:06:39.350 --> 00:06:42.290
in any territory gained from the conflict.

00:06:42.290 --> 00:06:44.853
The reaction to the Wilmot Priviso

00:06:44.853 --> 00:06:49.853
showed just how big the
sectional divide in the country

00:06:50.230 --> 00:06:53.570
was becoming, since party
lines broke down entirely.

00:06:53.570 --> 00:06:56.470
Northerners, Whig, and Democrat alike

00:06:56.470 --> 00:06:58.430
voted for the Wilmot Proviso,

00:06:58.430 --> 00:07:02.840
and Southerners, Whig, and
Democrat alike voted against it.

00:07:02.840 --> 00:07:05.670
Ultimately, the proviso
passed in the House

00:07:05.670 --> 00:07:07.770
was defeated in the Senate.

00:07:07.770 --> 00:07:10.560
And then gold was
discovered in California,

00:07:10.560 --> 00:07:12.220
just before the end of the war,

00:07:12.220 --> 00:07:14.740
sending hordes of prospectors West

00:07:14.740 --> 00:07:18.630
and making statehood for
California an urgent issue

00:07:18.630 --> 00:07:20.880
that would soon upset the balance of power

00:07:20.880 --> 00:07:23.700
between free and slave states in Congress.

00:07:23.700 --> 00:07:26.500
In other words, we can draw a direct line

00:07:26.500 --> 00:07:28.120
from the Mexican War

00:07:28.120 --> 00:07:30.760
to the breakdown of the
second party system,

00:07:30.760 --> 00:07:34.520
which was replaced by a solidly
Southern Democratic party

00:07:34.520 --> 00:07:36.950
and a new Northern Republican party,

00:07:36.950 --> 00:07:38.753
and from there to the Civil War.


Labor-leisure tradeoff

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:02.720
- [Instructor] So let's
keep talking about labor

00:00:02.720 --> 00:00:05.520
as a factor of production.

00:00:05.520 --> 00:00:07.242
In particular we're going to think

00:00:07.242 --> 00:00:09.470
about the supply curve of labor.

00:00:09.470 --> 00:00:12.250
So when you're thinking about
the supply or the demand curve

00:00:12.250 --> 00:00:14.080
for elite labor, when you're
thinking about quantity,

00:00:14.080 --> 00:00:17.090
you could just view that as hours worked

00:00:17.090 --> 00:00:18.810
in a certain time period.

00:00:18.810 --> 00:00:20.620
Hours worked.

00:00:20.620 --> 00:00:23.840
And then, for the price
of labor you could just do

00:00:23.840 --> 00:00:25.690
that as wages.

00:00:25.690 --> 00:00:28.490
And we've already thought
about what the demand curve

00:00:28.490 --> 00:00:30.310
for labor would look like.

00:00:30.310 --> 00:00:33.820
At high wages, not a lot
of folks will want to use

00:00:33.820 --> 00:00:35.100
that labor, it's going to be so expensive.

00:00:35.100 --> 00:00:36.600
They might not even be able to afford it,

00:00:36.600 --> 00:00:40.680
and then as wages come down,
more people will generally

00:00:40.680 --> 00:00:42.550
want, will demand that labor,

00:00:42.550 --> 00:00:44.460
and so they will want more hours

00:00:45.603 --> 00:00:49.977
for folks to work, and so this
would be our demand curve.

00:00:51.760 --> 00:00:54.930
Or we could call this
our labor demand curve.

00:00:54.930 --> 00:00:57.200
Now what about the labor supply curve?

00:00:57.200 --> 00:00:59.670
What do you think that's
going to look like?

00:00:59.670 --> 00:01:01.370
Well, not a trick question.

00:01:01.370 --> 00:01:04.960
When wages are low, a lot folks
might say hey, I have other

00:01:04.960 --> 00:01:08.110
things to do with my time,
but then as wages get higher

00:01:08.110 --> 00:01:10.996
and higher they might trade
off those other things.

00:01:10.996 --> 00:01:13.670
Doing those other things
in some ways has a higher

00:01:13.670 --> 00:01:15.910
opportunity cost, it gets more expensive.

00:01:15.910 --> 00:01:19.480
And so they might trade off
those other things for working.

00:01:19.480 --> 00:01:22.130
And so they might collectively
work more and more hours,

00:01:22.130 --> 00:01:25.140
and so as wages go up, generally speaking,

00:01:25.140 --> 00:01:26.340
hours worked goes up.

00:01:26.340 --> 00:01:30.660
So this is a fairly classic looking labor

00:01:30.660 --> 00:01:33.973
labor supply curve.

00:01:34.810 --> 00:01:37.410
And this dynamic, that
I just talked about,

00:01:37.410 --> 00:01:40.970
where people are trying to
trade off whether they work

00:01:40.970 --> 00:01:42.870
or whether they do other things,

00:01:42.870 --> 00:01:47.870
this is typically referred
to as the labor-leisure

00:01:48.350 --> 00:01:50.361
leisure trade off.

00:01:50.361 --> 00:01:53.940
Now, in everyday language,
when you use the word leisure,

00:01:53.940 --> 00:01:56.490
it's usually referred to
you're relaxing or spending

00:01:56.490 --> 00:01:59.560
time with friends or enjoying
yourself in some ways,

00:01:59.560 --> 00:02:01.980
but when people talk about
the labor-leisure trade off

00:02:01.980 --> 00:02:05.120
in economics, they're
really talking about labor

00:02:05.120 --> 00:02:07.090
or anything that is not labor.

00:02:07.090 --> 00:02:10.140
So, leisure would include
sleeping or eating

00:02:10.140 --> 00:02:13.770
or using the restroom, all
of those would be included,

00:02:13.770 --> 00:02:16.660
so it really should be
called the labor, not-labor

00:02:16.660 --> 00:02:19.400
trade off, but I guess
that doesn't sound as good

00:02:19.400 --> 00:02:21.970
as labor-leisure trade off.

00:02:21.970 --> 00:02:25.110
And so what you really see
happening here is this wages

00:02:25.110 --> 00:02:28.340
are higher and higher people
are willing to trade off

00:02:28.340 --> 00:02:31.860
leisure, I'll put that
in quotes for labor.

00:02:31.860 --> 00:02:34.960
Now, the effect that we often
talk about, why that is,

00:02:34.960 --> 00:02:36.820
and in a lot of ways that's common sense,

00:02:36.820 --> 00:02:38.590
that's the substitution effect.

00:02:38.590 --> 00:02:42.523
Substitution effect.

00:02:43.550 --> 00:02:47.720
As wages go higher, you could
view the opportunity cost

00:02:47.720 --> 00:02:51.070
of leisure gets more and more
expensive and if anything

00:02:51.070 --> 00:02:53.960
gets more expensive, you try
to substitute it with other

00:02:53.960 --> 00:02:56.190
things, in this case you
could substitute it with

00:02:56.190 --> 00:02:58.980
more labor, by just working more.

00:02:58.980 --> 00:03:02.090
Now there is an interesting
dynamic that some people

00:03:02.090 --> 00:03:05.160
talk about, which is the income effect.

00:03:05.160 --> 00:03:09.230
Which is the income effect.

00:03:09.230 --> 00:03:13.430
And the income effect is as
your wages go up you tend

00:03:13.430 --> 00:03:18.430
to want to buy or demand
more of everything.

00:03:18.490 --> 00:03:21.630
And you could view leisure
as a good that you,

00:03:21.630 --> 00:03:23.760
as a worker might want.

00:03:23.760 --> 00:03:26.710
So there might be dynamic
that if income gets above

00:03:26.710 --> 00:03:27.890
a certain level,

00:03:27.890 --> 00:03:30.750
that you actually might
not wanna work more.

00:03:30.750 --> 00:03:32.810
That you actually might want more leisure

00:03:32.810 --> 00:03:34.660
because you have more than enough

00:03:34.660 --> 00:03:36.920
to supply all of your needs.

00:03:36.920 --> 00:03:40.450
And so if you wanted to imagine
what a labor supply curve

00:03:40.450 --> 00:03:43.480
would look like if you could
imagine the income effect

00:03:43.480 --> 00:03:45.440
kicking in at higher wages,

00:03:45.440 --> 00:03:47.620
it actually could look
something like this.

00:03:47.620 --> 00:03:50.170
At low wages, it could look
very similar to what we just

00:03:50.170 --> 00:03:53.780
described, but then there
might be some wage where people

00:03:53.780 --> 00:03:57.060
are like you know what, I
have enough money and rather

00:03:57.060 --> 00:03:59.510
than just working that extra
hour I actually might want

00:03:59.510 --> 00:04:02.570
to spend that time with my
family or go on vacation

00:04:02.570 --> 00:04:05.440
and in a lot of ways it's
a very healthy mindset,

00:04:05.440 --> 00:04:07.410
as my personal opinion, I
don't think enough people

00:04:07.410 --> 00:04:11.320
have that mindset, but
if that were the case,

00:04:11.320 --> 00:04:13.260
at some point when wages
get to a certain point

00:04:13.260 --> 00:04:17.370
people actually might want to work less.

00:04:17.370 --> 00:04:18.450
And so you would have this

00:04:18.450 --> 00:04:21.330
backward bending labor supply curve.

00:04:21.330 --> 00:04:22.806
So let me write this.

00:04:22.806 --> 00:04:27.290
This is a labor supply curve supply curve

00:04:27.290 --> 00:04:31.430
with the income effect
after a certain point.

00:04:31.430 --> 00:04:33.880
Income effect.

00:04:33.880 --> 00:04:36.410
So it's an interesting
thing to think about.

00:04:36.410 --> 00:04:39.370
Is there a certain income
level above which people say,

00:04:39.370 --> 00:04:41.520
you know what, I have
enough and rather than work

00:04:41.520 --> 00:04:44.090
harder, I might work a little bit less.

00:04:44.090 --> 00:04:45.440
Interesting to think about.


Dividing by 0.1 and 0.01

WEBVTT
Kind: captions
Language: en

00:00:00.170 --> 00:00:02.700
- [Instructor] Let's say we're
trying to figure out what two

00:00:02.700 --> 00:00:04.840
divided by

00:00:04.840 --> 00:00:06.240
1/10 is.

00:00:06.240 --> 00:00:09.380
So pause this video and see
if you can have a go at that.

00:00:09.380 --> 00:00:11.210
All right, now there's a couple of ways

00:00:11.210 --> 00:00:13.150
that we could approach this.

00:00:13.150 --> 00:00:16.240
We could just try to think of
everything in terms of tenths,

00:00:16.240 --> 00:00:18.440
since we're dividing by 1/10.

00:00:18.440 --> 00:00:22.830
And so instead of two ones,
we could rewrite that two

00:00:22.830 --> 00:00:26.613
as the same thing as 20/10,

00:00:26.613 --> 00:00:28.350
20/10

00:00:28.350 --> 00:00:30.141
divided by,

00:00:30.141 --> 00:00:34.283
divided by, and I could
rewrite this as 1/10.

00:00:35.710 --> 00:00:39.120
And so, if you have 20 of something

00:00:39.120 --> 00:00:41.480
and if you want to divide it into groups

00:00:41.480 --> 00:00:43.040
of one of that something,

00:00:43.040 --> 00:00:45.110
how many groups are you going to have?

00:00:45.110 --> 00:00:47.530
Well, you're going to have 20 groups,

00:00:47.530 --> 00:00:49.830
so this is going to be equal to 20.

00:00:49.830 --> 00:00:50.780
So that's interesting.

00:00:50.780 --> 00:00:55.780
Two divided by 1/10 is equal to twenty.

00:00:56.480 --> 00:00:58.580
Now, what if I were to take, I don't know,

00:00:58.580 --> 00:01:00.120
I'm just gonna make up another number.

00:01:00.120 --> 00:01:03.760
Let's say if I were to
take seven divided by,

00:01:03.760 --> 00:01:06.470
divided by 1/10.

00:01:06.470 --> 00:01:08.293
What is that going to be equal to?

00:01:09.230 --> 00:01:10.850
Well, same idea.

00:01:10.850 --> 00:01:15.270
Seven ones is the same thing as seventy

00:01:15.270 --> 00:01:16.103
tenths.

00:01:16.103 --> 00:01:18.930
And if I divide that by

00:01:18.930 --> 00:01:19.880
one

00:01:19.880 --> 00:01:21.460
tenth,

00:01:21.460 --> 00:01:23.550
well, 70 of something divided into groups

00:01:23.550 --> 00:01:27.740
of one of that something is
going to be 70 equal groups.

00:01:27.740 --> 00:01:30.770
So this is going to be equal to

00:01:30.770 --> 00:01:31.890
70.

00:01:31.890 --> 00:01:35.400
So seven divided by 1/10 is equal to 70.

00:01:35.400 --> 00:01:37.720
Two divided by 1/10 is equal to 20.

00:01:37.720 --> 00:01:39.670
You might notice a pattern.

00:01:39.670 --> 00:01:42.100
Now there's other ways that
we could've approached this.

00:01:42.100 --> 00:01:46.600
We could've said something
like two divided by 1/10

00:01:46.600 --> 00:01:50.300
is the same thing as two over 1/10,

00:01:50.300 --> 00:01:53.090
and I could write 1/10 like that.

00:01:53.090 --> 00:01:55.910
And so when you divide by a fraction,

00:01:55.910 --> 00:01:59.020
that's the same thing as
multiplying by the reciprocal.

00:01:59.020 --> 00:02:01.280
So this is the same thing as

00:02:01.280 --> 00:02:03.410
two times,

00:02:03.410 --> 00:02:04.963
two times 10.

00:02:05.920 --> 00:02:07.800
And then that pattern might
be a little bit clearer,

00:02:07.800 --> 00:02:10.530
which is going to be equal to 20.

00:02:10.530 --> 00:02:12.040
When you divide by a tenth,

00:02:12.040 --> 00:02:14.680
it's the same thing as multiplying by 10.

00:02:14.680 --> 00:02:17.235
Same thing happened over here,
when we divided by a tenth,

00:02:17.235 --> 00:02:21.120
it was the same thing as
multiplying seven by 10

00:02:21.120 --> 00:02:22.750
to get to 70.

00:02:22.750 --> 00:02:24.490
So what do you think is going to happen

00:02:24.490 --> 00:02:27.290
if we divide by a hundredth?

00:02:27.290 --> 00:02:29.130
So let's say I have six.

00:02:29.130 --> 00:02:33.140
And I am going to...let me make
that into a different color.

00:02:33.140 --> 00:02:34.710
So let's say I have six,

00:02:34.710 --> 00:02:37.630
and I want to divide that by

00:02:37.630 --> 00:02:39.600
a hundredth.

00:02:39.600 --> 00:02:42.223
Pause this video, and see
if you can figure that out.

00:02:43.070 --> 00:02:44.770
Well, one way to approach it

00:02:44.770 --> 00:02:48.200
is we can express everything
in terms of hundredths.

00:02:48.200 --> 00:02:50.680
So... I really enjoy saying hundredths.

00:02:50.680 --> 00:02:52.510
Hundredths! (chuckles)

00:02:52.510 --> 00:02:54.770
So six could be rewritten as,

00:02:54.770 --> 00:02:57.560
each whole is equal to
a hundred hundredths,

00:02:57.560 --> 00:02:59.660
so six wholes, or six ones,

00:02:59.660 --> 00:03:02.269
is going to be equal to 600 hundredths.

00:03:02.269 --> 00:03:06.602
So this is going to be
equal to 600 hundred

00:03:06.602 --> 00:03:09.680
hundredths, hundredths (chuckles),

00:03:09.680 --> 00:03:12.120
and then that's going to be divided

00:03:12.120 --> 00:03:15.920
into equal groups of one hundredth.

00:03:15.920 --> 00:03:16.963
Hundredth.

00:03:18.544 --> 00:03:20.820
So if I have 600 of
something and I'm dividing it

00:03:20.820 --> 00:03:22.850
into equal groups of
one of that something,

00:03:22.850 --> 00:03:25.070
how many equal groups am I going to have?

00:03:25.070 --> 00:03:28.570
Well, I'm going to have 600 equal groups.

00:03:28.570 --> 00:03:31.700
Now another way that we
could've approached this

00:03:31.700 --> 00:03:33.560
is we could've approached
it with fractions.

00:03:33.560 --> 00:03:37.220
We could've said hey, this
thing is the same thing

00:03:37.220 --> 00:03:38.680
as six

00:03:38.680 --> 00:03:41.020
over one hundredth,

00:03:41.020 --> 00:03:43.510
which I could express like this,

00:03:43.510 --> 00:03:48.510
which is equal to six times the
reciprocal of one hundredth.

00:03:49.200 --> 00:03:53.460
So that's six times the
reciprocal of one hundredth,

00:03:53.460 --> 00:03:56.250
which is one hundred over one,

00:03:56.250 --> 00:03:59.670
or we could just write that as 100,

00:03:59.670 --> 00:04:03.280
which once again gets us to 600.

00:04:03.280 --> 00:04:05.360
So maybe you're seeing a pattern again.

00:04:05.360 --> 00:04:07.810
When you divide by a hundredth,

00:04:07.810 --> 00:04:11.310
that's equivalent to
multiplying by a hundred.

00:04:11.310 --> 00:04:13.940
When you divide by a tenth,

00:04:13.940 --> 00:04:17.650
that's the equivalent
as multiplying by 10.

00:04:17.650 --> 00:04:18.760
So I'll leave you there.

00:04:18.760 --> 00:04:20.733
I encourage you to think about that some.


Introduction to remainders

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:01.910
- [Instructor] We're
already somewhat familiar

00:00:01.910 --> 00:00:04.240
with the idea of division.

00:00:04.240 --> 00:00:08.510
If I were to say eight divided by two

00:00:08.510 --> 00:00:10.650
you could think of that as eight objects.

00:00:10.650 --> 00:00:12.440
So one, two, three,

00:00:12.440 --> 00:00:14.210
four, five, six,

00:00:14.210 --> 00:00:15.690
seven, eight

00:00:15.690 --> 00:00:19.940
divided into equal groups of two.

00:00:19.940 --> 00:00:22.640
And so how many equal groups
of two could you have?

00:00:22.640 --> 00:00:24.440
Well you could have one,

00:00:24.440 --> 00:00:25.620
two,

00:00:25.620 --> 00:00:26.610
three,

00:00:26.610 --> 00:00:29.030
or four groups of two

00:00:29.030 --> 00:00:31.900
and so you'd say eight divided
by two is equal to four.

00:00:31.900 --> 00:00:33.850
Another way we could've thought about that

00:00:33.850 --> 00:00:36.070
is you have one, two, three, four,

00:00:36.070 --> 00:00:38.119
five, six, seven, eight

00:00:38.119 --> 00:00:41.090
and if you were to divide
it into two equal groups

00:00:41.090 --> 00:00:43.660
well you could have one group of four.

00:00:43.660 --> 00:00:44.870
Let me get a little bit cleaner.

00:00:44.870 --> 00:00:48.560
One group of four and then
a second group of four.

00:00:48.560 --> 00:00:49.870
So two equal groups.

00:00:49.870 --> 00:00:51.650
How many in each of those equal groups?

00:00:51.650 --> 00:00:53.530
Well they're four in each of those groups.

00:00:53.530 --> 00:00:56.880
And so once again eight divided
by two is equal to four.

00:00:56.880 --> 00:01:00.080
Now we're going to extend
our knowledge of division

00:01:00.080 --> 00:01:03.690
by starting to think about
things that don't divide evenly.

00:01:03.690 --> 00:01:05.360
So what if we were to say

00:01:05.360 --> 00:01:06.860
what is

00:01:06.860 --> 00:01:11.250
eight divided by three?

00:01:11.250 --> 00:01:12.210
Pause this video

00:01:12.210 --> 00:01:15.120
and see if you can think
about that a little bit.

00:01:15.120 --> 00:01:18.690
All right so let's draw
eight objects again.

00:01:18.690 --> 00:01:20.990
One, two, three, four,

00:01:20.990 --> 00:01:23.830
five, six, seven, eight

00:01:23.830 --> 00:01:25.630
and one way to think about it

00:01:25.630 --> 00:01:27.090
is how we thought about it here

00:01:27.090 --> 00:01:30.380
is can we divide this into groups

00:01:30.380 --> 00:01:31.780
that all have three in them

00:01:31.780 --> 00:01:34.520
and how many groups would
we be able to make of three?

00:01:34.520 --> 00:01:35.830
Well let's try it out.

00:01:35.830 --> 00:01:38.930
I can make this group of three,

00:01:38.930 --> 00:01:42.320
I can make this second group of three

00:01:42.320 --> 00:01:45.050
but I can't make any more group of three

00:01:45.050 --> 00:01:49.260
and what I have left over are these two.

00:01:49.260 --> 00:01:51.720
And so the way that
you would describe this

00:01:51.720 --> 00:01:53.470
or one way to describe this is

00:01:53.470 --> 00:01:58.070
hey I was able to make two group of three

00:01:58.070 --> 00:02:00.040
so it's equal to two

00:02:00.040 --> 00:02:02.040
and there's some left over,

00:02:02.040 --> 00:02:03.860
there's a remainder.

00:02:03.860 --> 00:02:05.090
Let me write that down.

00:02:05.090 --> 00:02:06.480
Important concept.

00:02:06.480 --> 00:02:10.420
There is a remainder of two as well

00:02:10.420 --> 00:02:13.150
and so sometimes it's
written as just a lowercase r

00:02:13.150 --> 00:02:15.960
or remainder of two.

00:02:15.960 --> 00:02:17.690
Another way to think about it is

00:02:17.690 --> 00:02:20.690
two, this two times three is six

00:02:20.690 --> 00:02:22.620
and then if you were to
put back that remainder

00:02:22.620 --> 00:02:24.520
that's how you can get to eight.

00:02:24.520 --> 00:02:25.880
Now another way you can think about it

00:02:25.880 --> 00:02:27.520
is how we thought about
it in the second example

00:02:27.520 --> 00:02:29.100
with eight divided by two.

00:02:29.100 --> 00:02:30.490
Let me draw eight objects again.

00:02:30.490 --> 00:02:32.560
One, two, three, four,

00:02:32.560 --> 00:02:34.590
five, six, seven, eight

00:02:34.590 --> 00:02:35.770
and you could say hey

00:02:35.770 --> 00:02:39.780
let me divide that eight
into three equal groups.

00:02:39.780 --> 00:02:42.050
So pause this video and
see if you can divide this

00:02:42.050 --> 00:02:43.670
into three equal groups

00:02:43.670 --> 00:02:45.863
and then what might be left over.

00:02:47.070 --> 00:02:49.430
All right, so I'm going to try

00:02:49.430 --> 00:02:52.490
to divide this into three equal groups.

00:02:52.490 --> 00:02:55.550
I'm not going to be able to put
four in each of those groups

00:02:55.550 --> 00:02:58.270
'cause I can only make
two equal groups of four.

00:02:58.270 --> 00:03:01.070
I'm not going to be able
to put three into those,

00:03:01.070 --> 00:03:02.040
three equal groups

00:03:02.040 --> 00:03:04.910
'cause I would actually be
outing nine for doing that

00:03:04.910 --> 00:03:07.230
so each of my groups are
going to have to be two.

00:03:07.230 --> 00:03:09.960
So I could make one group of two,

00:03:09.960 --> 00:03:11.680
another group of two,

00:03:11.680 --> 00:03:13.870
and there you go three
equal groups of two.

00:03:13.870 --> 00:03:16.570
So I was able to sort out
three equal groups of two

00:03:16.570 --> 00:03:18.000
with just these six

00:03:18.000 --> 00:03:20.600
but once again I have a remainder.

00:03:20.600 --> 00:03:23.080
I'm not able to make use of these two,

00:03:23.080 --> 00:03:25.510
they're not able to fit into one of

00:03:25.510 --> 00:03:28.500
in this case one of
the three equal groups.

00:03:28.500 --> 00:03:30.720
If I said four equal groups
then they would fit in,

00:03:30.720 --> 00:03:32.190
but if I just said three equal groups

00:03:32.190 --> 00:03:33.510
'cause I'm dividing by three

00:03:33.510 --> 00:03:36.350
then I have this left over again.

00:03:36.350 --> 00:03:38.090
Let's do one more example.

00:03:38.090 --> 00:03:42.470
What if I were to ask you what is,

00:03:42.470 --> 00:03:43.930
what is

00:03:43.930 --> 00:03:45.560
13

00:03:45.560 --> 00:03:47.790
divided by

00:03:47.790 --> 00:03:49.790
divided by

00:03:50.650 --> 00:03:51.860
four?

00:03:51.860 --> 00:03:53.370
Pause this video and think about it

00:03:53.370 --> 00:03:55.090
and as you might imagine

00:03:55.090 --> 00:03:56.983
there will be a remainder involved.

00:03:58.270 --> 00:04:01.830
All right, well let's draw 13 objects,

00:04:01.830 --> 00:04:05.860
one, two, three, four,
five, six, seven, eight

00:04:05.860 --> 00:04:09.260
nine, ten, eleven, twelve, thirteen

00:04:09.260 --> 00:04:14.050
and we could try to divide
this into equal groups of four,

00:04:14.050 --> 00:04:16.250
that's one way to think
about it so let's see.

00:04:16.250 --> 00:04:19.780
That's a group of four,
I have one group of four.

00:04:19.780 --> 00:04:22.700
That's a group of four,
I have two groups of four

00:04:22.700 --> 00:04:24.700
and then that is a group of four

00:04:24.700 --> 00:04:28.650
so I'm able to find three
equal groups of four

00:04:28.650 --> 00:04:30.010
so this is equal to three.

00:04:30.010 --> 00:04:30.843
Another way to think about it

00:04:30.843 --> 00:04:33.760
four goes into 13 three times

00:04:33.760 --> 00:04:36.530
but then I have this
little lonely circle here,

00:04:36.530 --> 00:04:37.770
I have one left over

00:04:37.770 --> 00:04:40.560
I have a remainder of one

00:04:40.560 --> 00:04:43.710
because four times three
that gets you to 12

00:04:43.710 --> 00:04:44.970
but then if you wanna get to 13

00:04:44.970 --> 00:04:47.663
well then you gotta throw
in that remainder there.