Worked example finding area under density curves

WEBVTT
Kind: captions
Language: en

00:00:00.247 --> 00:00:02.887
- [Instructor] Consider
the density curve below

00:00:02.887 --> 00:00:05.080
and this density curve doesn't look like

00:00:05.080 --> 00:00:06.547
the ones we typically see

00:00:06.547 --> 00:00:08.133
that are a little bit curvier,

00:00:08.133 --> 00:00:09.549
but this is a little easier for us

00:00:09.549 --> 00:00:13.033
to work with and figure out areas.

00:00:13.033 --> 00:00:15.754
So they ask us to find the percent

00:00:15.754 --> 00:00:18.411
of the area under the density curve

00:00:18.411 --> 00:00:21.559
where x is more than two.

00:00:21.559 --> 00:00:25.726
So what area represents
when x is more than two?

00:00:27.005 --> 00:00:29.353
This is when x is equal to two.

00:00:29.353 --> 00:00:32.763
So they're talking about
this area right over here.

00:00:32.763 --> 00:00:34.626
And so we need to figure out the percent

00:00:34.626 --> 00:00:36.979
of the total area under the curve

00:00:36.979 --> 00:00:41.031
that this blue area actually represents.

00:00:41.031 --> 00:00:45.170
So first let's find the total
area under the density curve.

00:00:45.170 --> 00:00:47.325
The density only has area,

00:00:47.325 --> 00:00:48.993
the density curve only has area

00:00:48.993 --> 00:00:51.265
from x equals one to x equals three.

00:00:51.265 --> 00:00:54.148
So it does amount to finding the area

00:00:54.148 --> 00:00:56.101
of this larger trapezoid.

00:00:56.101 --> 00:00:59.835
Let me highlight this trapezoid in red.

00:00:59.835 --> 00:01:02.335
So we want to find the area of

00:01:03.810 --> 00:01:06.393
this trapezoid right over here.

00:01:08.118 --> 00:01:11.486
And then that should be equal to one

00:01:11.486 --> 00:01:13.231
because all density curves have

00:01:13.231 --> 00:01:16.392
an area of one under the total curve.

00:01:16.392 --> 00:01:18.719
So let's first verify that.

00:01:18.719 --> 00:01:21.070
There's a couple of
ways to think about it.

00:01:21.070 --> 00:01:23.485
We could split it up into two shapes

00:01:23.485 --> 00:01:25.026
or you could just use the formula

00:01:25.026 --> 00:01:27.870
for an area of a trapezoid.

00:01:27.870 --> 00:01:31.040
In fact let's use the formula
for an area of a trapezoid.

00:01:31.040 --> 00:01:33.637
The formula for an area of a trapezoid

00:01:33.637 --> 00:01:36.033
is you would take the average of,

00:01:36.033 --> 00:01:38.793
you would take the average of this length.

00:01:38.793 --> 00:01:40.971
Let me do that in another color.

00:01:40.971 --> 00:01:43.981
This length and this length

00:01:43.981 --> 00:01:47.843
and then multiply that times the base.

00:01:47.843 --> 00:01:50.159
So the average of this
length and this length,

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let's see this is 0.25.

00:01:52.550 --> 00:01:53.383
0.25

00:01:54.760 --> 00:01:56.502
plus this height,

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0.75,

00:01:59.286 --> 00:02:00.870
divided by two.

00:02:00.870 --> 00:02:03.994
So that's the average of those two sides

00:02:03.994 --> 00:02:05.244
times the base,

00:02:06.513 --> 00:02:10.166
times this right over here which is two.

00:02:10.166 --> 00:02:12.999
And so this is going to
give us as it should have,

00:02:12.999 --> 00:02:16.740
0.25 plus 0.75 which is equal to one.

00:02:16.740 --> 00:02:19.586
So the area under the
entire density curve is one

00:02:19.586 --> 00:02:24.049
which we need to be true for
this to be a density curve.

00:02:24.049 --> 00:02:26.766
Now let's think about what
percentage of that area

00:02:26.766 --> 00:02:29.524
is represented in blue right over here.

00:02:29.524 --> 00:02:31.473
Well we could do the same thing.

00:02:31.473 --> 00:02:33.666
We could say all right,
this is a trapezoid.

00:02:33.666 --> 00:02:37.809
We want to take the average of this side

00:02:37.809 --> 00:02:42.109
and this side and multiply
it times the base.

00:02:42.109 --> 00:02:44.192
So this side is 0.5 high.

00:02:45.492 --> 00:02:46.325
0.5

00:02:47.242 --> 00:02:48.075
plus 0.75,

00:02:49.949 --> 00:02:50.782
0.75 high,

00:02:53.479 --> 00:02:55.511
and we're going to take
the average of that

00:02:55.511 --> 00:02:58.640
divided by two times the base.

00:02:58.640 --> 00:03:00.708
Well the base going from two to three

00:03:00.708 --> 00:03:02.696
is only equal to,

00:03:02.696 --> 00:03:03.981
is equal to one.

00:03:03.981 --> 00:03:05.584
So times one.

00:03:05.584 --> 00:03:09.232
And so this is going to give us 1.25,

00:03:09.232 --> 00:03:10.065
1.25

00:03:11.692 --> 00:03:12.525
over two.

00:03:13.646 --> 00:03:16.485
And what is that going to be equal to?

00:03:16.485 --> 00:03:20.652
Well that would be the same
thing as zero point what?

00:03:21.934 --> 00:03:23.267
Let's see 0.625.

00:03:26.221 --> 00:03:27.349
Did I do that right, Yep.

00:03:27.349 --> 00:03:30.543
If I multiply two times
this, I would get 1.25.

00:03:30.543 --> 00:03:34.000
So the percent of the area
under the density curve

00:03:34.000 --> 00:03:36.207
where x is more than two,

00:03:36.207 --> 00:03:39.041
this is the decimal expression of it.

00:03:39.041 --> 00:03:40.748
If we wanted to write it as a percent,

00:03:40.748 --> 00:03:42.581
it would Be 62.5%

00:03:45.418 --> 00:03:47.501
Let's do another example.

00:03:49.930 --> 00:03:51.839
Consider the density curve below.

00:03:51.839 --> 00:03:53.589
All right we have another one of these

00:03:53.589 --> 00:03:55.825
somewhat angular density curves.

00:03:55.825 --> 00:03:58.749
Find the percent of the
area under the density curve

00:03:58.749 --> 00:04:01.832
where x is more than three.

00:04:01.832 --> 00:04:03.419
So we're talking about, let's see,

00:04:03.419 --> 00:04:04.599
this is where x is equal to three,

00:04:04.599 --> 00:04:06.143
x is more than three,

00:04:06.143 --> 00:04:10.275
we're talking about this
entire area right over here.

00:04:10.275 --> 00:04:13.041
Well this is actually
somewhat straightforward

00:04:13.041 --> 00:04:14.992
because if we're saying the area

00:04:14.992 --> 00:04:16.084
where x is more than three,

00:04:16.084 --> 00:04:19.089
that's the entire area
under the density curve.

00:04:19.089 --> 00:04:21.807
And the entire area
under any density curve

00:04:21.807 --> 00:04:23.513
needs to be equal to one.

00:04:23.513 --> 00:04:24.599
Or you could say find the percent

00:04:24.599 --> 00:04:25.655
of the area under the density curve.

00:04:25.655 --> 00:04:28.049
Well the whole density curve

00:04:28.049 --> 00:04:30.532
is where x is more than three.

00:04:30.532 --> 00:04:34.672
So 100%, we don't even have
to go through the trouble

00:04:34.672 --> 00:04:38.172
of trying to directly calculate the area.


Density curve worked example

WEBVTT
Kind: captions
Language: en

00:00:00.430 --> 00:00:02.926
- [Instructor] Consider
the density curve below.

00:00:02.926 --> 00:00:04.638
It's depicted right over here.

00:00:04.638 --> 00:00:05.863
It's a little unusual looking.

00:00:05.863 --> 00:00:07.047
It's more like a triangle

00:00:07.047 --> 00:00:08.679
than our standard density curves,

00:00:08.679 --> 00:00:09.992
but it's valid.

00:00:09.992 --> 00:00:12.112
Which of the following
statements are true?

00:00:12.112 --> 00:00:14.978
Choose all answers that apply.

00:00:14.978 --> 00:00:16.938
The mean of the density curve

00:00:16.938 --> 00:00:18.938
is less than the median.

00:00:19.924 --> 00:00:20.757
Pause this video

00:00:20.757 --> 00:00:21.590
and see if you can figure out

00:00:21.590 --> 00:00:23.803
whether that's true.

00:00:23.803 --> 00:00:25.270
Well, we don't know exactly

00:00:25.270 --> 00:00:26.657
where the mean and median are

00:00:26.657 --> 00:00:28.455
just by looking at this.

00:00:28.455 --> 00:00:29.314
But remember,

00:00:29.314 --> 00:00:30.373
the median is going to be

00:00:30.373 --> 00:00:31.763
the value for which the area

00:00:31.763 --> 00:00:32.946
to the right and the left

00:00:32.946 --> 00:00:34.577
are going to be equal.

00:00:34.577 --> 00:00:35.884
So, I would guess

00:00:35.884 --> 00:00:37.714
the median is going to be

00:00:37.714 --> 00:00:39.381
someplace like that.

00:00:40.973 --> 00:00:42.361
So that's my guess.

00:00:42.361 --> 00:00:44.158
My approximation.

00:00:44.158 --> 00:00:45.741
That is the median.

00:00:46.811 --> 00:00:48.602
And because our distribution

00:00:48.602 --> 00:00:50.967
goes off further to the left

00:00:50.967 --> 00:00:52.596
than it does to the right,

00:00:52.596 --> 00:00:53.429
you can view this

00:00:53.429 --> 00:00:54.924
as something of a tail.

00:00:54.924 --> 00:00:56.228
It's reasonable to say

00:00:56.228 --> 00:00:58.311
that this is left skewed.

00:00:59.570 --> 00:01:00.570
Left skewed.

00:01:03.651 --> 00:01:05.328
And generally speaking,

00:01:05.328 --> 00:01:08.263
if a distribution is left skewed

00:01:08.263 --> 00:01:10.300
the mean is to the left

00:01:10.300 --> 00:01:12.259
of the median.

00:01:12.259 --> 00:01:14.592
So, because it is left skewed

00:01:14.592 --> 00:01:16.387
the mean might be someplace

00:01:16.387 --> 00:01:18.225
like right over there.

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Another way,

00:01:19.058 --> 00:01:20.343
to even think about the mean

00:01:20.343 --> 00:01:21.653
is that the mean would be

00:01:21.653 --> 00:01:23.493
the balance point,

00:01:23.493 --> 00:01:24.759
where you'd place a fulcrum

00:01:24.759 --> 00:01:26.342
if this were a mass

00:01:27.249 --> 00:01:28.082
and you might say,

00:01:28.082 --> 00:01:29.706
why doesn't that happen at the median?

00:01:29.706 --> 00:01:30.539
Well remember,

00:01:30.539 --> 00:01:32.121
even when you're balancing something

00:01:32.121 --> 00:01:33.349
a smaller weight

00:01:33.349 --> 00:01:35.432
that is far away from the fulcrum

00:01:35.432 --> 00:01:37.792
can balance out a heavier weight

00:01:37.792 --> 00:01:40.412
that is closer in to the fulcrum.

00:01:40.412 --> 00:01:41.734
So, in terms of this first one

00:01:41.734 --> 00:01:43.042
the mean of the density curve

00:01:43.042 --> 00:01:44.614
is less than the median.

00:01:44.614 --> 00:01:45.447
In this case,

00:01:45.447 --> 00:01:46.280
or you could say,

00:01:46.280 --> 00:01:47.300
to the left of the median.

00:01:47.300 --> 00:01:50.568
We can consider this to be true.

00:01:50.568 --> 00:01:51.401
Now what about,

00:01:51.401 --> 00:01:54.723
the median of the density curve is three?

00:01:54.723 --> 00:01:55.743
Well, I already approximated

00:01:55.743 --> 00:01:57.131
where the median might be,

00:01:57.131 --> 00:01:57.964
saying hey

00:01:57.964 --> 00:01:59.043
this area looks roughly comparable

00:01:59.043 --> 00:02:00.269
to this area.

00:02:00.269 --> 00:02:02.027
The median definitely,

00:02:02.027 --> 00:02:03.046
I might not be right there,

00:02:03.046 --> 00:02:04.187
but the median is definitely

00:02:04.187 --> 00:02:05.985
not going to be three.

00:02:05.985 --> 00:02:07.214
This area right over here

00:02:07.214 --> 00:02:08.560
is for sure

00:02:08.560 --> 00:02:10.848
smaller than this area

00:02:10.848 --> 00:02:11.792
right over here.

00:02:11.792 --> 00:02:14.562
So we can rule that out.

00:02:14.562 --> 00:02:18.037
The area underneath the
density curve is one.

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Pause this video.

00:02:18.870 --> 00:02:20.777
Is that true?

00:02:20.777 --> 00:02:21.610
Yes.

00:02:21.610 --> 00:02:22.443
This is true.

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The area underneath any density curve

00:02:25.106 --> 00:02:26.171
is going to be one.

00:02:26.171 --> 00:02:27.234
If we look at the total area

00:02:27.234 --> 00:02:28.176
under the curve,

00:02:28.176 --> 00:02:30.509
it's always going to be one.

00:02:31.451 --> 00:02:32.683
So we answered this question.

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I'll leave you with one extra question

00:02:34.442 --> 00:02:35.911
that we can actually figure out

00:02:35.911 --> 00:02:38.284
from the information they've given us.

00:02:38.284 --> 00:02:39.224
What is the height

00:02:39.224 --> 00:02:40.774
of this point?

00:02:40.774 --> 00:02:42.455
Of this density curve right over here?

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What is this value?

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What is this height going to be?

00:02:46.304 --> 00:02:48.030
See if you can pause this video

00:02:48.030 --> 00:02:49.014
and figure it out

00:02:49.014 --> 00:02:50.563
and I'll give you a hint.

00:02:50.563 --> 00:02:51.396
The hint is

00:02:51.396 --> 00:02:52.648
this third statement.

00:02:52.648 --> 00:02:56.725
The area under the density curve is one.

00:02:56.725 --> 00:02:57.558
Alright,

00:02:57.558 --> 00:02:59.130
now let's try to work through it together.

00:02:59.130 --> 00:03:01.213
If we call this height h,

00:03:02.977 --> 00:03:05.111
we know how to find
the area of a triangle.

00:03:05.111 --> 00:03:07.132
It's 1/2 base times height.

00:03:07.132 --> 00:03:10.792
Area is equal to 1/2 base times height.

00:03:10.792 --> 00:03:12.781
We know that the area is one.

00:03:12.781 --> 00:03:14.217
This is a density curve

00:03:14.217 --> 00:03:15.456
so one is going to be equal

00:03:15.456 --> 00:03:16.847
to what's the length of the base?

00:03:16.847 --> 00:03:19.301
We go from one to six.

00:03:19.301 --> 00:03:20.134
So from one to six

00:03:20.134 --> 00:03:21.181
this base,

00:03:21.181 --> 00:03:22.780
the length of this base

00:03:22.780 --> 00:03:24.304
is five.

00:03:24.304 --> 00:03:27.418
1/2 times five times height.

00:03:27.418 --> 00:03:30.716
Or we could say one is equal to 5/2

00:03:30.716 --> 00:03:31.939
times height

00:03:31.939 --> 00:03:35.442
or multiply both sides by 2/5

00:03:35.442 --> 00:03:38.192
to solve for the height

00:03:38.192 --> 00:03:40.025
and what are gonna get?

00:03:40.025 --> 00:03:40.858
We're gonna get the height

00:03:40.858 --> 00:03:43.173
is equal to 2/5.

00:03:43.173 --> 00:03:44.006
So if you have

00:03:44.006 --> 00:03:46.073
a very clean triangular
density curve like this

00:03:46.073 --> 00:03:48.892
you can actually figure out the height

00:03:48.892 --> 00:03:51.475
with even if it was
not directly specified.


Median, mean and skew from density curves

WEBVTT
Kind: captions
Language: en

00:00:00.338 --> 00:00:02.515
- [Instructor] In other
videos we introduce ourselves

00:00:02.515 --> 00:00:04.659
to the idea of a density curve,

00:00:04.659 --> 00:00:06.547
which is a summary of a distribution,

00:00:06.547 --> 00:00:08.601
a distribution of data, and in the future

00:00:08.601 --> 00:00:11.428
we'll also look at things
like probability density.

00:00:11.428 --> 00:00:13.881
But what I want to talk
about in this video is

00:00:13.881 --> 00:00:16.169
think about what we can glean from them,

00:00:16.169 --> 00:00:19.532
the properties, how we can
describe density curves

00:00:19.532 --> 00:00:22.964
and the distributions they represent.

00:00:22.964 --> 00:00:25.051
And we have four of them right over here,

00:00:25.051 --> 00:00:27.113
and the first thing I
want to think about is

00:00:27.113 --> 00:00:30.883
if we can approximate what value would be

00:00:30.883 --> 00:00:33.666
the middle value or the
median for the data set

00:00:33.666 --> 00:00:36.499
described by these density curves.

00:00:37.477 --> 00:00:41.306
So just to remind ourselves,
if we have a set of numbers

00:00:41.306 --> 00:00:43.723
and we order them from least to greatest,

00:00:43.723 --> 00:00:46.146
the median would be the middle value,

00:00:46.146 --> 00:00:49.602
or the midway between
the middle two values.

00:00:49.602 --> 00:00:52.203
In a case like this, we
want to find the value

00:00:52.203 --> 00:00:55.898
for which half of the
values are above that value

00:00:55.898 --> 00:00:57.708
and half of the values are below.

00:00:57.708 --> 00:00:59.436
So in looking at a density curve,

00:00:59.436 --> 00:01:01.380
you'd want to look at the area,

00:01:01.380 --> 00:01:03.699
and you'd want to say, OK, at what value

00:01:03.699 --> 00:01:08.243
do we have equal area
above and below that value?

00:01:08.243 --> 00:01:10.618
And so for this one, just eyeballing it,

00:01:10.618 --> 00:01:14.535
this value right over
here would be the median.

00:01:15.794 --> 00:01:17.836
And in general, if you have a symmetric

00:01:17.836 --> 00:01:20.810
distribution like this, the median will be

00:01:20.810 --> 00:01:24.434
right along that line of symmetry.

00:01:24.434 --> 00:01:27.243
Here we have a slightly
more unusual distribution,

00:01:27.243 --> 00:01:29.729
this would be called
a bimodal distribution

00:01:29.729 --> 00:01:31.672
where you have two major lumps right

00:01:31.672 --> 00:01:33.891
over here, but it is symmetric.

00:01:33.891 --> 00:01:38.435
And that point of symmetry
is right over here,

00:01:38.435 --> 00:01:42.483
and so this value, once
again, would be the median.

00:01:42.483 --> 00:01:43.714
Another way to think about it is,

00:01:43.714 --> 00:01:47.121
the area to the left of that value

00:01:47.121 --> 00:01:48.754
is equal to the area to the right

00:01:48.754 --> 00:01:52.225
of that value, making it the median.

00:01:52.225 --> 00:01:55.458
But what if we're dealing with
non-symmetric distributions?

00:01:55.458 --> 00:01:57.449
Well we'd want to do the same principle.

00:01:57.449 --> 00:01:59.714
We would want to think,
at what value is the area

00:01:59.714 --> 00:02:03.201
on the right and the
area on the left equal?

00:02:03.201 --> 00:02:05.279
And once again this isn't
going to be super exact,

00:02:05.279 --> 00:02:07.746
but I'm going to try to approximate it.

00:02:07.746 --> 00:02:09.130
You might be tempted
to go right at the top

00:02:09.130 --> 00:02:11.811
of this lump right over here,
but if I were to do that

00:02:11.811 --> 00:02:14.026
it's pretty clear, even eyeballing it,

00:02:14.026 --> 00:02:16.435
that the right area right over here

00:02:16.435 --> 00:02:18.570
is larger than the left area.

00:02:18.570 --> 00:02:21.682
So that would not be the median.

00:02:21.682 --> 00:02:24.611
If I move the median a
little bit over to the right,

00:02:24.611 --> 00:02:28.778
this maybe right around here,
this looks a lot closer.

00:02:30.321 --> 00:02:32.115
Once again, I'm approximating it,

00:02:32.115 --> 00:02:34.626
but it's reasonable to
say that the area here

00:02:34.626 --> 00:02:38.137
looks pretty close to the
area right over there.

00:02:38.137 --> 00:02:39.401
And if that is the case,

00:02:39.401 --> 00:02:42.401
then this is going to be the median.

00:02:44.562 --> 00:02:46.649
Similarly, on this one right over here,

00:02:46.649 --> 00:02:49.732
maybe right over here, and once again

00:02:53.507 --> 00:02:56.492
I'm just approximating it,
but that seems reasonable,

00:02:56.492 --> 00:02:57.919
that this area is equal to that one,

00:02:57.919 --> 00:03:00.069
even though this is
longer it's much lower,

00:03:00.069 --> 00:03:01.698
this part of the curve is much higher

00:03:01.698 --> 00:03:05.018
even though it goes on less to the right.

00:03:05.018 --> 00:03:07.371
So that's the median for well behaved

00:03:07.371 --> 00:03:09.986
continuous distributions like this,

00:03:09.986 --> 00:03:12.509
it's going to be the value
for which the area to the left

00:03:12.509 --> 00:03:15.433
and the area to the right are equal.

00:03:15.433 --> 00:03:17.765
But what about the mean?

00:03:17.765 --> 00:03:21.356
Well the mean is, you take
each of the possible values

00:03:21.356 --> 00:03:25.102
and you weight it by their frequencies,

00:03:25.102 --> 00:03:26.372
you weight it by their frequencies

00:03:26.372 --> 00:03:28.862
and you add all of that up.

00:03:28.862 --> 00:03:32.617
And so for symmetric distributions
your mean and your median

00:03:32.617 --> 00:03:35.418
are actually going to be the same.

00:03:35.418 --> 00:03:37.441
So this is going to be your mean as well,

00:03:37.441 --> 00:03:40.034
this is going to be your mean as well.

00:03:40.034 --> 00:03:42.052
If you want to think about
it in terms of physics,

00:03:42.052 --> 00:03:44.115
the mean would be your balancing point,

00:03:44.115 --> 00:03:45.650
the point at which you would want to put

00:03:45.650 --> 00:03:47.067
a little fulcrum and you would

00:03:47.067 --> 00:03:49.159
want to balance the distribution.

00:03:49.159 --> 00:03:50.503
And so you could put a little fulcrum here

00:03:50.503 --> 00:03:51.866
and you could imagine that this thing

00:03:51.866 --> 00:03:55.882
would balance, this thing would balance.

00:03:55.882 --> 00:03:57.682
And that all comes out of this idea of the

00:03:57.682 --> 00:04:02.507
weighted average of all
of these possible values.

00:04:02.507 --> 00:04:05.602
What about for these less
symmetric distributions?

00:04:05.602 --> 00:04:07.732
Well let's think about it over here.

00:04:07.732 --> 00:04:09.372
Where would I have to put
the fulcrum, or what does

00:04:09.372 --> 00:04:13.529
our intuition say if we
wanted to balance this?

00:04:13.529 --> 00:04:16.834
Well, we have equal areas on either side,

00:04:16.834 --> 00:04:19.201
but when you have this long tail

00:04:19.201 --> 00:04:22.258
to the right it's going to pull the mean

00:04:22.258 --> 00:04:25.373
to the right of the median in this case.

00:04:25.373 --> 00:04:27.874
And so our balance point is probably

00:04:27.874 --> 00:04:30.434
going to be something closer to that.

00:04:30.434 --> 00:04:32.675
And once again, this
is me approximating it,

00:04:32.675 --> 00:04:35.592
but this would roughly be our mean.

00:04:36.770 --> 00:04:39.794
It would sit, in this case,
to the right of our median.

00:04:39.794 --> 00:04:42.410
Let me make it clear, this
median is referring to that,

00:04:42.410 --> 00:04:44.691
the mean is referring to this.

00:04:44.691 --> 00:04:47.099
In this case, because
I have this long tail

00:04:47.099 --> 00:04:49.155
to the left, it's likely
that I would have to

00:04:49.155 --> 00:04:52.244
balance it out right over here.

00:04:52.244 --> 00:04:56.411
So the mean would be this
value, right over there.

00:04:57.785 --> 00:04:59.418
And there's actually a term for these

00:04:59.418 --> 00:05:02.549
non-symmetric distributions where

00:05:02.549 --> 00:05:05.891
the mean is varying from the median.

00:05:05.891 --> 00:05:10.058
Distributions like this are
referred to as being skewed.

00:05:13.038 --> 00:05:15.753
And this distribution,
where you have the mean

00:05:15.753 --> 00:05:17.994
to the right of the median, where you have

00:05:17.994 --> 00:05:22.161
this long tail to the right,
this is called right skewed.

00:05:25.379 --> 00:05:28.202
Now, the technical idea
of skewness can get

00:05:28.202 --> 00:05:31.106
quite complicated, but generally speaking,

00:05:31.106 --> 00:05:33.007
you can spot it out when you have

00:05:33.007 --> 00:05:35.507
a long tail on one direction,

00:05:37.180 --> 00:05:39.603
that's the direction in
which it will be skewed,

00:05:39.603 --> 00:05:42.612
or if the mean is to that
direction of the median.

00:05:42.612 --> 00:05:44.046
So the mean is to the right of the median,

00:05:44.046 --> 00:05:45.796
so generally speaking, that's going

00:05:45.796 --> 00:05:48.629
to be a right skewed distribution.

00:05:49.469 --> 00:05:51.413
So the opposite of that,
here the mean is to the left

00:05:51.413 --> 00:05:53.911
of the median and we have this long tail

00:05:53.911 --> 00:05:57.465
on the left of our distribution,
so generally speaking

00:05:57.465 --> 00:06:01.632
we will describe these as
left skewed distributions.


Density Curves

WEBVTT
Kind: captions
Language: en

00:00:00.006 --> 00:00:02.290
- [Instructor] What we're
going to do in this video

00:00:02.290 --> 00:00:05.869
is think about how to visualize
distributions of data,

00:00:05.869 --> 00:00:08.368
and then to analyze those visualizations,

00:00:08.368 --> 00:00:10.143
and we will eventually get to something

00:00:10.143 --> 00:00:12.642
known as a density curve.

00:00:12.642 --> 00:00:14.609
But let's start with a simple example,

00:00:14.609 --> 00:00:16.882
just to review some concepts.

00:00:16.882 --> 00:00:18.952
Let's say I go to 16 students

00:00:18.952 --> 00:00:21.457
and I ask them to measure
how many glasses of water

00:00:21.457 --> 00:00:24.615
they drink per day for the last 30 days,

00:00:24.615 --> 00:00:26.181
and then to average it.

00:00:26.181 --> 00:00:28.079
And so this data point right over here

00:00:28.079 --> 00:00:30.173
tells us one student drank an average

00:00:30.173 --> 00:00:33.251
of 0.5 glasses of water per day.

00:00:33.251 --> 00:00:35.884
That person is probably very dehydrated.

00:00:35.884 --> 00:00:38.883
This person drank 8.1
glasses of water per day,

00:00:38.883 --> 00:00:42.607
on average, for the last 30
days, they are better hydrated.

00:00:42.607 --> 00:00:44.770
If we want to visualize that

00:00:44.770 --> 00:00:48.202
we can set up a frequency histogram,

00:00:48.202 --> 00:00:51.329
where we can create some categories.

00:00:51.329 --> 00:00:54.365
So this first category
would be for data points

00:00:54.365 --> 00:00:58.455
that are greater than or equal
to zero, and less than one,

00:00:58.455 --> 00:01:00.206
and we can see that two data points

00:01:00.206 --> 00:01:01.439
fall into that category,

00:01:01.439 --> 00:01:03.393
and that's why the bar right over here

00:01:03.393 --> 00:01:05.558
for that category is up to two.

00:01:05.558 --> 00:01:08.316
This category right over
here is greater than

00:01:08.316 --> 00:01:11.599
or equal to three, and less than four.

00:01:11.599 --> 00:01:16.360
Notice, there are four data
points in that category

00:01:16.360 --> 00:01:18.682
and on this frequency histogram

00:01:18.682 --> 00:01:22.340
the height of the bar is indeed four.

00:01:22.340 --> 00:01:25.514
So this is a nice way of
looking at a distribution.

00:01:25.514 --> 00:01:27.191
But you might be more concerned

00:01:27.191 --> 00:01:30.066
with what percentage of my data

00:01:30.066 --> 00:01:32.358
falls into each of these categories,

00:01:32.358 --> 00:01:33.834
and that becomes especially interesting

00:01:33.834 --> 00:01:35.968
if we have many, many, many data points,

00:01:35.968 --> 00:01:40.135
and if we had, you know,
1,600,432,507 data points,

00:01:43.519 --> 00:01:45.451
well just knowing the
absolute number that fit

00:01:45.451 --> 00:01:47.904
into each category isn't so useful,

00:01:47.904 --> 00:01:50.046
the percent that fits into each category

00:01:50.046 --> 00:01:51.793
is a lot more useful.

00:01:51.793 --> 00:01:53.985
And so for that, we could set up

00:01:53.985 --> 00:01:57.165
a relative frequency histogram.

00:01:57.165 --> 00:01:59.996
So notice, this is
representing the same data.

00:01:59.996 --> 00:02:01.554
But in that first category,

00:02:01.554 --> 00:02:04.205
instead of the bar height being two,

00:02:04.205 --> 00:02:07.399
the bar height is now 12.5%.

00:02:07.399 --> 00:02:08.630
Why is that?

00:02:08.630 --> 00:02:12.797
Because two of the 16 data
points fall into this category.

00:02:15.329 --> 00:02:20.228
2/16 is 1/8, which is 12.5%.

00:02:20.228 --> 00:02:22.678
And this one right over here, notice,

00:02:22.678 --> 00:02:24.273
instead of the height being four

00:02:24.273 --> 00:02:28.091
for four data points, it's now 25%.

00:02:28.091 --> 00:02:30.957
But these are saying the same thing.

00:02:30.957 --> 00:02:34.715
Four out of the 16 data points
fall into this category.

00:02:34.715 --> 00:02:39.121
4/16 is 1/4, which is 25%.

00:02:39.121 --> 00:02:42.279
So both of these types of
histograms are really useful

00:02:42.279 --> 00:02:45.632
and you will see them
used all of the time.

00:02:45.632 --> 00:02:47.377
But there are also cases where you have

00:02:47.377 --> 00:02:49.387
many, many, many more data points,

00:02:49.387 --> 00:02:52.554
and you want more granular categories.

00:02:54.164 --> 00:02:56.599
So what you could do, is, well,

00:02:56.599 --> 00:02:59.259
let's just make our categories
a little more granular.

00:02:59.259 --> 00:03:01.491
So for example, instead of them being

00:03:01.491 --> 00:03:03.354
one glass of water wide,

00:03:03.354 --> 00:03:05.810
maybe you make them half
a glass of water wide.

00:03:05.810 --> 00:03:08.053
So this first category could be

00:03:08.053 --> 00:03:12.136
greater than or equal to
zero, and less than 0.5,

00:03:14.063 --> 00:03:16.263
and that will give you a clearer picture,

00:03:16.263 --> 00:03:17.354
and I'm now assuming in a world

00:03:17.354 --> 00:03:19.266
where we have more than 16 data points,

00:03:19.266 --> 00:03:21.379
maybe we have 16 million data points,

00:03:21.379 --> 00:03:25.513
this would be percentages
on the left hand side.

00:03:25.513 --> 00:03:28.436
But maybe that isn't good enough for you,

00:03:28.436 --> 00:03:32.192
maybe you wanna get even more granular.

00:03:32.192 --> 00:03:34.751
So you make everything, each category,

00:03:34.751 --> 00:03:37.173
a quarter of a glass.

00:03:37.173 --> 00:03:38.623
But maybe that doesn't satisfy,

00:03:38.623 --> 00:03:41.513
you wanna get more and
more and more granular.

00:03:41.513 --> 00:03:43.930
Well, you could imagine
where this is going.

00:03:43.930 --> 00:03:45.763
You could get to a point
where you're approaching

00:03:45.763 --> 00:03:47.825
an infinite number of categories,

00:03:47.825 --> 00:03:50.844
and each category is infinitely thin,

00:03:50.844 --> 00:03:53.031
is super, super thin, to a point

00:03:53.031 --> 00:03:55.561
that if you just connect
the tops of the bars

00:03:55.561 --> 00:03:59.005
that you will actually get a curve.

00:03:59.005 --> 00:04:01.498
And this type of curve is something

00:04:01.498 --> 00:04:04.258
that we actually use in the statistics,

00:04:04.258 --> 00:04:07.220
and, as promised at the
beginning of the video,

00:04:07.220 --> 00:04:10.553
this is the density curve we talk about.

00:04:12.068 --> 00:04:14.596
And what's valuable about a density curve,

00:04:14.596 --> 00:04:17.843
it is a visualization of a distribution

00:04:17.843 --> 00:04:22.666
where the data points can take
on any value in a continuum.

00:04:22.666 --> 00:04:26.310
They're not just thrown
into these coarse buckets.

00:04:26.310 --> 00:04:29.463
So how would you interpret
something like this?

00:04:29.463 --> 00:04:31.813
If you look over the entire interval

00:04:31.813 --> 00:04:36.326
from zero, let's say, to
nine, assuming no one drank

00:04:36.326 --> 00:04:38.578
more than an average of
nine glasses per day,

00:04:38.578 --> 00:04:41.784
even in our 16 million data points,

00:04:41.784 --> 00:04:44.630
well then the area under
the curve over that interval

00:04:44.630 --> 00:04:47.380
is going to be 100%, or 1.0.

00:04:51.048 --> 00:04:54.822
This is going to be true
for any density curve,

00:04:54.822 --> 00:04:57.936
that the entire area of the curve is 100%,

00:04:57.936 --> 00:05:00.581
it represents all of the data points.

00:05:00.581 --> 00:05:04.418
A density curve will also
never take on a negative value,

00:05:04.418 --> 00:05:06.081
you won't see the curve dip down

00:05:06.081 --> 00:05:09.473
and do something strange like that.

00:05:09.473 --> 00:05:11.411
Now, with that out of the way,

00:05:11.411 --> 00:05:15.109
let's think about how
we would make use of it.

00:05:15.109 --> 00:05:18.920
If I wanted to know what
percentage of my data

00:05:18.920 --> 00:05:21.837
falls between two and four glasses,

00:05:23.004 --> 00:05:24.465
well I would look at that interval.

00:05:24.465 --> 00:05:26.465
I'd go from two to four,

00:05:27.368 --> 00:05:30.732
I would look at this
interval right over here,

00:05:30.732 --> 00:05:34.730
and I would try to figure out
the area under the curve here.

00:05:34.730 --> 00:05:39.342
And this area is going to be
greater than or equal to zero,

00:05:39.342 --> 00:05:42.769
and less than or equal to 100%.

00:05:42.769 --> 00:05:44.379
When I eyeball it right over here,

00:05:44.379 --> 00:05:47.680
it looks like it's about 40%

00:05:47.680 --> 00:05:50.273
of the entire area under the curve,

00:05:50.273 --> 00:05:54.440
so just eyeballing it, I would
say roughly 40% of my data

00:05:55.345 --> 00:05:57.755
falls into this interval.

00:05:57.755 --> 00:06:02.542
If I were to ask you what
percentage of the data

00:06:02.542 --> 00:06:04.375
is greater than three,

00:06:05.323 --> 00:06:08.531
well then you would be
looking at this area,

00:06:08.531 --> 00:06:12.505
and it looks like it is about 50%,

00:06:12.505 --> 00:06:14.510
but once again, I am estimating it.

00:06:14.510 --> 00:06:17.469
But you can start to see
how, even with estimation,

00:06:17.469 --> 00:06:19.931
a density curve could be useful.

00:06:19.931 --> 00:06:23.032
In the real world, statisticians
will often have tables

00:06:23.032 --> 00:06:25.655
that might represent the
information for the density curve,

00:06:25.655 --> 00:06:27.111
they might have computer programs

00:06:27.111 --> 00:06:28.921
or some type of automated tool,

00:06:28.921 --> 00:06:31.573
and there are also
well-known density curves.

00:06:31.573 --> 00:06:34.854
The famous Bell Curve that
we will study later on,

00:06:34.854 --> 00:06:36.455
where there's a lot of precise data

00:06:36.455 --> 00:06:40.511
and a lot of tools to
exactly figure out the areas.

00:06:40.511 --> 00:06:42.391
The last thing I'd like to cover

00:06:42.391 --> 00:06:46.484
is a key misconception for density curves.

00:06:46.484 --> 00:06:50.497
If I were to ask you,
approximately what percentage

00:06:50.497 --> 00:06:55.048
of my data is exactly three
glasses of water per day?

00:06:55.048 --> 00:06:57.626
And when I say exactly, I mean exactly

00:06:57.626 --> 00:07:02.198
the number 3.000 with zeroes
just going on and on forever,

00:07:02.198 --> 00:07:04.017
the exact number three.

00:07:04.017 --> 00:07:06.795
Well, you might be tempted to
just say okay, this is three.

00:07:06.795 --> 00:07:09.283
Let me see the corresponding
point on the curve.

00:07:09.283 --> 00:07:13.480
It looks like it is about 0.2,
or a little higher than that,

00:07:13.480 --> 00:07:17.625
so maybe you would say a
little bit more than 20%,

00:07:17.625 --> 00:07:20.387
or approximately 20%.

00:07:20.387 --> 00:07:23.715
And what I would say to
you, is this is wrong.

00:07:23.715 --> 00:07:27.497
Remember, the percentage
of the data in an interval

00:07:27.497 --> 00:07:30.014
is not the height of the curve,

00:07:30.014 --> 00:07:33.659
it is the area under the
curve in that interval.

00:07:33.659 --> 00:07:36.555
And if we're just talking
about one precise value,

00:07:36.555 --> 00:07:39.070
like exactly the number three,

00:07:39.070 --> 00:07:41.498
there is no area under the curve.

00:07:41.498 --> 00:07:43.489
This vertical line that I just drew

00:07:43.489 --> 00:07:46.136
over the number three has no width,

00:07:46.136 --> 00:07:49.141
and this actually makes
sense in the real world.

00:07:49.141 --> 00:07:52.207
Even if you were to look
at 16 million people,

00:07:52.207 --> 00:07:53.979
it is very unlikely that even anyone

00:07:53.979 --> 00:07:57.211
would drink exactly three
glasses of water per day.

00:07:57.211 --> 00:07:59.546
I'm talking about not one atom more

00:07:59.546 --> 00:08:02.079
or one atom less than three glasses.

00:08:02.079 --> 00:08:05.933
There might be many people
between 2.9 and 3.1,

00:08:05.933 --> 00:08:08.615
but no one is exactly three glasses a day.

00:08:08.615 --> 00:08:09.680
When someone says I'm drinking

00:08:09.680 --> 00:08:11.173
three glasses of water per day,

00:08:11.173 --> 00:08:12.459
that would be a rough estimate.

00:08:12.459 --> 00:08:15.376
They're probably 3.001, or 2.99999,

00:08:18.408 --> 00:08:20.575
or 3.15, or whatever else.

00:08:22.073 --> 00:08:24.403
And so instead, you could say

00:08:24.403 --> 00:08:27.575
what percentage falls in the interval,

00:08:27.575 --> 00:08:31.158
maybe, that is greater
than or equal to 2.9

00:08:32.789 --> 00:08:35.289
and less than or equal to 3.1.

00:08:36.740 --> 00:08:38.277
And so once you have an interval,

00:08:38.277 --> 00:08:40.442
then you actually can look at the area,

00:08:40.442 --> 00:08:43.815
so we're gonna go from 2.9 to 3.1,

00:08:43.815 --> 00:08:48.270
so now we have an interval
that actually has width,

00:08:48.270 --> 00:08:50.225
and so it'd be roughly the size

00:08:50.225 --> 00:08:53.427
of this yellow area that I'm
shading in right over here,

00:08:53.427 --> 00:08:56.229
and we can approximate it with a rectangle

00:08:56.229 --> 00:08:59.044
even though the top of
this curve isn't flat,

00:08:59.044 --> 00:09:01.414
but we can say, look, it's approximately

00:09:01.414 --> 00:09:04.247
like a rectangle that is 0.2 high,

00:09:05.220 --> 00:09:06.517
and what's the width?

00:09:06.517 --> 00:09:10.351
The width here, if we're
going from 2.9 to 3.1,

00:09:10.351 --> 00:09:13.184
the width is going to be 0.2 wide,

00:09:14.292 --> 00:09:16.785
and so we could approximate this area

00:09:16.785 --> 00:09:20.483
by approximating this rectangle,
the area of the rectangle.

00:09:20.483 --> 00:09:24.650
0.2 times 0.2, that would
give us an area of 0.04.

00:09:26.957 --> 00:09:30.747
Or we could say
approximately 4% of the data

00:09:30.747 --> 00:09:32.664
falls in this interval.


Worked example: distance and displacement from position-time graphs

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.702
- [Instructor] In other videos,
we've already talked about

00:00:01.702 --> 00:00:04.279
the difference between
distance and displacement.

00:00:04.279 --> 00:00:06.182
And we also saw what it meant

00:00:06.182 --> 00:00:08.750
to plot position versus time.

00:00:08.750 --> 00:00:10.384
What we're gonna do in this video

00:00:10.384 --> 00:00:12.045
is use all of those skills.

00:00:12.045 --> 00:00:15.095
We're going to look at
position versus time graphs,

00:00:15.095 --> 00:00:17.831
and use them in order to
figure out displacement

00:00:17.831 --> 00:00:20.176
and distance traveled.

00:00:20.176 --> 00:00:21.446
So this first question says,

00:00:21.446 --> 00:00:24.190
a 3.2 kilogram iguana runs back and forth

00:00:24.190 --> 00:00:25.862
along the ground.

00:00:25.862 --> 00:00:28.432
The following graph shows
the horizontal position

00:00:28.432 --> 00:00:31.637
of the iguana in meters over time.

00:00:31.637 --> 00:00:32.470
The first question is

00:00:32.470 --> 00:00:34.752
what is the displacement
of the iguana between

00:00:34.752 --> 00:00:38.117
zero seconds and five seconds?

00:00:38.117 --> 00:00:39.583
So, be very careful.

00:00:39.583 --> 00:00:42.542
It's not saying between zero
seconds and six seconds.

00:00:42.542 --> 00:00:45.116
It's saying between zero
seconds and five seconds.

00:00:45.116 --> 00:00:48.343
So pause the video, and see
if you can figure that out.

00:00:48.343 --> 00:00:52.479
Well, displacement is just
your change in position.

00:00:52.479 --> 00:00:53.758
And sign matters.

00:00:53.758 --> 00:00:55.461
We care about direction.

00:00:55.461 --> 00:00:57.806
So it's going to be your final position

00:00:57.806 --> 00:01:00.590
minus your initial position.

00:01:00.590 --> 00:01:02.814
Well, what is your final position?

00:01:02.814 --> 00:01:04.607
Well, we're not talking about the final,

00:01:04.607 --> 00:01:07.720
the last time that we have on the graph.

00:01:07.720 --> 00:01:10.447
Here our final is going
to be at 5 seconds.

00:01:10.447 --> 00:01:14.881
So our final position is
at a positive six meters.

00:01:14.881 --> 00:01:18.094
So our final position
is positive six meters.

00:01:18.094 --> 00:01:20.711
And from that you're gonna
subtract our initial position,

00:01:20.711 --> 00:01:24.314
where our initial position
was at negative two meters.

00:01:24.314 --> 00:01:25.719
Negative two meters.

00:01:25.719 --> 00:01:27.121
And so this is going to be equal to

00:01:27.121 --> 00:01:31.121
six minus negative two
is positive eight meters.

00:01:32.227 --> 00:01:34.069
So one way to think about it is

00:01:34.069 --> 00:01:37.761
net this iguana, shifts eight meters

00:01:37.761 --> 00:01:39.460
in the positive direction.

00:01:39.460 --> 00:01:42.093
And if we think positive
direction is, say, to the right,

00:01:42.093 --> 00:01:44.375
it would be eight meters to the right.

00:01:44.375 --> 00:01:46.431
We could draw a number line here.

00:01:46.431 --> 00:01:48.151
So if we, it's sometimes confusing

00:01:48.151 --> 00:01:49.946
'cause we're talking about
the horizontal direction,

00:01:49.946 --> 00:01:51.935
but we're plotting
position vertically here.

00:01:51.935 --> 00:01:53.736
But we could take the same number line

00:01:53.736 --> 00:01:55.158
and make it horizontal.

00:01:55.158 --> 00:01:58.439
And you'd have negative
six, and negative four,

00:01:58.439 --> 00:02:00.797
negative two, zero,

00:02:00.797 --> 00:02:04.970
two, four, six, eight.

00:02:04.970 --> 00:02:06.649
And so what's happening here

00:02:06.649 --> 00:02:10.795
is the iguana is starting at negative two,

00:02:10.795 --> 00:02:12.676
and then over the next three seconds,

00:02:12.676 --> 00:02:14.676
it goes to positive six.

00:02:17.089 --> 00:02:20.627
It goes over here, and so it
shifts to the right by eight.

00:02:20.627 --> 00:02:21.939
And that's what we saw right over there.

00:02:21.939 --> 00:02:25.354
Six minus negative two is eight.

00:02:25.354 --> 00:02:28.293
Now what about the distance
traveled by the iguana

00:02:28.293 --> 00:02:30.641
over that same time period?

00:02:30.641 --> 00:02:33.204
Pause the video, and see
if you can figure it out.

00:02:33.204 --> 00:02:36.713
Well, the distance is the
total length traveled.

00:02:36.713 --> 00:02:38.738
The total path length.

00:02:38.738 --> 00:02:41.403
But once again, it went from negative two

00:02:41.403 --> 00:02:44.596
all the way to positive six
over the first three seconds,

00:02:44.596 --> 00:02:46.416
and then it just stays there.

00:02:46.416 --> 00:02:48.470
So if we think about distance,

00:02:48.470 --> 00:02:50.326
we're actually going
to get the same result.

00:02:50.326 --> 00:02:53.070
Even though we don't care
about direction here,

00:02:53.070 --> 00:02:54.808
we just care about the magnitude,

00:02:54.808 --> 00:02:56.062
we still get the same thing.

00:02:56.062 --> 00:02:58.261
It traveled eight meters.

00:02:58.261 --> 00:02:59.912
So it travels eight meters.

00:02:59.912 --> 00:03:03.528
So we're gonna get the exact same result.

00:03:03.528 --> 00:03:05.345
Now what would be interesting is the thing

00:03:05.345 --> 00:03:07.335
what would be different if
instead of five seconds,

00:03:07.335 --> 00:03:09.606
if this said six seconds?

00:03:09.606 --> 00:03:12.216
If this was between zero and six seconds?

00:03:12.216 --> 00:03:14.078
Well, then for displacement, we'd say,

00:03:14.078 --> 00:03:18.130
hey, look, we went plus
eight, but then we're gonna go

00:03:18.130 --> 00:03:21.375
another negative six.

00:03:21.375 --> 00:03:23.298
So this right over here,

00:03:23.298 --> 00:03:24.615
lemme make that,

00:03:24.615 --> 00:03:27.560
you're gonna go another negative six,

00:03:27.560 --> 00:03:29.884
and so you actually have,

00:03:29.884 --> 00:03:31.895
you'd have the plus eight minus six.

00:03:31.895 --> 00:03:35.287
So if you did it for all
the way to the sixth second,

00:03:35.287 --> 00:03:38.435
then your displacement is
going to be plus two meters.

00:03:38.435 --> 00:03:40.771
You have the eight, and
then you subtract the six.

00:03:40.771 --> 00:03:42.243
Another way top think about it is

00:03:42.243 --> 00:03:43.947
you would finish at zero.

00:03:43.947 --> 00:03:46.211
You started at negative two.

00:03:46.211 --> 00:03:49.600
Zero minus negative two is two.

00:03:49.600 --> 00:03:51.760
If you wanted to think about the distance

00:03:51.760 --> 00:03:54.611
between zero seconds and six seconds,

00:03:54.611 --> 00:03:57.185
well, you would have this eight meters.

00:03:57.185 --> 00:03:58.857
And then, you would go,

00:03:58.857 --> 00:04:01.516
even though you're going back to zero.

00:04:01.516 --> 00:04:05.266
So you're going back to
zero right over here.

00:04:06.481 --> 00:04:07.661
Six meters.

00:04:07.661 --> 00:04:09.073
You wouldn't subtract it

00:04:09.073 --> 00:04:11.056
because the total path,
you traveled eight meters

00:04:11.056 --> 00:04:13.545
to the right and then
six meters to the left.

00:04:13.545 --> 00:04:14.864
And so you would add 'em.

00:04:14.864 --> 00:04:16.666
So if you said over the first six seconds,

00:04:16.666 --> 00:04:17.883
instead of the first five seconds,

00:04:17.883 --> 00:04:20.490
this would be 14 meters.

00:04:20.490 --> 00:04:21.893
Let me make that clear again.

00:04:21.893 --> 00:04:23.639
Displacement and distance,

00:04:23.639 --> 00:04:27.063
in both cases, you have plus
eight over that first leg,

00:04:27.063 --> 00:04:28.222
I guess you could say.

00:04:28.222 --> 00:04:30.279
And then when we talk about displacement,

00:04:30.279 --> 00:04:33.062
we subtract it 'cause we're
now moving to the left.

00:04:33.062 --> 00:04:35.558
We move to the left by six.

00:04:35.558 --> 00:04:38.344
So in the case of displacement,
you subtract the six,

00:04:38.344 --> 00:04:41.296
and you have a net displacement

00:04:41.296 --> 00:04:43.818
of plus two.

00:04:43.818 --> 00:04:46.328
But distance, the total path traveled,

00:04:46.328 --> 00:04:47.719
you have the eight to the right,

00:04:47.719 --> 00:04:49.187
and then six to the left.

00:04:49.187 --> 00:04:52.362
Which gives you a total
path traveled of 14.

00:04:52.362 --> 00:04:54.945
Let's do one more example here.

00:04:56.629 --> 00:05:00.049
A 2.7 kilogram armadillo

00:05:00.049 --> 00:05:01.945
rolls in a straight line in the desert.

00:05:01.945 --> 00:05:04.281
The following graph shows
the horizontal position

00:05:04.281 --> 00:05:05.156
in the...

00:05:05.156 --> 00:05:06.577
The horizontal position of the armadillo

00:05:06.577 --> 00:05:08.673
in meters over time.

00:05:08.673 --> 00:05:09.726
And so let's think about the same thing.

00:05:09.726 --> 00:05:11.750
Over the first 24 seconds,

00:05:11.750 --> 00:05:15.336
let's go all the way to the 24th second.

00:05:15.336 --> 00:05:18.049
Let's think about what the displacement is

00:05:18.049 --> 00:05:20.282
and what the distance traveled is.

00:05:20.282 --> 00:05:21.689
So, first pause this video,

00:05:21.689 --> 00:05:24.121
and see if you can figure
out the displacement

00:05:24.121 --> 00:05:27.881
over the first 24 seconds.

00:05:27.881 --> 00:05:30.601
Well, this is going to
be our final position

00:05:30.601 --> 00:05:32.615
minus our starting position.

00:05:32.615 --> 00:05:35.982
Our final position, we're at zero meters.

00:05:35.982 --> 00:05:39.006
Our starting position,
we were at time zero,

00:05:39.006 --> 00:05:40.839
we were at six meters.

00:05:42.258 --> 00:05:44.759
All right, lemme just
write the numbers down.

00:05:44.759 --> 00:05:46.344
So it is negative six.

00:05:46.344 --> 00:05:47.792
Let me say that one more time.

00:05:47.792 --> 00:05:52.256
At time 24, notice our
vertical coordinate.

00:05:52.256 --> 00:05:55.003
We are in a position of zero.

00:05:55.003 --> 00:05:56.573
That's where that came from.

00:05:56.573 --> 00:05:58.977
When we started, at time zero,

00:05:58.977 --> 00:06:01.169
our position was right over here.

00:06:01.169 --> 00:06:04.813
So our final, minus our
starting, is negative six.

00:06:04.813 --> 00:06:05.927
And you could also see that.

00:06:05.927 --> 00:06:07.653
If you just look along this line,

00:06:07.653 --> 00:06:11.805
we shifted from positive six to zero,

00:06:11.805 --> 00:06:14.537
which would be a shift of six to the left

00:06:14.537 --> 00:06:16.002
if you made this horizontal.

00:06:16.002 --> 00:06:18.386
Or a displacement of negative six.

00:06:18.386 --> 00:06:20.275
Lemme draw this on a horizontal line

00:06:20.275 --> 00:06:23.575
just to make this a little bit more clear.

00:06:23.575 --> 00:06:26.857
So if you have zero, four, oh...

00:06:26.857 --> 00:06:31.504
Zero, three,

00:06:31.504 --> 00:06:34.175
six, nine,

00:06:34.175 --> 00:06:37.940
12, 15, so on and so forth.

00:06:37.940 --> 00:06:42.357
We are starting at six,
do that purple color.

00:06:42.357 --> 00:06:43.837
We're starting at six,

00:06:43.837 --> 00:06:45.566
we do a bunch of stuff in between,

00:06:45.566 --> 00:06:49.439
but then we end up after
24 seconds at zero.

00:06:49.439 --> 00:06:50.742
So our shift,

00:06:50.742 --> 00:06:54.854
we went six to the left.

00:06:54.854 --> 00:06:57.087
Or we have a displacement of negative six.

00:06:57.087 --> 00:07:00.474
Zero minus six is negative six.

00:07:00.474 --> 00:07:03.666
So now let's try to figure out distance.

00:07:03.666 --> 00:07:06.183
Pause the video, and
figure out the distance

00:07:06.183 --> 00:07:10.266
that this armadillo travels
over this 24 seconds.

00:07:11.839 --> 00:07:13.839
So, this is interesting.

00:07:13.839 --> 00:07:15.038
So it's right here.

00:07:15.038 --> 00:07:17.646
It starts off at position of six.

00:07:17.646 --> 00:07:18.610
So let me do it right here.

00:07:18.610 --> 00:07:19.833
It starts off at position of six.

00:07:19.833 --> 00:07:23.024
It stays there for the
first eight seconds.

00:07:23.024 --> 00:07:25.960
Then, from the eighth
second to the 16th second,

00:07:25.960 --> 00:07:30.127
it's position increases
by nine to get to 15.

00:07:31.011 --> 00:07:32.731
So it does this.

00:07:32.731 --> 00:07:33.898
It goes to 15.

00:07:34.803 --> 00:07:37.270
So this is going to be plus.

00:07:37.270 --> 00:07:40.372
This is going to be plus nine.

00:07:40.372 --> 00:07:43.924
And then, on the 16th second,

00:07:43.924 --> 00:07:47.720
it goes from 15 back to zero.

00:07:47.720 --> 00:07:49.762
So it goes from 15

00:07:49.762 --> 00:07:53.901
back to zero.

00:07:53.901 --> 00:07:55.567
If we're thinking about displacement,

00:07:55.567 --> 00:07:57.381
we would write a minus 15 here,

00:07:57.381 --> 00:08:00.729
and then we would net these
out to get to a negative six.

00:08:00.729 --> 00:08:02.849
But we're thinking about distance.

00:08:02.849 --> 00:08:05.601
So we want, we think
about the absolute value

00:08:05.601 --> 00:08:08.592
of the various parts of the path traveled.

00:08:08.592 --> 00:08:10.048
So all of these are gonna be positive.

00:08:10.048 --> 00:08:12.280
We just say, hey, what
is the total journey?

00:08:12.280 --> 00:08:14.026
So this is gonna be plus 15.

00:08:14.026 --> 00:08:15.763
We just care about the
length of these arrows,

00:08:15.763 --> 00:08:17.371
not the direction.

00:08:17.371 --> 00:08:21.783
And so, nine plus 15 is 24.

00:08:21.783 --> 00:08:22.646
So this is interesting.

00:08:22.646 --> 00:08:26.783
Even though the armadillo
traveled a total of 24 meters,

00:08:26.783 --> 00:08:29.384
its entire path was 24 meters long.

00:08:29.384 --> 00:08:31.958
Its net shift, its displacement,

00:08:31.958 --> 00:08:36.144
is six meters to the left.

00:08:36.144 --> 00:08:38.344
And this is another thing to emphasize.

00:08:38.344 --> 00:08:40.004
This negative number,

00:08:40.004 --> 00:08:42.736
this is implying direction.

00:08:42.736 --> 00:08:45.366
It's saying, if we're
look at this number line,

00:08:45.366 --> 00:08:46.874
it's saying to the left.

00:08:46.874 --> 00:08:49.110
Even if it was a positive six,
because we're talking about

00:08:49.110 --> 00:08:51.060
displacement, it would imply,

00:08:51.060 --> 00:08:52.971
positive would mean to the right.

00:08:52.971 --> 00:08:54.957
Distance doesn't tell you about direction.

00:08:54.957 --> 00:08:57.683
It just tells you the absolute magnitude

00:08:57.683 --> 00:09:00.171
of the total distance traveled,

00:09:00.171 --> 00:09:02.421
or the length of the path.


Regrouping to add 1-digit number

WEBVTT
Kind: captions
Language: en

00:00:00.411 --> 00:00:05.171
So, we have the number 35. The 3 is in the tens place

00:00:05.171 --> 00:00:08.247
so it represents 30 or 3 tens—

00:00:08.247 --> 00:00:11.881
one 10, two groups of 10, three groups of 10

00:00:11.881 --> 00:00:14.108
and then the 5 is in the ones place

00:00:14.108 --> 00:00:17.638
so it represents five ones. We see them right over here

00:00:17.638 --> 00:00:19.740
—one, two, three, four, five.

00:00:19.740 --> 00:00:22.422
Now, we want to take that 35 or those 3 tens

00:00:22.422 --> 00:00:25.382
and 5 ones and add 6. 6 ones.

00:00:25.382 --> 00:00:26.845
The 6 is in the ones place—

00:00:26.845 --> 00:00:30.838
one, two three, four, five, six ones.

00:00:30.838 --> 00:00:32.382
And I encourage you to pause the video

00:00:32.382 --> 00:00:35.458
and try to do that. Add 35 to 6.

00:00:35.458 --> 00:00:37.479
So let's think about it now.

00:00:37.479 --> 00:00:39.162
So I'm gonna start with the ones.

00:00:39.162 --> 00:00:42.958
So I have 5 ones and I want to add 6 ones.

00:00:42.958 --> 00:00:46.544
So what's that going to be altogether?

00:00:46.544 --> 00:00:51.584
Well, 5 ones plus 6 ones, that's going to be 11 ones

00:00:51.584 --> 00:00:55.159
and I still have 3 tens so we could say it's going to be

00:00:55.159 --> 00:01:00.058
3 tens, 3 tens and 11 ones.

00:01:00.058 --> 00:01:03.157
3 tens, I could write, plus 11 ones here.

00:01:03.157 --> 00:01:05.397
Now this is a little bit of a problem because

00:01:05.397 --> 00:01:07.860
we can't write a two-digit number

00:01:07.860 --> 00:01:09.821
in the ones place or in any one of the places.

00:01:09.821 --> 00:01:13.583
I can't, this number isn't going to be 311.

00:01:13.583 --> 00:01:15.881
It's going to be 3 tens and 11 ones

00:01:15.881 --> 00:01:19.620
but how can I rewrite this or how can I regroup things

00:01:19.620 --> 00:01:22.174
so that I only have a single-digit number here?

00:01:22.174 --> 00:01:24.820
I have zero, one, two, three, four, five, six, seven, eight,

00:01:24.820 --> 00:01:29.080
or nine here instead of a two-digit number— the 1 one.

00:01:29.080 --> 00:01:30.637
Well, I can regroup. I can say,

00:01:30.637 --> 00:01:34.560
"Look, I have enough ones here to create a group of ten."

00:01:34.560 --> 00:01:38.078
I could take 10 of them, so let's take these 10,

00:01:38.078 --> 00:01:41.398
right over here, and put them together

00:01:41.398 --> 00:01:44.834
and make a new group of ten.

00:01:44.834 --> 00:01:48.073
So, a new group of ten right over there.

00:01:48.073 --> 00:01:51.440
So if I take, so just to be clear, what I just did

00:01:51.440 --> 00:01:54.656
I just took these 10, these 10 ones

00:01:54.656 --> 00:01:56.837
and I stuck them together and I turned it

00:01:56.837 --> 00:02:00.913
into this new group of, this new group of ten.

00:02:00.913 --> 00:02:02.561
So now what do we have?

00:02:02.561 --> 00:02:04.975
So when you regroup like this you see that you have

00:02:04.975 --> 00:02:09.678
one, two, three, four tens. 4 tens.

00:02:09.678 --> 00:02:12.580
And how many ones do you have now?

00:02:12.580 --> 00:02:16.156
4 tens plus? Well, I've regrouped all of these 10 ones

00:02:16.156 --> 00:02:20.335
and all I have left is this 1 one right over there.

00:02:20.335 --> 00:02:24.270
So I could write this as 4 tens plus 1 one.

00:02:24.270 --> 00:02:26.894
Once again, 3 tens and 11 ones,

00:02:26.894 --> 00:02:29.831
that's the same thing as 4 tens and 1 one.

00:02:29.831 --> 00:02:31.735
And so we can write that over here,

00:02:31.735 --> 00:02:40.266
we can write this as, 1 one and 4 tens.

00:02:40.386 --> 00:02:43.055
Now how could you get this if you weren't,

00:02:43.055 --> 00:02:45.655
if you didn't do it like this and drawing everything out

00:02:45.655 --> 00:02:47.755
and regrouping like this and this is actually

00:02:47.755 --> 00:02:49.591
what you should be doing in your head

00:02:49.591 --> 00:02:52.318
but another way of thinking about it, you could say,

00:02:52.318 --> 00:02:55.882
"Alright 5 plus 6," that's going to be 11 ones

00:02:55.882 --> 00:02:59.064
but I can't write an 11 here in the ones place

00:02:59.064 --> 00:03:01.594
so I could say, "That's going to be the same thing

00:03:01.594 --> 00:03:07.736
as 1 ten plus 1 one." 1 ten plus 1 one.

00:03:07.736 --> 00:03:11.461
Sometimes it's taught that 5 plus 6 is 11,

00:03:11.461 --> 00:03:14.620
carry the 1 but really what you're doing is you're saying,

00:03:14.620 --> 00:03:19.020
"5 plus 6 is 1 ten plus 1 one."

00:03:19.020 --> 00:03:22.420
And, in fact, an 11, the number 11 right over here.

00:03:22.420 --> 00:03:26.274
So if I were to write the number 11, the number 11

00:03:26.274 --> 00:03:27.818
has a 1 in the tens place and a 1

00:03:27.818 --> 00:03:31.093
in the ones place so it's 1 ten plus 1 one.

00:03:31.093 --> 00:03:33.032
So you're just saying that 5 plus 6 is 11

00:03:33.032 --> 00:03:36.270
which is the same thing as 1 ten and 1 one

00:03:36.270 --> 00:03:37.709
and then you add your tens together—

00:03:37.709 --> 00:03:43.375
1 ten plus another 3 tens is going to be 4 tens.

00:03:43.375 --> 00:03:45.673
But I really want you to appreciate what's going on.

00:03:45.673 --> 00:03:49.354
You're not just blindly saying, "Oh, 5 plus 6 is 11

00:03:49.354 --> 00:03:51.815
so I'm going to write 1 of the ones here

00:03:51.815 --> 00:03:53.533
and write the 1 in the tens place here."

00:03:53.533 --> 00:03:55.519
You're doing it because you're regrouping.

00:03:55.519 --> 00:03:58.793
You're regrouping that group of ten.

00:03:58.793 --> 00:04:01.334
You're saying, "Hey I could take 10 of these ones

00:04:01.334 --> 00:04:04.027
and I can turn them into a new group of ten

00:04:04.027 --> 00:04:06.675
and that would just leave me 1 in the ones place.

00:04:06.675 --> 00:04:09.032
1 in the ones place and I've just turned

00:04:09.032 --> 00:04:12.242
all the other ones into a new group of ten.


Breaking apart 2-digit addition problems

WEBVTT
Kind: captions
Language: en

00:00:00.245 --> 00:00:02.758
- Let's think about ways to break up addition problems.

00:00:02.758 --> 00:00:03.786
And this is useful,

00:00:03.786 --> 00:00:05.783
because if we break them up in the right way,

00:00:05.783 --> 00:00:09.612
it might be easier for us to actually compute the addition.

00:00:09.612 --> 00:00:11.704
So let's look at this first question.

00:00:11.704 --> 00:00:15.025
Lindsay isn't sure how to add 39 plus 61.

00:00:15.025 --> 00:00:17.544
Help Lindsay by choosing an addition problem that is

00:00:17.544 --> 00:00:20.944
the same as 39 plus 61.

00:00:20.944 --> 00:00:22.524
So let's look at these choices.

00:00:22.524 --> 00:00:25.296
This first choose I have 30 plus 60

00:00:25.296 --> 00:00:26.582
plus 90 plus ten.

00:00:26.582 --> 00:00:27.766
And I encourage you to pause the video

00:00:27.766 --> 00:00:30.378
and try to work this out actually before I do it.

00:00:30.378 --> 00:00:31.365
So this first choice,

00:00:31.365 --> 00:00:33.420
where do they get this 30 from?

00:00:33.420 --> 00:00:36.195
Well, 30, that's three tens,

00:00:37.305 --> 00:00:39.776
and I do have three tens right over here.

00:00:39.776 --> 00:00:41.133
The three in 39,

00:00:41.133 --> 00:00:44.662
that's in the tens place so it represents three tens or 30.

00:00:44.662 --> 00:00:47.350
And then we have 60,

00:00:47.355 --> 00:00:49.050
where did that come from?

00:00:49.050 --> 00:00:51.001
Well, in the number 61,

00:00:51.001 --> 00:00:52.789
the six is in the tens place,

00:00:52.789 --> 00:00:55.841
so it represents six tens or 60.

00:00:55.841 --> 00:00:57.908
And then we have plus 90.

00:00:57.908 --> 00:01:00.381
Now where is 90 coming from?

00:01:00.381 --> 00:01:03.248
I don't see the obvious 90 over here.

00:01:03.248 --> 00:01:04.246
It might be tempting to say,

00:01:04.246 --> 00:01:06.243
well I have a nine over here,

00:01:06.243 --> 00:01:07.346
but this is in the ones place,

00:01:07.346 --> 00:01:08.809
it's not in the tens place.

00:01:08.809 --> 00:01:12.408
This is nine, not 90.

00:01:12.408 --> 00:01:13.725
And over here I have one.

00:01:13.725 --> 00:01:15.565
I definitely don't have a 90.

00:01:15.565 --> 00:01:17.307
Or I definitely don't even have a ten.

00:01:17.307 --> 00:01:18.142
So this would make sense,

00:01:18.142 --> 00:01:19.605
instead of, if this didn't say 90,

00:01:19.605 --> 00:01:21.068
if this said nine,

00:01:21.068 --> 00:01:22.256
and instead of a ten,

00:01:22.256 --> 00:01:25.136
if this said one,

00:01:25.136 --> 00:01:26.901
'cause we have a one in the ones place,

00:01:26.901 --> 00:01:28.084
then it would make sense,

00:01:28.084 --> 00:01:28.885
but it didn't say that.

00:01:28.885 --> 00:01:31.637
It didn't say 30 plus 60 plus nine plus one,

00:01:31.637 --> 00:01:33.773
It's saying 30 plus 60 plus 90 plus ten.

00:01:33.773 --> 00:01:36.710
So we're not gonna pick this choice.

00:01:36.710 --> 00:01:37.836
The next choice,

00:01:37.836 --> 00:01:40.111
we have 30 plus 60 plus nine plus one,

00:01:40.111 --> 00:01:41.771
which makes complete sense,

00:01:41.771 --> 00:01:45.799
because we have the

00:01:45.799 --> 00:01:49.725
30 plus the nine,

00:01:49.725 --> 00:01:52.410
is going to be equal to 39.

00:01:52.410 --> 00:01:56.761
And then the 60 plus the one

00:01:56.761 --> 00:01:58.967
is going to be equal to 61.

00:01:58.967 --> 00:02:00.743
So these two things are equivalent.

00:02:00.743 --> 00:02:03.564
And the reason why it's useful to break up things this way,

00:02:03.564 --> 00:02:05.444
is 'cause you can compute in your head,

00:02:05.444 --> 00:02:06.140
30 plus 60,

00:02:06.140 --> 00:02:09.079
that's three tens plus six tens is gonna be nine tens.

00:02:09.079 --> 00:02:13.501
So these two pieces right over here are going to be,

00:02:13.501 --> 00:02:14.939
that's going to be 90.

00:02:14.939 --> 00:02:16.160
And then nine plus one,

00:02:16.160 --> 00:02:17.646
that's gonna be ten.

00:02:17.646 --> 00:02:18.645
And then 90 plus ten,

00:02:18.645 --> 00:02:20.304
well that's going to be equal to 100.

00:02:20.304 --> 00:02:21.801
Now they didn't ask us to compute that,

00:02:21.801 --> 00:02:22.522
they're just saying,

00:02:22.522 --> 00:02:25.151
hey which of these are the same as what we have up above,

00:02:25.151 --> 00:02:27.549
and this one is definitely going to be the case.

00:02:27.549 --> 00:02:28.895
And we can only pick one here,

00:02:28.895 --> 00:02:29.696
so we're done,

00:02:29.696 --> 00:02:31.855
but we can verify that this one isn't gonna be right.

00:02:31.855 --> 00:02:33.214
We have the nine plus one,

00:02:33.214 --> 00:02:35.454
then they have three and six.

00:02:35.454 --> 00:02:36.476
Now this three here,

00:02:36.476 --> 00:02:37.916
that doesn't represent just three,

00:02:37.916 --> 00:02:40.516
that's three tens, that's 30.

00:02:40.516 --> 00:02:41.816
This should be 30.

00:02:41.816 --> 00:02:43.314
And this is six tens,

00:02:43.314 --> 00:02:43.952
not just six,

00:02:43.952 --> 00:02:45.160
so that should be 60,

00:02:45.160 --> 00:02:47.760
but that's not what they originally wrote.

00:02:47.760 --> 00:02:49.818
So we can rule that one out as well.

00:02:49.818 --> 00:02:51.044
Let's do another one.

00:02:51.651 --> 00:02:55.917
Which addition problem is the same as 41 plus 57?

00:02:55.917 --> 00:02:57.389
And here they broke up everything into,

00:02:57.389 --> 00:02:58.817
looks like tens and ones.

00:02:58.817 --> 00:03:00.234
So even before I look at the choices,

00:03:00.234 --> 00:03:02.254
let me see if I can do that.

00:03:02.254 --> 00:03:05.991
So 41, in the tens place I have four.

00:03:05.991 --> 00:03:09.195
So that's going to be four tens,

00:03:09.195 --> 00:03:11.390
and in the ones place I have one.

00:03:11.390 --> 00:03:13.909
Plus one one.

00:03:13.909 --> 00:03:15.488
That's 41.

00:03:15.488 --> 00:03:17.635
And then 57,

00:03:17.635 --> 00:03:19.573
57 in the tens place,

00:03:19.573 --> 00:03:20.537
I have five,

00:03:20.537 --> 00:03:23.648
so plus five tens,

00:03:23.648 --> 00:03:25.890
and in the ones place I have seven.

00:03:25.890 --> 00:03:28.490
Plus seven ones.

00:03:28.490 --> 00:03:30.615
So let's see which of these choices is the same

00:03:30.615 --> 00:03:32.366
as what I just wrote here.

00:03:32.366 --> 00:03:33.737
So this first one,

00:03:33.737 --> 00:03:37.363
we have four tens

00:03:37.363 --> 00:03:38.603
and one one.

00:03:38.603 --> 00:03:40.608
Four tens and one one.

00:03:40.608 --> 00:03:42.163
Four tens and one one,

00:03:42.163 --> 00:03:44.103
that's 41, what I just...

00:03:44.103 --> 00:03:47.906
so this four tens plus one one would be 41.

00:03:47.906 --> 00:03:49.896
And then I also have,

00:03:49.896 --> 00:03:54.377
five tens and seven ones.

00:03:54.377 --> 00:03:57.265
Five tens and seven ones.

00:03:57.265 --> 00:03:59.760
So this first choice is exactly what I wrote down,

00:03:59.760 --> 00:04:02.466
it's just in a different order.

00:04:02.466 --> 00:04:03.209
If I write,

00:04:03.209 --> 00:04:04.602
four tens plus five tens

00:04:04.602 --> 00:04:07.063
plus one one plus seven ones.

00:04:07.063 --> 00:04:08.758
it's gonna be this first choice.

00:04:08.758 --> 00:04:10.128
And we know that we're done,

00:04:10.128 --> 00:04:11.903
but let's just look at the other choices

00:04:11.903 --> 00:04:13.518
to see why these don't make sense.

00:04:13.518 --> 00:04:16.107
So I see where the four tens come from,

00:04:16.107 --> 00:04:17.767
but then it has one ten.

00:04:17.767 --> 00:04:20.007
It looks like it's trying to take this one in a ones place

00:04:20.007 --> 00:04:21.517
and somehow turn it into a one ten,

00:04:21.517 --> 00:04:23.165
so that's definitely gonna be wrong.

00:04:23.165 --> 00:04:24.766
And then it says five ones,

00:04:24.766 --> 00:04:26.717
somehow this is five tens here,

00:04:26.717 --> 00:04:27.867
not five ones.

00:04:27.867 --> 00:04:29.805
So that doesn't make sense.

00:04:29.805 --> 00:04:31.965
And then this one says four ones,

00:04:31.965 --> 00:04:32.626
well this four,

00:04:32.626 --> 00:04:33.659
this is in the tens place,

00:04:33.659 --> 00:04:35.018
this is four tens.

00:04:35.018 --> 00:04:37.142
And then we have five ones,

00:04:37.142 --> 00:04:38.524
that five is in the tens place.

00:04:38.524 --> 00:04:39.917
It should be five tens.

00:04:39.917 --> 00:04:41.322
That should be tens,

00:04:41.322 --> 00:04:43.318
and that should be tens.

00:04:43.318 --> 00:04:46.447
So we feel good about the first choice.


Adding 2-digit numbers without regrouping 1

WEBVTT
Kind: captions
Language: en

00:00:00.331 --> 00:00:02.142
- Try to pause the video and figure out

00:00:02.142 --> 00:00:06.646
what 71 plus 24 is.

00:00:06.646 --> 00:00:08.968
All right, now let's do this together.

00:00:08.968 --> 00:00:13.217
So let's think about what 71 actually means.

00:00:13.217 --> 00:00:15.795
Well we have a one in the ones place.

00:00:15.795 --> 00:00:17.188
Let me make this clear.

00:00:17.188 --> 00:00:21.414
So this is the ones place and we have one one.

00:00:21.414 --> 00:00:23.921
So that's that one one right over there.

00:00:23.921 --> 00:00:27.544
And then the tens place - in the tens place

00:00:27.544 --> 00:00:30.887
we have seven so we have seven groups of ten.

00:00:30.887 --> 00:00:36.506
One, two, three, four, five, six, seven groups of ten.

00:00:36.506 --> 00:00:41.289
So this really means seven tens and one one.

00:00:41.289 --> 00:00:44.888
This right over here, 71 boxes.

00:00:44.888 --> 00:00:48.371
Now we're going to add to that two tens.

00:00:48.371 --> 00:00:50.716
We're gonna add to that two tens;

00:00:50.716 --> 00:00:51.854
you see them here.

00:00:51.854 --> 00:00:55.848
One ten, two tens and four ones.

00:00:55.848 --> 00:00:57.984
And four ones.

00:00:57.984 --> 00:00:59.934
So what is that going to be?

00:00:59.934 --> 00:01:04.206
Well lets first thing about the ones.

00:01:04.206 --> 00:01:06.900
We have one one and four more ones.

00:01:06.900 --> 00:01:09.268
Well that's going to be five ones.

00:01:09.268 --> 00:01:11.381
That's going to be five ones,

00:01:11.381 --> 00:01:12.937
so we can count them right over there.

00:01:12.937 --> 00:01:14.980
This is five ones right over there.

00:01:14.980 --> 00:01:18.625
One one plus four ones is five ones.

00:01:18.625 --> 00:01:19.972
And then when you look at the tens;

00:01:19.972 --> 00:01:22.782
seven tens plus two tens.

00:01:22.782 --> 00:01:25.289
Well if you combine all of these together -

00:01:25.289 --> 00:01:27.936
if you combine all of these together you

00:01:27.936 --> 00:01:30.867
are going to have nine of these groups of ten.

00:01:30.867 --> 00:01:32.980
We're gonna have nine tens.

00:01:32.980 --> 00:01:36.811
Seven tens plus two tens is nine tens.

00:01:36.811 --> 00:01:38.832
So we're left with nine tens which

00:01:38.832 --> 00:01:42.291
we see right over here and five ones.

00:01:42.291 --> 00:01:44.288
And five ones.

00:01:44.288 --> 00:01:46.308
Well what's another way of saying nine tens

00:01:46.308 --> 00:01:47.771
and five ones?

00:01:47.771 --> 00:01:52.083
Well that's 95, and we're done.


Adding 1 vs. adding 10

WEBVTT
Kind: captions
Language: en

00:00:00.236 --> 00:00:02.210
- [Voiceover] So, pause this video and, real quick,

00:00:02.210 --> 00:00:06.157
figure out what 27 plus 1 is, and then, if possible,

00:00:06.157 --> 00:00:09.982
figure out what 27 plus 10 is.

00:00:09.982 --> 00:00:12.788
Alright, so, let's think about it together.

00:00:12.788 --> 00:00:14.948
So, you might've been able to do this one pretty easily.

00:00:14.948 --> 00:00:16.782
You might've said, "Okay! If I'm adding 1,

00:00:16.782 --> 00:00:19.661
"it's just going to be the next highest number from 27.

00:00:19.661 --> 00:00:22.048
"I would go to 28 if I have 1 more."

00:00:22.048 --> 00:00:24.323
But, the reason why I'm doing this exercise is,

00:00:24.323 --> 00:00:27.418
we really think about place value.

00:00:27.418 --> 00:00:29.468
So, over here, the number 27,

00:00:29.468 --> 00:00:34.506
we have a 2 in the ten's place, this is in the ten's place,

00:00:34.506 --> 00:00:39.530
right over here, and then we have a 7 in the one's place.

00:00:39.530 --> 00:00:43.297
So, you could visualize the number 27 as 2 tens,

00:00:43.297 --> 00:00:45.363
and I have 2 groups of 10 here,

00:00:45.363 --> 00:00:48.521
and then 7 ones, and I have 7 ones right over here.

00:00:48.521 --> 00:00:50.077
And then, if I add 1 more,

00:00:50.077 --> 00:00:52.236
that's 1 more of these individual ones,

00:00:52.236 --> 00:00:54.001
and so, what am I going to be left with?

00:00:54.001 --> 00:00:56.880
Well, I'm still going to have 2 tens,

00:00:56.880 --> 00:00:59.713
I'm still going to have 2 tens,

00:00:59.713 --> 00:01:02.104
but how many ones am I going to have now?

00:01:02.104 --> 00:01:06.994
Oh, 1, 2, 3, 4, 5, 6, 7, 8 ones.

00:01:06.994 --> 00:01:09.149
So, it's going to be 28.

00:01:09.149 --> 00:01:11.819
So, that might not be surprising for you,

00:01:11.819 --> 00:01:14.002
but it's really valuable to think about

00:01:14.002 --> 00:01:16.440
in terms of place value because it can be very useful

00:01:16.440 --> 00:01:19.050
later on as you learn more math.

00:01:19.050 --> 00:01:21.443
Now, let's do the same thing with 27 plus 10.

00:01:21.443 --> 00:01:25.947
We already said 27, that's 2 tens and 7 ones,

00:01:25.947 --> 00:01:29.407
and now, with 10, the 1 isn't in the one's place,

00:01:29.407 --> 00:01:30.800
so it doesn't just represent 1,

00:01:30.800 --> 00:01:33.742
the 1 here is in the ten's place,

00:01:33.752 --> 00:01:38.765
so we have 1 10 and we have 0 ones.

00:01:38.765 --> 00:01:39.787
So, we just have--

00:01:39.787 --> 00:01:42.680
The number 10 is just a group of 10 here!

00:01:42.680 --> 00:01:44.676
So, when you add these together, what are you going to have?

00:01:44.676 --> 00:01:49.320
Well, now, you're going to have 1, 2, 3 tens,

00:01:49.320 --> 00:01:51.921
you're going to have 3 in the ten's place,

00:01:51.921 --> 00:01:55.339
and you're still going to have

00:01:55.339 --> 00:01:59.095
1, 2, 3, 4, 5, 6, 7 ones.

00:01:59.095 --> 00:02:01.472
7 ones.

00:02:01.472 --> 00:02:02.261
So, notice.

00:02:02.261 --> 00:02:04.003
Here, when you added a 1,

00:02:04.003 --> 00:02:06.882
you increased your one's place by 1.

00:02:06.882 --> 00:02:10.690
And, here, when you added 10, which is really 1 10,

00:02:10.690 --> 00:02:12.571
you didn't increase the one's place,

00:02:12.571 --> 00:02:15.212
you increased the ten's place.


Comparison word problems: roly-polies

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:03.663
- [Instructor] Lea has nine roly-polies.

00:00:03.663 --> 00:00:04.875
Let's write that down.

00:00:04.875 --> 00:00:08.375
Lea has nine roly-polies in her bug house.

00:00:09.774 --> 00:00:12.274
She has one more than Dingane,

00:00:14.916 --> 00:00:17.037
is I think how I would say that name.

00:00:17.037 --> 00:00:20.408
How many roly-polies does Dingane have?

00:00:20.408 --> 00:00:22.070
Dingane have.

00:00:22.070 --> 00:00:23.427
So that's what we need to figure out.

00:00:23.427 --> 00:00:25.636
So we need to figure
out how many roly-polies

00:00:25.636 --> 00:00:27.136
does Dingane have.

00:00:28.281 --> 00:00:31.823
So this right over here
is our question mark.

00:00:31.823 --> 00:00:33.271
Now it's always good.

00:00:33.271 --> 00:00:35.288
When you look at a word
problem like this you say,

00:00:35.288 --> 00:00:38.903
okay will Dingane have
more or less than Lea?

00:00:38.903 --> 00:00:43.033
Well it says she, Lea,
has one more than Dingane.

00:00:43.033 --> 00:00:45.349
So Lea is going to have more,

00:00:45.349 --> 00:00:46.578
and how much more?

00:00:46.578 --> 00:00:50.507
One more, so Dingane is going
to have one less than Lea.

00:00:50.507 --> 00:00:54.674
If Lea has one more, then
Dingane has one less than Lea.

00:00:55.679 --> 00:00:57.448
So how do we write one less?

00:00:57.448 --> 00:01:00.785
Well we could just subtract one.

00:01:00.785 --> 00:01:05.200
So this right over here,
this is one less, one less.

00:01:05.200 --> 00:01:09.329
This is saying that Dingane
has one less than Lea.

00:01:09.329 --> 00:01:11.217
So how much does Dingane have?

00:01:11.217 --> 00:01:14.025
Well he's going to have eight roly-polies.

00:01:14.025 --> 00:01:16.357
Eighth roly-polies.

00:01:16.357 --> 00:01:17.945
Now another way you could have done it,

00:01:17.945 --> 00:01:19.658
you could have drawn Lea's roly-polies.

00:01:19.658 --> 00:01:21.491
One, two, three, four,

00:01:22.562 --> 00:01:25.062
five, six, seven, eight, nine,

00:01:26.283 --> 00:01:28.027
and that would have been them rolled up,

00:01:28.027 --> 00:01:30.161
and you said okay that's
one more than Dingane.

00:01:30.161 --> 00:01:32.850
So Dingane has one less than this.

00:01:32.850 --> 00:01:34.284
So one less than this.

00:01:34.284 --> 00:01:36.169
You just take away one of these

00:01:36.169 --> 00:01:37.836
and you're left with

00:01:38.687 --> 00:01:42.951
one, two, three, four,
five, six, seven eight.

00:01:42.951 --> 00:01:45.207
You're left with these right over there.

00:01:45.207 --> 00:01:47.824
Let's do one more of these, woops.

00:01:47.824 --> 00:01:51.036
All right, let's do one more.

00:01:51.036 --> 00:01:54.667
So they tell us the mass of a small box

00:01:54.667 --> 00:01:58.483
is seven grams less than the large box.

00:01:58.483 --> 00:02:02.287
The mass of the small box is six grams.

00:02:02.287 --> 00:02:03.537
So let's write that.

00:02:03.537 --> 00:02:05.571
They tell us the small box,

00:02:05.571 --> 00:02:08.738
small box, it has a mass of six grams.

00:02:12.161 --> 00:02:15.023
And we need to figure out what is the mass

00:02:15.023 --> 00:02:16.440
of the large box.

00:02:18.284 --> 00:02:19.117
Large box.

00:02:21.168 --> 00:02:23.224
This is what we need to figure out.

00:02:23.224 --> 00:02:25.179
So this is a question mark.

00:02:25.179 --> 00:02:27.154
Now like we did in the last problem,

00:02:27.154 --> 00:02:30.422
is a large box gonna have a
larger mass or a smaller mass?

00:02:30.422 --> 00:02:31.956
And we shouldn't just look
at the names of the box,

00:02:31.956 --> 00:02:33.462
we should look at what they tell us.

00:02:33.462 --> 00:02:35.746
They tell us that the
mass of the small box

00:02:35.746 --> 00:02:38.996
is seven grams less than the large box.

00:02:40.076 --> 00:02:44.023
So this, the small box
is seven grams less than

00:02:44.023 --> 00:02:45.445
the large box.

00:02:45.445 --> 00:02:47.809
And that's another way of
saying that the large box

00:02:47.809 --> 00:02:50.942
is seven grams more than the small box.

00:02:50.942 --> 00:02:53.100
So another way to say seven grams more,

00:02:53.100 --> 00:02:56.640
the small box is six
grams plus seven grams.

00:02:56.640 --> 00:02:58.206
That's seven grams more.

00:02:58.206 --> 00:03:01.288
That's going to equal the
mass of the large box.

00:03:01.288 --> 00:03:03.626
So what's the mass of the large box?

00:03:03.626 --> 00:03:05.793
Well six plus seven is 13.

00:03:07.226 --> 00:03:10.393
Six plus seven is 13, and we're done.


Position-time graphs

WEBVTT
Kind: captions
Language: en

00:00:01.110 --> 00:00:03.182
- [Instructor] What we're
going to do in this video

00:00:03.182 --> 00:00:06.631
is think about different ways
to represent how position

00:00:06.631 --> 00:00:08.381
can change over time.

00:00:09.838 --> 00:00:11.210
So one of the more basic ways

00:00:11.210 --> 00:00:12.793
is through a table.

00:00:14.183 --> 00:00:16.463
For example right over
here in the left column

00:00:16.463 --> 00:00:18.920
I have time, maybe it's in seconds,

00:00:18.920 --> 00:00:21.170
and in the right column I have position

00:00:21.170 --> 00:00:24.810
and this could be in some
units, let's say it's in meters.

00:00:24.810 --> 00:00:26.831
So at time zero we're at three,

00:00:26.831 --> 00:00:30.150
after one second, we are still at three,

00:00:30.150 --> 00:00:32.546
after two seconds we're at negative one

00:00:32.546 --> 00:00:34.458
then after three seconds, we're at zero,

00:00:34.458 --> 00:00:36.147
after four seconds we're at zero,

00:00:36.147 --> 00:00:40.215
still at zero, after five
seconds we are at two,

00:00:40.215 --> 00:00:41.992
maybe two meters.

00:00:41.992 --> 00:00:44.650
Now this is somewhat useful,
but it's a little bit

00:00:44.650 --> 00:00:46.567
difficult to visualize.

00:00:47.434 --> 00:00:49.501
And it also doesn't
tell us what's happening

00:00:49.501 --> 00:00:52.398
in between these moments, what's happening

00:00:52.398 --> 00:00:54.481
at time half of a second.

00:00:55.454 --> 00:00:57.976
Did we just not move, did
our position just not change,

00:00:57.976 --> 00:01:00.745
or did it change and
then it got back to where

00:01:00.745 --> 00:01:02.972
it originally was after one second?

00:01:02.972 --> 00:01:05.990
We don't know when we
look at a table like this.

00:01:05.990 --> 00:01:07.834
But another way to think about it would be

00:01:07.834 --> 00:01:10.219
some type of animation.

00:01:10.219 --> 00:01:12.873
For example, let's say
we have our number line,

00:01:12.873 --> 00:01:17.355
and let's say the object
that's moving is a lemon.

00:01:17.355 --> 00:01:20.541
And so at time zero, it
starts at position three,

00:01:20.541 --> 00:01:22.997
so that's where it is
right now, and let's see

00:01:22.997 --> 00:01:24.199
if we can animate it.

00:01:24.199 --> 00:01:26.002
I'm just gonna try to
count off five seconds

00:01:26.002 --> 00:01:29.529
and move the lemon
accordingly to what we see

00:01:29.529 --> 00:01:33.696
on this position timetable
or time position table.

00:01:34.657 --> 00:01:35.490
Zero

00:01:36.395 --> 00:01:37.929
one

00:01:37.929 --> 00:01:38.762
two

00:01:41.267 --> 00:01:42.100
three

00:01:43.098 --> 00:01:43.931
four

00:01:44.901 --> 00:01:45.734
five.

00:01:47.688 --> 00:01:50.857
So that was somewhat useful,
but maybe even more useful

00:01:50.857 --> 00:01:53.243
thing would be to graph this somehow

00:01:53.243 --> 00:01:54.861
so that we don't have to keep looking

00:01:54.861 --> 00:01:56.699
at animation so that we can just look at

00:01:56.699 --> 00:02:00.574
with our eyes what happens over time.

00:02:00.574 --> 00:02:02.796
So for that, we can construct what's known

00:02:02.796 --> 00:02:04.879
as a position time graph.

00:02:06.438 --> 00:02:09.798
Typically, time is on your horizontal axis

00:02:09.798 --> 00:02:12.912
and position is on your vertical axis.

00:02:12.912 --> 00:02:15.345
So let's think about this a little bit.

00:02:15.345 --> 00:02:18.982
So at time equals zero,
our position is at three.

00:02:18.982 --> 00:02:22.558
So at time zero, our position is at three,

00:02:22.558 --> 00:02:26.641
and then at time equal
one, we're at three again,

00:02:27.821 --> 00:02:30.821
at time two, we are at negative one,

00:02:31.763 --> 00:02:35.546
at time two, our position is negative one,

00:02:35.546 --> 00:02:38.546
at time three, our position is zero,

00:02:40.232 --> 00:02:41.388
so our position is zero.

00:02:41.388 --> 00:02:42.968
Remember, even though we're thinking about

00:02:42.968 --> 00:02:46.815
left right here, here position is up down.

00:02:46.815 --> 00:02:51.115
So here our position
is zero at time three,

00:02:51.115 --> 00:02:55.785
and then at time four, our
position is still zero,

00:02:55.785 --> 00:02:59.618
and then at time five,
our position is at two.

00:03:02.459 --> 00:03:04.956
Our position is at two.

00:03:04.956 --> 00:03:08.554
So for the first second,
I don't have a change

00:03:08.554 --> 00:03:10.515
in position or at least
that's what I assumed

00:03:10.515 --> 00:03:14.328
when I animated the
lemon, and then as I go

00:03:14.328 --> 00:03:17.575
from the first second
to the second second,

00:03:17.575 --> 00:03:20.044
my position went from
three to negative one,

00:03:20.044 --> 00:03:22.293
from three to negative
one, and if we do that

00:03:22.293 --> 00:03:24.764
at a constant rate we would have a line

00:03:24.764 --> 00:03:27.113
that looks something
like this, I'm trying,

00:03:27.113 --> 00:03:29.343
that's supposed to be a straight line,

00:03:29.343 --> 00:03:32.636
and then from time two to
three, we go from position

00:03:32.636 --> 00:03:37.239
negative one to zero,
from negative one to zero.

00:03:37.239 --> 00:03:39.895
Here, it would've been
going from negative one

00:03:39.895 --> 00:03:43.737
to zero moving one to
the right, but over here,

00:03:43.737 --> 00:03:46.343
since we're plotting our
position on the vertical axis,

00:03:46.343 --> 00:03:48.702
it looks like we went up
but this is just really

00:03:48.702 --> 00:03:52.342
going from position negative
one to position zero

00:03:52.342 --> 00:03:55.470
from time two seconds to three seconds.

00:03:55.470 --> 00:03:57.503
Now from three to four, at
least the way I depicted it,

00:03:57.503 --> 00:04:00.604
our position does not change,
and then from time four

00:04:00.604 --> 00:04:04.271
to five, our position
goes from zero to two,

00:04:05.272 --> 00:04:06.689
from zero to two.

00:04:08.052 --> 00:04:10.166
And so what I have
constructed here is known

00:04:10.166 --> 00:04:12.858
as a position time graph, and from this,

00:04:12.858 --> 00:04:15.549
without an animation, you can immediately

00:04:15.549 --> 00:04:18.163
get an understanding of
how the thing's position

00:04:18.163 --> 00:04:20.142
has changed over time.

00:04:20.142 --> 00:04:22.359
So let's do the animation one more time,

00:04:22.359 --> 00:04:25.275
and just try to follow along
on the position time graph,

00:04:25.275 --> 00:04:27.669
and maybe I'll slow it down a little bit.

00:04:27.669 --> 00:04:29.769
So for the first second
we're gonna be stationary,

00:04:29.769 --> 00:04:33.186
so we can just count off one Mississippi.

00:04:35.186 --> 00:04:39.090
And then we go to, our
position goes to negative one

00:04:39.090 --> 00:04:43.257
over the next second, so then
we would go two Mississippi.

00:04:45.359 --> 00:04:49.378
And then we would go three Mississippi,

00:04:49.378 --> 00:04:53.045
four Mississippi, and
then five Mississippi.

00:04:56.117 --> 00:04:57.117
But hopefully you get an appreciation

00:04:57.117 --> 00:04:59.335
that this is just the way
of immediately glancing

00:04:59.335 --> 00:05:00.393
and seeing what's happening.


Distance and displacement in one dimension

WEBVTT
Kind: captions
Language: en

00:00:00.208 --> 00:00:01.148
- [Instructor] Previous videos

00:00:01.148 --> 00:00:03.488
we've talked a little bit
about distance traveled

00:00:03.488 --> 00:00:04.958
versus displacement.

00:00:04.958 --> 00:00:07.027
What I'm gonna do in
this video is discuss it

00:00:07.027 --> 00:00:09.777
on a one dimensional number line.

00:00:10.914 --> 00:00:14.335
And we'll get a little bit
more mathy in this video.

00:00:14.335 --> 00:00:17.128
So here is my number line.

00:00:17.128 --> 00:00:19.664
Let's say that this is a zero,

00:00:19.664 --> 00:00:20.497
one, two

00:00:21.795 --> 00:00:22.868
three,

00:00:22.868 --> 00:00:24.108
four and it keeps going on

00:00:24.108 --> 00:00:25.390
and then in the negative direction,

00:00:25.390 --> 00:00:26.992
negative one,

00:00:26.992 --> 00:00:28.705
negative two,

00:00:28.705 --> 00:00:30.983
and negative three.

00:00:30.983 --> 00:00:33.811
And let's say that I start off

00:00:33.811 --> 00:00:34.894
with a lemon.

00:00:36.591 --> 00:00:40.582
Let's say my lemon starts
off right over here at zero

00:00:40.582 --> 00:00:42.533
on my number line.

00:00:42.533 --> 00:00:46.810
And let's say it first
moves two to the right

00:00:46.810 --> 00:00:48.781
so at first moves two to the right.

00:00:48.781 --> 00:00:51.286
I'll denote that by plus two

00:00:51.286 --> 00:00:53.619
and then from there, it moves

00:00:53.619 --> 00:00:55.036
three to the left

00:00:56.016 --> 00:00:58.530
and then it moves three to the left.

00:00:58.530 --> 00:01:01.379
And I will use negative for the left.

00:01:01.379 --> 00:01:03.838
So it moves three to the left.

00:01:03.838 --> 00:01:06.269
And then let's say that it then

00:01:06.269 --> 00:01:07.102
moves

00:01:08.227 --> 00:01:10.227
another one to the left.

00:01:11.849 --> 00:01:13.939
So then it goes another one to the left

00:01:13.939 --> 00:01:15.639
and I'll denote negative one

00:01:15.639 --> 00:01:17.965
as moving one to the left.

00:01:17.965 --> 00:01:20.466
So based on what we know
about distance traveled

00:01:20.466 --> 00:01:21.866
and displacement,

00:01:21.866 --> 00:01:25.449
what is the distance
traveled for this dot?

00:01:26.659 --> 00:01:28.616
Distance traveled.

00:01:28.616 --> 00:01:29.731
Pause the video

00:01:29.731 --> 00:01:32.648
and see if you can figure that out.

00:01:33.924 --> 00:01:34.774
Or remember,

00:01:34.774 --> 00:01:38.654
distance traveled is
the entire path length.

00:01:38.654 --> 00:01:42.569
Or the entire length of the dot's journey.

00:01:42.569 --> 00:01:45.425
So this is going to be equal to

00:01:45.425 --> 00:01:46.435
two to the right.

00:01:46.435 --> 00:01:47.787
So plus two

00:01:47.787 --> 00:01:50.510
and then three to the left.

00:01:50.510 --> 00:01:52.823
Now this is an important notion.

00:01:52.823 --> 00:01:54.962
When we talk about distances,

00:01:54.962 --> 00:01:58.069
we wouldn't say positive or negative.

00:01:58.069 --> 00:02:01.402
We just care about the absolute value of

00:02:02.422 --> 00:02:04.714
the amount that we are traveling.

00:02:04.714 --> 00:02:07.243
So we won't specify a direction.

00:02:07.243 --> 00:02:08.169
Now you might say, "hey,

00:02:08.169 --> 00:02:09.918
"where is the direction being specified?"

00:02:09.918 --> 00:02:13.122
well implicitly, whether
something is positive or negative

00:02:13.122 --> 00:02:14.062
on this number line

00:02:14.062 --> 00:02:15.539
is giving a direction.

00:02:15.539 --> 00:02:17.100
But if we're talking about distances,

00:02:17.100 --> 00:02:19.545
we wouldn't pay attention
to the direction.

00:02:19.545 --> 00:02:22.993
We only care about the magnitude.

00:02:22.993 --> 00:02:24.888
So this would be two

00:02:24.888 --> 00:02:26.399
plus three

00:02:26.399 --> 00:02:27.709
plus one.

00:02:27.709 --> 00:02:29.520
Doesn't matter if this is one to the left

00:02:29.520 --> 00:02:30.353
or one to the right.

00:02:30.353 --> 00:02:32.383
Doesn't even matter if it's
positive one or negative one.

00:02:32.383 --> 00:02:34.423
We care about its absolute value.

00:02:34.423 --> 00:02:36.506
We care about its magnitude.

00:02:36.506 --> 00:02:39.069
So the distance traveled in this example

00:02:39.069 --> 00:02:41.939
is going to be six units.

00:02:41.939 --> 00:02:45.353
Whatever the units are on my
number line right over here.

00:02:45.353 --> 00:02:46.906
If these are in meters

00:02:46.906 --> 00:02:48.870
then this would be six meters.

00:02:48.870 --> 00:02:50.870
Now what is the displacement?

00:02:50.870 --> 00:02:55.725
Now remember, displacement
is net change in position.

00:02:55.725 --> 00:02:56.808
Displacement.

00:02:58.783 --> 00:03:00.880
What is that going to be?

00:03:00.880 --> 00:03:04.432
Pause the video and see
if you can figure it out.

00:03:04.432 --> 00:03:07.114
Well displacement is going to be

00:03:07.114 --> 00:03:09.093
you could view this as equal to

00:03:09.093 --> 00:03:11.148
your final position

00:03:11.148 --> 00:03:12.116
and we'll use X.

00:03:12.116 --> 00:03:14.045
Let's say this is the X-axis.

00:03:14.045 --> 00:03:16.762
So we'll say X final, your final position

00:03:16.762 --> 00:03:19.270
minus your initial position.

00:03:19.270 --> 00:03:22.934
It's really just your change in position!

00:03:22.934 --> 00:03:25.184
So what is your change in position here?

00:03:25.184 --> 00:03:27.490
Well your final position is

00:03:27.490 --> 00:03:31.251
you are at negative two
at x equals negative two.

00:03:31.251 --> 00:03:33.121
And then what was your initial position?

00:03:33.121 --> 00:03:35.601
Your initial position,
you started at zero.

00:03:35.601 --> 00:03:40.121
So negative two minus zero
is equal to negative two.

00:03:40.121 --> 00:03:43.169
So how would we visualize
that on our drawing here?

00:03:43.169 --> 00:03:44.767
Well we started here.

00:03:44.767 --> 00:03:47.644
Just think about what is
your net change in position?

00:03:47.644 --> 00:03:48.765
You started here.

00:03:48.765 --> 00:03:50.820
And regardless of what your path was,

00:03:50.820 --> 00:03:53.237
you ended up two to the left.

00:03:55.042 --> 00:03:56.588
So your

00:03:56.588 --> 00:03:59.421
your displacement is negative two.

00:04:00.294 --> 00:04:02.962
Now displacement, we care

00:04:02.962 --> 00:04:04.439
not just about the magnitude.

00:04:04.439 --> 00:04:07.305
We care about the magnitude
and the direction.

00:04:07.305 --> 00:04:09.008
So you might be saying, well,

00:04:09.008 --> 00:04:10.673
where is the direction specified

00:04:10.673 --> 00:04:12.958
if I just say negative two?

00:04:12.958 --> 00:04:15.556
Well, the sign in a one dimensional case

00:04:15.556 --> 00:04:18.029
is giving us our direction.

00:04:18.029 --> 00:04:21.004
So the sign is giving us a direction.

00:04:21.004 --> 00:04:24.681
I started off implicitly with this notion

00:04:24.681 --> 00:04:27.417
that negative is to the left

00:04:27.417 --> 00:04:29.242
and positive is to the right.

00:04:29.242 --> 00:04:30.984
And we're in this one dimensional world.

00:04:30.984 --> 00:04:33.909
And those are the only two
directions that I can travel in.

00:04:33.909 --> 00:04:35.811
So if I'm in this one dimensional world

00:04:35.811 --> 00:04:38.277
or if I'm thinking about
just one dimension,

00:04:38.277 --> 00:04:41.050
the sign gives me my direction.

00:04:41.050 --> 00:04:42.965
So that's why displacement

00:04:42.965 --> 00:04:46.323
where I care about the
magnitude and the direction,

00:04:46.323 --> 00:04:48.420
I do care about the sign.

00:04:48.420 --> 00:04:49.628
While distance,

00:04:49.628 --> 00:04:52.369
where I only care about the magnitude,

00:04:52.369 --> 00:04:55.253
I don't care about the sign.

00:04:55.253 --> 00:04:57.803
So I just keep adding up the magnitudes

00:04:57.803 --> 00:05:00.401
while over here, another
way you can think about it,

00:05:00.401 --> 00:05:04.037
you first get displaced
by two to the right

00:05:04.037 --> 00:05:05.549
so that's plus two.

00:05:05.549 --> 00:05:07.290
The plus says to the right.

00:05:07.290 --> 00:05:09.520
Then you get displaced
by three to the left.

00:05:09.520 --> 00:05:11.853
So that is minus three.

00:05:11.853 --> 00:05:15.071
And then you get displaced
by one to the left again.

00:05:15.071 --> 00:05:16.513
So that's minus one.

00:05:16.513 --> 00:05:17.830
That's why we're talking
about displacement.

00:05:17.830 --> 00:05:19.815
That's why we care about the sign.

00:05:19.815 --> 00:05:22.393
And if you are to add
all of these together,

00:05:22.393 --> 00:05:25.186
you are going to get a net displacement

00:05:25.186 --> 00:05:26.363
of negative two.

00:05:26.363 --> 00:05:28.934
But an easier way was just
what's your final position

00:05:28.934 --> 00:05:31.267
minus your initial position?


Distance and displacement introduction

WEBVTT
Kind: captions
Language: en

00:00:00.187 --> 00:00:03.428
- [Instructor] So let's
say we have a sheep

00:00:03.428 --> 00:00:05.500
and it is hungry.

00:00:05.500 --> 00:00:07.836
So that is my sheep.

00:00:07.836 --> 00:00:10.626
My best, quick drawing of a sheep

00:00:10.626 --> 00:00:12.748
and it is just following the grass

00:00:12.748 --> 00:00:15.419
wherever it finds good grass to eat

00:00:15.419 --> 00:00:18.687
and so in pursuit of tasty grass,

00:00:18.687 --> 00:00:22.020
it first goes 10 kilometers to the east,

00:00:25.441 --> 00:00:26.941
10 kilometers east

00:00:28.553 --> 00:00:31.053
and then it takes a right turn

00:00:32.067 --> 00:00:35.590
and then it goes five kilometers south,

00:00:35.590 --> 00:00:37.340
five kilometers south

00:00:39.269 --> 00:00:41.609
and then it takes another right turn

00:00:41.609 --> 00:00:44.192
and it goes 10 kilometers west,

00:00:46.771 --> 00:00:48.354
10 kilometers west.

00:00:50.873 --> 00:00:55.040
My question to you is how
far has this sheep traveled?

00:00:57.756 --> 00:01:00.860
And pause the video and see
if you can figure that out.

00:01:00.860 --> 00:01:03.527
How far has that sheep traveled?

00:01:05.614 --> 00:01:08.242
Alright, now let's try
to answer this together

00:01:08.242 --> 00:01:10.272
and it's a little bit of a trick question

00:01:10.272 --> 00:01:13.761
because depending how
you interpret the idea

00:01:13.761 --> 00:01:16.167
how far something has traveled,

00:01:16.167 --> 00:01:19.391
there could actually be
two valid answers here.

00:01:19.391 --> 00:01:20.371
One answer is you could say

00:01:20.371 --> 00:01:23.608
what is the total distance traveled?

00:01:23.608 --> 00:01:25.120
So let me write this one down.

00:01:25.120 --> 00:01:27.134
This is an important concept.

00:01:27.134 --> 00:01:28.551
Distance traveled

00:01:31.303 --> 00:01:34.375
and this would be the
total length of its journey

00:01:34.375 --> 00:01:37.032
or the total length of its path.

00:01:37.032 --> 00:01:40.546
So the distance traveled in this situation

00:01:40.546 --> 00:01:44.213
would be 10 kilometers
plus five kilometers,

00:01:46.865 --> 00:01:50.198
plus five kilometers plus 10 kilometers,

00:01:51.995 --> 00:01:56.162
plus 10 kilometers which
would give us 25 kilometers.

00:01:58.441 --> 00:02:00.070
That would be 25 kilometers.

00:02:00.070 --> 00:02:01.659
So some of you all might have said

00:02:01.659 --> 00:02:05.386
that the sheep traveled 25 kilometers.

00:02:05.386 --> 00:02:06.693
If that's what you came up with,

00:02:06.693 --> 00:02:11.556
what you calculated is the
total distance traveled.

00:02:11.556 --> 00:02:14.334
Now, some of you might have
said, wait, hold on a second,

00:02:14.334 --> 00:02:16.214
the sheep was here before

00:02:16.214 --> 00:02:19.355
and then it ends up right over here

00:02:19.355 --> 00:02:21.155
at the end of its journey

00:02:21.155 --> 00:02:24.067
and so its change in position
is it would have moved

00:02:24.067 --> 00:02:26.412
a net five kilometers south.

00:02:26.412 --> 00:02:28.462
From this point to this point

00:02:28.462 --> 00:02:31.629
would have been five kilometers south.

00:02:32.809 --> 00:02:35.089
It doesn't matter what
it did along the way,

00:02:35.089 --> 00:02:36.673
what its path was.

00:02:36.673 --> 00:02:40.016
At the end of the day, it
ended up five kilometers south

00:02:40.016 --> 00:02:44.265
of where it started out and
this would also be a valid way

00:02:44.265 --> 00:02:46.557
of saying how far it traveled

00:02:46.557 --> 00:02:49.974
and this notion is known as displacement.

00:02:51.881 --> 00:02:56.141
So the displacement would
be in this situation

00:02:56.141 --> 00:02:57.974
five kilometers south.

00:03:00.765 --> 00:03:01.916
So pause this video.

00:03:01.916 --> 00:03:05.768
What do you think is the
definition or the difference

00:03:05.768 --> 00:03:09.161
between saying the
distance something traveled

00:03:09.161 --> 00:03:11.161
versus its displacement?

00:03:12.851 --> 00:03:15.997
Well, as we mentioned, the
distance something travels

00:03:15.997 --> 00:03:18.703
is the entire length of its journey.

00:03:18.703 --> 00:03:22.870
It's the entire path and this
thing will never be negative.

00:03:24.053 --> 00:03:28.745
You could travel zero kilometers
but as long as you are,

00:03:28.745 --> 00:03:32.444
your position keeps changing,
this thing will only increase

00:03:32.444 --> 00:03:35.589
because it's the total
length of your path.

00:03:35.589 --> 00:03:39.844
Your displacement is
your change in position

00:03:39.844 --> 00:03:41.965
and since it's a change in position,

00:03:41.965 --> 00:03:45.245
not only will we mention
the five kilometers,

00:03:45.245 --> 00:03:47.081
we'll say in what direction.

00:03:47.081 --> 00:03:50.558
The position of the sheep
has changed five kilometers

00:03:50.558 --> 00:03:52.027
to the south.

00:03:52.027 --> 00:03:53.819
So let me write this down.

00:03:53.819 --> 00:03:56.236
This is total length of path,

00:03:57.491 --> 00:04:01.241
total length of path is
the distance traveled

00:04:04.943 --> 00:04:09.737
while the displacement is
the change in position,

00:04:09.737 --> 00:04:11.320
change in position.