Solve a scale drawing word problem

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:02.690
Sally is an
architect who creates

00:00:02.690 --> 00:00:05.320
a blueprint of a
rectangular dining room.

00:00:05.320 --> 00:00:07.640
The area of the
actual dining room

00:00:07.640 --> 00:00:14.470
is 1,600 times
larger than the area

00:00:14.470 --> 00:00:16.690
of the dining room
on the blueprint.

00:00:16.690 --> 00:00:21.070
The length of the dining room
on the blueprint is 3 inches.

00:00:21.070 --> 00:00:25.160
What is the length of the
actual dining room in feet?

00:00:25.160 --> 00:00:27.760
So there's a couple of really
interesting things going on

00:00:27.760 --> 00:00:28.260
here.

00:00:28.260 --> 00:00:30.660
They give us the dimensions
of the blueprint in inches.

00:00:30.660 --> 00:00:33.130
We want the actual
length in feet.

00:00:33.130 --> 00:00:36.010
And then they tell us that the
area of the actual dining room

00:00:36.010 --> 00:00:38.120
is 1,600 times larger.

00:00:38.120 --> 00:00:40.590
So they're not saying that
the scale of the blueprint

00:00:40.590 --> 00:00:42.540
is at 1/1600.

00:00:42.540 --> 00:00:44.290
It's going to be
something less than that,

00:00:44.290 --> 00:00:48.956
and let's think about what
that scale is going to be.

00:00:48.956 --> 00:00:50.830
Let's just think about
some different scales.

00:00:54.990 --> 00:00:57.670
Let's say that this
is my blueprint,

00:00:57.670 --> 00:01:02.480
and this is the actual
reality of the dining room

00:01:02.480 --> 00:01:05.069
that we're thinking about.

00:01:05.069 --> 00:01:07.500
And my blueprint
is let's just say 1

00:01:07.500 --> 00:01:11.320
by 1, just for the
sake of argument.

00:01:11.320 --> 00:01:19.740
Now, if this was a 1 by 1 square
and we increased the dimensions

00:01:19.740 --> 00:01:28.170
by a factor of 2, so
it's a 2 by 2 square,

00:01:28.170 --> 00:01:30.370
what's the area going to be?

00:01:30.370 --> 00:01:32.690
Well, this area
is going to be 4.

00:01:32.690 --> 00:01:35.080
This area is 1, this area is 4.

00:01:35.080 --> 00:01:38.265
So you notice that if we
increase by a factor of 2,

00:01:38.265 --> 00:01:41.960
it increase our area
by a factor of 4.

00:01:41.960 --> 00:01:44.722
Or another way of saying, if we
increase each of our dimensions

00:01:44.722 --> 00:01:46.430
by a factor of 2,
we're going to increase

00:01:46.430 --> 00:01:48.510
our area by a factor of 4.

00:01:48.510 --> 00:01:52.530
If instead we increased
each of our dimensions

00:01:52.530 --> 00:01:59.180
by a factor of 3, this
would be a 3 by 3 square,

00:01:59.180 --> 00:02:04.670
and we would increase our
area by a factor of 9.

00:02:04.670 --> 00:02:08.539
So notice, whatever factor
we're increasing the area by,

00:02:08.539 --> 00:02:10.250
it's going to be the
factor that we're

00:02:10.250 --> 00:02:13.740
increasing the
dimensions by squared.

00:02:13.740 --> 00:02:15.700
So let's just think
about it that way.

00:02:15.700 --> 00:02:17.670
So they're telling us
that we're increasing

00:02:17.670 --> 00:02:21.450
the area by 1,600 times.

00:02:21.450 --> 00:02:24.610
Actually, let me just clean
this thing up a little bit.

00:02:24.610 --> 00:02:32.810
So one way we could imagine
it, if our drawing did

00:02:32.810 --> 00:02:36.100
have an area of 1,
which we can't assume,

00:02:36.100 --> 00:02:39.260
but we could for the
sake of just figuring out

00:02:39.260 --> 00:02:41.870
what the scale of
the drawing is.

00:02:41.870 --> 00:02:43.640
Let me clear all of this here.

00:02:49.470 --> 00:02:57.100
So the area of the actual dining
room is 1,600 times larger,

00:02:57.100 --> 00:03:00.220
and so if the drawing
had an area of 1,

00:03:00.220 --> 00:03:02.010
then the area of the
actual dining room

00:03:02.010 --> 00:03:04.850
would be 1,600 So
what would I have

00:03:04.850 --> 00:03:07.420
to multiply each of
the dimensions by

00:03:07.420 --> 00:03:11.160
to get an area factor of 1,600?

00:03:11.160 --> 00:03:13.180
Well, if I multiply
this dimension

00:03:13.180 --> 00:03:19.350
by 40 and this dimension by 40,
we see 40 times 40 is 1,600.

00:03:19.350 --> 00:03:22.010
You might say, hey, Sal,
how did you figure out 40?

00:03:22.010 --> 00:03:24.160
Well, the 16 is a big clue.

00:03:24.160 --> 00:03:27.870
We know that 4 times
4 is equal to 16,

00:03:27.870 --> 00:03:31.020
and so if you gave a 0 to each
of these 4's, if you made it

00:03:31.020 --> 00:03:35.500
40 times 40, then that
is going to be 1,600.

00:03:35.500 --> 00:03:37.790
So this information
right over here

00:03:37.790 --> 00:03:47.170
tells us that the scale
factor of the lengths is 40.

00:03:47.170 --> 00:03:52.400
That would result in an scale
factor for the area of 1,600.

00:03:52.400 --> 00:03:54.030
So that's a good starting point.

00:03:54.030 --> 00:03:57.820
Now let's go to the actual
dining room on the blueprint.

00:03:57.820 --> 00:03:59.570
So the actual dining
room on the blueprint

00:03:59.570 --> 00:04:00.819
doesn't have these dimensions.

00:04:00.819 --> 00:04:03.530
We just used that to figure
out the scaling factor.

00:04:03.530 --> 00:04:05.430
The actual dining
room on the blueprint

00:04:05.430 --> 00:04:07.217
has a length of 3 inches.

00:04:07.217 --> 00:04:08.800
So maybe it looks
something like this.

00:04:08.800 --> 00:04:10.758
They don't give us any
of the other dimensions,

00:04:10.758 --> 00:04:15.870
so we can even imagine a 3 inch
by 2 inch, 1 inch, whatever

00:04:15.870 --> 00:04:16.390
we want.

00:04:16.390 --> 00:04:20.649
We could even imagine a
3 inch by 3 inch square.

00:04:20.649 --> 00:04:22.520
They only care about the length.

00:04:22.520 --> 00:04:25.865
Now let's multiply both of
these by a factor of 40.

00:04:25.865 --> 00:04:27.490
And we only care
about the length here.

00:04:27.490 --> 00:04:29.823
They actually say what's the
length of the actual dining

00:04:29.823 --> 00:04:30.390
room.

00:04:30.390 --> 00:04:33.200
So let's multiply it,
and obviously, this

00:04:33.200 --> 00:04:35.460
is not drawn to scale.

00:04:35.460 --> 00:04:37.960
Let's multiply this
times a factor of 40.

00:04:41.300 --> 00:04:46.860
So 3 times 40 is 120,
and this, of course,

00:04:46.860 --> 00:04:49.270
is what we're referring
to as the length.

00:04:49.270 --> 00:04:51.270
Now, you might be tempted
to say OK, we're done.

00:04:51.270 --> 00:04:52.500
This will be 120.

00:04:52.500 --> 00:04:56.125
But remember, this
is 120 inches.

00:04:59.280 --> 00:05:03.190
So what is 120 inches
in terms of feet?

00:05:03.190 --> 00:05:09.440
Well, 1 foot is
equal to 12 inches.

00:05:09.440 --> 00:05:12.110
If we were to multiply
both of these times 10,

00:05:12.110 --> 00:05:17.612
we know that 10 feet
is equal to 120 inches.

00:05:17.612 --> 00:05:19.570
Or another way you could
have thought about it,

00:05:19.570 --> 00:05:28.270
you have 120 inches divided
by 12 inches per foot

00:05:28.270 --> 00:05:29.690
is going to give you 10.

00:05:29.690 --> 00:05:35.555
So 120 divided by-- 120 inches--
let me write it this way.

00:05:35.555 --> 00:05:42.645
120 inches divided
by 12 inches per foot

00:05:42.645 --> 00:05:46.980
is going to give you 10 feet.

00:05:46.980 --> 00:05:51.010
So that's the actual length
of the dining room in feet.


Interpreting a scale drawing

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:05.560
Maya and Mabel are
inspecting a 80 to 1

00:00:05.560 --> 00:00:08.280
scale floor plan
of their new house.

00:00:08.280 --> 00:00:11.450
The dimensions of the living
room in the scaled plan

00:00:11.450 --> 00:00:21.160
are 4 centimeters by 5
centimeters right over here.

00:00:21.160 --> 00:00:28.169
What is the area of the
living room in the real world?

00:00:28.169 --> 00:00:30.210
So they gave us these
dimensions right over here.

00:00:30.210 --> 00:00:32.000
This is the scale
plan, and then we

00:00:32.000 --> 00:00:34.850
could figure out these
dimensions in the real world

00:00:34.850 --> 00:00:36.840
by looking at the scale
factor right over here.

00:00:36.840 --> 00:00:40.040
It's an 80 to 1
scale floor plan.

00:00:40.040 --> 00:00:41.720
And we can assume
that the house is

00:00:41.720 --> 00:00:44.360
much bigger than the floor plan.

00:00:44.360 --> 00:00:50.620
So the 80, for every
80 units in the house,

00:00:50.620 --> 00:00:55.510
that represents 1 unit
on the floor plan.

00:00:55.510 --> 00:00:57.950
So if we had 80
meters in the house,

00:00:57.950 --> 00:01:01.220
that would be represented as
1 meter on the floor plan.

00:01:01.220 --> 00:01:04.590
If we had 80 centimeters
in the house,

00:01:04.590 --> 00:01:09.270
that would be represented by 1
centimeter in the floor plan.

00:01:09.270 --> 00:01:10.790
And it goes the
other way around.

00:01:10.790 --> 00:01:12.930
1 centimeter on the
floor plan would

00:01:12.930 --> 00:01:15.009
represent 80 centimeters
in the house.

00:01:15.009 --> 00:01:17.300
And it's always important to
do-- if this confuses you,

00:01:17.300 --> 00:01:20.710
just always do a reality check
that the house should be bigger

00:01:20.710 --> 00:01:22.340
than the floor plan.

00:01:22.340 --> 00:01:24.900
So if the floor plan for this
dimension of our living room

00:01:24.900 --> 00:01:28.850
is 4 centimeters, the actual
house will be 80 times that.

00:01:28.850 --> 00:01:34.350
And 80 times 4 is 320-- let
me do that in a blue color--

00:01:34.350 --> 00:01:36.475
is equal to 320 centimeters.

00:01:39.640 --> 00:01:42.780
And we can do the same thing for
the length of the living room.

00:01:42.780 --> 00:01:47.270
So 80 times 5
centimeters is going

00:01:47.270 --> 00:01:53.820
to get us to-- is going to be--
80 times 5 is 400 centimeters.

00:01:53.820 --> 00:01:55.880
So we could figure out
the area of this room

00:01:55.880 --> 00:01:58.290
in centimeters, if we
like, and I guess, why not?

00:01:58.290 --> 00:02:01.200
It might be easier to
convert it to meters later.

00:02:01.200 --> 00:02:04.960
So let's see, 400 centimeters
times 320 centimeters.

00:02:04.960 --> 00:02:06.360
Let me write this down.

00:02:06.360 --> 00:02:11.980
400 times 320.

00:02:11.980 --> 00:02:13.240
Let's think about it.

00:02:13.240 --> 00:02:19.900
4 times 32 is going to
be 120, plus 8, 128.

00:02:19.900 --> 00:02:21.410
And I have 1, 2, 3 zeroes.

00:02:21.410 --> 00:02:22.970
1, 2, 3.

00:02:22.970 --> 00:02:30.080
So it's going to be 128,000
centimeters squared.

00:02:34.940 --> 00:02:37.730
Now that's a lot of
square centimeters.

00:02:37.730 --> 00:02:41.260
What would we do if we wanted
to convert it into meters?

00:02:41.260 --> 00:02:44.970
Well, we just have to figure out
how many square centimeters are

00:02:44.970 --> 00:02:47.820
there in a square meter.

00:02:47.820 --> 00:02:49.760
So let's think
about it this way.

00:02:49.760 --> 00:02:57.190
A meter is equal to-- 1 meter
is equal to 100 centimeters.

00:02:57.190 --> 00:03:01.810
So a square meter, so
that's right over there.

00:03:01.810 --> 00:03:06.420
1 meter squared would be
1 meter by 1 meter, which

00:03:06.420 --> 00:03:12.200
is the same thing as 100
centimeters by 100 centimeters.

00:03:12.200 --> 00:03:15.650
And so if you were to calculate
this area in centimeters,

00:03:15.650 --> 00:03:22.480
100 times 100 is 10,000, is
equal to 10,000 centimeters

00:03:22.480 --> 00:03:22.980
squared.

00:03:27.710 --> 00:03:30.190
So you have 10,000
square centimeters

00:03:30.190 --> 00:03:32.030
for every square meter.

00:03:32.030 --> 00:03:36.230
And so, if you want to convert
128,000 centimeters squared

00:03:36.230 --> 00:03:39.760
to meters squared, you
would divide by 10,000.

00:03:39.760 --> 00:03:53.077
So dividing that by 10,000 would
give us 12.8 square meters.

00:03:53.077 --> 00:03:54.660
Now, another way you
could've done it,

00:03:54.660 --> 00:03:56.243
and maybe this would
have been easier,

00:03:56.243 --> 00:03:57.560
is to convert it up here.

00:03:57.560 --> 00:03:58.970
Instead of saying
400 centimeters

00:03:58.970 --> 00:04:01.770
times 320 centimeters,
you would say, well,

00:04:01.770 --> 00:04:03.825
400 centimeters, that's
going to be 4 meters.

00:04:06.640 --> 00:04:10.585
And 320 centimeters,
well, that's 3.2 meters.

00:04:13.420 --> 00:04:15.330
And you would say,
OK, 4 times 3.2,

00:04:15.330 --> 00:04:18.350
that is 12.8 square meters.

00:04:18.350 --> 00:04:20.420
But either way, the
area of the living

00:04:20.420 --> 00:04:23.560
room in the real world in meters
squared, or square meters,

00:04:23.560 --> 00:04:25.280
is 12.8.


Banana proportionality

WEBVTT
Kind: captions
Language: en

00:00:00.404 --> 00:00:02.981
- [Voiceover] Today, Nate has 100 bananas.

00:00:02.981 --> 00:00:06.050
He will eat two of them every day.

00:00:06.050 --> 00:00:10.454
Is the number of bananas Nate has left

00:00:10.454 --> 00:00:14.499
proportional to the number of days that pass?

00:00:14.499 --> 00:00:16.030
And I encourage you to pause this video

00:00:16.030 --> 00:00:17.741
and think about this.

00:00:17.741 --> 00:00:18.700
And what's interesting here,

00:00:18.700 --> 00:00:20.976
they're not saying, is the number of bananas eaten,

00:00:20.976 --> 00:00:23.304
they're saying the number of bananas Nate has left,

00:00:23.304 --> 00:00:26.244
proportional to the number of days that pass.

00:00:26.244 --> 00:00:27.777
So let's draw a little table here to think

00:00:27.777 --> 00:00:29.913
about this a little bit more.

00:00:29.913 --> 00:00:32.025
So I'm gonna make three columns.

00:00:32.025 --> 00:00:34.416
I'm gonna make three columns.

00:00:35.926 --> 00:00:37.157
So in the first column,

00:00:37.157 --> 00:00:39.688
this is gonna be the number of days that pass.

00:00:39.688 --> 00:00:43.449
So number of days...

00:00:44.709 --> 00:00:47.479
that pass.

00:00:47.479 --> 00:00:48.650
So that's this right over here,

00:00:48.650 --> 00:00:50.532
the number of days that pass.

00:00:50.532 --> 00:00:51.553
And this middle column,

00:00:51.553 --> 00:00:54.316
I'm gonna write the number of bananas Nate has left.

00:00:54.316 --> 00:00:57.428
Number of bananas...

00:00:58.588 --> 00:01:01.881
bananas left.

00:01:02.981 --> 00:01:05.044
And over here, I'm gonna make the ratio between the two.

00:01:05.044 --> 00:01:07.140
In order for this to be a proportional relationship,

00:01:07.140 --> 00:01:10.267
the ratio between these two has to be constant.

00:01:10.267 --> 00:01:12.260
So bananas left.

00:01:13.380 --> 00:01:16.072
So I'm gonna divide the second column by the first column.

00:01:16.072 --> 00:01:18.658
Bananas left...

00:01:19.798 --> 00:01:24.013
left, divided by days passed.

00:01:24.013 --> 00:01:28.146
Days passed.

00:01:28.146 --> 00:01:31.049
All right, so let's think about it a little bit.

00:01:31.049 --> 00:01:34.531
When one day has passed,

00:01:34.531 --> 00:01:36.621
how many bananas will he have left?

00:01:36.621 --> 00:01:39.245
Well, in that one day he will have eaten two bananas,

00:01:39.245 --> 00:01:42.449
so you're going to have 98 bananas left.

00:01:42.449 --> 00:01:44.748
And so what's the ratio of bananas left to days passed?

00:01:44.748 --> 00:01:49.718
Well, it's 98 over one, which is going to be equal to 98.

00:01:49.718 --> 00:01:50.715
All right.

00:01:50.715 --> 00:01:53.107
When two days have passed,

00:01:53.107 --> 00:01:55.034
how many bananas is he gonna have left?

00:01:55.034 --> 00:01:57.148
Well, he's going to consume two more bananas,

00:01:57.148 --> 00:01:59.206
so he's going to have 96 left,

00:01:59.206 --> 00:02:00.444
and so what's the ratio?

00:02:00.444 --> 00:02:03.323
It's going to be 96 to two,

00:02:03.323 --> 00:02:07.387
which is equal to 48.

00:02:07.387 --> 00:02:09.500
So clearly this ratio is not constant.

00:02:09.500 --> 00:02:12.357
It changed just from going to one day to the next day.

00:02:12.357 --> 00:02:14.074
So we don't have a constant ratio

00:02:14.074 --> 00:02:15.956
of bananas left to days passed,

00:02:15.956 --> 00:02:17.557
so this is not,

00:02:17.557 --> 00:02:19.971
this is not a proportional,

00:02:19.971 --> 00:02:23.872
proportional relationship.

00:02:23.872 --> 00:02:25.521
Now, things might've been a little bit different

00:02:25.521 --> 00:02:29.259
if they said the number of bananas Nate has eaten,

00:02:29.259 --> 00:02:32.852
is that proportional to the number of days that passed?

00:02:32.852 --> 00:02:34.228
Well, yeah, sure, because then,

00:02:34.228 --> 00:02:37.363
if this was the number of bananas eaten,

00:02:37.363 --> 00:02:39.313
if this was the number of bananas eaten,

00:02:39.313 --> 00:02:41.078
then it would always be two times the number

00:02:41.078 --> 00:02:43.145
of days that pass, so that would be two,

00:02:43.145 --> 00:02:44.537
and then that would be four,

00:02:44.537 --> 00:02:47.625
and then these ratios would always be two.

00:02:47.625 --> 00:02:50.040
But that's not what they asked for.

00:02:50.040 --> 00:02:52.362
They wanted us to compare number of bananas left

00:02:52.362 --> 00:02:54.545
to number of days that pass.


Movie ticket proportionality

WEBVTT
Kind: captions
Language: en

00:00:00.284 --> 00:00:03.721
- A movie theater charges $10.50 per ticket

00:00:03.721 --> 00:00:06.693
and $5 per bucket of popcorn.

00:00:06.693 --> 00:00:08.655
You never buy the popcorn because you think

00:00:08.655 --> 00:00:10.756
it's too expensive.

00:00:10.756 --> 00:00:13.754
Is the total price you pay proportional to the number

00:00:13.754 --> 00:00:16.155
of tickets you buy?

00:00:16.155 --> 00:00:18.777
This is interesting, because they tell us the price

00:00:20.047 --> 00:00:22.023
of a bucket of popcorn, but then they tell us

00:00:22.023 --> 00:00:23.927
that we never buy the popcorn.

00:00:23.927 --> 00:00:24.995
So I guess, I don't know they even told us

00:00:24.995 --> 00:00:27.386
what the price is; maybe to confuse us.

00:00:27.386 --> 00:00:29.256
Is the total price you pay proportional

00:00:29.256 --> 00:00:30.985
to the number of tickets you buy?

00:00:30.985 --> 00:00:32.309
Well, you're only buying tickets;

00:00:32.309 --> 00:00:34.155
you're not gonna be buying any popcorn,

00:00:34.155 --> 00:00:38.091
so yeah, you're gonna spend $10.50 for every ticket,

00:00:38.091 --> 00:00:39.519
so it should be proportional.

00:00:39.519 --> 00:00:41.551
Just to see that a little bit clearer,

00:00:41.551 --> 00:00:44.488
let's draw a little table here.

00:00:44.488 --> 00:00:46.236
So, number of tickets;

00:00:49.387 --> 00:00:52.394
and then price, the total price.

00:00:52.394 --> 00:00:54.461
We're assuming I never buy the popcorn.

00:00:54.461 --> 00:00:58.617
So, if I buy one ticket, the price is going to be 10.50.

00:00:58.617 --> 00:01:01.717
If I buy two tickets, it's gonna be two times 10.50,

00:01:01.717 --> 00:01:06.187
or $21.00.

00:01:06.187 --> 00:01:08.660
If I buy three tickets, gonna be three times 10.50,

00:01:08.660 --> 00:01:09.322
which is what?

00:01:09.322 --> 00:01:11.510
31.50.

00:01:13.559 --> 00:01:16.712
And so, you can see the ratio between price

00:01:17.652 --> 00:01:19.528
and number of tickets,

00:01:21.152 --> 00:01:22.754
it's always going to be 10.50.

00:01:22.754 --> 00:01:23.892
10.50 divided by one;

00:01:23.892 --> 00:01:25.320
21 divided by two;

00:01:25.320 --> 00:01:27.050
31.50 divided by three;

00:01:27.050 --> 00:01:29.627
it's always going to be 10.50.

00:01:29.627 --> 00:01:32.356
That's 'cause the price, put another way,

00:01:32.356 --> 00:01:35.009
the price is just going to be 10.50

00:01:35.827 --> 00:01:38.341
times the number of tickets.

00:01:41.760 --> 00:01:44.506
So, clearly a proportional relationship.


Testing whether area is proportional to side length

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.450
- So let's draw a square.

00:00:03.450 --> 00:00:06.520
So it's not perfectly drawn but this is a

00:00:06.520 --> 00:00:09.920
reasonable attempt at a square, and of course

00:00:09.920 --> 00:00:11.720
one of the things that we know about a square

00:00:11.720 --> 00:00:15.020
is all of the sides are going to have the same length.

00:00:15.020 --> 00:00:17.090
So if the length of this side is x,

00:00:17.090 --> 00:00:19.050
well all of them are going to have,

00:00:19.050 --> 00:00:21.490
all of them are going to have length x.

00:00:21.490 --> 00:00:24.490
We also know this is a, these are going to be

00:00:24.490 --> 00:00:27.110
right angles, that's what makes it a square.

00:00:27.110 --> 00:00:29.190
Now the question I want to ask,

00:00:29.190 --> 00:00:31.390
are the lengths of the side of a square

00:00:31.390 --> 00:00:33.890
proportional to its area?

00:00:33.890 --> 00:00:35.450
And I encourage you to pause the video and think about it,

00:00:35.450 --> 00:00:37.890
are the lengths of a side, are the lengths

00:00:37.890 --> 00:00:42.180
of the sides of a square proportional to the square's area?

00:00:43.370 --> 00:00:45.250
Well to think about that let's draw,

00:00:45.250 --> 00:00:46.750
let's have a little table here.

00:00:46.750 --> 00:00:51.730
So side length, side length

00:00:52.210 --> 00:00:55.450
and that's going to be given as x,

00:00:55.450 --> 00:00:58.290
so that's going to be x, and then

00:00:58.290 --> 00:00:59.850
I'll have another column, I can draw

00:00:59.850 --> 00:01:02.150
that a little bit neater so let me

00:01:02.150 --> 00:01:04.520
use this tool, so there you go,

00:01:04.520 --> 00:01:07.490
and I'm going to make two columns,

00:01:07.490 --> 00:01:09.881
two columns right over here,

00:01:09.881 --> 00:01:12.170
and then over here I'm going to put area.

00:01:13.490 --> 00:01:15.750
Area, and what's the area going to be?

00:01:15.750 --> 00:01:19.090
Well the area is going to be one of the sides squared,

00:01:19.090 --> 00:01:21.800
I suppose you could view it as the width times the height,

00:01:21.800 --> 00:01:25.650
so its going to be x times x or x squared.

00:01:25.650 --> 00:01:27.490
So let's just pick some values for x

00:01:27.490 --> 00:01:29.850
and then figure out what the area is going to be.

00:01:29.850 --> 00:01:32.790
So if x is going to be equal to one,

00:01:32.790 --> 00:01:34.550
the area's going to be one times one,

00:01:34.550 --> 00:01:36.020
which is still one.

00:01:36.020 --> 00:01:38.590
If x is going to be equal to two,

00:01:38.590 --> 00:01:40.780
the area's going to be two times two,

00:01:40.780 --> 00:01:42.790
which is equal to four.

00:01:42.790 --> 00:01:44.920
If x is going to be equal to three,

00:01:44.920 --> 00:01:47.770
the area's going to be three times three,

00:01:47.770 --> 00:01:51.430
which is equal to nine, and I could keep going

00:01:51.430 --> 00:01:53.190
but I think this is enough, these are enough points

00:01:53.190 --> 00:01:56.450
to think about is the side length proportional to the area,

00:01:56.450 --> 00:01:58.590
or is the area proportional to the side length?

00:01:58.590 --> 00:01:59.890
Now one of the ways you can think about

00:01:59.890 --> 00:02:02.840
proportionality is, are the ratios between

00:02:02.840 --> 00:02:05.050
the side length and the corresponding area

00:02:05.050 --> 00:02:07.650
always the same, so we want to look at

00:02:07.650 --> 00:02:10.250
the ratio between side length and area,

00:02:10.250 --> 00:02:13.160
or area and side length, as long as they're always constant.

00:02:13.160 --> 00:02:17.050
Area and side length, here let me do those

00:02:17.050 --> 00:02:21.380
as two separate, and side length.

00:02:21.380 --> 00:02:23.150
So let's make an extra column here.

00:02:23.150 --> 00:02:25.070
Let's make an extra column here,

00:02:26.340 --> 00:02:29.350
so there we go, and let's look at that ratio,

00:02:29.350 --> 00:02:33.710
the ratio of area to side,

00:02:33.710 --> 00:02:35.940
to side length.

00:02:35.940 --> 00:02:38.620
So here the ratio's one over one,

00:02:38.620 --> 00:02:41.281
one over one, which equals one.

00:02:41.281 --> 00:02:42.850
Now that seems reasonable.

00:02:42.850 --> 00:02:44.620
For this next one the ratio of

00:02:44.620 --> 00:02:47.820
area over side length is four over two,

00:02:47.820 --> 00:02:51.220
four over two, which equals two.

00:02:51.220 --> 00:02:52.590
I don't even have to go to this third one,

00:02:52.590 --> 00:02:54.800
I can, I can say this ratio is going to be

00:02:54.800 --> 00:02:58.520
area over side length is going to be equal to three.

00:02:58.520 --> 00:03:01.380
And notice I get a different value every time.

00:03:01.380 --> 00:03:03.350
This is not equal to a constant, I don't have

00:03:03.350 --> 00:03:06.680
a constant ratio between area and side length

00:03:06.680 --> 00:03:08.720
so that tells us that these two, that

00:03:08.720 --> 00:03:11.790
the side length and the area are not proportional.

00:03:11.790 --> 00:03:14.500
Not, not proportional.

00:03:15.550 --> 00:03:17.650
And if you look at it, it makes a lot of sense

00:03:17.650 --> 00:03:20.250
that the ratio is actually going to be equal to

00:03:20.250 --> 00:03:22.290
the actual side length, that it changes

00:03:22.290 --> 00:03:23.750
depending on the side length and that's because

00:03:23.750 --> 00:03:26.080
we're squaring things, it's literally,

00:03:26.080 --> 00:03:28.420
you're literally taking x squared,

00:03:28.420 --> 00:03:30.020
if, you know, you have x here,

00:03:30.020 --> 00:03:31.510
this is going to be x squared

00:03:31.510 --> 00:03:35.190
and the ratio is going to be x squared over x,

00:03:35.190 --> 00:03:36.890
which of course is always just going to be

00:03:36.890 --> 00:03:38.990
equal, this is x times x divided by x,

00:03:38.990 --> 00:03:40.590
that's always going to be equal to x,

00:03:40.590 --> 00:03:41.590
and we see that right over here.

00:03:41.590 --> 00:03:43.931
But it's definitely not proportional.


Proportionality between side length and perimeter

WEBVTT
Kind: captions
Language: en

00:00:00.128 --> 00:00:01.405
- [Voiceover] We have a square here

00:00:01.405 --> 00:00:03.481
and all the sides have length x.

00:00:03.481 --> 00:00:04.688
And what I want to think about here

00:00:04.688 --> 00:00:06.635
is whether the perimeter of the square

00:00:06.635 --> 00:00:10.627
is proportional to the length of a side of the square.

00:00:10.627 --> 00:00:12.202
So let's think about it a little bit.

00:00:12.202 --> 00:00:13.887
I'm going to draw a table,

00:00:13.887 --> 00:00:17.101
so let me make some columns here.

00:00:17.101 --> 00:00:20.444
So there you have that, we have that,

00:00:20.444 --> 00:00:22.343
and we'll make three columns

00:00:22.343 --> 00:00:24.101
and on the first column

00:00:24.101 --> 00:00:26.887
we're going to think about the Side length.

00:00:26.887 --> 00:00:30.376
Side length,

00:00:30.376 --> 00:00:32.116
and that's measured as x.

00:00:32.116 --> 00:00:36.211
And in the next column I want to think about the Perimeter.

00:00:36.211 --> 00:00:37.177
Well what is that?

00:00:37.177 --> 00:00:39.696
That's going to be x plus x plus x plus x

00:00:39.696 --> 00:00:41.472
which is, of course, four x.

00:00:41.472 --> 00:00:43.352
And then I want to think about the ratio.

00:00:43.352 --> 00:00:48.550
The ratio between Perimeter,

00:00:50.340 --> 00:00:55.273
Perimeter and Side length.

00:00:56.173 --> 00:00:59.867
So let's see, when the Side length is equal to one,

00:00:59.867 --> 00:01:00.865
what's the Perimeter?

00:01:00.865 --> 00:01:02.118
It's going to be four times one.

00:01:02.118 --> 00:01:03.836
One plus one plus one plus one.

00:01:03.836 --> 00:01:05.120
It's going to be four.

00:01:05.120 --> 00:01:06.207
And what's our ratio?

00:01:06.207 --> 00:01:07.564
It's going to be four to one

00:01:07.564 --> 00:01:09.599
which, of course, is equal to four.

00:01:09.599 --> 00:01:11.566
Now what if the Side length is two?

00:01:11.566 --> 00:01:12.807
Well then the Perimeter is going to be

00:01:12.807 --> 00:01:13.985
two plus two plus two plus two.

00:01:13.985 --> 00:01:15.726
It's going to be four times two.

00:01:15.726 --> 00:01:17.176
That's going to be eight.

00:01:17.176 --> 00:01:19.649
So the ratio's going to be eight to two

00:01:19.649 --> 00:01:21.367
which is going to be equal to four.

00:01:21.367 --> 00:01:23.168
I think we see a pattern here.

00:01:23.168 --> 00:01:24.688
The ratio of Perimeter to Side length,

00:01:24.688 --> 00:01:27.556
it looks like it's always going to be four.

00:01:27.556 --> 00:01:28.810
Now we could keep going here.

00:01:28.810 --> 00:01:31.368
If our Side length is three,

00:01:31.368 --> 00:01:33.069
then our Perimeter is going to be 12.

00:01:33.069 --> 00:01:34.207
What's the ratio?

00:01:34.207 --> 00:01:36.934
12 divided by three is equal to four.

00:01:36.934 --> 00:01:39.092
Now this isn't some magical thing here,

00:01:39.092 --> 00:01:40.047
because to figure out the Perimeter

00:01:40.047 --> 00:01:41.984
we're multiplying it by four.

00:01:41.984 --> 00:01:44.143
So if the ratio of Perimeter to Side length,

00:01:44.143 --> 00:01:46.785
it's always going to be,

00:01:46.785 --> 00:01:48.280
the Perimeter is going to be four x

00:01:48.280 --> 00:01:49.688
if the Side length is x,

00:01:49.688 --> 00:01:51.243
and four x divided by x,

00:01:51.243 --> 00:01:53.805
well that's always going to be four.

00:01:53.805 --> 00:01:55.972
Well this definitely meets our conditions

00:01:55.972 --> 00:01:58.001
for being a proportional relationship.

00:01:58.001 --> 00:02:00.419
The ratio between Perimeter and Side length

00:02:00.419 --> 00:02:02.137
is always going to be four.

00:02:02.137 --> 00:02:04.157
We didn't have to do this table here,

00:02:04.157 --> 00:02:06.084
but this makes it a little bit more concrete.

00:02:06.084 --> 00:02:08.242
Or we could write it like this,

00:02:08.242 --> 00:02:09.538
we could write that

00:02:09.538 --> 00:02:14.165
Perimeter

00:02:14.165 --> 00:02:15.575
for a square

00:02:15.575 --> 00:02:19.184
is equal to four times the Side length.

00:02:23.014 --> 00:02:24.421
And so, once again, it becomes clear

00:02:24.421 --> 00:02:25.717
it's a proportional relationship.

00:02:25.717 --> 00:02:28.621
The Perimeter is equal to a constant times the Side length,

00:02:28.621 --> 00:02:30.745
or the Perimeter divided by the Side length

00:02:30.745 --> 00:02:31.904
is equal to four.

00:02:31.904 --> 00:02:34.436
So this is definitely a proportional relationship

00:02:34.436 --> 00:02:37.245
between Side length and Perimeter.


Rate problem using fractions

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:03.520
Odie the octopus swam home.

00:00:03.520 --> 00:00:06.010
He swam at a constant speed.

00:00:06.010 --> 00:00:08.800
He plotted how far
he'd gone every 1/7

00:00:08.800 --> 00:00:14.110
of a second on the graph below.

00:00:14.110 --> 00:00:16.280
Not all octopi
can create graphs,

00:00:16.280 --> 00:00:19.690
but Odie was no
ordinary octopus.

00:00:19.690 --> 00:00:22.210
And actually, octopi are some
of the smartest creatures

00:00:22.210 --> 00:00:23.400
on this planet.

00:00:23.400 --> 00:00:26.830
What was Odie's speed--
after probably dolphins

00:00:26.830 --> 00:00:30.170
and great apes and
probably most primates,

00:00:30.170 --> 00:00:32.800
but they're still fairly
smart, smarter than most fish.

00:00:32.800 --> 00:00:36.300
What was Odie's speed
in meters per second?

00:00:36.300 --> 00:00:42.740
Note we use m/s to show the
units of meters per second.

00:00:42.740 --> 00:00:43.500
All right.

00:00:43.500 --> 00:00:49.120
So to figure out speed
in meters per second,

00:00:49.120 --> 00:00:51.450
we essentially need
to divide distance

00:00:51.450 --> 00:00:58.505
in meters divided
by time in seconds.

00:01:02.850 --> 00:01:05.530
So let's see if we
can pull that off.

00:01:05.530 --> 00:01:09.010
So the easiest thing-- let me
find a point that I can clearly

00:01:09.010 --> 00:01:12.017
read how far Odie has traveled.

00:01:12.017 --> 00:01:13.850
So it looks like this
point right over here.

00:01:13.850 --> 00:01:18.020
It's clearly that the
distance in meters is 4/7.

00:01:18.020 --> 00:01:18.960
So let me do that.

00:01:18.960 --> 00:01:20.840
So the distance
in meters is 4/7,

00:01:20.840 --> 00:01:22.215
so I'll do that
in the numerator.

00:01:22.215 --> 00:01:23.620
the distance in meters.

00:01:23.620 --> 00:01:25.465
Gee, I don't like
that con-- well, I'll

00:01:25.465 --> 00:01:25.830
just go with this color.

00:01:25.830 --> 00:01:27.830
You know, the contrast
with the white background

00:01:27.830 --> 00:01:29.494
is a little hard on the eye.

00:01:29.494 --> 00:01:37.540
And It took him 5/7 of
a second to get there.

00:01:37.540 --> 00:01:41.480
So 4/7 meters-- actually, let me
make sure I put the units here.

00:01:41.480 --> 00:01:50.600
4/7 meters in 5/7 seconds.

00:01:50.600 --> 00:01:55.080
And so literally, we just
need to divide 4/7 by 5/7,

00:01:55.080 --> 00:01:57.900
and we're going to get
it in meters per second.

00:01:57.900 --> 00:01:59.980
So this is going to
be the same thing

00:01:59.980 --> 00:02:10.675
as 4/7 over 5/7
meters per second.

00:02:13.290 --> 00:02:18.950
And 4/7 divided by 5/7 is the
same thing as 4/7 times 7/5.

00:02:18.950 --> 00:02:20.130
So let me write that down.

00:02:20.130 --> 00:02:24.760
So 4/7 times 7/5.

00:02:24.760 --> 00:02:26.470
Dividing by a fraction
is the same thing

00:02:26.470 --> 00:02:28.620
as multiplying by
its reciprocal.

00:02:28.620 --> 00:02:36.910
So times 7/5, and we get
this-- well, this 7 and this 7

00:02:36.910 --> 00:02:39.787
cancel out, and
you get 4 over 5.

00:02:39.787 --> 00:02:41.620
You could have also
done that by multiplying

00:02:41.620 --> 00:02:43.730
the numerator and the
denominator here by 7.

00:02:43.730 --> 00:02:46.550
If you multiply it
by 7, multiply by 7,

00:02:46.550 --> 00:02:47.950
you would get 4/5.

00:02:47.950 --> 00:02:51.230
But either way, his speed
in meters per second

00:02:51.230 --> 00:02:55.620
is 4/5 meters per second.


Understanding variables representing integers on number line

WEBVTT
Kind: captions
Language: en

00:00:00.267 --> 00:00:03.667
- Let's get some practice
understanding the variables

00:00:03.667 --> 00:00:07.066
and the negative numbers
that they might represent

00:00:07.066 --> 00:00:08.766
or the positive numbers.

00:00:08.766 --> 00:00:10.500
So we're told to order
the following expressions

00:00:10.500 --> 00:00:12.566
by their values from least to greatest.

00:00:12.566 --> 00:00:14.200
And they've given us
these three expressions

00:00:14.200 --> 00:00:17.267
q minus n, n, and n minus q

00:00:17.267 --> 00:00:19.534
and then they plot n and
q on the number line.

00:00:19.534 --> 00:00:20.467
So just to get our bearings,

00:00:20.467 --> 00:00:23.100
let's see, three hash
marks to the left of zero

00:00:23.100 --> 00:00:23.900
is negative three.

00:00:23.900 --> 00:00:26.900
So each hash mark we must
be going down by one.

00:00:26.900 --> 00:00:29.733
So this must be negative
one, negative two,

00:00:29.733 --> 00:00:31.066
and this is negative three.

00:00:31.066 --> 00:00:32.600
And then so as we go to the right,

00:00:32.600 --> 00:00:34.366
each hash mark must increase by one.

00:00:34.366 --> 00:00:37.867
So zero, one, two, and then three.

00:00:37.867 --> 00:00:39.933
And then this just helps us
get a little more bearings.

00:00:39.933 --> 00:00:42.633
But let's just think about
each of these expressions.

00:00:42.633 --> 00:00:44.967
So this first one is q minus n.

00:00:44.967 --> 00:00:47.433
And q is to the right
of n on the number line.

00:00:47.433 --> 00:00:51.300
We know that q is greater than n.

00:00:51.300 --> 00:00:54.367
So if q is greater than n and
you're subtracting n from q

00:00:54.367 --> 00:00:56.629
it actually doesn't matter
if they're both negative

00:00:56.629 --> 00:00:58.900
or both positive or one's
negative and one's positive.

00:00:58.900 --> 00:01:03.833
Just the fact that we know
that q is greater than n

00:01:03.833 --> 00:01:08.033
that means that q minus n
is going to be positive.

00:01:08.033 --> 00:01:09.873
And if you actually want to look at this

00:01:09.873 --> 00:01:14.366
particular circumstance, q
is positive, n is negative.

00:01:14.366 --> 00:01:16.233
If you subtract a negative,

00:01:16.233 --> 00:01:19.433
you're going to
essentially add a positive.

00:01:19.433 --> 00:01:21.266
So this value right over here,

00:01:21.266 --> 00:01:22.866
not only is it going to be positive,

00:01:22.866 --> 00:01:26.433
it's going to be a positive
value greater than q.

00:01:26.433 --> 00:01:27.800
And if we had to compare it versus q,

00:01:27.800 --> 00:01:29.400
we would know that it's greater than q,

00:01:29.400 --> 00:01:31.233
but they don't ask us to do that.

00:01:31.233 --> 00:01:33.033
Now we have n.

00:01:33.033 --> 00:01:36.800
n is a negative value.

00:01:36.800 --> 00:01:37.867
It's a negative number.

00:01:37.867 --> 00:01:39.433
And it's a negative number
between negative one

00:01:39.433 --> 00:01:40.233
and negative two.

00:01:40.233 --> 00:01:44.600
It looks like it's
approximately negative 1.8.

00:01:44.600 --> 00:01:46.700
We don't know for sure
but if we just eyeball it,

00:01:46.700 --> 00:01:49.300
this thing is negative
and it looks like it's

00:01:49.300 --> 00:01:53.066
approximately negative 1.8.

00:01:53.066 --> 00:01:55.233
Now what's n minus q?

00:01:55.233 --> 00:01:57.466
n minus q?

00:01:57.466 --> 00:02:01.400
So it's going to be the
negative of q minus n.

00:02:01.400 --> 00:02:04.067
So n minus q, we have the smaller number

00:02:04.067 --> 00:02:08.400
and from that we're
subtracting the larger number.

00:02:08.400 --> 00:02:11.800
So this thing right over
here is going to be negative.

00:02:11.800 --> 00:02:13.769
So the largest of these values
is definitely going to be

00:02:13.769 --> 00:02:15.900
q minus n which is going to be positive.

00:02:15.900 --> 00:02:17.400
And then we have to figure
out which is going to be

00:02:17.400 --> 00:02:18.766
more negative.

00:02:18.766 --> 00:02:23.300
This n value or this n minus q value?

00:02:23.300 --> 00:02:24.766
Well let's think about it a little bit.

00:02:24.766 --> 00:02:27.576
We could just try to
approximate what q is.

00:02:27.576 --> 00:02:30.281
And if we look at it, q
looks like it's approximately

00:02:30.281 --> 00:02:34.577
this looks like roughly
0.8 and this looks like

00:02:34.592 --> 00:02:38.700
it's approximately, we've
already said, negative 1.8.

00:02:38.700 --> 00:02:40.400
So if we make those
assumptions right over here

00:02:40.400 --> 00:02:41.633
this thing is going to be

00:02:41.633 --> 00:02:46.366
approximately negative 1.8

00:02:46.366 --> 00:02:50.633
minus 0.8 which is

00:02:50.633 --> 00:02:54.100
equal to negative 2.6.

00:02:54.100 --> 00:02:55.733
So when you look at it like
this you clearly see that

00:02:55.733 --> 00:02:59.000
this is going to be more negative
than this right over here.

00:02:59.000 --> 00:03:02.698
So this is the smallest,

00:03:02.698 --> 00:03:06.933
and this is the largest, or the
greatest I should say maybe.

00:03:06.933 --> 00:03:08.800
Maybe let me call this the least.

00:03:08.800 --> 00:03:11.900
That might be better because
sometimes when people say

00:03:11.900 --> 00:03:14.634
small and large they're
referring to absolute value.

00:03:14.634 --> 00:03:16.400
But let's say this is the least and

00:03:16.400 --> 00:03:20.667
this is the greatest.

00:03:20.667 --> 00:03:21.966
So if we wanted to order them

00:03:21.966 --> 00:03:26.266
we would go n minus q, and then

00:03:26.266 --> 00:03:28.000
if you're doing this on
Khan Academy exercise,

00:03:28.000 --> 00:03:30.700
you can actually click on
these and move them around,

00:03:30.700 --> 00:03:33.366
but if we can't, it will be n minus q

00:03:33.366 --> 00:03:35.134
which is the most negative,

00:03:35.134 --> 00:03:37.933
then you have n which is still
negative but not as negative.

00:03:37.933 --> 00:03:41.700
This is roughly negative 1.8,
this is roughly negative 2.6.

00:03:41.700 --> 00:03:45.933
And then q minus n which is going to be

00:03:45.933 --> 00:03:49.067
roughly positive 2.6.

00:03:49.067 --> 00:03:53.000
So this is the greatest.

00:03:53.000 --> 00:03:56.641
And so let's do the next question.

00:03:57.933 --> 00:04:02.200
Order the, whoops, order
the following expressions

00:04:02.200 --> 00:04:05.100
by their values from least to greatest.

00:04:05.100 --> 00:04:07.467
So once again the kind of
same drill although here

00:04:07.467 --> 00:04:09.066
each hash mark looks like it's a half

00:04:09.066 --> 00:04:10.300
because it takes two to get to one,

00:04:10.300 --> 00:04:11.766
so this is half.

00:04:11.766 --> 00:04:15.100
This is negative 1/2 right over here.

00:04:15.100 --> 00:04:18.288
And we want to compare a minus b,

00:04:18.288 --> 00:04:21.876
to a, to a minus 0.5.

00:04:21.876 --> 00:04:23.466
So all of them were either a,

00:04:23.466 --> 00:04:27.013
you can even think of
this as a minus zero.

00:04:27.013 --> 00:04:28.931
Right that's the same thing as a.

00:04:28.931 --> 00:04:29.855
So let's see.

00:04:29.855 --> 00:04:32.300
In all these we have an a and
we're subtracting something.

00:04:32.300 --> 00:04:34.333
Where in here we're subtracting zero.

00:04:34.333 --> 00:04:35.700
I guess we're subtracting nothing.

00:04:35.700 --> 00:04:37.100
Here you're subtracting b.

00:04:37.100 --> 00:04:39.966
And here you're subtracting 0.5.

00:04:39.966 --> 00:04:42.205
So in general the more you subtract,

00:04:42.205 --> 00:04:45.752
the more that you subtract,

00:04:45.752 --> 00:04:48.233
the smaller it's going to be.

00:04:48.233 --> 00:04:49.266
So let's see.

00:04:49.266 --> 00:04:52.200
In which of these cases
am I subtracting the most?

00:04:52.200 --> 00:04:54.966
Well here I'm subtracting
a positive number.

00:04:54.966 --> 00:04:57.033
Here I'm subtracting zero.

00:04:57.033 --> 00:05:01.467
Here I am subtracting a
b is a negative number.

00:05:01.467 --> 00:05:03.567
Here I am subtracting a negative number.

00:05:03.567 --> 00:05:06.266
b is negative.

00:05:06.266 --> 00:05:08.400
This is clearly positive.

00:05:08.400 --> 00:05:12.900
So if you subtract a
positive number from a,

00:05:12.900 --> 00:05:14.233
you're going to get a lower value

00:05:14.233 --> 00:05:15.566
than if you subtract a negative number.

00:05:15.566 --> 00:05:17.166
In fact if you subtract a negative number

00:05:17.166 --> 00:05:18.533
you're going to add to a.

00:05:18.533 --> 00:05:20.500
You're going to get a
number greater than a.

00:05:20.500 --> 00:05:23.500
So the least is when you
subtract the largest value

00:05:23.500 --> 00:05:25.500
or the greatest value.

00:05:25.500 --> 00:05:26.900
So a minus 0.5.

00:05:26.900 --> 00:05:30.433
We're subtracting a positive number there.

00:05:30.433 --> 00:05:32.200
We're subtracting 0.5.

00:05:32.200 --> 00:05:34.800
Followed by a where we're
not subtracting anything.

00:05:34.800 --> 00:05:37.667
And then you have a minus b.

00:05:37.667 --> 00:05:39.433
This right over here is
going to be the greatest.

00:05:39.433 --> 00:05:40.400
Why is this?

00:05:40.400 --> 00:05:42.300
Well we know that b is a negative number.

00:05:42.300 --> 00:05:45.033
Notice it's below zero right over here so

00:05:45.033 --> 00:05:48.667
if b is a negative number, you
subtract a negative number,

00:05:48.667 --> 00:05:50.066
you're actually going
to get a value that is

00:05:50.066 --> 00:05:51.133
greater than a.

00:05:51.133 --> 00:05:52.133
Let me make it very clear.

00:05:52.133 --> 00:05:55.533
This value right over here
is going to be less than a.

00:05:55.533 --> 00:05:57.601
This value over here clearly equals a.

00:05:57.601 --> 00:05:59.400
And this value over here
is actually going to be

00:05:59.400 --> 00:06:00.866
greater than a.

00:06:00.866 --> 00:06:02.700
So now we've ordered it
from least to greatest.

00:06:02.700 --> 00:06:04.567
Once again if we were doing it
on the Khan Academy exercises

00:06:04.567 --> 00:06:06.567
we would have a little
tool where we could click

00:06:06.567 --> 00:06:07.900
and move these around.

00:06:07.900 --> 00:06:12.033
But this is the least followed
by this, followed by that.

00:06:12.033 --> 00:06:13.333
Now another way that you could do it,

00:06:13.333 --> 00:06:14.633
just like we did in the last example,

00:06:14.633 --> 00:06:18.133
you could try to estimate
roughly what these values are.

00:06:18.133 --> 00:06:20.667
b looks like it's, I don't
know, it's not exactly,

00:06:20.667 --> 00:06:23.845
it looks like it's about negative .2,

00:06:23.845 --> 00:06:27.166
so approximately negative 0.2.

00:06:27.166 --> 00:06:28.967
Remember this is negative half.

00:06:28.967 --> 00:06:30.809
This is negative .5.

00:06:30.809 --> 00:06:34.200
a looks like it is
approximately, I don't know,

00:06:34.200 --> 00:06:38.953
negative .7, negative 0.7.

00:06:39.800 --> 00:06:40.767
And so you could actually do it

00:06:40.767 --> 00:06:45.100
with the actual values if you like,

00:06:45.100 --> 00:06:49.089
but you would get the same result.


Substitution with negative numbers

WEBVTT
Kind: captions
Language: en

00:00:00.348 --> 00:00:02.043
- [Voiceover] Let's give
ourselves some practice

00:00:02.043 --> 00:00:04.899
substituting positive and negative values

00:00:04.899 --> 00:00:06.756
for variables.

00:00:06.756 --> 00:00:08.359
So we're told to evaluate X,

00:00:08.359 --> 00:00:12.561
we're told to evaluate X minus negative Y,

00:00:12.561 --> 00:00:15.905
where X is equal to negative two

00:00:15.905 --> 00:00:18.969
and Y is equal to five.

00:00:18.969 --> 00:00:20.618
So everywhere we see an X, we can replace

00:00:20.618 --> 00:00:21.849
with a negative two.

00:00:21.849 --> 00:00:23.845
Everywhere we see a Y,
we replace with a five.

00:00:23.845 --> 00:00:25.820
So this is a Y right over here,

00:00:25.820 --> 00:00:27.375
and then of course,

00:00:27.375 --> 00:00:29.210
and then of course

00:00:29.210 --> 00:00:30.416
this is an X.

00:00:30.416 --> 00:00:31.229
Let's do that.

00:00:31.229 --> 00:00:34.016
So instead of that X,
let's write negative two.

00:00:34.016 --> 00:00:36.802
So we have negative two minus

00:00:36.802 --> 00:00:38.358
and then we have in the parentheses

00:00:38.358 --> 00:00:40.307
a negative Y.

00:00:40.307 --> 00:00:42.838
Y is equal to,

00:00:42.838 --> 00:00:44.906
Y is equal to five.

00:00:44.906 --> 00:00:47.135
Now what is this going to evaluate to?

00:00:47.135 --> 00:00:49.039
Well, this is the same thing as

00:00:49.039 --> 00:00:50.687
negative two,

00:00:50.687 --> 00:00:53.240
now subtracting a negative five,

00:00:53.240 --> 00:00:54.494
so all of this business here,

00:00:54.494 --> 00:00:55.610
subtracting a negative five,

00:00:55.610 --> 00:00:57.700
that's the same thing as adding a five.

00:00:57.700 --> 00:01:00.393
So it's going to be
negative two plus five,

00:01:00.393 --> 00:01:02.320
which is equal to three.

00:01:02.320 --> 00:01:04.978
This is going to be equal to three.

00:01:05.826 --> 00:01:08.301
And there's several ways
to think about this.

00:01:08.301 --> 00:01:10.972
my brain thinks okay, I'm
starting at negative two.

00:01:10.972 --> 00:01:13.920
If I add two I get to zero and then

00:01:13.920 --> 00:01:15.266
I would have to add another,

00:01:15.266 --> 00:01:16.868
and then I have to add another three

00:01:16.868 --> 00:01:18.634
so that gets me to three.

00:01:18.634 --> 00:01:20.213
You can even view this as,

00:01:20.213 --> 00:01:21.233
you can even view this,

00:01:21.233 --> 00:01:23.021
my brain kind of says negative two,

00:01:23.021 --> 00:01:24.741
well let's see, I have to add,

00:01:24.741 --> 00:01:27.364
i have to add two to get back to zero

00:01:27.364 --> 00:01:29.314
and then I have to add another three

00:01:29.314 --> 00:01:31.219
if I want to add a total of five.

00:01:31.219 --> 00:01:33.772
So this is going to be zero plus three

00:01:33.772 --> 00:01:35.654
gets me to three.

00:01:35.654 --> 00:01:37.024
Another way to think about it,

00:01:37.024 --> 00:01:38.810
negative two plus five is the same thing

00:01:38.810 --> 00:01:41.598
as five minus two.

00:01:41.598 --> 00:01:43.618
Five minus two,

00:01:43.618 --> 00:01:45.730
which is of course equal to three.

00:01:45.730 --> 00:01:47.403
Or you could of course
draw it on a number line.

00:01:47.403 --> 00:01:48.703
And you would say if you
start at negative two

00:01:48.703 --> 00:01:50.536
and you take five steps to the right,

00:01:50.536 --> 00:01:52.511
you get to positive three.

00:01:52.511 --> 00:01:53.347
Let's do another one of these.

00:01:53.347 --> 00:01:55.645
This one's a little bit more,

00:01:55.645 --> 00:01:57.943
a little bit more complex.

00:01:57.943 --> 00:01:58.687
So let's see.

00:01:58.687 --> 00:02:01.218
We're told to evaluate
three minus negative six

00:02:01.218 --> 00:02:05.815
plus negative H plus negative four,

00:02:05.815 --> 00:02:08.740
where H is equal to negative seven.

00:02:08.740 --> 00:02:09.831
Well there's two ways you could do it.

00:02:09.831 --> 00:02:11.875
I could just take the
negative seven and replace it,

00:02:11.875 --> 00:02:14.615
the H with that negative seven,

00:02:14.615 --> 00:02:17.982
or I could actually try to
simplify this expression first

00:02:17.982 --> 00:02:20.583
and then do the substitution for H.

00:02:20.583 --> 00:02:21.603
Let's actually do that.

00:02:21.603 --> 00:02:23.996
My brain feels like doing that.

00:02:23.996 --> 00:02:25.482
So this expression,

00:02:25.482 --> 00:02:29.289
I have the three, but instead
of subtracting a negative six,

00:02:29.289 --> 00:02:30.637
instead of subtracting a negative six,

00:02:30.637 --> 00:02:32.831
that's gonna be the same
thing as just adding six.

00:02:33.436 --> 00:02:35.363
So three plus six.

00:02:35.363 --> 00:02:37.709
And adding a negative H,

00:02:37.709 --> 00:02:39.660
adding a negative H,

00:02:39.660 --> 00:02:41.958
adding a negative H, that's the same thing

00:02:41.958 --> 00:02:43.327
as just subtracting H.

00:02:43.327 --> 00:02:45.650
So three plus six minus H,

00:02:45.650 --> 00:02:48.064
and then adding a negative four,

00:02:48.064 --> 00:02:49.969
adding a negative four,

00:02:49.969 --> 00:02:52.268
that's the same thing as subtracting four.

00:02:52.268 --> 00:02:55.216
That's the same thing as subtracting four.

00:02:55.216 --> 00:02:57.955
And now of course we can do this in any,

00:02:57.955 --> 00:03:00.534
you know, addition and subtraction we have

00:03:00.534 --> 00:03:03.227
we can change the order in which we do it.

00:03:03.227 --> 00:03:06.570
So let's do that just to
kind of simplify all of this.

00:03:06.570 --> 00:03:08.475
So, actually first of all,

00:03:08.475 --> 00:03:10.402
I can figure out what three plus six is.

00:03:10.402 --> 00:03:12.421
Three plus six is equal to nine.

00:03:12.421 --> 00:03:15.648
And then I have this minus four here.

00:03:15.648 --> 00:03:18.644
So I could say nine minus four,

00:03:18.644 --> 00:03:21.106
actually I want to be careful
not to skip any steps.

00:03:21.106 --> 00:03:23.009
So three plus six,

00:03:23.009 --> 00:03:26.376
three plus six, in a
color that you can see,

00:03:26.376 --> 00:03:28.536
so three plus six is nine.

00:03:28.536 --> 00:03:29.946
So that's nine

00:03:30.876 --> 00:03:33.608
minus H minus four,

00:03:34.290 --> 00:03:35.817
minus four.

00:03:35.817 --> 00:03:39.020
Now I could change the order in which I do

00:03:39.020 --> 00:03:40.970
this addition or subtraction, so

00:03:40.970 --> 00:03:43.131
this is going to be the equivalent of

00:03:43.131 --> 00:03:45.283
nine minus four.

00:03:46.079 --> 00:03:49.225
Nine minus four minus H,

00:03:49.924 --> 00:03:51.294
minus H.

00:03:51.294 --> 00:03:53.105
And I just did that so I can simplify

00:03:53.105 --> 00:03:55.381
and figure out what nine minus four is.

00:03:55.381 --> 00:03:57.308
Nine minus four, of course,

00:03:57.308 --> 00:03:59.142
let me do this in blue, navy blue.

00:03:59.142 --> 00:04:01.371
Nine minus four is five.

00:04:01.371 --> 00:04:03.624
So this whole thing
simplified to five minus H

00:04:03.624 --> 00:04:05.620
before I even did the substitution.

00:04:05.620 --> 00:04:09.568
And now I can substitute
H with negative seven.

00:04:09.568 --> 00:04:11.774
So this is going to be equal to,

00:04:11.774 --> 00:04:13.619
this is going to be equal to,

00:04:14.814 --> 00:04:16.708
when I do the substitution,
I'll write it up here,

00:04:16.708 --> 00:04:19.565
it's going to be five minus,

00:04:19.565 --> 00:04:21.701
I'll do the minus in that magenta color,

00:04:21.701 --> 00:04:24.742
minus and now where I see an H,

00:04:24.742 --> 00:04:26.531
I'm gonna replace it with negative seven.

00:04:26.531 --> 00:04:28.319
Five minus negative seven.

00:04:28.319 --> 00:04:29.270
You want to be very careful there,

00:04:29.270 --> 00:04:30.478
you might be tempted to say,

00:04:30.478 --> 00:04:31.963
oh I have a negative here, negative here,

00:04:31.963 --> 00:04:33.983
let me just replace H with a seven.

00:04:33.983 --> 00:04:37.559
Remember, H is negative seven,
so you're subtracting H.

00:04:37.559 --> 00:04:40.114
You're gonna subtract negative seven.

00:04:40.114 --> 00:04:42.343
So this is five minus negative seven,

00:04:42.343 --> 00:04:44.317
which is the same thing,

00:04:44.317 --> 00:04:47.666
which is the same thing
as five plus seven.

00:04:48.422 --> 00:04:50.826
Five plus seven,

00:04:50.826 --> 00:04:54.936
which we all know is equal to 12.

00:04:54.936 --> 00:04:56.376
And we're done.

00:04:56.376 --> 00:04:57.605
Let's do a few more of these.

00:04:57.605 --> 00:04:59.207
You can't really get enough practice here,

00:04:59.207 --> 00:05:01.670
this is some important foundational skills

00:05:01.670 --> 00:05:04.270
for the rest of your mathematical lives.

00:05:04.270 --> 00:05:05.477
(laughing) Alright.

00:05:05.477 --> 00:05:08.170
So consider, I don't make
you too stressed about it.

00:05:08.170 --> 00:05:10.284
Consider the following number line.

00:05:10.284 --> 00:05:11.724
Alright, so we've got a number line here

00:05:11.724 --> 00:05:14.996
and let's see, they didn't
mark off all the numbers here.

00:05:14.996 --> 00:05:17.435
This is negative four, E is at this point,

00:05:17.435 --> 00:05:19.200
this is then we go to two.

00:05:19.200 --> 00:05:21.638
So it looks like we're
counting by twos here,

00:05:21.638 --> 00:05:24.214
that this is negative
two, this is zero, yep.

00:05:24.214 --> 00:05:25.864
Negative four, if you increase by two,

00:05:25.864 --> 00:05:28.928
negative four, negative
two, zero, two, four,

00:05:28.928 --> 00:05:30.299
this would be a six.

00:05:30.299 --> 00:05:31.854
This would be a negative six.

00:05:31.854 --> 00:05:33.363
They intentionally left those numbers off,

00:05:33.363 --> 00:05:34.965
so we had to figure out that hey, look,

00:05:34.965 --> 00:05:37.078
between negative four and two,

00:05:37.078 --> 00:05:39.075
to go from negative four to two,

00:05:39.075 --> 00:05:40.515
you have to increase by six

00:05:40.515 --> 00:05:43.070
and we only have one,
two, three hashmarks.

00:05:43.070 --> 00:05:45.856
So each of those hashmarks
must be increasing by two.

00:05:45.856 --> 00:05:47.342
Well anyway, now we know,

00:05:47.342 --> 00:05:48.712
now we know what all the points

00:05:48.712 --> 00:05:49.663
in the number line are.

00:05:49.663 --> 00:05:51.939
Evaluate E minus F.

00:05:51.939 --> 00:05:55.120
Well we know, we know that E

00:05:55.120 --> 00:05:58.092
is equal to negative two.

00:05:58.092 --> 00:06:01.366
And we know, we know that F

00:06:01.366 --> 00:06:03.188
is equal to four.

00:06:03.944 --> 00:06:05.442
So this is going to be the same thing,

00:06:05.442 --> 00:06:08.988
E is equal to negative two minus,

00:06:09.803 --> 00:06:12.449
minus positive four.

00:06:12.449 --> 00:06:14.888
Minus positive four.

00:06:14.888 --> 00:06:16.669
Let me do it in that same blue color.

00:06:17.507 --> 00:06:19.016
Minus four.

00:06:19.016 --> 00:06:21.268
Well negative two minus four,

00:06:21.268 --> 00:06:22.592
we have the number line in front of us,

00:06:22.592 --> 00:06:24.566
this is just going to be negative six.

00:06:24.566 --> 00:06:26.121
If you start at negative two,

00:06:26.121 --> 00:06:28.885
you subtract, if you subtract
two you get to negative four,

00:06:28.885 --> 00:06:31.926
you subtract another two,
you get to negative six.

00:06:31.926 --> 00:06:33.691
So we are done.

00:06:33.691 --> 00:06:35.239
Let's do one more of these.

00:06:37.280 --> 00:06:39.787
Let's evaluate T plus negative U.

00:06:39.787 --> 00:06:42.482
Where once again T and U
are on this number line

00:06:42.482 --> 00:06:45.152
and it looks like each of these hashmarks

00:06:45.152 --> 00:06:47.566
we're incrementing by three as we go up.

00:06:47.566 --> 00:06:49.930
Zero, three, six.

00:06:49.930 --> 00:06:53.810
And if we go down, zero,
negative three, negative six.

00:06:54.537 --> 00:06:59.111
So it's clear that U is equal to three.

00:06:59.111 --> 00:07:03.058
U is equal to three and it is also clear

00:07:03.058 --> 00:07:05.915
that T is equal to negative six.

00:07:05.915 --> 00:07:08.515
T is equal to negative six.

00:07:08.515 --> 00:07:11.919
So T plus negative U is going to be,

00:07:12.431 --> 00:07:15.798
so it's going to be negative six plus,

00:07:15.798 --> 00:07:20.798
negative six plus negative U, U is three.

00:07:22.177 --> 00:07:23.548
U is three.

00:07:23.548 --> 00:07:24.453
You want to be very careful,

00:07:24.453 --> 00:07:25.846
you might say oh well U is positive

00:07:25.846 --> 00:07:27.030
so I'm just gonna put
a positive number here,

00:07:27.030 --> 00:07:28.772
but remember, it's negative U.

00:07:28.772 --> 00:07:31.255
Wherever you see the U,
replace it with the three.

00:07:31.255 --> 00:07:32.951
So it's negative three.

00:07:32.951 --> 00:07:35.134
So this is gonna be six
plus negative three,

00:07:35.134 --> 00:07:37.781
or sorry, negative six
plus negative three,

00:07:37.781 --> 00:07:42.544
which we can rewrite as
negative six minus three.

00:07:43.237 --> 00:07:45.437
All of this business can be rewritten as

00:07:45.437 --> 00:07:47.201
negative six minus three.

00:07:47.201 --> 00:07:49.175
Negative six minus three.

00:07:49.175 --> 00:07:51.311
If you're at negative six
and you go three to the left,

00:07:51.311 --> 00:07:52.984
you go three more negative,

00:07:52.984 --> 00:07:55.677
you're going to end up at negative nine.

00:07:55.677 --> 00:08:00.677
And we're all done.


Equivalent expressions with negative numbers

WEBVTT
Kind: captions
Language: en

00:00:00.359 --> 00:00:02.414
- What I hope to do in this
video is several examples

00:00:02.414 --> 00:00:04.213
from the Khan Academy exercise on

00:00:04.213 --> 00:00:06.350
negative number addition and subtraction

00:00:06.350 --> 00:00:07.993
equivalent expressions.

00:00:07.993 --> 00:00:09.668
This is just to really make sure that we

00:00:10.006 --> 00:00:12.606
fully digested how we can compute with

00:00:12.606 --> 00:00:14.162
negative and positive numbers,

00:00:14.162 --> 00:00:15.589
and also the properties

00:00:17.558 --> 00:00:20.561
of arithmetic that we can apply
when we compute with them.

00:00:20.561 --> 00:00:21.722
Let's get started.

00:00:21.722 --> 00:00:24.067
It says Raul tried to evaluate

00:00:24.067 --> 00:00:25.867
an expression step by step.

00:00:26.081 --> 00:00:27.863
All right, they start with the expression,

00:00:27.863 --> 00:00:29.000
they give us the steps.

00:00:29.000 --> 00:00:31.380
They say find Raul's mistake.

00:00:31.380 --> 00:00:33.221
We have Step 1, Step 2, Step 3,

00:00:33.221 --> 00:00:35.635
or maybe Raul did not make a mistake.

00:00:36.219 --> 00:00:38.289
Like always, I encourage
you to pause the video

00:00:38.289 --> 00:00:40.773
and try to solve it on your
own before I work through it.

00:00:41.665 --> 00:00:43.255
All right, so let's
look at this first step.

00:00:43.255 --> 00:00:46.703
It says 3 minus negative 5 minus 7.

00:00:47.133 --> 00:00:49.153
It looks like he correctly recognized

00:00:49.153 --> 00:00:52.229
that subtracting a negative
5 is the same thing

00:00:52.229 --> 00:00:54.380
as adding a positive 5.

00:00:54.380 --> 00:00:57.096
He replaced the minus negative 5

00:00:57.096 --> 00:01:00.138
with the plus 5 right over there.

00:01:00.861 --> 00:01:02.042
That seems right.

00:01:02.042 --> 00:01:02.901
All right.

00:01:03.685 --> 00:01:05.873
Step 1 seems all right.

00:01:06.365 --> 00:01:08.090
Then, what does he do next?

00:01:08.090 --> 00:01:09.274
He has two options.

00:01:09.274 --> 00:01:12.618
He could say 3 plus 5 is equal to 8,

00:01:12.618 --> 00:01:14.649
and then subtract 7 from that,

00:01:14.649 --> 00:01:15.996
an you would get 1.

00:01:16.857 --> 00:01:20.384
He could even say 5 minus 7 is negative 2,

00:01:20.384 --> 00:01:22.880
and then add that to 3, and get 1.

00:01:23.726 --> 00:01:26.664
It looks like Raul tried
to go that second way,

00:01:26.664 --> 00:01:28.905
but he made a slight mistake.

00:01:28.905 --> 00:01:33.862
He said 5 minus 7 is
equal to 2 not negative 2.

00:01:33.862 --> 00:01:35.452
5 minus 7 should be negative 2,

00:01:35.452 --> 00:01:37.484
but he wrote a positive 2 right over here.

00:01:37.484 --> 00:01:40.242
His mistake is in Step 2.

00:01:40.242 --> 00:01:42.517
Then obviously, Step 3 becomes wrong,

00:01:42.517 --> 00:01:43.725
because Step 2.

00:01:46.155 --> 00:01:49.117
This is where he first makes the mistake.

00:01:50.054 --> 00:01:52.010
Let's do several more of these.

00:01:53.163 --> 00:01:55.572
Which of the following
expressions are equivalent

00:01:55.572 --> 00:01:59.693
to 6 plus negative 4 minus 5.

00:02:00.077 --> 00:02:02.294
All right, so let's look
at all the options here.

00:02:03.094 --> 00:02:04.418
This first one, let's see,

00:02:04.418 --> 00:02:05.834
has a negative 5.

00:02:05.834 --> 00:02:07.111
Well, the negative 5,

00:02:07.111 --> 00:02:09.155
this minus 5 if you put it out front,

00:02:09.155 --> 00:02:11.476
you could represent it as a negative 5.

00:02:12.383 --> 00:02:14.634
Then it has a negative 4 here.

00:02:15.264 --> 00:02:16.886
Here, we're adding a negative 4,

00:02:17.362 --> 00:02:18.599
or I guess you could say,

00:02:18.599 --> 00:02:21.501
we have minus 4 here,

00:02:21.501 --> 00:02:24.520
which is the same thing
as adding a negative 4.

00:02:24.520 --> 00:02:25.866
That seems reasonable.

00:02:25.866 --> 00:02:27.341
Then, you have plus 6,

00:02:27.341 --> 00:02:29.918
and you just had a
positive 6 right over here.

00:02:29.918 --> 00:02:32.019
This one actually seem equivalent.

00:02:32.019 --> 00:02:32.901
It just the order,

00:02:32.901 --> 00:02:34.167
they just switched the order.

00:02:34.751 --> 00:02:36.588
Instead of adding the 6 first,

00:02:36.588 --> 00:02:38.410
they add the positive 6 last.

00:02:38.410 --> 00:02:40.628
Instead of having the minus 5 at the end,

00:02:40.628 --> 00:02:42.555
they put a negative 5 in the beginning.

00:02:42.555 --> 00:02:44.729
Instead of saying adding negative 4,

00:02:44.729 --> 00:02:46.147
they just said minus 4.

00:02:46.147 --> 00:02:47.249
These are all equivalent.

00:02:47.249 --> 00:02:49.815
This one is going to be
an equivalent expression.

00:02:49.815 --> 00:02:51.469
Now, let's see, this one has ...

00:02:52.453 --> 00:02:54.702
it has the 6 and the minus 4,

00:02:54.702 --> 00:02:55.486
they just changed the order.

00:02:55.486 --> 00:02:56.926
Now, it has positive 5.

00:02:56.926 --> 00:02:58.157
That's not going to work,

00:02:58.157 --> 00:03:00.943
because we're subtracting 5 out here.

00:03:01.481 --> 00:03:03.404
If we want to put the 5 out front,

00:03:03.404 --> 00:03:05.621
it needs to be a negative 5.

00:03:05.621 --> 00:03:06.898
This isn't going to be right.

00:03:06.898 --> 00:03:08.163
I'm not going to select none of the above,

00:03:08.163 --> 00:03:09.858
because I found a choice
that is equivalent.

00:03:10.672 --> 00:03:12.361
Let's keep going, it's a lot of fun.

00:03:12.361 --> 00:03:14.717
Which of the following expressions are

00:03:14.717 --> 00:03:18.514
equivalent to negative 4
plus 4 plus 5 in parenthesis.

00:03:18.514 --> 00:03:19.593
If I were to compute this,

00:03:19.593 --> 00:03:20.963
I would say negative 4 plus,

00:03:20.963 --> 00:03:23.656
and I would add the 4 and the 5 first.

00:03:23.656 --> 00:03:25.282
I would get 9.

00:03:25.282 --> 00:03:27.336
It would be negative 4
plus 9 which is equal to 5.

00:03:28.028 --> 00:03:28.904
Let's see.

00:03:28.904 --> 00:03:33.161
Zero plus 5, well that's
going to be equal to 5.

00:03:33.161 --> 00:03:34.277
That's equivalent.

00:03:34.830 --> 00:03:36.161
I mean, I could just do it again.

00:03:38.714 --> 00:03:41.283
Negative 4 plus 9 is 5,

00:03:41.283 --> 00:03:43.557
which is the same thing as 9 minus 4.

00:03:43.557 --> 00:03:45.298
This right here is 5.

00:03:45.651 --> 00:03:47.554
Now, this is negative 4 minus 4,

00:03:47.554 --> 00:03:49.516
which would be negative 8.

00:03:49.516 --> 00:03:52.256
Then, plus 5 would be negative 3.

00:03:53.009 --> 00:03:54.078
That's not equivalent.

00:03:54.078 --> 00:03:54.937
In order to be equivalent,

00:03:54.937 --> 00:03:56.575
this should evaluate out to be 5.

00:03:56.575 --> 00:03:57.620
I'm not going to select that,

00:03:57.620 --> 00:03:58.739
and I'm not going to
select none of the above,

00:03:58.739 --> 00:04:00.538
because I all ready found
a choice that I like.

00:04:01.353 --> 00:04:02.709
Let's do a few more of these.

00:04:02.709 --> 00:04:04.625
One again, which are equivalent?

00:04:04.625 --> 00:04:08.048
Negative 5 plus 7 minus negative 3.

00:04:09.032 --> 00:04:12.530
Here you have negative 5 plus 7 plus 3.

00:04:12.530 --> 00:04:13.173
Well, that make sense.

00:04:13.173 --> 00:04:15.053
If you subtract a negative 3,

00:04:15.053 --> 00:04:17.096
that's the same thing as adding a 3.

00:04:17.096 --> 00:04:19.429
They didn't change the negative 5 plus 7.

00:04:19.736 --> 00:04:21.530
They didn't change the negative 5 plus 7.

00:04:21.530 --> 00:04:23.167
I definitely select this one.

00:04:23.167 --> 00:04:24.525
Definitely select that.

00:04:24.525 --> 00:04:25.732
Now, let's look at this one.

00:04:27.716 --> 00:04:30.806
Negative 5, here they're subtracting a 5,

00:04:30.806 --> 00:04:31.966
so that seems reasonable.

00:04:31.966 --> 00:04:34.091
Adding 7, they're adding 7.

00:04:34.783 --> 00:04:36.889
Here, they're subtracting a negative 3,

00:04:36.889 --> 00:04:39.628
which is the same thing
as adding a positive 3.

00:04:39.628 --> 00:04:42.927
Here, they just have a negative 3 here.

00:04:42.927 --> 00:04:44.668
If this was a positive 3,

00:04:44.668 --> 00:04:45.933
then we would be in business.

00:04:45.933 --> 00:04:46.769
This is a negative 3,

00:04:46.769 --> 00:04:47.988
so this isn't going to work.

00:04:47.988 --> 00:04:49.728
I will not select that.

00:04:49.753 --> 00:04:50.914
I won't select none of the above,

00:04:50.914 --> 00:04:52.156
because I found a choice.

00:04:53.217 --> 00:04:54.426
Let's do one more of these.

00:04:55.041 --> 00:04:56.875
Which of the following expressions

00:04:56.875 --> 00:04:59.371
are equivalent to X plus Y,

00:04:59.371 --> 00:05:00.718
and since there's parenthesis around it

00:05:00.718 --> 00:05:01.879
I would do that first,

00:05:01.879 --> 00:05:04.015
and then I would subtract Z.

00:05:04.015 --> 00:05:05.257
Select all that apply.

00:05:06.010 --> 00:05:06.801
All right.

00:05:08.200 --> 00:05:13.065
Here, they have Z plus X plus Y.

00:05:13.772 --> 00:05:15.654
They have the X plus Y part,

00:05:16.361 --> 00:05:19.821
but then they have a Z here,

00:05:19.821 --> 00:05:22.247
while over here they have minus Z.

00:05:22.247 --> 00:05:23.861
If there were a negative Z here,

00:05:23.861 --> 00:05:25.661
this and this would of been equivalent,

00:05:25.661 --> 00:05:27.065
but they don't have a negative Z here.

00:05:27.065 --> 00:05:29.817
I would not select that one.

00:05:30.617 --> 00:05:34.449
Now, this one you have X plus Y minus Z.

00:05:34.849 --> 00:05:36.968
Well this, they just
changed the parenthesis.

00:05:36.968 --> 00:05:38.663
Instead of doing the X plus Y first,

00:05:38.663 --> 00:05:41.286
they're dong the Y minus Z first.

00:05:42.239 --> 00:05:45.291
This goes back to
addition and subtraction.

00:05:45.291 --> 00:05:47.323
You have the associative property.

00:05:47.323 --> 00:05:48.948
You could associate these.

00:05:48.948 --> 00:05:50.202
You could start with

00:05:50.202 --> 00:05:51.630
adding these two first,

00:05:51.630 --> 00:05:53.069
and then you do the subtracting,

00:05:53.069 --> 00:05:54.544
or you could do the subtraction first.

00:05:54.544 --> 00:05:56.320
Then, you add this one.

00:05:56.320 --> 00:05:57.527
If you don't believe me,

00:05:57.527 --> 00:05:58.944
you could try it out with some numbers,

00:05:58.944 --> 00:06:00.000
and I encourage you to try it out

00:06:00.000 --> 00:06:01.567
with some negative numbers.

00:06:01.567 --> 00:06:03.401
It will indeed all work out.

00:06:03.401 --> 00:06:05.526
All they do is they change
where the parenthesis are.

00:06:05.526 --> 00:06:07.662
They did the Y minus Z first,

00:06:07.662 --> 00:06:11.145
and then they do the Y minus Z,

00:06:11.145 --> 00:06:12.526
and then they add it to the X,

00:06:12.526 --> 00:06:14.639
as opposed to doing the X plus Y first,

00:06:14.639 --> 00:06:16.520
then subtracting the Z.

00:06:16.520 --> 00:06:18.621
I would go with that one.


Justification for

WEBVTT
Kind: captions
Language: en

00:00:00.618 --> 00:00:02.418
- [Voiceover] What I
hope to do in this video

00:00:02.418 --> 00:00:05.721
is a reasonably quick
justification of the idea

00:00:05.721 --> 00:00:08.570
that the absolute value of a minus b

00:00:08.570 --> 00:00:12.371
is equal to the absolute
value of b minus a

00:00:12.371 --> 00:00:15.050
for any two numbers a or b.

00:00:15.050 --> 00:00:18.321
And before I even break into this,

00:00:18.321 --> 00:00:20.969
I wanna start with a simpler proposition.

00:00:20.969 --> 00:00:23.989
The idea that if I have some number x,

00:00:23.989 --> 00:00:27.744
and I'm just using x because
I already used up a and b,

00:00:27.744 --> 00:00:30.344
that the absolute value of x
is going to be the same thing

00:00:30.344 --> 00:00:33.076
as the absolute value of negative x.

00:00:33.076 --> 00:00:36.188
And to feel good about this
we could draw a number line.

00:00:36.188 --> 00:00:38.747
So we could draw

00:00:38.747 --> 00:00:41.172
a number line,

00:00:41.172 --> 00:00:43.849
and this is zero.

00:00:43.849 --> 00:00:47.115
So there's one circumstance
in which x is positive.

00:00:47.115 --> 00:00:49.108
Obviously if x was zero
these are both gonna be

00:00:49.108 --> 00:00:51.270
equal to zero, and it would be true.

00:00:51.270 --> 00:00:53.569
But one world where x is positive,

00:00:53.569 --> 00:00:55.467
then negative x would be the same distance

00:00:55.467 --> 00:00:57.316
to the left of zero,

00:00:57.316 --> 00:01:00.341
so negative x would be here.

00:01:00.341 --> 00:01:03.611
And the absolute value of
x would be this distance.

00:01:03.611 --> 00:01:06.772
And the absolute value
of negative x would be

00:01:06.772 --> 00:01:10.258
that same distance but just to the left.

00:01:10.258 --> 00:01:13.435
So these two things would be equivalent.

00:01:13.435 --> 00:01:16.875
Now if x were negative,
so let me draw that.

00:01:17.948 --> 00:01:21.405
So if x were negative,

00:01:21.405 --> 00:01:23.413
so if this was x,

00:01:23.413 --> 00:01:25.453
then what would negative x be?

00:01:25.453 --> 00:01:27.436
Well negative x would
be the opposite of it,

00:01:27.436 --> 00:01:30.500
so it would be the same
distance to the right of zero.

00:01:30.500 --> 00:01:32.206
So I know that this looks weird,

00:01:32.206 --> 00:01:33.989
me putting the negative x out here,

00:01:33.989 --> 00:01:36.100
but remember, x could be negative two.

00:01:36.100 --> 00:01:37.933
And if x is negative two,

00:01:37.933 --> 00:01:41.376
then negative negative two
is going to be positive.

00:01:41.376 --> 00:01:44.220
But either way you see
that the absolute value

00:01:44.220 --> 00:01:46.523
of these two things are the same.

00:01:46.523 --> 00:01:49.179
The absolute value is just
the distance from zero.

00:01:49.179 --> 00:01:51.565
This distance is still

00:01:51.565 --> 00:01:53.948
going to be the same as this distance.

00:01:53.948 --> 00:01:56.483
This distance is the absolute value of x.

00:01:56.483 --> 00:01:57.915
This distance, in this case,

00:01:57.915 --> 00:02:00.451
is the absolute value of negative x.

00:02:00.451 --> 00:02:03.932
So hopefully you feel
good that these two things

00:02:03.932 --> 00:02:06.203
are equivalent.

00:02:06.203 --> 00:02:08.580
Now, if you agree to that,

00:02:08.580 --> 00:02:12.310
now let's work on this a little bit.

00:02:12.310 --> 00:02:16.510
So we can start

00:02:16.510 --> 00:02:18.863
with the absolute value of

00:02:18.863 --> 00:02:21.250
a minus b.

00:02:21.250 --> 00:02:23.242
And there's a bunch of

00:02:23.242 --> 00:02:25.653
ways that we could tackle this.

00:02:25.653 --> 00:02:28.986
But one way we could approach it,

00:02:28.986 --> 00:02:30.876
and actually let me write a little lower,

00:02:30.876 --> 00:02:33.242
so I don't bump into this
stuff right over here.

00:02:33.242 --> 00:02:37.254
So let me start with the
absolute value of a minus b.

00:02:37.254 --> 00:02:41.689
Now a, we could rewrite, this is just a,

00:02:41.689 --> 00:02:43.226
I don't wanna say it's
positive or negative,

00:02:43.226 --> 00:02:45.002
a could be a negative number.

00:02:45.002 --> 00:02:46.474
This is just a right over here,

00:02:46.474 --> 00:02:48.779
but we can rewrite this
as being the same thing

00:02:48.779 --> 00:02:51.434
as negative negative a.

00:02:51.434 --> 00:02:55.343
Or I guess negative negative a plus,

00:02:55.343 --> 00:02:58.622
minus b.

00:03:00.142 --> 00:03:03.787
So let's just think
about this a little bit,

00:03:03.787 --> 00:03:05.814
and make sure we feel comfortable

00:03:05.814 --> 00:03:08.958
with what I just did.

00:03:08.958 --> 00:03:10.981
So this a

00:03:12.735 --> 00:03:15.242
is the same thing as all of this business.

00:03:15.242 --> 00:03:17.969
The negative negative a.

00:03:17.969 --> 00:03:20.005
The negative of a negative
is going to be a positive,

00:03:20.005 --> 00:03:21.129
this is positive a.

00:03:21.129 --> 00:03:24.234
So I haven't changed
this in any dramatic way.

00:03:24.234 --> 00:03:27.296
But what's this going
to be the same thing as?

00:03:27.296 --> 00:03:29.993
Well we can factor out a negative sign.

00:03:29.993 --> 00:03:32.184
So let's do that, and actually let me,

00:03:32.184 --> 00:03:35.512
we could factor out, using
a color I haven't used yet,

00:03:35.512 --> 00:03:39.080
we could factor out the negative signs.

00:03:39.080 --> 00:03:43.203
So this is going to be
equal to the absolute value

00:03:43.203 --> 00:03:46.314
of, I can factor out the negatives,

00:03:46.314 --> 00:03:48.604
the negative of,

00:03:48.604 --> 00:03:51.127
and then I have open parentheses,

00:03:51.127 --> 00:03:54.328
open parentheses negative a.

00:03:54.328 --> 00:03:56.162
And if you factor out a negative here,

00:03:56.162 --> 00:03:58.424
this is going to be positive b.

00:03:58.424 --> 00:04:00.271
So or plus b.

00:04:00.271 --> 00:04:03.479
Let me just close the
parentheses in the same color,

00:04:03.479 --> 00:04:05.119
just to make them match.

00:04:05.119 --> 00:04:07.094
Now why is this interesting?

00:04:07.094 --> 00:04:09.950
Well we just said the
absolute value of a negative

00:04:09.950 --> 00:04:13.026
is the same thing as, the
absolute value of negative x

00:04:13.026 --> 00:04:15.219
is the same thing as
the absolute value of x.

00:04:15.219 --> 00:04:17.058
So the absolute value of the negative

00:04:17.058 --> 00:04:19.927
of negative a plus b is
going to be equal to,

00:04:19.927 --> 00:04:22.497
this is going to be equal to

00:04:22.497 --> 00:04:26.041
the absolute value, let
me do that yellow color

00:04:26.041 --> 00:04:29.362
is going to be equal to the
absolute value of this thing

00:04:29.362 --> 00:04:31.081
without the negative right here.

00:04:31.081 --> 00:04:32.033
You see that right over here.

00:04:32.033 --> 00:04:33.548
The absolute value of negative x

00:04:33.548 --> 00:04:35.481
is the same thing as
the absolute value of x.

00:04:35.481 --> 00:04:38.314
So the absolute value of
negative negative a plus b,

00:04:38.314 --> 00:04:41.289
is the same thing as the absolute value of

00:04:41.289 --> 00:04:45.112
negative a plus b.

00:04:49.761 --> 00:04:52.160
And negative a plus b,
well that's the same thing

00:04:52.160 --> 00:04:55.193
as b minus a.

00:04:55.193 --> 00:04:57.753
I'm just swapping the
order right over here.

00:04:57.753 --> 00:05:01.753
B minus a.

00:05:01.753 --> 00:05:03.697
And we're done.

00:05:03.697 --> 00:05:05.337
We've just shown that the absolute value

00:05:05.337 --> 00:05:07.304
of a minus b is the same thing

00:05:07.304 --> 00:05:10.121
as the absolute value of b minus a.

00:05:10.121 --> 00:05:12.120
And obviously you hopefully
felt pretty good about it

00:05:12.120 --> 00:05:13.536
by just trying out a bunch of numbers,

00:05:13.536 --> 00:05:14.806
and thinking about the number line,

00:05:14.806 --> 00:05:17.392
and thinking about what the
difference of two numbers mean,

00:05:17.392 --> 00:05:18.968
if one is larger than the other.

00:05:18.968 --> 00:05:21.378
But this is we're literally just really

00:05:21.378 --> 00:05:23.696
factoring out a negative sign here,

00:05:23.696 --> 00:05:25.649
and recognizing this property up here,

00:05:25.649 --> 00:05:27.859
to make the same conclusion.


Interpreting absolute value as distance

WEBVTT
Kind: captions
Language: en

00:00:00.232 --> 00:00:01.370
- [Voiceover] What I
hope to do in this video

00:00:01.370 --> 00:00:02.855
is get a little bit more practice

00:00:02.855 --> 00:00:05.874
thinking about absolute value
of the difference of numbers

00:00:05.874 --> 00:00:08.561
as the distance between those two numbers.

00:00:09.328 --> 00:00:11.489
So in this first question, we are asked

00:00:11.489 --> 00:00:13.114
which of the following expressions

00:00:13.114 --> 00:00:15.900
are equivalent to the absolute value of

00:00:15.900 --> 00:00:18.222
A minus B.

00:00:18.222 --> 00:00:19.639
And just as a reminder, the absolute value

00:00:19.639 --> 00:00:23.074
of A minus B, this expression,

00:00:23.074 --> 00:00:24.933
this is going to give us the distance

00:00:24.933 --> 00:00:27.185
between A and B.

00:00:27.185 --> 00:00:30.018
So it's going to give us this
distance right over here.

00:00:30.018 --> 00:00:32.013
It's going to give us this
distance right over here.

00:00:32.013 --> 00:00:34.660
That is the absolute value of A minus B,

00:00:34.660 --> 00:00:35.962
which is of course the same thing

00:00:35.962 --> 00:00:38.422
as the absolute value of B minus A.

00:00:38.422 --> 00:00:40.860
So which of these
expressions are equivalent?

00:00:40.860 --> 00:00:44.761
So this first one has
the absolute value of A

00:00:44.761 --> 00:00:47.432
minus the absolute value of B.

00:00:47.432 --> 00:00:49.382
Well what is the absolute value of A?

00:00:49.382 --> 00:00:52.145
Well that's the distance
that A is from zero.

00:00:52.145 --> 00:00:54.235
So that's going to be this distance.

00:00:54.235 --> 00:00:58.878
This distance right over here
is the absolute value of A.

00:00:58.878 --> 00:01:01.247
That's the absolute value
of A right over there

00:01:01.247 --> 00:01:03.591
and then the absolute value of B is

00:01:03.591 --> 00:01:05.426
going to be this distance.

00:01:05.426 --> 00:01:07.841
That's the distance that B is from zero.

00:01:07.841 --> 00:01:09.141
So that right over there is going to be

00:01:09.141 --> 00:01:11.719
the absolute value of B.

00:01:11.719 --> 00:01:12.949
That's the absolute value of B.

00:01:12.949 --> 00:01:15.852
So if you take, if you say
the absolute value of A

00:01:15.852 --> 00:01:17.616
minus the absolute value of B,

00:01:17.616 --> 00:01:19.451
what are you going to be left with?

00:01:19.451 --> 00:01:21.309
Well, you're going to be left with,

00:01:21.309 --> 00:01:24.002
you're going to be left
with this distance.

00:01:24.002 --> 00:01:27.275
You're going to be left with
this distance right over here.

00:01:27.275 --> 00:01:29.900
This distance is the absolute value of A

00:01:29.900 --> 00:01:32.199
minus the absolute value of B.

00:01:32.199 --> 00:01:34.776
Absolute value of A minus this distance

00:01:34.776 --> 00:01:37.121
is gonna give you this green distance.

00:01:37.121 --> 00:01:38.816
Well that's exactly what we have up here.

00:01:38.816 --> 00:01:40.581
The absolute value of A minus B

00:01:40.581 --> 00:01:42.392
is the distance between A and B

00:01:42.392 --> 00:01:44.899
and that's what this
green distance is as well.

00:01:44.899 --> 00:01:45.868
So,

00:01:46.751 --> 00:01:48.888
this is going to be equivalent

00:01:48.888 --> 00:01:51.066
to the absolute value of A minus B.

00:01:51.066 --> 00:01:52.251
And if you wanna really verify it,

00:01:52.251 --> 00:01:53.667
you could try it with some numbers.

00:01:53.667 --> 00:01:55.176
I mean what they tell us about A and B

00:01:55.176 --> 00:01:56.986
is that both of them are
going to be negative.

00:01:56.986 --> 00:01:58.798
They're both to the left of zero.

00:01:58.798 --> 00:02:01.863
And we also see that B is greater than A,

00:02:01.863 --> 00:02:03.396
or it's less negative than A.

00:02:03.396 --> 00:02:05.067
So you could even try
it with some numbers.

00:02:05.067 --> 00:02:06.762
You could say well maybe B is negative one

00:02:06.762 --> 00:02:10.083
and A is negative five and then verify

00:02:10.083 --> 00:02:11.893
that this would be true.

00:02:11.893 --> 00:02:13.960
Now what about the absolute value of A

00:02:13.960 --> 00:02:16.515
plus the absolute value of B?

00:02:16.515 --> 00:02:19.092
Well that would be taking this distance,

00:02:19.092 --> 00:02:21.484
this magenta distance,
absolute value of A,

00:02:21.484 --> 00:02:24.478
and then adding it to this blue distance,

00:02:24.478 --> 00:02:26.336
that absolute value of B.

00:02:26.336 --> 00:02:28.774
So this would give you a larger distance

00:02:28.774 --> 00:02:32.512
than the distance
between those two points.

00:02:32.512 --> 00:02:33.999
Or, you could try it with numbers.

00:02:33.999 --> 00:02:36.134
I mean, imagine a world, just like I said,

00:02:36.134 --> 00:02:40.547
imagine a world where A
is equal to negative five

00:02:40.547 --> 00:02:43.588
and B is equal to negative one.

00:02:43.588 --> 00:02:46.282
Well in this world, the
distance between the two,

00:02:46.282 --> 00:02:48.672
the absolute value of A minus B

00:02:48.672 --> 00:02:51.854
would be equal to the absolute
value of negative five.

00:02:51.854 --> 00:02:55.964
Negative five minus negative one.

00:02:55.964 --> 00:02:57.613
Minus negative one.

00:02:57.613 --> 00:02:59.533
Which is the same thing as

00:03:00.319 --> 00:03:03.384
the absolute value of
negative five plus one.

00:03:03.384 --> 00:03:05.729
Which is equal to the absolute
value of negative four.

00:03:05.729 --> 00:03:07.099
Which is equal to four.

00:03:07.099 --> 00:03:08.399
So for these particular numbers,

00:03:08.399 --> 00:03:09.699
and I just picked them, I just picked two

00:03:09.699 --> 00:03:12.579
negative numbers where A
is more negative than B,

00:03:12.579 --> 00:03:13.924
the way it's drawn,

00:03:13.924 --> 00:03:15.457
this distance in green,

00:03:15.457 --> 00:03:16.851
or this distance right over here,

00:03:16.851 --> 00:03:18.012
would be four.

00:03:18.012 --> 00:03:19.869
Now the absolute value of A,

00:03:19.869 --> 00:03:23.167
absolute value of A plus
the absolute value of B,

00:03:23.167 --> 00:03:25.998
in this circumstance,
is going to be equal to

00:03:25.998 --> 00:03:26.882
five.

00:03:26.882 --> 00:03:28.554
It's going to be the absolute
value of negative five,

00:03:28.554 --> 00:03:30.201
which would just be five

00:03:30.201 --> 00:03:31.920
plus the absolute value of negative one,

00:03:31.920 --> 00:03:33.058
which would be one.

00:03:33.058 --> 00:03:34.428
This would be equal to six.

00:03:34.428 --> 00:03:36.007
So for these numbers,
once again I just picked

00:03:36.007 --> 00:03:37.724
two random numbers that
met the constraints

00:03:37.724 --> 00:03:39.233
that both are negative and that A

00:03:39.233 --> 00:03:41.881
is more negative than B is,

00:03:41.881 --> 00:03:43.204
it didn't hold up.

00:03:43.204 --> 00:03:44.923
So this is not going to be the case.

00:03:44.923 --> 00:03:46.154
And I'm not going to
select none of the above,

00:03:46.154 --> 00:03:47.941
because I found a choice that I know

00:03:47.941 --> 00:03:50.217
is going to be true.

00:03:50.217 --> 00:03:51.703
Let's do another one of these.

00:03:51.703 --> 00:03:53.955
Let's do several more of these.

00:03:53.955 --> 00:03:55.208
Which of the following expressions

00:03:55.208 --> 00:03:58.691
are equivalent to the
absolute value of A minus B?

00:03:58.691 --> 00:04:01.269
Once again, absolute value of A minus B,

00:04:01.269 --> 00:04:03.591
that is the distance between A and B.

00:04:03.591 --> 00:04:07.051
That is this distance that
I'm drawing right now.

00:04:07.051 --> 00:04:09.279
That is this distance right over here.

00:04:09.279 --> 00:04:12.686
That is the absolute value of A minus B.

00:04:13.825 --> 00:04:15.311
Well what is this first choice?

00:04:15.311 --> 00:04:18.655
Just A minus B without the absolute value.

00:04:18.655 --> 00:04:21.441
Well we see that A is less than B,

00:04:21.441 --> 00:04:22.231
it's more to the left.

00:04:22.231 --> 00:04:24.669
In fact, A is negative and B is positive.

00:04:24.669 --> 00:04:25.852
So if you take a negative number

00:04:25.852 --> 00:04:27.988
and then you subtract a
positive number from it,

00:04:27.988 --> 00:04:29.661
you're going to get a negative number.

00:04:29.661 --> 00:04:31.975
This thing right over here
is going to be negative.

00:04:33.023 --> 00:04:36.110
Or if you subtract a larger
number from a smaller number,

00:04:36.110 --> 00:04:38.572
you're going to get a negative value.

00:04:38.572 --> 00:04:40.450
But the distance between
these two numbers,

00:04:40.450 --> 00:04:42.886
we took the absolute value,
this is a positive value.

00:04:42.886 --> 00:04:44.348
This is just a distance.

00:04:44.348 --> 00:04:46.462
So this isn't going to be the case.

00:04:46.462 --> 00:04:47.879
Now let's look at this choice.

00:04:47.879 --> 00:04:50.635
The negative of B minus A.

00:04:51.380 --> 00:04:53.957
Well B minus A is going to be positive.

00:04:53.957 --> 00:04:55.304
How do we know that?

00:04:55.304 --> 00:04:57.185
Well B is larger than A.

00:04:57.185 --> 00:04:58.206
B is greater than A.

00:04:58.206 --> 00:05:00.086
So if B is greater than A, B minus A

00:05:00.086 --> 00:05:01.619
is going to be positive.

00:05:01.619 --> 00:05:03.175
But then we're taking the negative of it,

00:05:03.175 --> 00:05:06.287
so this whole expression
is going to be negative.

00:05:06.287 --> 00:05:08.400
It is going to be negative again.

00:05:08.400 --> 00:05:09.955
Another way to think about it,

00:05:09.955 --> 00:05:11.464
B is a positive number,

00:05:11.464 --> 00:05:13.345
you subtract a negative number from it,

00:05:13.345 --> 00:05:14.576
that's the same thing as adding

00:05:14.576 --> 00:05:16.016
the absolute value of
that negative number.

00:05:16.016 --> 00:05:17.594
This part is going to be positive.

00:05:17.594 --> 00:05:19.406
But then you have this
negative out front of it,

00:05:19.406 --> 00:05:21.495
it is going to be negative.

00:05:21.495 --> 00:05:22.842
And, like in the last example,

00:05:22.842 --> 00:05:24.699
you could try out numbers
that meet these constraints.

00:05:24.699 --> 00:05:28.669
Maybe B is positive three
and A is negative two.

00:05:28.669 --> 00:05:30.364
And I encourage you, try this out.

00:05:30.364 --> 00:05:32.941
Figure out what the absolute
value of A minus B is,

00:05:32.941 --> 00:05:34.102
it'll be five.

00:05:34.102 --> 00:05:35.311
And figure out which of these

00:05:35.311 --> 00:05:36.379
give you that same result.

00:05:36.379 --> 00:05:38.120
And neither of them will.

00:05:38.120 --> 00:05:41.557
So the answer here is none of the above.

00:05:41.557 --> 00:05:43.338
Let's keep going, this is a lot of fun.

00:05:44.175 --> 00:05:46.869
Alright, select the best interpretation

00:05:46.869 --> 00:05:48.820
of the following equation.

00:05:48.820 --> 00:05:51.327
So we have the absolute
value of 11 minus X.

00:05:51.327 --> 00:05:53.741
So this is the distance between 11 and X

00:05:53.741 --> 00:05:56.783
equals the absolute
value of Y minus three.

00:05:56.783 --> 00:05:58.594
So this is the distance
between Y and three.

00:05:58.594 --> 00:05:59.894
So this is telling us
that the distance between

00:05:59.894 --> 00:06:04.493
11 and X is the same as the
distance between Y and three.

00:06:04.493 --> 00:06:06.303
So they say the distance between 11 and X

00:06:06.303 --> 00:06:08.462
is equal to the distance
between Y and three.

00:06:08.462 --> 00:06:11.249
Yeah, that's exactly, that's
exactly what I just said.

00:06:11.249 --> 00:06:12.293
So I would select that.

00:06:12.293 --> 00:06:13.850
But let's look at the other choices.

00:06:13.850 --> 00:06:16.055
The distance between 11 and a negative x

00:06:16.055 --> 00:06:20.095
is qual to the distance
between Y and negative three.

00:06:20.095 --> 00:06:21.322
Well the distance between,

00:06:21.322 --> 00:06:23.203
let me underline this,

00:06:23.203 --> 00:06:26.198
the distance between 11 and negative X,

00:06:26.198 --> 00:06:27.653
let me just in a different color.

00:06:28.886 --> 00:06:30.431
The distance,

00:06:30.431 --> 00:06:32.849
I'm having trouble changing colors.

00:06:32.849 --> 00:06:37.238
So, the distance between
11 and negative X,

00:06:37.238 --> 00:06:38.609
that's not going to be this over here,

00:06:38.609 --> 00:06:40.024
that's going to be, you could take

00:06:40.024 --> 00:06:42.300
the absolute value of
11 and then from that

00:06:42.300 --> 00:06:45.156
you would subtract negative X.

00:06:45.156 --> 00:06:47.711
That's this thing right over here.

00:06:47.711 --> 00:06:50.333
So this would actually simplify
to the absolute value of 11

00:06:50.333 --> 00:06:54.212
plus X, which is not
what we have over here.

00:06:54.212 --> 00:06:57.322
And then the distance
between Y and negative three,

00:06:57.322 --> 00:06:58.205
same idea.

00:06:58.205 --> 00:07:02.175
That's gonna be Y minus negative three.

00:07:02.175 --> 00:07:03.825
Which is not what we have over here.

00:07:03.825 --> 00:07:05.241
So this is not,

00:07:05.241 --> 00:07:08.718
this is not what this equation represents.

00:07:09.578 --> 00:07:11.847
The distance between 11 and Y,

00:07:12.504 --> 00:07:13.660
okay so now they're really mixing.

00:07:13.660 --> 00:07:16.399
They're saying 11 and Y
is equal to the distance

00:07:16.399 --> 00:07:17.885
between negative X and negative three.

00:07:17.885 --> 00:07:20.022
So now they've just completely
mixed everything up.

00:07:20.022 --> 00:07:21.809
So that's not gonna be the case.

00:07:21.809 --> 00:07:23.335
Let's do one more of these.

00:07:24.358 --> 00:07:26.031
Let's do one more.

00:07:26.031 --> 00:07:29.512
So we are asked, which of
the following expressions

00:07:29.512 --> 00:07:32.268
is equal to the rectangle's area?

00:07:33.171 --> 00:07:34.401
Alright, so if we want to figure out

00:07:34.401 --> 00:07:35.679
the area of a rectangle,

00:07:35.679 --> 00:07:39.277
just multiply the width times the height.

00:07:39.277 --> 00:07:42.319
Or you could say the
length times the height.

00:07:42.319 --> 00:07:44.333
So let's see which of
these represent that.

00:07:44.986 --> 00:07:47.765
So the absolute value of J minus L.

00:07:48.625 --> 00:07:51.273
So, let me get a color here.

00:07:51.273 --> 00:07:53.176
So J

00:07:53.176 --> 00:07:55.731
minus L.

00:07:55.731 --> 00:07:59.655
The absolute value of J minus L.

00:07:59.655 --> 00:08:01.002
So J minus L,

00:08:01.002 --> 00:08:03.021
so J is this X coordinate,

00:08:03.021 --> 00:08:04.833
it's gonna be negative six,

00:08:04.833 --> 00:08:07.340
and L is this X coordinate,

00:08:07.340 --> 00:08:08.665
it's going to be,

00:08:08.665 --> 00:08:10.894
it's going to be positive six.

00:08:10.894 --> 00:08:12.774
So the absolute value of J minus L

00:08:12.774 --> 00:08:14.701
is going to be our,

00:08:14.701 --> 00:08:17.465
is going to the the difference
in the horizontal axis,

00:08:18.340 --> 00:08:21.638
or it's going to be the
distance on the horizontal axis

00:08:21.638 --> 00:08:23.217
between this point and that point.

00:08:23.217 --> 00:08:27.047
Or you say the horizontal
distance between those two points.

00:08:27.047 --> 00:08:30.391
So it would be the length
of this line segment.

00:08:30.391 --> 00:08:35.128
So that is the absolute
value of J minus L.

00:08:35.128 --> 00:08:37.496
Once again, the X coordinate
here is negative six,

00:08:37.496 --> 00:08:39.866
the X coordinate here is positive six,

00:08:39.866 --> 00:08:40.770
and you can even figure it out.

00:08:40.770 --> 00:08:43.279
It's going to be negative six minus six,

00:08:43.279 --> 00:08:45.461
which would be negative 12,

00:08:45.461 --> 00:08:47.551
and then you take the
absolute value of that,

00:08:47.551 --> 00:08:48.851
this is going to be 12.

00:08:48.851 --> 00:08:49.849
And you don't even have
to figure that out here,

00:08:49.849 --> 00:08:51.986
we just know that the length of this line

00:08:51.986 --> 00:08:55.580
is the absolute value of J minus L.

00:08:56.646 --> 00:08:57.739
So that's that.

00:08:57.739 --> 00:09:01.128
And then they have the
absolute value of M minus Q.

00:09:01.128 --> 00:09:03.474
So the absolute value,

00:09:03.474 --> 00:09:05.980
the absolute value of M minus Q.

00:09:05.980 --> 00:09:09.161
So they have M over here,
that's the Y coordinate here,

00:09:09.161 --> 00:09:11.902
and Q is the Y coordinate down here.

00:09:11.902 --> 00:09:14.014
So the absolute value of M minus Q

00:09:14.014 --> 00:09:15.687
is going to be the distance,

00:09:15.687 --> 00:09:18.263
the vertical distance
between these two points,

00:09:18.263 --> 00:09:19.471
which is really just,

00:09:19.471 --> 00:09:21.422
'cause the X value isn't changing,

00:09:21.422 --> 00:09:23.466
this is actually going to be the length.

00:09:23.466 --> 00:09:26.716
This is going to be the
length of that side.

00:09:26.716 --> 00:09:31.360
That's going to be the
absolute value of M minus Q.

00:09:31.360 --> 00:09:34.611
So yeah, if you multiply this
length times this length,

00:09:34.611 --> 00:09:37.188
you're going to get the
area of the rectangle.

00:09:37.188 --> 00:09:39.099
So I didn't even have to
look at the other choices,

00:09:39.099 --> 00:09:41.143
I would definitely go with this one.

00:09:41.143 --> 00:09:42.652
But let's see where the other ones

00:09:42.652 --> 00:09:44.347
probably aren't correct.

00:09:44.347 --> 00:09:47.226
So this is the absolute
value of J minus M.

00:09:47.226 --> 00:09:48.689
So here you're taking the difference of

00:09:48.689 --> 00:09:52.821
the X coordinate here and
the Y coordinate over there.

00:09:52.821 --> 00:09:54.494
So that's kind of bizarre.

00:09:54.494 --> 00:09:57.465
This this already looks suspicious.

00:09:57.465 --> 00:10:00.159
Here you're saying the
absolute value of J minus N.

00:10:00.159 --> 00:10:02.458
Absolute value of J,

00:10:02.458 --> 00:10:04.826
absolute value of J minus N.

00:10:04.826 --> 00:10:07.356
Well, their X coordinates are the same,

00:10:07.356 --> 00:10:08.890
so this is actually going to be,

00:10:08.890 --> 00:10:10.467
this we actually know is going to be zero.

00:10:10.467 --> 00:10:13.975
J is equal to N, they're
both equal to negative six.

00:10:13.975 --> 00:10:15.971
That's not gonna give you
the length of this line,

00:10:15.971 --> 00:10:19.060
because we have no change along X here.

00:10:19.060 --> 00:10:20.987
All the change is along Y.

00:10:20.987 --> 00:10:22.324
If we wanted to figure out the length

00:10:22.324 --> 00:10:24.112
of this line right over here

00:10:24.112 --> 00:10:29.112
we would have to find
the absolute value of K,

00:10:29.777 --> 00:10:32.842
of the change in our Y coordinates.

00:10:32.842 --> 00:10:35.048
So absolute value of K minus O

00:10:35.048 --> 00:10:36.929
would give you the length of this line.

00:10:36.929 --> 00:10:38.740
And if you wanted the length
of this right over here,

00:10:38.740 --> 00:10:40.458
you'd want your change in X,

00:10:40.458 --> 00:10:44.336
so that would be the
absolute value of N minus P,

00:10:44.336 --> 00:10:48.770
or you could say the
absolute value of P minus N.

00:10:48.770 --> 00:10:49.978
But they didn't use those choices.

00:10:49.978 --> 00:10:51.918
So yeah, we feel good about that.