Finding area by breaking up the shape

WEBVTT
Kind: captions
Language: en

00:00:00.704 --> 00:00:02.370
We have this strange
looking shape here,

00:00:02.370 --> 00:00:04.360
and then we're given
some of its dimensions.

00:00:04.360 --> 00:00:09.240
We know that this side right
over here has a length of 3.5.

00:00:09.240 --> 00:00:11.650
This side over here is 6.5.

00:00:11.650 --> 00:00:13.745
Then we know from
here to here is 2,

00:00:13.745 --> 00:00:16.585
and then from here to here is 7.

00:00:16.585 --> 00:00:18.960
And then they're giving us
this dimension right over here

00:00:18.960 --> 00:00:20.840
is 3.5.

00:00:20.840 --> 00:00:22.580
So given that,
let's see if we can

00:00:22.580 --> 00:00:26.460
find the area of
this entire figure.

00:00:26.460 --> 00:00:28.970
And I encourage you to pause
the video right now and try

00:00:28.970 --> 00:00:30.900
this on your own.

00:00:30.900 --> 00:00:32.570
I assume you've
given a go at it.

00:00:32.570 --> 00:00:34.070
And there might be
a few things that

00:00:34.070 --> 00:00:35.560
jump out at you immediately.

00:00:35.560 --> 00:00:36.510
The first thing
is that they have

00:00:36.510 --> 00:00:37.760
these two triangles up here.

00:00:37.760 --> 00:00:39.310
And they give us all of
the dimensions for them,

00:00:39.310 --> 00:00:41.370
or at least they give us
the base and the height

00:00:41.370 --> 00:00:43.720
for it, which is enough
to figure out the area.

00:00:46.370 --> 00:00:54.200
If I had a rectangle that was 2
units wide and 3.5 units high,

00:00:54.200 --> 00:01:02.010
if we know that it would
have an area of 2 times 3.5.

00:01:02.010 --> 00:01:03.907
Now a triangle is
just going to be,

00:01:03.907 --> 00:01:05.990
especially a triangle like
this, a right triangle,

00:01:05.990 --> 00:01:08.590
is just going to be half
of a rectangle like this.

00:01:08.590 --> 00:01:11.660
We just care about
half of its area.

00:01:11.660 --> 00:01:18.230
So this area is going to
be 1/2 times 2 times 3.5.

00:01:18.230 --> 00:01:20.580
1/2 times 2 is equal to 1.

00:01:20.580 --> 00:01:25.180
1 times 3.5 is 3.5 square units.

00:01:25.180 --> 00:01:29.500
So the area of that part is
going to be 3.5 square units.

00:01:29.500 --> 00:01:33.530
Let's think about the area of
this triangle right over here.

00:01:33.530 --> 00:01:35.970
Well, once again we
have its height is 3.5.

00:01:35.970 --> 00:01:37.950
Its base is 7.

00:01:37.950 --> 00:01:44.680
So its area is going to
be 1/2 times 7 times 3.5.

00:01:44.680 --> 00:01:48.950
1/2 times 7 is 3.5 times 3.5.

00:01:48.950 --> 00:01:50.380
So this part is
3.5, and I'm going

00:01:50.380 --> 00:01:52.770
to multiply that
times 3.5 again.

00:01:52.770 --> 00:01:55.370
Let's figure out what
that product is equal to.

00:01:55.370 --> 00:02:00.694
3.5 times 3.5.

00:02:00.694 --> 00:02:03.920
5 times 5 is 25.

00:02:03.920 --> 00:02:08.380
3 times 5 is 15, plus 2 is 17.

00:02:08.380 --> 00:02:09.870
Let's cross that out.

00:02:09.870 --> 00:02:11.830
Move one place over to the left.

00:02:11.830 --> 00:02:14.560
3 times 5 is 15.

00:02:14.560 --> 00:02:19.410
3 times 3 is 9, plus 1 is 10.

00:02:19.410 --> 00:02:22.630
So that gets us
to 5 plus 0 is 5.

00:02:22.630 --> 00:02:24.920
7 plus 5 is 12, carry the 1.

00:02:24.920 --> 00:02:26.260
1 plus 1 is 2.

00:02:26.260 --> 00:02:27.780
And we have a 1.

00:02:27.780 --> 00:02:30.515
We have two digits to the
right of the decimal, one, two.

00:02:30.515 --> 00:02:31.890
So we're going to
have two digits

00:02:31.890 --> 00:02:34.160
to the right of the
decimal in the answer.

00:02:34.160 --> 00:02:38.870
The area here is
12.25 square units.

00:02:38.870 --> 00:02:43.920
Now this region may be a
little bit more difficult,

00:02:43.920 --> 00:02:47.456
because it's kind of us this
weird trapezoid looking thing.

00:02:47.456 --> 00:02:49.080
But one thing that
might pop out at you

00:02:49.080 --> 00:02:51.230
is that you can
divide it very easily

00:02:51.230 --> 00:02:55.050
into a rectangle and a triangle.

00:02:55.050 --> 00:02:56.970
And we can actually
figure out the dimensions

00:02:56.970 --> 00:02:59.980
that we need to figure out
the areas of each of these.

00:02:59.980 --> 00:03:02.496
We know what the width
of this rectangle

00:03:02.496 --> 00:03:04.370
is, or the length of
this rectangle, whatever

00:03:04.370 --> 00:03:05.500
you want to call it.

00:03:05.500 --> 00:03:08.000
It's going to be 2
units plus 7 units.

00:03:08.000 --> 00:03:10.270
So this is going to be 9.

00:03:10.270 --> 00:03:14.130
We know that this
distance is 3.5.

00:03:14.130 --> 00:03:17.310
If this distance right
over here is 3.5,

00:03:17.310 --> 00:03:22.780
then this distance down here
has to add up with 3.5 to 6.5,

00:03:22.780 --> 00:03:26.680
so this must be 3.

00:03:26.680 --> 00:03:30.660
Now we can actually
figure out the area.

00:03:30.660 --> 00:03:32.200
The area of this
rectangle is just

00:03:32.200 --> 00:03:38.092
going to be its height times
its length, or 9 times 3.5.

00:03:38.092 --> 00:03:40.859
9 times 3.5.

00:03:40.859 --> 00:03:42.650
And one way you could
do it-- we could even

00:03:42.650 --> 00:03:44.983
try to do this in our head--
this is going to be 9 times

00:03:44.983 --> 00:03:47.580
3 plus 9 times 0.5.

00:03:47.580 --> 00:03:49.660
9 times 3 is 27.

00:03:49.660 --> 00:03:54.630
9 times 0.5, that's just half of
nine, so it's going to be 4.5.

00:03:54.630 --> 00:03:58.070
27 plus 4 will get
us to 31, so that's

00:03:58.070 --> 00:04:01.050
going to be equal to 31.5.

00:04:01.050 --> 00:04:03.300
Or you could multiply it
out like this, if you like.

00:04:03.300 --> 00:04:05.977
But the area of
this region is 31.5.

00:04:05.977 --> 00:04:08.060
And then the area of this
triangle right over here

00:04:08.060 --> 00:04:10.860
is going to be 9
times 3 times 1/2.

00:04:10.860 --> 00:04:12.320
We're looking at a triangle.

00:04:12.320 --> 00:04:14.440
9 times 3 is 27.

00:04:14.440 --> 00:04:19.974
27 times 1/2 is 13.5.

00:04:19.974 --> 00:04:21.640
So to find the area
of the entire thing,

00:04:21.640 --> 00:04:23.750
we just have to
sum up these areas.

00:04:23.750 --> 00:04:43.030
We have 31.5 plus 13.5
plus 12.25 plus 3.5.

00:04:43.030 --> 00:04:44.930
So we just have a 5
here in the hundredths.

00:04:44.930 --> 00:04:46.080
That's the only one.

00:04:46.080 --> 00:04:52.580
5 plus 5 is 10, plus 7 is 17.

00:04:52.580 --> 00:05:00.920
1 plus 1 is 2, plus 3 is 5,
plus 2 is 7, plus 3 is 10.

00:05:00.920 --> 00:05:06.160
1 plus 3 is 4, plus
1 is 5, plus 1 is 6.

00:05:06.160 --> 00:05:14.059
So we get a total area for this
figure of 60.75 square units.


Finding area by rearranging parts

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.700
We have four quadrilaterals
drawn right over here.

00:00:03.700 --> 00:00:05.980
And what I want
us to think about

00:00:05.980 --> 00:00:09.105
is looking at this green
quadrilateral here.

00:00:09.105 --> 00:00:10.730
I want you to pause
the video and think

00:00:10.730 --> 00:00:13.030
about which of
these figures have

00:00:13.030 --> 00:00:17.500
the same area as the
green quadrilateral?

00:00:17.500 --> 00:00:20.360
And so pause the video
now and think about that.

00:00:20.360 --> 00:00:22.040
So I'm assuming you
gave a shot at it.

00:00:22.040 --> 00:00:22.980
Now let's think about it.

00:00:22.980 --> 00:00:24.480
And the way I'm
going to think about

00:00:24.480 --> 00:00:27.350
is to really rearrange parts
of this green quadrilateral

00:00:27.350 --> 00:00:29.040
to make it look
more like maybe some

00:00:29.040 --> 00:00:30.800
of these other quadrilaterals.

00:00:30.800 --> 00:00:33.040
So for example, if
we were to if we

00:00:33.040 --> 00:00:35.860
were to put a little
dotted line right over here

00:00:35.860 --> 00:00:38.170
and a dotted line
right over here,

00:00:38.170 --> 00:00:43.360
we see that our green
shape is actually made up,

00:00:43.360 --> 00:00:46.560
you could imagine it being made
up of, a triangle, and then

00:00:46.560 --> 00:00:48.810
a rectangle, and then
another triangle.

00:00:48.810 --> 00:00:51.320
And what's interesting
about the two triangles

00:00:51.320 --> 00:00:54.310
is that they represent
the exact same area.

00:00:54.310 --> 00:00:55.870
They essentially
both represent, they

00:00:55.870 --> 00:01:00.080
each represent half of this
rectangle right over here.

00:01:03.890 --> 00:01:05.519
Let me do that in a color.

00:01:05.519 --> 00:01:09.130
They represent half
of this entire thing

00:01:09.130 --> 00:01:13.150
if I were to color it all in.

00:01:13.150 --> 00:01:15.020
And if you have
trouble visualizing it,

00:01:15.020 --> 00:01:17.690
imagine taking this top
part right over here

00:01:17.690 --> 00:01:19.600
and then flipping it over.

00:01:19.600 --> 00:01:20.600
It would look like this.

00:01:20.600 --> 00:01:23.200
If you flip it over, this
line right over here,

00:01:23.200 --> 00:01:24.650
it would look
something like this.

00:01:27.710 --> 00:01:29.930
My best attempt to draw it.

00:01:29.930 --> 00:01:33.790
So take that top section, it
would look something like that.

00:01:33.790 --> 00:01:38.520
And then move it down right
over here to fit in here.

00:01:38.520 --> 00:01:42.680
And then this plus this will
fill in this entire region

00:01:42.680 --> 00:01:43.950
right over here.

00:01:43.950 --> 00:01:48.190
So that original green trapezoid
that we were looking at,

00:01:48.190 --> 00:01:52.740
if you take that top
part out, it essentially

00:01:52.740 --> 00:01:57.610
has the exact same area
as a rectangle that

00:01:57.610 --> 00:02:03.960
has a height of 4
and a length of 5.

00:02:03.960 --> 00:02:06.275
So this right over here
has the exact same area

00:02:06.275 --> 00:02:07.550
as our trapezoid.

00:02:07.550 --> 00:02:09.449
And once again,
how did we do that?

00:02:09.449 --> 00:02:11.180
Well, we just took
this top part,

00:02:11.180 --> 00:02:13.234
flipped it over, and
relocated it down here.

00:02:13.234 --> 00:02:14.650
And we said hey,
we could actually

00:02:14.650 --> 00:02:17.090
construct a rectangle that way.

00:02:17.090 --> 00:02:20.280
So essentially, and if
you want to know its area,

00:02:20.280 --> 00:02:22.460
we could either just
count the squares here.

00:02:22.460 --> 00:02:26.520
So we have, let me do
this in an easier to see.

00:02:26.520 --> 00:02:35.642
So we have 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15,

00:02:35.642 --> 00:02:41.517
16, 17, 18, 19, 20 of these
unit squares right over here.

00:02:41.517 --> 00:02:43.600
And we know that there's
an easier way to do that.

00:02:43.600 --> 00:02:46.600
We could have just multiplied
the height times the width.

00:02:46.600 --> 00:02:50.110
We could have just said,
look this thing is 1, 2, 3,

00:02:50.110 --> 00:02:54.440
4 high and 1, 2, 3, 4, 5 wide.

00:02:54.440 --> 00:02:58.710
So 4 times 5 is going to give
us 20 of these units squares.

00:02:58.710 --> 00:03:02.890
So that's the area in terms of
unit squares, or square units,

00:03:02.890 --> 00:03:05.560
of that original
green trapezoid.

00:03:05.560 --> 00:03:08.470
Now let's see which one
of these match that.

00:03:08.470 --> 00:03:10.350
So this pink one
right over here.

00:03:10.350 --> 00:03:12.720
If you don't even
count this bottom part,

00:03:12.720 --> 00:03:15.590
if you were to just separate
this top part right over here.

00:03:15.590 --> 00:03:20.730
This top part is
4 high by 5 wide.

00:03:20.730 --> 00:03:23.580
So just this top
part alone is 20.

00:03:23.580 --> 00:03:25.940
And then it has this
extra right over here.

00:03:25.940 --> 00:03:27.970
So the pink has a
larger area than

00:03:27.970 --> 00:03:30.370
our original green trapezoid.

00:03:30.370 --> 00:03:34.370
The blue rectangle is 3 by 5.

00:03:34.370 --> 00:03:39.140
So it has an area
of 15 square units.

00:03:39.140 --> 00:03:40.660
Now the red one is interesting.

00:03:40.660 --> 00:03:47.230
It is 1, 2, 3, 4 high and 1,
2, 3, 4, 5 long or 5 wide.

00:03:47.230 --> 00:03:50.570
4 times 5 is 20 squares,
and you can validate that.

00:03:50.570 --> 00:03:52.930
And so the red rectangle
has the same area

00:03:52.930 --> 00:03:55.770
as our original green trapezoid.


How to find the area of a strange quadrilateral on a grid

WEBVTT
Kind: captions
Language: en

00:00:00.970 --> 00:00:03.060
I want you to pause this
video and figure out

00:00:03.060 --> 00:00:05.741
if you can figure out the area
of this quadrilateral right

00:00:05.741 --> 00:00:06.240
over here.

00:00:06.240 --> 00:00:07.281
And I'll give you a hint.

00:00:07.281 --> 00:00:11.800
Try to break it up into
shapes where it is easier

00:00:11.800 --> 00:00:15.510
to find the area, especially
given the grid of these unit

00:00:15.510 --> 00:00:18.620
squares that we have
right over here.

00:00:18.620 --> 00:00:20.380
So I assume you gave it a go.

00:00:20.380 --> 00:00:22.115
Now let's try to do it together.

00:00:22.115 --> 00:00:25.895
So I'm going to
start at points where

00:00:25.895 --> 00:00:32.520
it would be very easy to measure
the dimensions of whatever

00:00:32.520 --> 00:00:33.970
figures we might break it up at.

00:00:33.970 --> 00:00:35.750
And so in general,
I want to go to any

00:00:35.750 --> 00:00:39.090
of the whole numbers
of these units,

00:00:39.090 --> 00:00:41.107
so that point right over there.

00:00:41.107 --> 00:00:41.690
And let's see.

00:00:41.690 --> 00:00:43.675
I could start to move
in this direction.

00:00:46.954 --> 00:00:48.870
And it looks like I might
be able to construct

00:00:48.870 --> 00:00:51.550
a triangle down here.

00:00:51.550 --> 00:00:54.377
And it would be tempting
to go all the way across,

00:00:54.377 --> 00:00:56.210
but that wouldn't be
too useful because this

00:00:56.210 --> 00:00:58.646
gets me to halfway
through a unit.

00:00:58.646 --> 00:01:01.020
And I'm just eyeballing it to
say halfway through a unit.

00:01:01.020 --> 00:01:04.099
It might not be exactly
halfway through a unit.

00:01:04.099 --> 00:01:06.540
And so instead, let me
see if I can get away

00:01:06.540 --> 00:01:10.190
with making a triangle
just like that.

00:01:10.190 --> 00:01:11.360
Now, I just did that.

00:01:11.360 --> 00:01:15.310
So let me try to
raise this up to see

00:01:15.310 --> 00:01:18.746
if I can make another
triangle where it would

00:01:18.746 --> 00:01:20.625
be easy to figure
out its dimensions.

00:01:20.625 --> 00:01:22.500
So once again, I don't
want to go all the way

00:01:22.500 --> 00:01:26.490
up here because now I'm
not at a whole unit.

00:01:26.490 --> 00:01:30.990
Instead, let me take a right
and go right over here.

00:01:30.990 --> 00:01:33.354
And notice, both
of these are very

00:01:33.354 --> 00:01:34.770
easy to figure out
its dimensions.

00:01:34.770 --> 00:01:42.000
This is 1, 2, 3, 4, 5
units long and 1 unit high.

00:01:42.000 --> 00:01:47.070
This one right over here
is 1, 2, 3, 4 units long

00:01:47.070 --> 00:01:49.880
and 1, 2 units wide.

00:01:49.880 --> 00:01:57.570
So let's see if we can cover
the entire quadrilateral,

00:01:57.570 --> 00:02:00.620
if we can break it up, I should
say, into a bunch of figures

00:02:00.620 --> 00:02:01.480
like this.

00:02:01.480 --> 00:02:04.460
So it seems like we have
another one just like that.

00:02:04.460 --> 00:02:07.510
And then I could drop this
down, and then we're done.

00:02:07.510 --> 00:02:09.919
All of these are pretty
straightforward to figure out

00:02:09.919 --> 00:02:11.130
what their dimensions are.

00:02:11.130 --> 00:02:12.640
This is 5 by 1.

00:02:12.640 --> 00:02:13.900
This is 4 by 2.

00:02:13.900 --> 00:02:19.160
This is 1, 2, 3,
4, 5, 6 by 1, 2.

00:02:19.160 --> 00:02:24.160
And this is 1 by 1, 2, 3, 4, 5.

00:02:24.160 --> 00:02:26.217
So what is the area
of this figure?

00:02:26.217 --> 00:02:28.300
And of course, we have
this center rectangle right

00:02:28.300 --> 00:02:29.630
over here.

00:02:29.630 --> 00:02:36.040
Well, a triangle that is 5 units
long and 1 unit high, its area

00:02:36.040 --> 00:02:42.425
is going to be 1/2
times 1 times 5.

00:02:42.425 --> 00:02:46.430
Or I could write it
1/2 times 1 times

00:02:46.430 --> 00:02:48.510
5, depending on what
multiplication symbol

00:02:48.510 --> 00:02:50.130
you are more comfortable with.

00:02:50.130 --> 00:02:53.040
Well that's just going
to be 1/2 times 5, which

00:02:53.040 --> 00:02:56.100
is going to be equal to 2.5.

00:02:56.100 --> 00:02:58.640
So that's 2.5 right over there.

00:02:58.640 --> 00:03:02.470
This one is going to
be 1/2 times 4 times 2.

00:03:02.470 --> 00:03:05.470
Well, that's just going
to be 2 times 2 or 4.

00:03:05.470 --> 00:03:11.210
This one is going to be 1/2
times 2 times 1, 2, 3, 4, 5, 6.

00:03:11.210 --> 00:03:14.600
Well, 1/2 half times 2
is 1 times 6 is just 6.

00:03:14.600 --> 00:03:16.600
And then this one's
going to be 1/2 times

00:03:16.600 --> 00:03:20.680
1 times 1, 2, 3, 4, 5.

00:03:20.680 --> 00:03:24.350
So once again, the area of
this one is going to be 2.5.

00:03:24.350 --> 00:03:29.750
And then finally, this
is a 3 by 4 rectangle.

00:03:29.750 --> 00:03:32.060
And you could even count
the unit squares in here.

00:03:32.060 --> 00:03:33.610
But it has 12 of
those unit square,

00:03:33.610 --> 00:03:35.437
so it has an area of 12.

00:03:35.437 --> 00:03:38.020
So if we want to find the total
area, we just add all of these

00:03:38.020 --> 00:03:39.090
together.

00:03:39.090 --> 00:03:49.200
So 2.5 plus 2.5 is 5, plus 4 is
9, plus 6 is 15, plus 12 is 27.

00:03:49.200 --> 00:03:53.290
So it has a total area of 27.


How to describe real-world situations with inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.076 --> 00:00:05.076
- The speed of sound is
approximately 768 miles per hour.

00:00:05.540 --> 00:00:09.604
When an object travels faster
than the speed of sound,

00:00:09.604 --> 00:00:12.068
it creates a sonic boom.

00:00:12.068 --> 00:00:13.446
And there's actually a whole video

00:00:13.446 --> 00:00:17.249
on how sonic booms are
created, on Khan Academy.

00:00:17.249 --> 00:00:19.008
I encourage you to watch it,
it is actually fascinating.

00:00:19.008 --> 00:00:20.508
But anyway, back to this question.

00:00:20.508 --> 00:00:23.643
Write an inequality to
represent, s, the speeds

00:00:23.643 --> 00:00:27.042
at which a moving object
creates a sonic boom.

00:00:27.042 --> 00:00:29.068
So we'll start an inequality, we'll do

00:00:29.068 --> 00:00:29.898
what they told us to.

00:00:29.898 --> 00:00:31.880
S, let's see. S has to be,

00:00:31.880 --> 00:00:34.089
they say when an object travels faster

00:00:34.089 --> 00:00:35.790
than the speed of sound.

00:00:35.790 --> 00:00:39.183
So our speed has to be greater
than the speed of sound.

00:00:39.183 --> 00:00:41.909
Our speed has to be greater
than the speed of sound,

00:00:41.909 --> 00:00:46.381
which is 768 miles per hour,

00:00:46.381 --> 00:00:51.014
so s has to be set greater than 768.

00:00:51.014 --> 00:00:52.148
Now another way we could have written that

00:00:52.148 --> 00:00:54.838
is that 768 needs to be less than s.

00:00:54.838 --> 00:00:59.146
We could have written 768
needs to be less than s.

00:00:59.219 --> 00:01:00.629
That's another way we could write it.

00:01:00.629 --> 00:01:02.381
In either case, our speed is going

00:01:02.381 --> 00:01:06.660
to be greater than 768 miles per hour.

00:01:06.660 --> 00:01:07.939
We check our answer.

00:01:07.939 --> 00:01:09.731
Let's do a few more of these.

00:01:09.731 --> 00:01:11.809
To cook a steak to medium-rare,

00:01:11.809 --> 00:01:13.980
it needs to have an internal temperature

00:01:13.980 --> 00:01:16.566
of 135 degrees Fahrenheit.

00:01:16.566 --> 00:01:20.211
A lower internal temperature
will undercook the steak.

00:01:20.211 --> 00:01:22.234
Write an inequality to represent all

00:01:22.234 --> 00:01:24.632
of the possible temperatures, t,

00:01:24.632 --> 00:01:26.586
of an undercooked steak.

00:01:27.110 --> 00:01:28.706
Well, to be an undercooked steak

00:01:28.706 --> 00:01:33.706
your temperature is going
to be less than 135 degrees.

00:01:34.270 --> 00:01:37.056
So your temperature, t,
is going to be less than.

00:01:37.056 --> 00:01:39.994
You could either just use the
less than on your keyboard,

00:01:39.994 --> 00:01:41.869
or you could click over here on less than.

00:01:41.869 --> 00:01:44.995
Your temperature is going
to be less than 135.

00:01:44.995 --> 00:01:47.552
We're assuming your temperature
is in Fahrenheit degrees.

00:01:47.552 --> 00:01:49.407
So your temperature needs to be less than

00:01:49.407 --> 00:01:52.347
Fahrenheit, less then 135 degrees,

00:01:52.347 --> 00:01:55.763
if you are going to have
an undercooked steak.

00:01:56.222 --> 00:01:58.556
So, you got that one right.

00:01:58.564 --> 00:02:01.249
Let's do a couple more.
Or one, let's do one more.

00:02:01.249 --> 00:02:06.249
Simone is going snowboarding tomorrow

00:02:06.630 --> 00:02:10.415
if the temperature is equal to or colder

00:02:10.415 --> 00:02:13.193
than five degrees celsius.

00:02:13.193 --> 00:02:14.902
Write an inequality to represent

00:02:14.902 --> 00:02:18.350
the temperatures at which
Simone will go snowboarding.

00:02:18.350 --> 00:02:21.200
So you might want to say "Oh, well

00:02:21.200 --> 00:02:25.226
"maybe you know that t is
less than five degrees."

00:02:25.226 --> 00:02:26.120
Remember, they're telling us that

00:02:26.120 --> 00:02:27.528
the temperature is equal to or colder,

00:02:27.528 --> 00:02:29.031
so she's even willing to go snowboarding

00:02:29.031 --> 00:02:31.594
if it's less than or
equal to five degrees.

00:02:31.594 --> 00:02:32.775
So instead of just saying less than,

00:02:32.775 --> 00:02:35.501
let's say less than or, whoops.

00:02:35.501 --> 00:02:40.361
Let's say less than or
equal to five degrees.

00:02:40.361 --> 00:02:41.872
So this inequality represented here,

00:02:41.872 --> 00:02:45.187
this shows us all of the temperatures

00:02:45.187 --> 00:02:50.187
at which she is willing
to go snowboarding.

00:02:50.982 --> 00:02:52.854
We check our answer and we got it right.


How to find mistakes in one step equations

WEBVTT
Kind: captions
Language: en

00:00:00.423 --> 00:00:04.370
- [Voiceover] Carly tried to
solve an equation step by step.

00:00:04.370 --> 00:00:06.042
And they tell us, they see, we see

00:00:06.042 --> 00:00:07.667
how she tried to solve the equation.

00:00:07.667 --> 00:00:10.082
They say find Carly's mistake.

00:00:10.082 --> 00:00:11.289
So let's see what Carly did.

00:00:11.289 --> 00:00:15.817
She started with 7a = 28.

00:00:15.817 --> 00:00:19.578
Then on the left hand
side she divides it by a

00:00:19.578 --> 00:00:23.201
and on the right hand
side she divides by 7.

00:00:23.201 --> 00:00:24.663
Well this seems strange.

00:00:24.663 --> 00:00:25.824
When you're manipulating an equation

00:00:25.824 --> 00:00:27.589
whatever you do to one
side, you have to do

00:00:27.589 --> 00:00:28.727
to the other side.

00:00:28.727 --> 00:00:31.722
Over here she decides to
divide the left side by a,

00:00:31.722 --> 00:00:35.042
on the right side she
should divide by a as well.

00:00:35.042 --> 00:00:37.527
Or if she wants to divide
the right side by 7,

00:00:37.527 --> 00:00:40.104
she should divide the
left side by 7 as well,

00:00:40.104 --> 00:00:42.542
but she's dividing both sides
by two different things.

00:00:42.542 --> 00:00:46.211
So step 1 is where she makes the mistake.

00:00:46.211 --> 00:00:47.302
The right thing for her to do,

00:00:47.302 --> 00:00:49.392
or I guess maybe the
most reasonable thing,

00:00:49.392 --> 00:00:52.758
if she wants to solve for
a, divide both sides by 7.

00:00:52.758 --> 00:00:54.848
Then she would have
been left with just an a

00:00:54.848 --> 00:00:56.682
on the left hand side,
because it would have been

00:00:56.682 --> 00:00:58.679
7a/7 and a 4 over there.

00:00:58.679 --> 00:01:00.978
She would have said, oh,
a must be equal to 4.

00:01:00.978 --> 00:01:02.232
So let's keep going.

00:01:02.232 --> 00:01:05.181
Let's do a few more of these.

00:01:05.181 --> 00:01:07.781
Trent tried to solve an
equation step by step.

00:01:07.781 --> 00:01:09.592
All right. Find Trent's mistake.

00:01:09.592 --> 00:01:11.101
So a lot of mistakes happening

00:01:11.101 --> 00:01:12.913
in algebra problems right now.

00:01:12.913 --> 00:01:16.511
So g/3 = 4/3.

00:01:16.511 --> 00:01:17.997
Now let's see. The first step.

00:01:17.997 --> 00:01:22.223
g/3 x 3, so he's multiplying
the left hand side times 3

00:01:22.223 --> 00:01:27.006
and on the right hand side,
he's multiplying by 1/3.

00:01:27.006 --> 00:01:29.398
So once again, he's doing
two different things

00:01:29.398 --> 00:01:30.930
to the left and the right hand side

00:01:30.930 --> 00:01:32.324
even though you're supposed
to do the same thing.

00:01:32.324 --> 00:01:34.669
If you do two different
things, the equality

00:01:34.669 --> 00:01:37.548
will not hold anymore.

00:01:37.548 --> 00:01:38.918
Notice, if these two things are...

00:01:38.918 --> 00:01:42.586
If g/3 = 4, if you multiply this times 3

00:01:42.586 --> 00:01:45.628
and you only multiply this times 1/3,

00:01:45.628 --> 00:01:47.184
well then this thing is
going to become larger,

00:01:47.184 --> 00:01:49.134
because is you multiply by
3 that's going to be larger

00:01:49.134 --> 00:01:51.920
if you take the same thing
and multiply it by 1/3.

00:01:51.920 --> 00:01:54.219
Then the equality won't hold true anymore.

00:01:54.219 --> 00:01:55.635
In order for it to hold true,
if you're going to multiply

00:01:55.635 --> 00:01:58.608
the left by 3 you have to
multiply the right by 3.

00:01:58.608 --> 00:02:01.835
So he made a mistake on step 1.

00:02:01.835 --> 00:02:02.532
All right.

00:02:02.532 --> 00:02:04.993
Ling tried to solve an
equation step by step.

00:02:04.993 --> 00:02:07.013
All right. Find Ling's mistake.

00:02:07.013 --> 00:02:10.496
Let's see 12 = p + 6.2.

00:02:10.496 --> 00:02:12.794
All right. So now it looks like,

00:02:12.794 --> 00:02:17.554
on the left hand side Ling adds 6.2

00:02:17.554 --> 00:02:19.644
and on the right hand side, so there was

00:02:19.644 --> 00:02:21.594
p + 6.2 is the old right hand side,

00:02:21.594 --> 00:02:26.192
but it looks they then
try to subtract 6.2.

00:02:26.192 --> 00:02:28.606
So it's the same number, but over here

00:02:28.606 --> 00:02:32.391
they're adding it and over
here they're subtracting it.

00:02:32.391 --> 00:02:34.852
So they're not doing the
same thing to both sides.

00:02:34.852 --> 00:02:36.687
If you want to add 6.2
to the left hand side

00:02:36.687 --> 00:02:38.846
you need to add 6.2 to
the right hand side.

00:02:38.846 --> 00:02:40.982
If you want to subtract 6.2
from the right hand side

00:02:40.982 --> 00:02:44.651
you have to subtract 6.2
from the left hand side.

00:02:44.651 --> 00:02:47.646
So a lot of mistakes going on in step 1.

00:02:47.646 --> 00:02:51.361
Let me see one where there's
not a mistake in step 1.

00:02:51.361 --> 00:02:52.034
All right.

00:02:52.034 --> 00:02:54.472
Alanna tried to solve an
equation step by step.

00:02:54.472 --> 00:02:59.472
4c = 12, divides the left hand side by 4

00:03:00.254 --> 00:03:02.529
and then multiplies the
right hand side by 4.

00:03:02.529 --> 00:03:04.619
No if you're going to divide
the left hand side by 4

00:03:04.619 --> 00:03:06.987
you have to divide the right
hand side by 4 as well.

00:03:06.987 --> 00:03:08.427
You don't multiply it by 4.

00:03:08.427 --> 00:03:11.260
So a mistake in step 1.

00:03:11.260 --> 00:03:13.489
Let's do one more of these.

00:03:13.489 --> 00:03:18.411
All right. n + 12 = 18.3.

00:03:18.411 --> 00:03:20.431
So over here you had n + 12

00:03:20.431 --> 00:03:23.705
and then Rico subtracts 12.

00:03:23.705 --> 00:03:25.493
So if he subtracts 12
from the left hand side

00:03:25.493 --> 00:03:27.931
he needs to subtract 12
from the right hand side.

00:03:27.931 --> 00:03:28.929
It looks like he does that.

00:03:28.929 --> 00:03:30.996
He had 18.3 and he subtracts 12.

00:03:30.996 --> 00:03:33.736
So he subtracts 12 from both sides.

00:03:33.736 --> 00:03:37.846
So the left side is now
n + 12 - 12, was just n,

00:03:37.846 --> 00:03:39.471
which is why he subtracted 12,

00:03:39.471 --> 00:03:41.770
so you're just left with
an n on the left hand side,

00:03:41.770 --> 00:03:46.762
and on the right hand
side, let's see, 18.3 -12.

00:03:46.762 --> 00:03:51.336
Well 18 - 12 is 6, so this should be 6.3.

00:03:51.336 --> 00:03:53.542
So he made a little bit
of an arithmetic mistake

00:03:53.542 --> 00:03:58.542
in step, he made an arithmetic mistake

00:03:58.673 --> 00:04:03.673
and I think we are all done.


How to write one-step equations for word problems

WEBVTT
Kind: captions
Language: en

00:00:00.678 --> 00:00:02.327
- [Voiceover] Anna wants
to celebrate her birthday

00:00:02.327 --> 00:00:04.509
by eating pizza with her friends.

00:00:04.509 --> 00:00:09.509
For $42.50 total, they
can buy p boxes of pizza.

00:00:09.525 --> 00:00:12.659
Each box of pizza costs $8.50.

00:00:12.659 --> 00:00:16.026
Select the equation that
matches this situation.

00:00:16.026 --> 00:00:17.094
So before I even look at these,

00:00:17.094 --> 00:00:19.439
let's see if I can make
sense of the sentence here.

00:00:19.439 --> 00:00:22.653
So for $42.50 total, and
I'll just write 42.5,

00:00:22.653 --> 00:00:24.060
especially because in all these choices

00:00:24.060 --> 00:00:26.521
they didn't write 42.50,
they just wrote 42.5

00:00:26.521 --> 00:00:28.526
which is equivalent.

00:00:28.526 --> 00:00:33.526
So 42.50 that's the total
amount they spent on pizza

00:00:35.692 --> 00:00:39.149
and if I wanted to
figure out how many boxes

00:00:39.149 --> 00:00:41.563
of pizza they could buy, I could divide

00:00:41.563 --> 00:00:44.582
the total amount they
spend, I could divide that

00:00:44.582 --> 00:00:47.252
by the price per box.

00:00:47.252 --> 00:00:50.201
That would give me the number of boxes.

00:00:50.201 --> 00:00:54.357
So this is the total, total dollars.

00:00:54.357 --> 00:00:59.047
This right over here is the dollar per box

00:00:59.047 --> 00:01:01.834
and then this would give
me the number of boxes.

00:01:01.834 --> 00:01:05.061
# of boxes.

00:01:05.061 --> 00:01:07.220
Now other ways that I
could think about it.

00:01:07.220 --> 00:01:09.728
I could say, well what's
the total that they spend?

00:01:09.728 --> 00:01:13.072
So 42.50, but what's another way

00:01:13.072 --> 00:01:15.254
of thinking about the total they spend?

00:01:15.254 --> 00:01:17.089
Well you could have the amount they spend

00:01:17.089 --> 00:01:20.270
per box, times the number of boxes.

00:01:20.270 --> 00:01:21.802
So this is the total they spend

00:01:21.802 --> 00:01:23.009
and this another way of thinking about

00:01:23.009 --> 00:01:25.842
the total they spend, so these
two things must be equal.

00:01:25.842 --> 00:01:27.862
So let's see, if I can see anything here

00:01:27.862 --> 00:01:30.741
that looks like this, well
actually this first choice,

00:01:30.741 --> 00:01:35.741
this, is exactly, is exactly
what I wrote over here.

00:01:36.546 --> 00:01:38.032
Let's see this choice right over here.

00:01:38.032 --> 00:01:41.677
P is equal to 8.5 x 42.5.

00:01:41.677 --> 00:01:43.906
Well we've already been
able to write an equation

00:01:43.906 --> 00:01:47.320
that has explicitly, that
has just a p on one side

00:01:47.320 --> 00:01:49.270
and so when you solve
for just a p on one side,

00:01:49.270 --> 00:01:50.594
you get this thing over here,

00:01:50.594 --> 00:01:53.287
not this thing, so we could rule that out.

00:01:53.287 --> 00:01:56.259
Over here it looks kind of like this,

00:01:56.259 --> 00:01:58.024
except the p is on the wrong side.

00:01:58.024 --> 00:02:01.112
This has 8.5p is equal to 42.5,

00:02:01.112 --> 00:02:04.525
not 42.5p is equal to 8.5.

00:02:04.525 --> 00:02:06.591
If we try to get the p
on the other side here,

00:02:06.591 --> 00:02:09.331
you could divide both sides by p,

00:02:09.331 --> 00:02:11.653
but then you would get
p divided by p is one.

00:02:11.653 --> 00:02:16.653
You would get 42.5 is equal
to 8.5/p which is not true.

00:02:18.015 --> 00:02:20.778
We have 8.5 times p is equal to 42.5,

00:02:20.778 --> 00:02:23.704
so this is, this is not
going to be the case.

00:02:23.704 --> 00:02:25.933
One thing to realize, no
matter what you come up with,

00:02:25.933 --> 00:02:28.464
if you came up with this
first, or if you came up

00:02:28.464 --> 00:02:30.971
with this first, you
can go between these two

00:02:30.971 --> 00:02:35.128
with some algebraic manipulations.

00:02:35.128 --> 00:02:36.846
So for example, to go from this blue one

00:02:36.846 --> 00:02:38.123
to what I wrote in red up here,

00:02:38.123 --> 00:02:40.863
you just divide both sides by 8.5.

00:02:40.863 --> 00:02:42.883
So you divide by 8.5 on the left,

00:02:42.883 --> 00:02:44.508
you divide 8.5 on the right.

00:02:44.508 --> 00:02:46.551
Obviously to keep the equal sign

00:02:46.551 --> 00:02:48.502
you have to do the same
thing to the left and right,

00:02:48.502 --> 00:02:53.502
but now you would have 42.5/8.5

00:02:53.610 --> 00:02:56.303
is equal to, is equal to p.

00:02:56.303 --> 00:02:58.579
Which is exactly what we have over there.

00:02:58.579 --> 00:03:00.320
Let's do one more of these.

00:03:00.320 --> 00:03:01.899
Good practice.

00:03:01.899 --> 00:03:05.916
Mr. Herman's class is selling
candy for a school fundraiser.

00:03:05.916 --> 00:03:08.818
The class has a goal of raising $500

00:03:08.818 --> 00:03:11.535
by selling c boxes of candy.

00:03:11.535 --> 00:03:15.250
For every box they sell, they make $2.75.

00:03:15.250 --> 00:03:17.781
Write an equation that
the students could solve

00:03:17.781 --> 00:03:20.033
to figure out how many boxes of candy

00:03:20.033 --> 00:03:21.728
they need to sell.

00:03:21.728 --> 00:03:25.536
How many boxes of candy they need to sell.

00:03:25.536 --> 00:03:27.904
Well there's a couple of ways
you could think about it.

00:03:27.904 --> 00:03:31.271
They have a goal of raising
$500 and so they want

00:03:31.271 --> 00:03:34.963
to get a total of $500,
and if each box is $2.75,

00:03:34.963 --> 00:03:36.635
divide the total by the amount they get

00:03:36.635 --> 00:03:38.771
each per box and then this is going

00:03:38.771 --> 00:03:43.275
to be equal to the number of
boxes that they need to sell.

00:03:43.275 --> 00:03:44.529
So this we've done.

00:03:44.529 --> 00:03:46.247
This is an equation that
the students could solve

00:03:46.247 --> 00:03:47.733
to figure out how many boxes of candy

00:03:47.733 --> 00:03:49.359
they need to sell.

00:03:49.359 --> 00:03:50.752
Another way you could think about it,

00:03:50.752 --> 00:03:55.752
it's 2.75 per box times c boxes.

00:03:57.927 --> 00:04:00.782
This is the total amount
of money they will raise.

00:04:00.782 --> 00:04:02.919
Whoops, this is the amount.

00:04:02.919 --> 00:04:05.426
So this is the amount that they will raise

00:04:05.426 --> 00:04:08.654
and their goal is that,
their goal is to raise $500.

00:04:08.654 --> 00:04:12.229
So they want this to be equal to $500.

00:04:12.229 --> 00:04:14.435
So this also could be an equation

00:04:14.435 --> 00:04:16.089
that the students could
solve to figure out

00:04:16.089 --> 00:04:19.793
how many boxes of candy they need to sell.


How to solve one-step equations with fractions and decimals

WEBVTT
Kind: captions
Language: en

00:00:00.283 --> 00:00:01.862
- [Voiceover] Let's give
ourselves some practice

00:00:01.862 --> 00:00:03.650
solving equations.

00:00:03.650 --> 00:00:08.421
So let's say we had
the equation 1/3 plus A

00:00:08.421 --> 00:00:10.905
is equal to 5/3.

00:00:10.905 --> 00:00:13.924
What is the A that makes
this equation true?

00:00:13.924 --> 00:00:17.604
If I had 1/3 plus this
A, what does A need to be

00:00:17.604 --> 00:00:22.604
in order for 1/3 plus
that to be equal to 5/3?

00:00:23.873 --> 00:00:27.135
So there's a bunch of
different ways of doing this,

00:00:27.135 --> 00:00:29.735
and this is one of the fun
things about equations is

00:00:29.735 --> 00:00:32.835
there's no exactly one right way to do it.

00:00:32.835 --> 00:00:34.831
But let's think about
what at least I think

00:00:34.831 --> 00:00:36.004
might be the simplest way.

00:00:36.004 --> 00:00:38.105
And before I work through
anything, you should always try to

00:00:38.105 --> 00:00:40.891
pause the video, and do it on your own.

00:00:40.891 --> 00:00:45.891
So what I like to think
about is can I have just my A

00:00:46.313 --> 00:00:47.892
on one side of the equation?

00:00:47.892 --> 00:00:49.536
And since it's already
on the left-hand side,

00:00:49.536 --> 00:00:51.440
let's see if I can keep
it on the left-hand side,

00:00:51.440 --> 00:00:53.831
but get rid of this 1/3 somehow.

00:00:53.831 --> 00:00:55.930
Well the easiest was I can
think of getting rid of

00:00:55.930 --> 00:00:59.314
this 1/3 is to subtract
1/3 from the left-hand

00:00:59.314 --> 00:01:00.757
side of the equation.

00:01:00.757 --> 00:01:02.661
Now I can't just do that from the

00:01:02.661 --> 00:01:04.298
left-hand side of the equation.

00:01:04.298 --> 00:01:06.759
If 1/3 plus A is equal to 5/3,

00:01:06.759 --> 00:01:09.719
and if I just subtract 1/3
from the left-hand side,

00:01:09.719 --> 00:01:11.715
then they're not going
to be equal anymore.

00:01:11.715 --> 00:01:13.699
Then this thing is going to be 1/3 less,

00:01:13.699 --> 00:01:15.059
which this thing isn't going to change.

00:01:15.059 --> 00:01:16.800
So then this thing on
the left would become

00:01:16.800 --> 00:01:18.321
less than 5/3.

00:01:18.321 --> 00:01:20.862
So in order to hold the
equality, whatever I do on the

00:01:20.862 --> 00:01:23.986
left-hand side I have to do on
the right-hand side as well.

00:01:23.986 --> 00:01:26.819
So I have to subtract 1/3 from both sides.

00:01:26.819 --> 00:01:30.313
And if I do that, then
on the left-hand side,

00:01:30.313 --> 00:01:32.391
1/3 minus 1/3, that's the whole reason why

00:01:32.391 --> 00:01:35.190
I subtracted 1/3 was
to get rid of the 1/3,

00:01:35.190 --> 00:01:40.190
and I am left with A is
equal to 5/3 minus 1/3,

00:01:41.829 --> 00:01:46.055
5/3 minus 1/3,

00:01:46.055 --> 00:01:49.762
minus 1/3, and what is
that going to be equal to?

00:01:49.762 --> 00:01:53.268
I have five of something,
in this case I have 5/3,

00:01:53.268 --> 00:01:55.893
and I'm gonna subtract 1/3.

00:01:55.893 --> 00:01:58.902
So I'm gonna be left with 4/3.

00:01:58.902 --> 00:02:03.902
So I could write A is equal to 4/3.

00:02:05.856 --> 00:02:07.482
And you could check to
make sure that works.

00:02:07.482 --> 00:02:12.482
1/3 plus 4/3 is indeed equal to 5/3.

00:02:12.972 --> 00:02:14.853
Let's do another one of these.

00:02:14.853 --> 00:02:19.853
So let's say that we have
the equation K minus eight

00:02:21.240 --> 00:02:24.920
is equal to 11.8.

00:02:24.920 --> 00:02:27.522
So once again I wanna solve for K.

00:02:27.522 --> 00:02:29.437
I wanna have just a K
on the left-hand side.

00:02:29.437 --> 00:02:33.477
I don't want this subtracting
this eight right over here.

00:02:33.477 --> 00:02:36.011
So in order to get rid of
this eight, let's add eight

00:02:36.011 --> 00:02:36.795
on the left-hand side.

00:02:36.795 --> 00:02:38.095
And of course, if I do
it on the left-hand side,

00:02:38.095 --> 00:02:40.278
I have to do it on the
right-hand side as well.

00:02:40.278 --> 00:02:42.775
So we're gonna add eight to both sides.

00:02:42.775 --> 00:02:45.524
The left-hand side, you
are substracting eight

00:02:45.524 --> 00:02:46.441
and then you're adding eight.

00:02:46.441 --> 00:02:47.615
That's just going to cancel out,

00:02:47.615 --> 00:02:49.761
and you're just going to be left with K.

00:02:49.761 --> 00:02:53.175
And on the right-hand
side, 11.8 plus eight.

00:02:53.175 --> 00:02:58.175
Well, 11 plus eight is 19,
so it's going to be 19.8.

00:02:58.758 --> 00:03:01.115
And we're done, and once again,
what's neat about equations,

00:03:01.115 --> 00:03:03.413
you can always check to see
if you got the right answer.

00:03:03.413 --> 00:03:07.803
19.8 minus eight is 11.8.

00:03:07.803 --> 00:03:10.542
Let's do another one,
this is too much fun.

00:03:10.542 --> 00:03:15.542
Alright, so let's say that
I had 5/13 is equal to

00:03:16.462 --> 00:03:20.259
T minus 6/13.

00:03:20.259 --> 00:03:22.362
Alright, this is interesting
'cause now I have my variable

00:03:22.362 --> 00:03:23.878
on the right-hand side.

00:03:23.878 --> 00:03:24.841
But let's just leave it there.

00:03:24.841 --> 00:03:27.545
Let's just see if we can
solve for T by getting rid of

00:03:27.545 --> 00:03:29.263
everything else on the right-hand side.

00:03:29.263 --> 00:03:33.663
And like we've done in the
past, if I'm subtracting 6/13,

00:03:33.663 --> 00:03:35.602
so why don't I just add it?

00:03:35.602 --> 00:03:37.424
Why don't I just add 6/13?

00:03:37.424 --> 00:03:38.776
I can't just do that
on the right-hand side.

00:03:38.776 --> 00:03:41.110
Then the two sides won't be equal anymore,

00:03:41.110 --> 00:03:43.106
so I gotta do it on the
left-hand side if I wanna

00:03:43.106 --> 00:03:45.289
hold the equality.

00:03:45.289 --> 00:03:47.007
So what happens?

00:03:47.007 --> 00:03:47.832
So what happens?

00:03:47.832 --> 00:03:50.466
On the left-hand side I have,

00:03:50.466 --> 00:03:52.126
let me give myself a
little bit more space,

00:03:52.126 --> 00:03:56.615
I have 5/13 plus 6/13,

00:03:57.548 --> 00:04:01.880
plus 6/13 are equal to,

00:04:01.880 --> 00:04:03.702
are equal to...

00:04:03.702 --> 00:04:06.745
Well, I was subtracting
6/13, now I add 6/13.

00:04:06.745 --> 00:04:09.135
Those are just going to add to zero.

00:04:09.135 --> 00:04:12.943
6/13 minus 6/13 is just
zero, so you're left with T.

00:04:12.943 --> 00:04:14.278
So T is equal to this.

00:04:14.278 --> 00:04:17.401
If I have 5/13 and I add to that 6/13,

00:04:17.401 --> 00:04:19.304
well I'm gonna have 11/13.

00:04:19.304 --> 00:04:24.018
So this is going to be
11/13 is equal to T,

00:04:24.018 --> 00:04:25.319
or I could write that
the other way around.

00:04:25.319 --> 00:04:30.319
I could write T is equal to 11/13.


How to use the distributive property with variables

WEBVTT
Kind: captions
Language: en

00:00:00.524 --> 00:00:02.984
We're asked to apply the
distributive property.

00:00:02.984 --> 00:00:04.626
And we have 1/2 times the expression

00:00:04.626 --> 00:00:07.189
2a-6b+8.

00:00:07.189 --> 00:00:08.791
So, to figure this out,
I've actually already

00:00:08.791 --> 00:00:11.453
copy and pasted this
problem onto my scratch pad.

00:00:11.453 --> 00:00:12.529
I have it right over here.

00:00:12.529 --> 00:00:15.693
1/2(2a-6b+8).

00:00:15.693 --> 00:00:17.193
So, lemme just rewrite it.

00:00:17.193 --> 00:00:18.751
So, I'm gonna take, and
lemme color code it, too,

00:00:18.751 --> 00:00:21.263
just for fun, so it's going to be 1/2

00:00:21.263 --> 00:00:24.430
times, give myself some space,

00:00:24.430 --> 00:00:29.430
1/2(2a-6b),

00:00:30.823 --> 00:00:35.223
so, 2a-6b,

00:00:35.223 --> 00:00:38.891
minus six, lemme write it this way

00:00:38.891 --> 00:00:43.712
- 6b, and then we have plus eight.

00:00:44.627 --> 00:00:48.572
Plus, and I will do eight in this color.

00:00:49.727 --> 00:00:51.197
+8.

00:00:51.197 --> 00:00:53.273
And so, i just need to distribute the 1/2.

00:00:53.273 --> 00:00:55.498
If I multiplying 1/2 times
this entire expression,

00:00:55.498 --> 00:00:59.025
that means I multiply 1/2
times each of these terms.

00:00:59.025 --> 00:01:01.860
So, I'm gonna multiply 1/2 times this,

00:01:01.860 --> 00:01:05.887
1/2 times this, and 1/2 times that.

00:01:05.887 --> 00:01:08.324
So, 1/2 times 2a, so this is going to be

00:01:08.324 --> 00:01:12.602
1/2 times 2a, times,

00:01:12.602 --> 00:01:14.058
lemme do it in that same color

00:01:14.058 --> 00:01:16.061
so you see where the 2a came from.

00:01:16.061 --> 00:01:19.244
1/2(2a) minus,

00:01:19.630 --> 00:01:22.445
minus 1/2(6b).

00:01:22.924 --> 00:01:27.008
Minus 1/2(6b).

00:01:28.397 --> 00:01:33.397
Times 6b+1/2(8).

00:01:34.562 --> 00:01:38.835
1/2(8).

00:01:39.290 --> 00:01:40.930
And so, what's this going to be?

00:01:40.930 --> 00:01:45.014
Well, let's see, I have 1/2(2a).

00:01:45.326 --> 00:01:47.229
1/2(2) is just one, so you're just

00:01:47.229 --> 00:01:48.725
going to be left with A.

00:01:48.725 --> 00:01:53.031
And then you have minus 1/2(6b).

00:01:53.031 --> 00:01:53.929
Well, we could just think about what

00:01:53.929 --> 00:01:56.150
1/2(6) is going to be.

00:01:56.150 --> 00:01:58.055
1/2(6) is going to be three, and then you

00:01:58.055 --> 00:01:59.848
still are multiplying times B.

00:01:59.848 --> 00:02:01.787
So, it's gonna be 3b.

00:02:01.787 --> 00:02:05.700
And then we have plus 1/2(8).

00:02:05.700 --> 00:02:07.720
Half of eight is four.

00:02:07.720 --> 00:02:10.493
Or as you can say, eight
halves is equal to four wholes.

00:02:10.493 --> 00:02:13.293
Alright, so this is going to be four.

00:02:13.293 --> 00:02:17.390
So, it's a-3b+4,

00:02:17.390 --> 00:02:19.659
a-3b+4.

00:02:19.659 --> 00:02:20.923
So, let's type that in.

00:02:20.923 --> 00:02:21.802
It's going to be

00:02:21.802 --> 00:02:26.802
a-3b+4.

00:02:27.488 --> 00:02:28.955
And notice, it's just literally half

00:02:28.955 --> 00:02:29.798
of each of these terms.

00:02:29.798 --> 00:02:32.855
Half of 2a is A, half of 6b is 3b,

00:02:32.855 --> 00:02:35.663
so we have minus 6b, so
it's gonna be minus 3b,

00:02:35.663 --> 00:02:37.459
and then plus eight, instead of that,

00:02:37.459 --> 00:02:38.686
half of that plus four.

00:02:38.686 --> 00:02:40.261
So, let's check our answer.

00:02:40.261 --> 00:02:41.293
And we got it right.

00:02:41.293 --> 00:02:43.647
Let's do another one of these.

00:02:44.056 --> 00:02:45.239
So, let's say,

00:02:45.867 --> 00:02:47.261
so, they say apply the
distributive property

00:02:47.261 --> 00:02:50.296
to factor out the greatest common factor.

00:02:50.296 --> 00:02:52.827
And here, we have 60m-40,

00:02:52.827 --> 00:02:55.089
so lemme get my scratch pad out again.

00:02:55.089 --> 00:02:58.564
So, I'm running out of space that way.

00:02:58.564 --> 00:03:02.746
So, we have, write it like this.

00:03:03.762 --> 00:03:05.679
We have 60,

00:03:06.365 --> 00:03:09.781
60m-40.

00:03:10.483 --> 00:03:12.469
Minus 40.

00:03:13.493 --> 00:03:15.459
So, what is the greatest common factor

00:03:15.459 --> 00:03:18.194
of 60m and 40?

00:03:18.194 --> 00:03:20.069
Well, 10 might jump out at us.

00:03:20.069 --> 00:03:21.650
We might say, okay, look, you know?

00:03:21.650 --> 00:03:25.089
60 is 10, so we could say
this is the same thing

00:03:25.089 --> 00:03:28.630
as 10 times six and actually, and then,

00:03:28.630 --> 00:03:29.825
of course, you have the M there,

00:03:29.825 --> 00:03:31.790
so you could do this 10 times 6m.

00:03:31.790 --> 00:03:35.222
And then you could view,
you could view this as

00:03:35.222 --> 00:03:38.550
10 times four.

00:03:38.550 --> 00:03:41.029
But we, 10 still isn't the
greatest common factor.

00:03:41.029 --> 00:03:42.784
You'll say, well, how do you know that?

00:03:42.784 --> 00:03:45.922
Well, because four and six still share

00:03:45.922 --> 00:03:47.295
a factor in common.

00:03:47.295 --> 00:03:49.288
They still share two.

00:03:49.288 --> 00:03:50.958
So, if you're actually factoring out

00:03:50.958 --> 00:03:52.628
the greatest common factor, what's left

00:03:52.628 --> 00:03:54.858
should not share a factor with each other.

00:03:54.858 --> 00:03:56.655
So, let me think even harder about

00:03:56.655 --> 00:04:00.958
what a greatest common
factor of 60 and 40 is.

00:04:00.958 --> 00:04:03.984
Well, two times 10 is 20.

00:04:03.984 --> 00:04:07.093
So, you could actually factor out a 20.

00:04:07.093 --> 00:04:10.583
So, you have 20 and 30m.

00:04:10.583 --> 00:04:12.828
Sorry, 20 and 3m.

00:04:12.828 --> 00:04:17.353
And 40 could be factored out into 20 and,

00:04:17.353 --> 00:04:18.935
20 and two.

00:04:18.935 --> 00:04:23.317
And now 3m and two, 3m and
two share no common factors.

00:04:23.317 --> 00:04:24.996
So, you know that you have fully factored

00:04:24.996 --> 00:04:26.526
these two things out.

00:04:26.526 --> 00:04:27.761
Now, if you think this is something

00:04:27.761 --> 00:04:29.597
kind of a strange art that I just did,

00:04:29.597 --> 00:04:32.623
one way to think about
greatest common factors,

00:04:32.623 --> 00:04:34.124
you say, okay, 60, you could literally do

00:04:34.124 --> 00:04:35.152
a prime factorization.

00:04:35.152 --> 00:04:39.748
You could say 60 is two times
30, which is two times 15,

00:04:40.595 --> 00:04:42.697
which is three times five.

00:04:42.697 --> 00:04:44.630
So, that's 60's prime factorization.

00:04:44.630 --> 00:04:47.670
Two times two times three times five.

00:04:47.670 --> 00:04:50.380
And then 40's prime factorization

00:04:50.380 --> 00:04:54.912
is two times 20, 20 is two times 10.

00:04:55.426 --> 00:04:58.762
10 is two times five.

00:04:58.762 --> 00:05:02.230
So, that right over here, this
is 40's prime factorization.

00:05:02.230 --> 00:05:03.897
And to get out the greatest common factor,

00:05:03.897 --> 00:05:06.723
you wanna get out as many
common prime factors.

00:05:06.723 --> 00:05:10.014
So you have, here, you
have two twos and a five.

00:05:10.014 --> 00:05:11.749
Here, you have two twos and a five.

00:05:11.749 --> 00:05:13.139
You can't go to three twos and a five

00:05:13.139 --> 00:05:15.467
'cause there aren't three
twos and a five over here.

00:05:15.467 --> 00:05:18.487
So, we have two twos and a five here.

00:05:18.487 --> 00:05:21.218
Two twos and a five here.

00:05:21.218 --> 00:05:23.323
So, two times two times
five is going to be

00:05:23.323 --> 00:05:25.866
the greatest common factor.

00:05:25.866 --> 00:05:28.720
So, two times two times
five, that's four times five.

00:05:28.720 --> 00:05:30.860
Four times five, that is 20.

00:05:30.860 --> 00:05:33.028
That's one way of kind
of very systematically

00:05:33.028 --> 00:05:35.121
figuring out a greatest common factor.

00:05:35.121 --> 00:05:36.696
But anyway, now that we know that 20

00:05:36.696 --> 00:05:39.396
is the greatest common
factor, let's factor it out.

00:05:39.396 --> 00:05:43.184
So, this is going to be equal to 20 times,

00:05:43.184 --> 00:05:47.476
so 60m divided by 20, you're
just going to be left with 3m.

00:05:47.852 --> 00:05:49.687
Just going to be left with 3m.

00:05:49.687 --> 00:05:54.190
And then minus, minus 40 divided by 20,

00:05:54.190 --> 00:05:55.922
you're just left with the two.

00:05:55.922 --> 00:05:58.149
Minus two.

00:05:58.149 --> 00:06:00.955
Minus two, so let's type that in.

00:06:01.685 --> 00:06:06.006
So, this is going to be 20 times,

00:06:06.006 --> 00:06:11.006
20(3m-2).

00:06:11.621 --> 00:06:12.924
And once again, we feel good that we,

00:06:12.924 --> 00:06:16.551
literally, we did take out
the greatest common factor

00:06:16.551 --> 00:06:19.858
because 3m and two,
especially three and two

00:06:19.858 --> 00:06:21.886
are now relatively prime.

00:06:21.886 --> 00:06:23.455
Relatively prime just
means they don't share

00:06:23.455 --> 00:06:26.236
any factors in common other than one.


How to use the distributive property to factor out the greatest common factor

WEBVTT
Kind: captions
Language: en

00:00:00.411 --> 00:00:01.850
- [Voiceover] We're asked
to apply the distributive

00:00:01.850 --> 00:00:04.648
property to factor out the
greatest common factor,

00:00:04.648 --> 00:00:07.504
and we have 35 plus 50 is equal to,

00:00:07.504 --> 00:00:10.162
so let me get my scratch pad out.

00:00:10.162 --> 00:00:15.162
So we have 35 plus 50 is equal to,

00:00:16.745 --> 00:00:20.204
now what is the greatest
common factor of 35 and 50.

00:00:20.204 --> 00:00:22.549
So what's the largest whole number that's

00:00:22.549 --> 00:00:24.883
divisible into both of these.

00:00:24.883 --> 00:00:27.518
Well I could write 35 as, let's see,

00:00:27.518 --> 00:00:31.024
I could write that as five times seven,

00:00:31.024 --> 00:00:34.390
and I could write 50 as five times ten,

00:00:34.390 --> 00:00:38.384
and so we see five is the
greatest common factor.

00:00:38.384 --> 00:00:41.066
Seven and ten don't have
any factors in common.

00:00:41.066 --> 00:00:43.062
So I could rewrite this,

00:00:43.062 --> 00:00:48.062
I could write 35 as
equal to five times seven

00:00:50.469 --> 00:00:54.102
and I could rewrite 50 as equal to,

00:00:54.102 --> 00:00:55.681
get another color here,

00:00:55.681 --> 00:00:58.642
I could rewrite 50 as five times ten,

00:00:59.482 --> 00:01:01.892
and of course, I'm adding them,

00:01:01.892 --> 00:01:03.958
so I have plus right over here.

00:01:03.958 --> 00:01:05.560
If I want, I could put parentheses,

00:01:05.560 --> 00:01:07.197
but order of operations would make me do

00:01:07.197 --> 00:01:09.380
the multiplication first, anyway.

00:01:09.380 --> 00:01:12.433
But now I want to factor out
that greatest common factor.

00:01:12.433 --> 00:01:16.502
I want to factor out the five.

00:01:16.635 --> 00:01:18.377
So what I'm really doing right over here

00:01:18.377 --> 00:01:21.395
is I'm unwinding the
distributive property.

00:01:21.395 --> 00:01:24.611
So if I factor out a five,
this is going to be equal to,

00:01:24.611 --> 00:01:28.895
this is going to be equal to
let's factor out the five.

00:01:28.895 --> 00:01:32.772
Five times, so you do 35 divided by five,

00:01:32.772 --> 00:01:34.130
you're just left with the seven.

00:01:34.130 --> 00:01:36.475
You're just left with the seven over here.

00:01:36.475 --> 00:01:38.228
So you're just left with the seven

00:01:38.228 --> 00:01:39.830
after you've factored out the five,

00:01:39.830 --> 00:01:42.588
and over here, you're
just left with the ten.

00:01:42.628 --> 00:01:47.628
So five, or seven plus ten,

00:01:49.837 --> 00:01:50.534
and we're done.

00:01:50.534 --> 00:01:55.534
35 plus 50 is equal to
five times seven plus ten.

00:01:55.677 --> 00:01:59.217
So let me now go and type that in.

00:01:59.217 --> 00:02:04.217
So this is the same thing as five,

00:02:05.846 --> 00:02:10.791
five times seven plus ten.

00:02:10.791 --> 00:02:11.987
And you know you've factored out

00:02:11.987 --> 00:02:13.612
the greatest common factor because

00:02:13.612 --> 00:02:16.735
seven and ten don't have any
factors in common anymore.

00:02:16.735 --> 00:02:18.546
They're called relatively prime.

00:02:18.546 --> 00:02:21.530
They have no factors in
common other than one.

00:02:21.530 --> 00:02:24.467
So we could now check that.

00:02:24.467 --> 00:02:27.666
Let's do a couple more of these.

00:02:28.008 --> 00:02:30.074
Apply the distributive
property to factor out

00:02:30.074 --> 00:02:31.386
the greatest common factor.

00:02:31.386 --> 00:02:34.114
So let's see if we can just do
this one a little bit faster.

00:02:34.114 --> 00:02:36.552
So let's see, the largest
number it's divisible,

00:02:36.552 --> 00:02:41.097
in both 75 and 20, I don't
know, let me try five.

00:02:41.207 --> 00:02:46.207
So if I say five times,
so 75 divided by five,

00:02:46.269 --> 00:02:50.192
let's see is going to
be, it's going to be 15.

00:02:50.192 --> 00:02:51.388
Is that right?

00:02:51.388 --> 00:02:55.950
Yeah, cause five times ten
is 50, five times five is 25.

00:02:55.950 --> 00:02:57.866
Yeah, so it's 15, and I got that 15

00:02:57.866 --> 00:03:00.025
by dividing 75 by five.

00:03:00.025 --> 00:03:05.025
So 15 plus, and then 20
divided by five would be,

00:03:06.851 --> 00:03:09.022
20 divided by five would be four.

00:03:09.022 --> 00:03:10.729
15 plus four.

00:03:10.729 --> 00:03:12.110
Let's see, did I do this right?

00:03:12.110 --> 00:03:14.780
15 and four don't have
any factors in common,

00:03:14.780 --> 00:03:16.684
and if I were to apply the
distributive property here,

00:03:16.684 --> 00:03:19.308
I'd have five times 15 is 75.

00:03:19.308 --> 00:03:22.210
Five times four is 20.

00:03:22.210 --> 00:03:23.650
Yeah, I'm feeling good about that.

00:03:23.650 --> 00:03:25.374
We got it right.


How to write algebraic expressions from word problems

WEBVTT
Kind: captions
Language: en

00:00:00.249 --> 00:00:03.686
- [Voiceover] The price of a
visit to the dentist is $50.

00:00:03.686 --> 00:00:05.822
If the dentist fills any cavities,

00:00:05.822 --> 00:00:08.852
an additional charge of $100 per cavity

00:00:08.852 --> 00:00:10.687
gets added to the bill.

00:00:10.687 --> 00:00:13.450
If the dentist finds n cavities,

00:00:13.450 --> 00:00:15.888
what will the cost of the visit be?

00:00:15.888 --> 00:00:17.595
Write your answer as an expression.

00:00:17.595 --> 00:00:18.454
Alright, so we're talking about

00:00:18.454 --> 00:00:19.591
the cost of the visit.

00:00:19.591 --> 00:00:22.070
So you're gonna spend $50 no matter what.

00:00:22.530 --> 00:00:24.874
And then you're gonna
get an additional charge

00:00:24.874 --> 00:00:27.057
of $100 per cavity it tells us.

00:00:27.057 --> 00:00:29.785
An additional charge of $100 per cavity.

00:00:29.785 --> 00:00:32.386
So lets see, if you have n cavities

00:00:32.386 --> 00:00:35.288
it's gonna be $100 times n.

00:00:35.288 --> 00:00:37.958
I can just write this as 100n.

00:00:37.958 --> 00:00:39.386
Now lets see if this makes sense.

00:00:39.386 --> 00:00:41.627
If you have no cavities, if n is zero,

00:00:41.627 --> 00:00:44.019
then you're just gonna pay your $50.

00:00:44.019 --> 00:00:45.911
But if you have one cavity,

00:00:45.911 --> 00:00:48.756
you're gonna pay your
$50 plus 100 times one.

00:00:48.756 --> 00:00:49.754
If you have two cavities,

00:00:49.754 --> 00:00:52.494
you're gonna pay $50
plus 100 times 2, yeah.

00:00:52.494 --> 00:00:54.120
This seems to make sense.

00:00:54.120 --> 00:00:55.559
So lets check our answer.

00:00:55.559 --> 00:00:58.125
We got it right, lets
do another one of these.

00:00:58.125 --> 00:01:01.689
Sunny earns $12 per hour delivering cakes.

00:01:01.689 --> 00:01:04.649
She worked of x hours this week.

00:01:04.649 --> 00:01:06.929
She worked for X hours this week.

00:01:06.929 --> 00:01:10.199
Unfortunately, she was charged $15

00:01:10.199 --> 00:01:12.359
for a late delivery on Tuesday.

00:01:12.359 --> 00:01:15.552
How much money did Sunny earn this week?

00:01:15.552 --> 00:01:18.326
So if you see that she earns $12 per hour

00:01:18.326 --> 00:01:19.987
and she worked for x hours.

00:01:19.987 --> 00:01:24.352
Okay, she got $12 per hour times x hours.

00:01:24.352 --> 00:01:25.864
So that's how much she would've made

00:01:25.864 --> 00:01:27.789
except for the fact that she also had,

00:01:27.789 --> 00:01:29.856
she was charged, I guess her employer,

00:01:29.856 --> 00:01:31.528
charges her for late delivery.

00:01:31.528 --> 00:01:34.628
So she had to take $15
out of her paychecks.

00:01:34.628 --> 00:01:35.951
So this is what she would've gotten paid

00:01:35.951 --> 00:01:37.623
based on her hourly wage

00:01:37.623 --> 00:01:39.086
and the number of hours she worked.

00:01:39.086 --> 00:01:41.919
But then she has that late
fee that she has to pay.

00:01:41.919 --> 00:01:44.357
So it's gonna be 12 x minus 15

00:01:44.357 --> 00:01:47.654
is how much she actually earned this week.

00:01:47.654 --> 00:01:50.611
Lets do a few more of these.

00:01:51.261 --> 00:01:55.491
There are c players on
the Cougars hockey team.

00:01:55.491 --> 00:01:59.624
The team scored a total
of 36 goals this season.

00:01:59.624 --> 00:02:03.583
One of the players, Matthew,
scored two more goals

00:02:03.583 --> 00:02:06.125
than the average per player.

00:02:06.125 --> 00:02:08.923
How many goals did Matthew score?

00:02:08.923 --> 00:02:11.478
Alright, well lets think
about the average per player.

00:02:11.478 --> 00:02:12.640
Cause we know that he scored two more

00:02:12.640 --> 00:02:14.125
than the average player.

00:02:14.125 --> 00:02:17.039
So the average is going to
be the total number of goals

00:02:17.039 --> 00:02:19.523
divided by the number of players.

00:02:19.523 --> 00:02:21.288
And they tell us that they're c players.

00:02:21.288 --> 00:02:23.355
So this expression right over here,

00:02:23.355 --> 00:02:25.154
that will tell us the average,

00:02:25.154 --> 00:02:29.195
that that's the average goals per player.

00:02:29.195 --> 00:02:31.052
Now we know Matthew scored two more

00:02:31.052 --> 00:02:32.422
than this thing over here.

00:02:32.422 --> 00:02:34.791
So we can just add two to that.

00:02:34.791 --> 00:02:37.821
And that's how many goals Matthew scores.

00:02:37.821 --> 00:02:39.887
Lets check our answer.

00:02:39.887 --> 00:02:41.153
Now the key here, this might
seem a little confusing,

00:02:41.153 --> 00:02:44.020
but remember, this says
Matthew scored two more goals.

00:02:44.020 --> 00:02:47.585
Two more goals than the average player.

00:02:47.585 --> 00:02:51.149
The average player, the
average goals per player

00:02:51.149 --> 00:02:54.795
is 36 divided by c.

00:02:54.795 --> 00:02:57.935
Lets do one more of these.

00:02:58.881 --> 00:03:02.120
Hannah has 127 books in her collection.

00:03:02.120 --> 00:03:04.547
Her school is hosting a book donation.

00:03:04.547 --> 00:03:07.055
There are z students at her school

00:03:07.055 --> 00:03:09.423
and they each plan to donate
the same amount of books

00:03:09.423 --> 00:03:13.754
and reach a total donation of 300 books.

00:03:13.754 --> 00:03:17.655
How many books will Hannah
have in her collection

00:03:17.655 --> 00:03:19.884
after her donation?

00:03:19.884 --> 00:03:21.254
Alright, so lets think about it.

00:03:21.254 --> 00:03:23.819
Hannah has 127 books in her collection.

00:03:23.819 --> 00:03:26.292
That's how much she has in her collection.

00:03:26.292 --> 00:03:28.452
Her school is hosting a book donation.

00:03:28.452 --> 00:03:30.251
There's z students at her school

00:03:30.251 --> 00:03:32.945
and they each plan to donate
the same amount of books

00:03:32.945 --> 00:03:36.254
and reach a total of 300 books.

00:03:36.254 --> 00:03:38.994
How many books will Hannah
have in her collection

00:03:38.994 --> 00:03:40.085
after her donation?

00:03:40.085 --> 00:03:42.094
So 127, and I know I just reread it

00:03:42.094 --> 00:03:43.487
cause that first time
I was like, okay wait,

00:03:43.487 --> 00:03:45.484
is 127 the amount of books that Hannah has

00:03:45.484 --> 00:03:48.810
or the amount that her school
has would be very clear.

00:03:48.810 --> 00:03:51.892
127 is the number of book Hannah has.

00:03:51.892 --> 00:03:54.052
Then she's gonna donate
some of those books.

00:03:54.052 --> 00:03:56.525
So she's gonna donate some
of those books to her school.

00:03:56.525 --> 00:03:57.593
And we have to figure out how much

00:03:57.593 --> 00:03:59.358
is she going to donate to her school.

00:03:59.358 --> 00:04:02.527
And we see that the
students, the z students

00:04:02.527 --> 00:04:04.826
plan on together, they each want to donate

00:04:04.826 --> 00:04:06.149
the same number of books

00:04:06.149 --> 00:04:08.750
and together they're
gonna donate 300 books.

00:04:08.750 --> 00:04:11.797
So how much is each
student going to donate?

00:04:12.197 --> 00:04:15.925
Well, if z students
together are donating 300,

00:04:15.925 --> 00:04:19.721
each student is going to
donate 300 divided by z.

00:04:19.721 --> 00:04:22.972
300 divided by z.

00:04:22.972 --> 00:04:26.084
And so Hannah starts with 127 books.

00:04:26.084 --> 00:04:29.398
She's going to donate her
share of books to the school.

00:04:29.868 --> 00:04:33.073
And each student is going to
donate 300 divided by z books.

00:04:33.073 --> 00:04:35.058
So Hannah is going to be left with

00:04:35.058 --> 00:04:37.856
127 minus 300 over z books

00:04:37.856 --> 00:04:39.801
after her donation.


How to write basic algebraic expressions from word problems

WEBVTT
Kind: captions
Language: en

00:00:00.120 --> 00:00:02.720
- Let's do a few examples
of writing expressions

00:00:02.720 --> 00:00:05.620
to help us solve word problems.

00:00:05.620 --> 00:00:07.990
So they tell us Susie ran a race.

00:00:07.990 --> 00:00:10.950
She ran five miles an hour, and the

00:00:10.950 --> 00:00:13.890
race took her t hours to complete.

00:00:13.890 --> 00:00:15.950
How long was the race?

00:00:15.950 --> 00:00:18.480
Write your answer as an expression.

00:00:18.480 --> 00:00:20.590
Alright, if we're trying to
figure out the number of miles,

00:00:20.590 --> 00:00:24.290
you'd want to take her speed,
which is in miles per hour,

00:00:24.290 --> 00:00:28.490
so five miles per hour
times the number of hours.

00:00:28.490 --> 00:00:30.550
So the number of hours is t, so

00:00:30.550 --> 00:00:33.250
you can literally just
write that as five t.

00:00:33.250 --> 00:00:35.950
Five t literally means five times t.

00:00:35.950 --> 00:00:37.420
Now another way you could have done it

00:00:37.420 --> 00:00:39.190
is you could have used this
little dot symbol here.

00:00:39.190 --> 00:00:41.720
This is not a decimal, it
sits higher than a decimal,

00:00:41.720 --> 00:00:46.290
and it means multiplication,
five times t, either of these,

00:00:46.290 --> 00:00:50.690
five dot t or five t means
five times t, it makes sense.

00:00:50.690 --> 00:00:52.650
You take your speed times your time,

00:00:52.650 --> 00:00:55.750
and you'll figure out how
far you actually went.

00:00:55.750 --> 00:00:57.080
Let me check my answer here.

00:00:57.080 --> 00:00:59.390
Let's do a few more of these.

00:00:59.390 --> 00:01:03.390
The Running Aces card team won $548

00:01:03.390 --> 00:01:06.250
playing poker in tournaments last year.

00:01:06.250 --> 00:01:10.450
The winnings were split
evenly among the p players.

00:01:10.450 --> 00:01:14.250
How much of money did each, I guess you'd

00:01:14.250 --> 00:01:15.520
say how much money, not how much

00:01:15.520 --> 00:01:17.320
of money, it's a grammatical error.

00:01:17.320 --> 00:01:20.120
How much money did each player receive?

00:01:20.120 --> 00:01:22.720
Write your answer as an expression.

00:01:22.720 --> 00:01:26.950
So they had $548, and they
had to split it evenly

00:01:26.950 --> 00:01:29.610
among the p players, so
they're gonna split it,

00:01:29.610 --> 00:01:31.320
so I'm just gonna go backslash here,

00:01:31.320 --> 00:01:33.720
or I literally could have just clicked

00:01:33.720 --> 00:01:37.420
on this as well, 548 divided by p.

00:01:37.420 --> 00:01:38.950
I'm gonna divide the
amount of money we got

00:01:38.950 --> 00:01:41.090
amongst the number of players.

00:01:41.090 --> 00:01:44.790
So that's how much money each
player actually receives.

00:01:44.790 --> 00:01:47.190
Let's check the answer,
we've got it right.

00:01:47.190 --> 00:01:49.350
Let's do a few more of these.

00:01:49.350 --> 00:01:53.420
Phil received a prize of x
dollars from a poker tournament.

00:01:53.420 --> 00:01:57.500
The tournament cost him $100 to enter.

00:01:57.500 --> 00:02:00.750
What were Phil's net
winnings from the tournament?

00:02:00.750 --> 00:02:02.350
Write your answer as an expression.

00:02:02.350 --> 00:02:03.510
Alright, his net winnings, okay.

00:02:03.510 --> 00:02:05.950
He gets x dollars, but
it's not like he just

00:02:05.950 --> 00:02:08.090
walks away with x, I means,
he does get the x dollars,

00:02:08.090 --> 00:02:10.350
but he had to pay something
to enter the tournament.

00:02:10.350 --> 00:02:13.350
He had to pay $100 to
enter the tournament,

00:02:13.350 --> 00:02:17.950
so for example, if x
was $105, he wins $105,

00:02:17.950 --> 00:02:20.920
but if he had to spend $100
to enter the tournament,

00:02:20.920 --> 00:02:23.120
then his net would be 105 minus 100,

00:02:23.120 --> 00:02:25.480
or five dollars, so this
is giving us the net.

00:02:25.480 --> 00:02:26.850
We're getting how much he got,

00:02:26.850 --> 00:02:31.000
but we're subtracting how
much he actually had to spend.

00:02:31.210 --> 00:02:32.880
Let's do one more of these.

00:02:32.880 --> 00:02:37.880
Hillary made 48 chocolate chip
cookies and y sugar cookies.

00:02:37.950 --> 00:02:40.890
How many total cookies did Hillary make?

00:02:40.890 --> 00:02:44.150
Well, 48 chocolate chip cookies, and then

00:02:44.150 --> 00:02:46.550
you add that, the number of sugar cookies

00:02:46.550 --> 00:02:47.850
she made, and there you have it.

00:02:47.850 --> 00:02:49.290
That's the total number
of cookies she made.

00:02:49.290 --> 00:02:52.320
If she made two sugar
cookies, y would be two,

00:02:52.320 --> 00:02:53.990
and she would've made
a total of 50 cookies.

00:02:53.990 --> 00:02:56.950
If she made 100 sugar
cookies, then this would be

00:02:56.950 --> 00:03:01.450
100 plus 48 which would
be 148 total cookies.

00:03:01.450 --> 00:03:04.080
So hopefully that gives you
a little bit of practice,

00:03:04.080 --> 00:03:04.980
and I encourage you to do this exercise

00:03:04.980 --> 00:03:07.810
so you get a lot more practice.


How to write basic expressions with variables

WEBVTT
Kind: captions
Language: en

00:00:00.059 --> 00:00:01.833
- [Voiceover] Let's do some examples

00:00:01.833 --> 00:00:05.001
of the writing expressions
with variables exercise.

00:00:05.001 --> 00:00:06.286
So it says "Write an expression

00:00:06.286 --> 00:00:08.703
to represent 11 more than a."

00:00:10.277 --> 00:00:12.955
Well you could just have a

00:00:12.955 --> 00:00:14.790
but if you want 11 more than a,

00:00:14.790 --> 00:00:16.623
you would wanna add 11

00:00:17.766 --> 00:00:20.054
so you could write that as a plus 11.

00:00:20.054 --> 00:00:22.406
You could also write that as 11 plus a.

00:00:22.406 --> 00:00:26.041
Both of them would be 11 more than a.

00:00:26.041 --> 00:00:27.697
So let's check our answer here.

00:00:27.697 --> 00:00:28.716
We got it right.

00:00:28.716 --> 00:00:30.537
Let's do a few more of these.

00:00:30.537 --> 00:00:32.845
"Write an expression to represent the sum

00:00:32.845 --> 00:00:33.845
of d and 9."

00:00:35.770 --> 00:00:37.569
So the sum of d and 9,

00:00:37.569 --> 00:00:39.623
that means you're gonna add d and 9.

00:00:39.623 --> 00:00:41.345
So I could write that as d plus 9

00:00:41.345 --> 00:00:44.178
or I could write that as 9 plus d.

00:00:46.630 --> 00:00:47.900
And check our answer.

00:00:47.900 --> 00:00:49.161
Got that right.

00:00:49.161 --> 00:00:50.705
Let's do a few more of these.

00:00:50.705 --> 00:00:54.691
"Write an expression to
represent j minus 15."

00:00:54.691 --> 00:00:57.574
Well, I could just write
it with math symbols

00:00:57.574 --> 00:00:59.420
instead of writing the word minus.

00:00:59.420 --> 00:01:01.227
Instead of writing M-I-N-U-S,

00:01:01.227 --> 00:01:03.310
I could write j minus 15.

00:01:04.370 --> 00:01:06.858
And then I check my answer.

00:01:06.858 --> 00:01:07.898
Got it right.

00:01:07.898 --> 00:01:09.839
Let's do a few more of these.

00:01:09.839 --> 00:01:11.018
This is a lot of fun.

00:01:11.018 --> 00:01:14.768
"Write an expression to
represent 7 times r."

00:01:15.775 --> 00:01:16.721
There's a couple ways I could do it.

00:01:16.721 --> 00:01:18.269
I could use this little
dot right over here,

00:01:18.269 --> 00:01:20.186
do 7 times r like that.

00:01:21.291 --> 00:01:22.674
That would be correct.

00:01:22.674 --> 00:01:24.469
I could literally just write 7r.

00:01:24.469 --> 00:01:28.060
If I just wrote 7r that would also count.

00:01:28.060 --> 00:01:29.399
Let me check my answer.

00:01:29.399 --> 00:01:31.117
That's right.

00:01:31.117 --> 00:01:32.611
Let me do a couple of other of these

00:01:32.611 --> 00:01:35.438
just so you can see that I could've

00:01:35.438 --> 00:01:38.038
just done 10 and this is not a decimal,

00:01:38.038 --> 00:01:40.291
it sits a little bit
higher than a decimal.

00:01:40.291 --> 00:01:42.487
It's multiplication and the reason why

00:01:42.487 --> 00:01:44.840
once you start doing
algebra, you use this symbol

00:01:44.840 --> 00:01:47.403
instead of that kind of
cross for multiplication

00:01:47.403 --> 00:01:51.061
is that x-looking thing
gets confused with x

00:01:51.061 --> 00:01:53.064
when you're using x as a variable

00:01:53.064 --> 00:01:54.694
so that's why this is a lot more useful.

00:01:54.694 --> 00:01:57.111
So we wanna write 10 times u,

00:01:57.990 --> 00:01:59.904
10 times u, let's check our answer.

00:01:59.904 --> 00:02:00.978
We got it right.

00:02:00.978 --> 00:02:02.966
Let's do one more.

00:02:02.966 --> 00:02:07.455
"Write an expression to
represent 8 divided by d."

00:02:07.455 --> 00:02:10.206
So we could write it as
8 and then I could write

00:02:10.206 --> 00:02:13.039
a slash like that, 8 divided by d.

00:02:14.617 --> 00:02:15.732
And there you go.

00:02:15.732 --> 00:02:17.510
This is 8 divided by d.

00:02:17.510 --> 00:02:20.081
Let me check, let me check the answer.

00:02:20.081 --> 00:02:21.637
I'll do one more of these.

00:02:21.637 --> 00:02:23.341
Oh, it's 6 divided by b.

00:02:23.341 --> 00:02:24.470
Alright, same thing.

00:02:24.470 --> 00:02:26.867
So 6, I could use this
tool right over here.

00:02:26.867 --> 00:02:27.995
It does the same thing

00:02:27.995 --> 00:02:30.078
as if I were to press the backslash.

00:02:30.078 --> 00:02:32.157
So 6 divided by b.

00:02:32.157 --> 00:02:33.364
Check my answer.

00:02:33.364 --> 00:02:34.697
We got it right.


What are terms, factors, and coefficients in algebraic expressions?

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:03.970
What I want to do in this video
is think about how expressions

00:00:03.970 --> 00:00:06.640
are formed and the
words we use to describe

00:00:06.640 --> 00:00:08.596
the different parts
of an expression.

00:00:08.596 --> 00:00:09.970
And the reason
why this is useful

00:00:09.970 --> 00:00:12.220
is when you hear other people
refer to some expression

00:00:12.220 --> 00:00:14.370
and say, oh, I don't agree
with the second term,

00:00:14.370 --> 00:00:16.670
or the third term
has four factors,

00:00:16.670 --> 00:00:18.736
or why is the coefficient
on that term 6,

00:00:18.736 --> 00:00:20.360
you'll know what
they're talking about,

00:00:20.360 --> 00:00:22.040
and you can communicate
in the same way.

00:00:22.040 --> 00:00:24.270
So let's think about what
those words actually mean.

00:00:24.270 --> 00:00:25.310
So we have an expression here.

00:00:25.310 --> 00:00:27.018
And the first thing
I want to think about

00:00:27.018 --> 00:00:32.450
are the terms of an
expression or what a term is.

00:00:32.450 --> 00:00:37.900
And one way to think about it
is the terms are the things that

00:00:37.900 --> 00:00:39.720
are getting added
and subtracted.

00:00:39.720 --> 00:00:42.110
So, for example, in this
expression right over here,

00:00:42.110 --> 00:00:44.690
you have three things that are
getting added and subtracted.

00:00:44.690 --> 00:00:47.190
The first thing, you're
taking 2 times 3.

00:00:47.190 --> 00:00:48.760
You're adding that to 4.

00:00:48.760 --> 00:00:51.970
And then from that,
you're subtracting 7y.

00:00:51.970 --> 00:00:56.290
So in this example,
you have three terms.

00:00:56.290 --> 00:00:59.050
The first term is 2 times 3.

00:00:59.050 --> 00:01:01.280
The second term is
just the number 4.

00:01:01.280 --> 00:01:07.530
And the third term is 7 times y.

00:01:07.530 --> 00:01:10.339
Now, let's think about
the term "factor."

00:01:10.339 --> 00:01:12.130
And when people are
talking about a factor,

00:01:12.130 --> 00:01:13.780
especially in terms
of an expression,

00:01:13.780 --> 00:01:15.321
they're talking
about the things that

00:01:15.321 --> 00:01:17.830
are getting multiplied
in each term.

00:01:17.830 --> 00:01:19.350
So, for example,
if you said, what

00:01:19.350 --> 00:01:22.020
are the factors
of the first term?

00:01:22.020 --> 00:01:25.100
The first term refers to this
one right over here-- 2 times

00:01:25.100 --> 00:01:25.730
3.

00:01:25.730 --> 00:01:27.200
And there's two factors.

00:01:27.200 --> 00:01:29.280
There's a 2 and
a 3, and they are

00:01:29.280 --> 00:01:31.470
being multiplied by each other.

00:01:31.470 --> 00:01:34.580
So here you have two
factors in the first term.

00:01:34.580 --> 00:01:35.720
What about the second term?

00:01:39.010 --> 00:01:40.160
This was the first term.

00:01:40.160 --> 00:01:45.190
The second term here has
only one factor, just the 4.

00:01:45.190 --> 00:01:47.030
It's not being
multiplied by anything.

00:01:47.030 --> 00:01:50.540
And the third term here,
once again, has two factors.

00:01:50.540 --> 00:01:54.710
It's the product of 7 times y.

00:01:54.710 --> 00:01:56.570
So we have two factors here.

00:01:56.570 --> 00:01:59.630
We have a 7 and a y.

00:01:59.630 --> 00:02:02.920
And this constant factor
here, this number 7

00:02:02.920 --> 00:02:04.960
that is multiplying
the variable,

00:02:04.960 --> 00:02:06.420
also has a special name.

00:02:06.420 --> 00:02:10.125
It is called the coefficient
of this term-- coefficient.

00:02:14.360 --> 00:02:16.320
And the coefficient
is the nonvariable

00:02:16.320 --> 00:02:18.391
that multiplies the
rest of the term.

00:02:18.391 --> 00:02:19.890
That's one way of
thinking about it.

00:02:19.890 --> 00:02:20.820
So here's 7y.

00:02:20.820 --> 00:02:24.170
Even if it was 7xy
or 7xyz or 7xyz

00:02:24.170 --> 00:02:27.590
squared, that nonvariable that's
multiplying everything else,

00:02:27.590 --> 00:02:30.730
we would consider to
be the coefficient.

00:02:30.730 --> 00:02:32.147
Now, let's do a
few more examples.

00:02:32.147 --> 00:02:34.104
And in each of these--
I encourage you actually

00:02:34.104 --> 00:02:36.780
right now to pause the video--
think about what the terms are.

00:02:36.780 --> 00:02:39.960
How many terms are there
in each expression, how

00:02:39.960 --> 00:02:45.470
many factors in each term,
and what are the coefficients?

00:02:45.470 --> 00:02:47.160
So let's look at this first one.

00:02:47.160 --> 00:02:49.850
It's clear that we have three
things being added together.

00:02:49.850 --> 00:02:51.000
This is the first term.

00:02:51.000 --> 00:02:52.260
This is the second term.

00:02:52.260 --> 00:02:53.490
And this is the third term.

00:02:53.490 --> 00:02:54.810
So this is the first term.

00:02:54.810 --> 00:02:55.810
This is the second term.

00:02:55.810 --> 00:02:57.220
This is the third term.

00:02:57.220 --> 00:02:59.020
And they each have two factors.

00:02:59.020 --> 00:03:03.160
This first one has
the factors 3 and x.

00:03:03.160 --> 00:03:06.310
The second one here has
the factors x and y.

00:03:06.310 --> 00:03:10.280
And this third one has
the factors y and z.

00:03:10.280 --> 00:03:11.830
Now, what are the
coefficients here?

00:03:11.830 --> 00:03:13.746
Well, remember, coefficients
was a nonvariable

00:03:13.746 --> 00:03:15.920
multiplying a bunch
of other variables.

00:03:15.920 --> 00:03:19.300
And so here, the coefficient in
this first term right over here

00:03:19.300 --> 00:03:20.380
is a 3.

00:03:20.380 --> 00:03:21.880
Now, you might be
saying, well, what

00:03:21.880 --> 00:03:24.201
about the coefficients on
these terms right over here?

00:03:24.201 --> 00:03:25.700
Depending on how
you think about it,

00:03:25.700 --> 00:03:27.660
one way to say it
is, well, xy is

00:03:27.660 --> 00:03:29.790
the same thing as 1 times xy.

00:03:29.790 --> 00:03:31.845
So some people would
say that, hey, you

00:03:31.845 --> 00:03:34.130
have a coefficient
of 1 here on the xy.

00:03:34.130 --> 00:03:35.320
Or it's implicitly there.

00:03:35.320 --> 00:03:38.892
It wasn't written, but you're
multiplying everything by 1.

00:03:38.892 --> 00:03:40.600
And that might be
subject to a little bit

00:03:40.600 --> 00:03:43.390
of interpretation
one way or another.

00:03:43.390 --> 00:03:45.600
Now, this one is
really interesting,

00:03:45.600 --> 00:03:47.680
because if we look at
the bigger expression,

00:03:47.680 --> 00:03:50.800
if we look at the
whole thing, it's

00:03:50.800 --> 00:03:53.150
clearly made up of three terms.

00:03:53.150 --> 00:03:55.860
The first term is xyz.

00:03:55.860 --> 00:04:01.050
The second term is x plus
1, that whole thing times y.

00:04:01.050 --> 00:04:03.019
And then the third term is 4x.

00:04:03.019 --> 00:04:05.560
And if you look at that level,
if you look at the first term,

00:04:05.560 --> 00:04:07.710
and you say, well, how many
factors does that have?

00:04:07.710 --> 00:04:11.650
Well, you would say that it has
three factors-- x, y, and z.

00:04:11.650 --> 00:04:13.570
How many factors does
the second term have?

00:04:13.570 --> 00:04:15.486
Well, you could say,
well, it has two factors.

00:04:15.486 --> 00:04:20.459
One factor is x plus y, and
then the other factor is y.

00:04:20.459 --> 00:04:22.050
The first factor is x plus 1.

00:04:22.050 --> 00:04:23.130
And the second one is y.

00:04:23.130 --> 00:04:26.610
It's multiplying
this expression.

00:04:26.610 --> 00:04:29.620
This smaller expression
itself is one of the factors.

00:04:29.620 --> 00:04:31.040
And the other one is y.

00:04:31.040 --> 00:04:34.970
And then this third one also
has two factors, a 4 and an x.

00:04:34.970 --> 00:04:37.710
And if someone said, hey, what's
the coefficient on this term?

00:04:37.710 --> 00:04:42.577
You would say, hey, look,
the coefficient is the 4.

00:04:42.577 --> 00:04:44.160
Now, let's look at
this one over here.

00:04:44.160 --> 00:04:46.785
Actually, before I look at that
one, what was interesting about

00:04:46.785 --> 00:04:49.960
this is that here you had
a little smaller expression

00:04:49.960 --> 00:04:52.740
itself acting as
one of the factors.

00:04:52.740 --> 00:04:54.540
So then you can go
and then zoom in

00:04:54.540 --> 00:04:56.120
on this expression
right over here.

00:04:56.120 --> 00:04:57.750
And you can ask
the same question.

00:04:57.750 --> 00:05:00.740
On this smaller expression,
how many terms does it have?

00:05:00.740 --> 00:05:03.580
Well, it has two
terms-- an x and a 1.

00:05:03.580 --> 00:05:05.910
Those are the two things
being added or subtracted.

00:05:05.910 --> 00:05:08.390
And each of them have
exactly one factor.

00:05:08.390 --> 00:05:11.440
So when we're giving
these, you can

00:05:11.440 --> 00:05:14.660
keep nesting these expressions
to think about when you talk

00:05:14.660 --> 00:05:17.570
about terms or factors
or factors of terms,

00:05:17.570 --> 00:05:20.164
you have to really specify
what part of the nesting

00:05:20.164 --> 00:05:21.080
you're thinking about.

00:05:21.080 --> 00:05:23.690
If you're talking about the
terms of this whole expression,

00:05:23.690 --> 00:05:25.540
there's one, two, three.

00:05:25.540 --> 00:05:28.730
But then you could look at this
subexpression, which itself

00:05:28.730 --> 00:05:30.440
is a factor of a term,
and say, oh, well,

00:05:30.440 --> 00:05:32.900
there's only two
terms in this one.

00:05:32.900 --> 00:05:34.460
Now, let's look at this one.

00:05:34.460 --> 00:05:37.554
How many terms?

00:05:37.554 --> 00:05:39.220
Well, once again,
there's clearly three.

00:05:39.220 --> 00:05:40.969
Actually, let me add
one more, because I'm

00:05:40.969 --> 00:05:42.850
tired of expressions
with three terms.

00:05:42.850 --> 00:05:45.220
So I'm just going
to add a 1 here.

00:05:45.220 --> 00:05:47.130
So now, we clearly
have four terms.

00:05:47.130 --> 00:05:50.040
This is the first term, second
term, third term, fourth term.

00:05:50.040 --> 00:05:52.690
And how many factors
are in each of them?

00:05:52.690 --> 00:05:53.900
Well, this is interesting.

00:05:53.900 --> 00:05:55.430
You might say, well, the
factors are the things

00:05:55.430 --> 00:05:56.513
that are being multiplied.

00:05:56.513 --> 00:05:58.580
But here I'm dividing by y.

00:05:58.580 --> 00:06:00.730
Remember, dividing by
y is the same thing

00:06:00.730 --> 00:06:02.695
as multiplying by
its reciprocal.

00:06:02.695 --> 00:06:04.070
So it would usually
be considered

00:06:04.070 --> 00:06:05.570
to have three
factors here, where

00:06:05.570 --> 00:06:10.000
the factors are 3x and 1/y.

00:06:10.000 --> 00:06:12.880
If you multiplied 3
times x times 1/y,

00:06:12.880 --> 00:06:16.184
you're going to get exactly
what you have right over here.

00:06:16.184 --> 00:06:17.850
So you would say this
has three factors.

00:06:17.850 --> 00:06:19.724
If someone asks, what's
the coefficient here?

00:06:19.724 --> 00:06:22.430
Well, you'd say, well,
that 3 is the coefficient.

00:06:22.430 --> 00:06:24.029
Here how many
factors do you have?

00:06:24.029 --> 00:06:26.320
And this is a little bit
tricky, because you might say,

00:06:26.320 --> 00:06:30.690
well, isn't 5x squared times
y, isn't that equal to 5 times

00:06:30.690 --> 00:06:33.760
x times x times y?

00:06:33.760 --> 00:06:35.480
And you'd be right.

00:06:35.480 --> 00:06:38.770
So it would be very tempting to
say that you have four factors.

00:06:38.770 --> 00:06:42.130
But the convention, the
tradition that most people use,

00:06:42.130 --> 00:06:44.640
is that they
consider the exponent

00:06:44.640 --> 00:06:48.560
with x as a base as just
one factor, this as just one

00:06:48.560 --> 00:06:49.350
factor.

00:06:49.350 --> 00:06:52.670
So traditionally, people will
say this has three factors.

00:06:52.670 --> 00:06:56.980
It has a 5x squared
and a y. x squared

00:06:56.980 --> 00:06:58.150
is just considered a factor.

00:06:58.150 --> 00:07:00.180
And once again, what's
the coefficient?

00:07:00.180 --> 00:07:01.570
It's the 5.

00:07:01.570 --> 00:07:04.300
So with that in mind,
how many factors here?

00:07:04.300 --> 00:07:06.340
Well, you have
three factors here.

00:07:06.340 --> 00:07:07.650
You have an x.

00:07:07.650 --> 00:07:09.450
You have a y squared.

00:07:09.450 --> 00:07:11.630
And you have a z to the fifth.

00:07:11.630 --> 00:07:15.200
And then, finally, this last
term, it's a constant term.

00:07:15.200 --> 00:07:16.450
How many factors does it have?

00:07:16.450 --> 00:07:17.060
Well, it's just 1.

00:07:17.060 --> 00:07:18.393
It's just got a 1 sitting there.

00:07:18.393 --> 00:07:20.190
It's not being
multiplied by anything.


Thinking about the changing values of variables and expressions

WEBVTT
Kind: captions
Language: en

00:00:00.319 --> 00:00:01.924
- [Voiceover] What I wanna do
in this video is think about

00:00:01.924 --> 00:00:04.260
how does the value of an expression change

00:00:04.260 --> 00:00:07.409
as a variable that makes
up that expression changes?

00:00:07.409 --> 00:00:09.930
So, let's start with a
pretty simple expression.

00:00:09.930 --> 00:00:13.566
Let's say we have the
expression 100 minus x.

00:00:13.566 --> 00:00:15.699
We wanna know how does
this expression change

00:00:15.699 --> 00:00:17.917
as x increases?

00:00:17.917 --> 00:00:20.619
As x increases and like always,

00:00:20.619 --> 00:00:22.265
try to work it out for yourself.

00:00:22.265 --> 00:00:25.217
Pause the video before I
try to have a go at it.

00:00:25.922 --> 00:00:27.547
Well, there's a couple of
ways you could think about it.

00:00:27.547 --> 00:00:30.961
You say, "Okay, I have 100
and I'm subtracting x."

00:00:30.961 --> 00:00:35.127
So, as x increases, as
x increases over here,

00:00:35.127 --> 00:00:39.169
I'm going to be subtracting
larger and larger values.

00:00:39.169 --> 00:00:42.054
So, if I'm subtracting
larger and larger values,

00:00:42.054 --> 00:00:44.451
I'm going to get smaller
and smaller values.

00:00:44.451 --> 00:00:47.763
So, this whole thing, this whole
thing is going to decrease.

00:00:47.763 --> 00:00:50.262
This whole thing is going to decrease

00:00:50.262 --> 00:00:53.546
and if you wanna make it a
little bit more concrete,

00:00:53.546 --> 00:00:55.517
you can actually try
out some values there.

00:00:55.517 --> 00:00:56.878
So, you can make a little table here.

00:00:56.878 --> 00:01:00.474
X and then, what is 100 - x going to be?

00:01:00.474 --> 00:01:02.140
I'm just gonna pick some values of x.

00:01:02.140 --> 00:01:03.684
I'm gonna make them increase.

00:01:03.684 --> 00:01:08.032
Say, x is zero, x is 50, x is 100.

00:01:08.032 --> 00:01:11.425
Well, when an x is zero,
100 - x is 100 - zero.

00:01:11.425 --> 00:01:12.807
So, it's 100.

00:01:12.807 --> 00:01:14.818
When x is 50, it's going be 100 - 50,

00:01:14.818 --> 00:01:16.525
so it's going to be 50.

00:01:16.525 --> 00:01:19.836
When x is 100, it's 100
- 100, so it's zero.

00:01:19.836 --> 00:01:23.534
So, it's pretty clear here
that as x is increasing,

00:01:23.534 --> 00:01:28.491
as x is increasing, I'll just
write incr. for increasing,

00:01:28.491 --> 00:01:33.042
we see that 100 - x is decreasing.

00:01:33.042 --> 00:01:35.318
We see that that is decreasing.

00:01:35.318 --> 00:01:37.451
Let's do this with a
couple more expressions

00:01:37.451 --> 00:01:39.138
that have different forms.

00:01:39.138 --> 00:01:42.165
So, let's say that I have the expression.

00:01:42.165 --> 00:01:47.165
Let's say that I have the
expression five over x

00:01:47.366 --> 00:01:51.368
plus five and x is decreasing,

00:01:51.368 --> 00:01:55.310
x is decreasing, but we also know that

00:01:55.310 --> 00:01:58.256
it is positive and even
while it's decreasing,

00:01:58.256 --> 00:02:00.735
it's staying above zero.

00:02:00.735 --> 00:02:05.407
So, we're saying x is
staying greater than zero.

00:02:05.407 --> 00:02:07.602
So, in this case, so ya know,
this could be a situation

00:02:07.602 --> 00:02:10.731
where x is decreasing from 10 to nine

00:02:10.731 --> 00:02:15.731
or a million to 100,000,
but it's staying positive

00:02:15.831 --> 00:02:17.255
while it's decreasing.

00:02:17.255 --> 00:02:18.901
Let's think about that.

00:02:18.901 --> 00:02:23.309
We are going to be dividing
by smaller and smaller

00:02:23.309 --> 00:02:25.321
positive values.

00:02:25.321 --> 00:02:29.384
So, as you have smaller
and smaller positive values

00:02:29.384 --> 00:02:31.693
of the denominator, you're
dividing by smaller and smaller

00:02:31.693 --> 00:02:33.073
positive values.

00:02:33.073 --> 00:02:35.858
So, if you're dividing by
smaller positive values,

00:02:35.858 --> 00:02:38.722
you're going to know that this
thing is going to get larger.

00:02:38.722 --> 00:02:41.038
This entire expression
is going to get larger

00:02:41.038 --> 00:02:43.558
as you divide by smaller
and smaller positive values.

00:02:43.558 --> 00:02:44.776
So, if that expression gets larger,

00:02:44.776 --> 00:02:46.382
then you're just adding five to it.

00:02:46.382 --> 00:02:48.575
The whole thing is going to increase.

00:02:48.575 --> 00:02:52.639
The whole thing is going to increase as,

00:02:52.639 --> 00:02:55.322
the whole thing is going
to increase as x decreases

00:02:55.322 --> 00:02:56.987
while staying positive.

00:02:56.987 --> 00:02:58.917
And once again, we can
make a little table to

00:02:58.917 --> 00:03:00.360
take a look at that.

00:03:00.360 --> 00:03:05.360
So, this is x, this is
five over x plus five.

00:03:06.556 --> 00:03:08.609
Now, let's see. I'll just try out some.

00:03:08.609 --> 00:03:13.609
Let's see, I'll go from
x 100 to x is five,

00:03:16.267 --> 00:03:17.853
to x is one.

00:03:17.853 --> 00:03:20.453
So, this is clearly x is decreasing.

00:03:20.453 --> 00:03:21.834
X is decreasing.

00:03:21.834 --> 00:03:23.460
When x is 100, you're gonna have five

00:03:23.460 --> 00:03:25.938
divided by 100, which
would be five hundredths

00:03:25.938 --> 00:03:29.595
plus five, so it would be 5.05.

00:03:29.595 --> 00:03:33.456
When x is five, you're gonna
have five divided by five,

00:03:33.456 --> 00:03:37.803
which is one plus five, which is six.

00:03:37.803 --> 00:03:41.115
When x is one, you're gonna
have five divided by one,

00:03:41.115 --> 00:03:42.212
which is five plus five.

00:03:42.212 --> 00:03:43.472
You just, it's 10.

00:03:43.472 --> 00:03:46.499
So notice, when x is staying
positive, but decreasing,

00:03:46.499 --> 00:03:48.571
the whole expression,
five over x plus five,

00:03:48.571 --> 00:03:52.168
this thing right over here is increasing.

00:03:53.428 --> 00:03:55.653
Let's do one more of these.

00:03:55.653 --> 00:03:58.274
So, let's say that we have the expression

00:03:58.274 --> 00:04:00.082
and we'll change up the variable here.

00:04:00.082 --> 00:04:04.511
3y over 2y and I'm curious what happens

00:04:04.511 --> 00:04:06.461
to the value of this expression

00:04:06.461 --> 00:04:11.461
as y increases, as y increases.

00:04:13.846 --> 00:04:18.634
Well, as you have larger
y's here on the numerator,

00:04:18.634 --> 00:04:20.585
you're also going to have larger y's here

00:04:20.585 --> 00:04:21.905
in the denominator

00:04:21.905 --> 00:04:23.205
and one way you could think about this,

00:04:23.205 --> 00:04:25.278
this is the same thing as three halves

00:04:25.278 --> 00:04:28.407
times y over y.

00:04:28.407 --> 00:04:30.377
And no matter what y you have here,

00:04:30.377 --> 00:04:32.978
as long as it's not equal to zero

00:04:32.978 --> 00:04:34.179
and actually, let's just
assume that it doesn't

00:04:34.179 --> 00:04:35.724
cross zero because at that point,

00:04:35.724 --> 00:04:39.382
at that point, this thing
would become undefined.

00:04:39.382 --> 00:04:40.701
Let's just say that y increases

00:04:40.701 --> 00:04:43.342
and y is greater than
zero just for simplicity.

00:04:43.342 --> 00:04:47.223
So, y is a positive value here

00:04:47.223 --> 00:04:50.494
and actually, for any non-zero value,

00:04:50.494 --> 00:04:52.059
you take y, you divide by y,

00:04:52.059 --> 00:04:53.379
you're just gonna get one.

00:04:53.379 --> 00:04:54.720
So, it doesn't matter what y is.

00:04:54.720 --> 00:04:56.081
Y could be a million over a million.

00:04:56.081 --> 00:04:57.158
This thing is just gonna be one.

00:04:57.158 --> 00:04:58.316
Y could be five over five,

00:04:58.316 --> 00:04:59.840
it's still just going to be one.

00:04:59.840 --> 00:05:01.628
The value of the expression
is not going to change.

00:05:01.628 --> 00:05:03.721
It's just going to be three halves.

00:05:03.721 --> 00:05:06.585
So, the expression is
just going to be the same.

00:05:06.585 --> 00:05:08.698
You're not going to have any change

00:05:08.698 --> 00:05:13.073
as y increases, as y is
positive and it increases.