Visually adding fractions with unlike denominators

WEBVTT
Kind: captions
Language: en

00:00:00.284 --> 00:00:02.989
- [Voiceover] Let's see if
we can calculate what 5/6

00:00:02.989 --> 00:00:05.392
plus 1/4 is, and to help us,

00:00:05.392 --> 00:00:08.713
I have a visual representation of 5/6,

00:00:08.713 --> 00:00:11.360
and a visual representation of 1/4.

00:00:11.360 --> 00:00:14.425
Notice I have this whole whole,

00:00:14.425 --> 00:00:15.156
I guess you could say,

00:00:15.156 --> 00:00:18.686
broken up into one, two,
three, four, five, six

00:00:18.686 --> 00:00:21.124
sections, and we've
shaded in five of them,

00:00:21.124 --> 00:00:22.889
so this is 5/6,

00:00:22.889 --> 00:00:24.955
and then down here, we have another whole,

00:00:24.955 --> 00:00:28.252
and we have one out of the
four equal sections shaded in

00:00:28.252 --> 00:00:30.992
so this is 1/4, and I want to add them,

00:00:30.992 --> 00:00:32.688
and I encourage you at any
point, pause the video,

00:00:32.688 --> 00:00:35.218
and see if you could
figure it out on your own.

00:00:35.218 --> 00:00:37.552
Well, whenever we're adding
fractions, we like to think

00:00:37.552 --> 00:00:40.060
in terms of fractions that
have the same denominator,

00:00:40.060 --> 00:00:42.521
and these clearly don't
have the same denominator,

00:00:42.521 --> 00:00:45.052
but in order to rewrite them,
with a common denominator,

00:00:45.052 --> 00:00:48.059
we just have to think of a
common multiple of six and four,

00:00:48.059 --> 00:00:51.322
and ideally, the smallest
common multiple of six and four,

00:00:51.322 --> 00:00:52.993
and the way that I like to do that is

00:00:52.993 --> 00:00:55.826
I like to take the larger
of the two, which is six,

00:00:55.826 --> 00:00:57.858
and then think about its multiples.

00:00:57.858 --> 00:00:59.925
So I could first think about six itself.

00:00:59.925 --> 00:01:02.188
Six is clearly divisible
by six, but it's not

00:01:02.188 --> 00:01:06.461
perfectly divisible by
four, so now, let's multiply

00:01:06.461 --> 00:01:09.634
by two, so then we get to 12.

00:01:09.634 --> 00:01:12.960
12 is divisible by both six and four.

00:01:12.960 --> 00:01:16.420
So 12 is a good common denominator here.

00:01:16.420 --> 00:01:19.195
It's the least common
multiple of six and four.

00:01:19.195 --> 00:01:22.260
So we can rewrite both of
these fractions as something

00:01:22.260 --> 00:01:23.723
over 12.

00:01:23.723 --> 00:01:26.184
So, something over 12

00:01:26.184 --> 00:01:28.589
plus something,

00:01:28.589 --> 00:01:32.653
plus something over 12

00:01:32.653 --> 00:01:34.789
is equal to.

00:01:34.789 --> 00:01:36.519
Now, there's a bunch of ways to tackle it,

00:01:36.519 --> 00:01:39.015
but what I want to do is I
just want to visualize it here

00:01:39.015 --> 00:01:41.581
on this drawing.

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So, if I go,

00:01:43.288 --> 00:01:45.482
if I were to go from,

00:01:45.482 --> 00:01:47.792
if I were to go from six equal sections

00:01:47.792 --> 00:01:50.486
to 12 equal sections,
which is what I'm doing

00:01:50.486 --> 00:01:52.552
if I'm going from six
in the denominator to 12

00:01:52.552 --> 00:01:53.957
in the denominator.

00:01:53.957 --> 00:01:57.127
I'm essentially multiplying
each of these sections by,

00:01:57.127 --> 00:01:59.553
or, I'm essentially multiplying
the number of sections

00:01:59.553 --> 00:02:00.923
I have by two,

00:02:00.923 --> 00:02:03.826
or I'm taking each of
these existing sections

00:02:03.826 --> 00:02:05.788
and I'm turning them into two sections,

00:02:05.788 --> 00:02:07.251
so let's do that.

00:02:07.251 --> 00:02:08.551
Let's do that.

00:02:08.551 --> 00:02:12.684
Let me see if I can do it pretty neatly,

00:02:12.684 --> 00:02:16.422
so, I can do it a little
bit neater than that.

00:02:16.422 --> 00:02:18.419
So, it'll look like that.

00:02:18.419 --> 00:02:20.091
And,

00:02:20.091 --> 00:02:21.415
whoops.

00:02:21.415 --> 00:02:23.656
Let me do this one.

00:02:23.656 --> 00:02:26.790
I want to divide them
fairly close to evenly.

00:02:26.790 --> 00:02:30.320
I'm doing it by eye so it's
not going to be perfect.

00:02:30.320 --> 00:02:32.723
So, and you have that one.

00:02:32.723 --> 00:02:34.917
And then last not,

00:02:34.917 --> 00:02:36.250
last but not least,

00:02:36.250 --> 00:02:38.354
you have that one there, and then notice,

00:02:38.354 --> 00:02:41.790
I had six sections, but
now I've doubled the number

00:02:41.790 --> 00:02:42.417
of sections.

00:02:42.417 --> 00:02:45.088
I've turned the six sections
into 12 sections by turning

00:02:45.088 --> 00:02:48.420
each of the original six
into two, so now I have

00:02:48.420 --> 00:02:51.392
one, two, three, four,

00:02:51.392 --> 00:02:54.492
five, six, seven, eight,

00:02:54.492 --> 00:02:58.892
nine, 10, 11, 12 sections.

00:02:58.892 --> 00:03:01.585
So if I have 12 sections
now, how many of those 12

00:03:01.585 --> 00:03:02.955
are now shaded in?

00:03:02.955 --> 00:03:04.975
Instead of having five
of the six, I now have

00:03:04.975 --> 00:03:08.493
10 of the 12 that are shaded in.

00:03:08.493 --> 00:03:10.722
So I now have 10/12.

00:03:10.722 --> 00:03:13.787
5/6 is the same thing as 10/12.

00:03:13.787 --> 00:03:15.390
Another way you could
have thought about that,

00:03:15.390 --> 00:03:18.547
to go from six to 12, I
had to multiply by two,

00:03:18.547 --> 00:03:20.556
so then I have to do the
same thing in the numerator.

00:03:20.556 --> 00:03:22.959
Five times two is 10.

00:03:22.959 --> 00:03:24.991
But hopefully you see
that those two fractions

00:03:24.991 --> 00:03:26.291
are equivalent,

00:03:26.291 --> 00:03:29.124
that I didn't change
how much is shaded in,

00:03:29.124 --> 00:03:31.423
I just took each of the
original six and I turned it

00:03:31.423 --> 00:03:34.720
into two, or I multiplied
the total number of sections

00:03:34.720 --> 00:03:36.589
by two to get 12,

00:03:36.589 --> 00:03:38.261
and then instead of having 5/6,

00:03:38.261 --> 00:03:41.152
I now have 10/12 shaded in.

00:03:41.152 --> 00:03:43.521
Now let's do the same thing with the four,

00:03:43.521 --> 00:03:44.786
with the 1/4.

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Right here, I've depicted 1/4,
but I want to turn this into

00:03:47.654 --> 00:03:49.860
something over 12.

00:03:49.860 --> 00:03:53.018
So to turn it into something over 12,

00:03:53.018 --> 00:03:56.187
each section has to be
turned into three sections.

00:03:56.187 --> 00:03:57.093
So let's do that.

00:03:57.093 --> 00:04:00.123
Let's turn each section
into three sections.

00:04:00.123 --> 00:04:04.918
So, that's one, two, and three.

00:04:04.918 --> 00:04:09.887
So then I have one, two,

00:04:09.887 --> 00:04:11.384
and three.

00:04:11.384 --> 00:04:13.091
I have,

00:04:13.091 --> 00:04:14.124
I think you can see where this is going.

00:04:14.124 --> 00:04:15.645
One,

00:04:15.645 --> 00:04:17.120
two and three.

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I have one, two, and three.

00:04:21.125 --> 00:04:24.225
And so notice, all I did is I multiplied,

00:04:24.225 --> 00:04:26.059
before I had four equal sections.

00:04:26.059 --> 00:04:28.486
Now I turned each of those
four sections into three

00:04:28.486 --> 00:04:32.155
sections, so now I have 12 equal sections.

00:04:32.155 --> 00:04:34.419
And I did that, essentially,
by multiplying the number

00:04:34.419 --> 00:04:37.054
of sections I had by three.

00:04:37.054 --> 00:04:39.655
So now what fraction is shaded in?

00:04:39.655 --> 00:04:41.721
Well, now, this original

00:04:41.721 --> 00:04:44.300
that was one out of the
four, we can now see is three

00:04:44.300 --> 00:04:47.888
out of the 12 equal sections.

00:04:47.888 --> 00:04:52.161
It's now three out of
the 12 equal sections,

00:04:52.161 --> 00:04:53.891
and so what is this going to be?

00:04:53.891 --> 00:04:58.651
Well, if I have 10/12,
and I'm adding it to 3/12,

00:04:58.651 --> 00:05:00.427
well how many twelfths do I have?

00:05:00.427 --> 00:05:04.583
I'm going to have 13/12.

00:05:04.583 --> 00:05:07.381
And you could see it
visually over here as well.

00:05:07.381 --> 00:05:10.528
Up here in green, I have 10/12 shaded in.

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Each of these boxes are a twelfth.

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Let me write that down.

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Each of these boxes are 1/12.

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That's 1/12.

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This is 1/12.

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So how many twelfths do I have shaded in?

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I have the 10 that are shaded in in green,

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and then I have an 11/12,

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a 12/12 and then finally,

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the 13/12 is one way to think about it.


Adding and subtracting multiple fractions with unlike denominators

WEBVTT
Kind: captions
Language: en

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- [Voiceover] So we want
to figure out what 1/4

00:00:01.990 --> 00:00:05.585
plus 3/5 minus 3/10 is.

00:00:05.585 --> 00:00:06.960
And I encourage you to pause the video

00:00:06.960 --> 00:00:09.555
and see if you could figure
this out on your own.

00:00:09.555 --> 00:00:11.760
All right, I'm assuming
you've had an attempt,

00:00:11.760 --> 00:00:13.460
let's work through this together now.

00:00:13.460 --> 00:00:15.455
So the first thing that
might jump out at you

00:00:15.455 --> 00:00:16.760
is look, I have these fractions

00:00:16.760 --> 00:00:17.760
that I'm adding and subtracting,

00:00:17.760 --> 00:00:19.665
but they all have different denominators.

00:00:19.665 --> 00:00:21.855
So in order to add and subtract
them in a reasonable way,

00:00:21.855 --> 00:00:24.630
we'd want to rewrite them so they all have

00:00:24.630 --> 00:00:26.220
the same denominator.

00:00:26.220 --> 00:00:28.260
So what we really want to
do is find a common multiple

00:00:28.260 --> 00:00:31.380
of four, five and 10.

00:00:31.380 --> 00:00:34.020
And like I always say, we
can use any common multiple,

00:00:34.020 --> 00:00:36.120
but we simplify things
a little bit if we use

00:00:36.120 --> 00:00:39.555
the smallest common multiple,
or the least common multiple.

00:00:39.555 --> 00:00:42.240
And one way to find the
least common multiple

00:00:42.240 --> 00:00:44.550
is take the largest of these numbers

00:00:44.550 --> 00:00:47.490
and look at their multiples
and keep increasing

00:00:47.490 --> 00:00:48.990
the multiples until you find one

00:00:48.990 --> 00:00:51.115
that is divisible by the other two.

00:00:51.115 --> 00:00:53.755
So for example, I could start at 10.

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10 is divisible by five but
is not divisible by four.

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So then we can go 20.

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20 is divisible by both five and four,

00:01:02.285 --> 00:01:07.285
so the least common multiple
of four, five and 10 is 20.

00:01:07.920 --> 00:01:09.720
So let's rewrite all of these fractions

00:01:09.720 --> 00:01:12.060
as something over 20.

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So let's start with 1/4.

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1/4 is what over 20?

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Well, to go from four to 20,
you have to multiply by five,

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so you have to do the same
thing to the numerator.

00:01:22.720 --> 00:01:24.055
You have to multiply it by five.

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One times five is five.

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One over four is the same thing as 5/20.

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Four is four times one,
20 is four times five.

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Then we want to think about,
well, what happens to 3/5

00:01:39.760 --> 00:01:43.720
If I write it as something over 20?

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Well to go in the
denominator from five to 20,

00:01:46.425 --> 00:01:48.090
you have to multiply it by four,

00:01:48.090 --> 00:01:50.430
so you have to multiply the
numerator by four as well.

00:01:50.430 --> 00:01:54.375
So three times four is
going to be equal to 12.

00:01:54.375 --> 00:01:58.020
And then we're going to subtract 3/10,

00:01:58.020 --> 00:02:01.050
but how do we write that
as something over 20?

00:02:01.050 --> 00:02:02.730
Well, to go from 10 to 20,

00:02:02.730 --> 00:02:04.620
you multiply the denominator by two,

00:02:04.620 --> 00:02:05.985
so if we want to have the same fraction,

00:02:05.985 --> 00:02:08.580
we need to multiply the
numerator by two as well.

00:02:08.580 --> 00:02:11.155
So three times two is six.

00:02:11.155 --> 00:02:13.885
So what is this going to be equal to?

00:02:13.885 --> 00:02:15.460
Well now I have 5/20

00:02:15.460 --> 00:02:19.810
plus 12/20 minus 6/20.

00:02:19.810 --> 00:02:21.745
So what is this going to be?

00:02:21.745 --> 00:02:23.995
Well, there's a bunch of ways
you could think about it.

00:02:23.995 --> 00:02:26.020
You could say, well,
this is just going to be,

00:02:26.020 --> 00:02:30.540
this is going to be 5/20

00:02:31.295 --> 00:02:36.230
plus 12/20 plus 12/20

00:02:36.230 --> 00:02:40.460
minus 6/20, minus 6/20.

00:02:40.460 --> 00:02:41.690
Or another way of thinking about it,

00:02:41.690 --> 00:02:42.950
the way I just wrote it here,

00:02:42.950 --> 00:02:46.955
five plus 12 minus six,
all of that, over 20.

00:02:46.955 --> 00:02:50.885
This is how many 20ths
are going to result.

00:02:50.885 --> 00:02:52.950
So what is that equal to?

00:02:52.950 --> 00:02:55.660
Well that's going to be
equal to all of this,

00:02:55.660 --> 00:02:58.460
we are counting 20ths, is
one way to think about it.

00:02:58.460 --> 00:03:01.250
So five plus 12 is 17

00:03:01.250 --> 00:03:03.195
minus six is 11.

00:03:03.195 --> 00:03:06.645
So we get 11/20.

00:03:06.645 --> 00:03:10.320
I have 5/20, I add 12/20
and I take away 6/20,

00:03:10.320 --> 00:03:13.660
I'm going to be left with 11/20.

00:03:13.660 --> 00:03:17.645
Let's do one more example of
this, just for good practice.

00:03:17.645 --> 00:03:22.520
So here I have 4/9 minus 1/6 plus 1/3.

00:03:22.520 --> 00:03:24.350
And like before, I encourage
you to puase the video

00:03:24.350 --> 00:03:26.200
and see if you can work this out.

00:03:26.925 --> 00:03:28.485
Well, what I want to
do is I want to rewrite

00:03:28.485 --> 00:03:29.860
all of these fractions so that they have

00:03:29.860 --> 00:03:32.155
a common denominator.

00:03:32.155 --> 00:03:33.795
And to find a common
denominator, you need to find

00:03:33.795 --> 00:03:38.310
a common multiple of nine, six and three.

00:03:38.310 --> 00:03:40.275
And like I did before, what I could do is

00:03:40.275 --> 00:03:42.385
I could start with a nine and say okay,

00:03:42.385 --> 00:03:44.830
nine, it's not divisible by six.

00:03:44.830 --> 00:03:47.125
It is divisible by three,
but that's not good enough.

00:03:47.125 --> 00:03:50.525
I want to find the common
multiple of nine, six and three,

00:03:50.525 --> 00:03:52.790
and as small as one as possible.

00:03:52.790 --> 00:03:55.745
So let's look at the next
multiple of nine, we get to 18.

00:03:55.745 --> 00:04:00.160
And 18 is divisible by nine
of course, six and three.

00:04:00.160 --> 00:04:01.505
So let's go with 18.

00:04:01.505 --> 00:04:04.210
Let's write everything
as something over 18.

00:04:04.210 --> 00:04:07.120
So 4/9 is what over 18?

00:04:07.120 --> 00:04:09.230
Well, nine times two is 18,

00:04:09.230 --> 00:04:10.655
that's what I did to the denominator,

00:04:10.655 --> 00:04:12.755
so I have to multiply the
numerator by two as well.

00:04:12.755 --> 00:04:15.720
So this is going to be 8/18.

00:04:15.720 --> 00:04:19.530
Well, what's 1/6 over 18?

00:04:19.530 --> 00:04:21.920
Well, to go from six to
18 in the denominator,

00:04:21.920 --> 00:04:23.420
I have to multiply it by three,

00:04:23.420 --> 00:04:25.895
so I have to multiply the
numerator by three as well.

00:04:25.895 --> 00:04:28.490
So one times three is three.

00:04:28.490 --> 00:04:30.960
And then last but not least,

00:04:30.960 --> 00:04:33.820
what is 1/3 as something over 18?

00:04:33.820 --> 00:04:36.055
Well, three times six is 18,

00:04:36.055 --> 00:04:39.220
so one times six is going to be six.

00:04:39.220 --> 00:04:40.790
Or you could do it another
way, you could say,

00:04:40.790 --> 00:04:43.220
well, what's 1/3 of 18?
It's going to be six.

00:04:43.220 --> 00:04:45.695
What's 1/6 of 18? It's going to be three.

00:04:45.695 --> 00:04:48.725
4/9 of 18? That's a little bit
harder to think in your brain

00:04:48.725 --> 00:04:50.295
but that's going to be 8/18.

00:04:50.295 --> 00:04:52.995
Either way, I've just
rewritten these fractions,

00:04:52.995 --> 00:04:55.695
and I've rewritten them in
the corresponding color.

00:04:55.695 --> 00:04:57.820
So this up here is the exact same thing

00:04:57.820 --> 00:05:02.820
as 8/18 minus 3/18 plus 6/18.

00:05:03.285 --> 00:05:05.525
So what is this going to be equal to?

00:05:05.525 --> 00:05:07.925
It's going to be a
certain number of 18ths,

00:05:07.925 --> 00:05:11.190
and so if I have 8/18 minus 3/18,

00:05:11.190 --> 00:05:13.395
that's going to be 5/18,

00:05:13.395 --> 00:05:18.360
plus 6/18 is going to be 11/18.

00:05:18.360 --> 00:05:21.525
And we are done.


Subtracting fractions with unlike denominators

WEBVTT
Kind: captions
Language: en

00:00:00.055 --> 00:00:01.822
- [VOICEOVER] Let's see
if we can figure out what

00:00:01.822 --> 00:00:06.822
4/3 minus 1/5 is, and

00:00:08.650 --> 00:00:10.377
if you think you know how to do it,

00:00:10.377 --> 00:00:12.984
I encourage you to pause
the video and give it a go.

00:00:13.790 --> 00:00:15.821
So, when you first look
at this, the thing that

00:00:15.821 --> 00:00:16.724
might jump out at you is

00:00:16.724 --> 00:00:18.785
we have different denominators here.

00:00:18.795 --> 00:00:23.704
So, it's not obvious how to
subtract 1/5 from 4/3 when

00:00:23.704 --> 00:00:25.939
you have these different denominators and

00:00:25.939 --> 00:00:28.418
the key is, is to rewrite each
of these fractions so that

00:00:28.418 --> 00:00:30.212
they have the same denominator.

00:00:30.693 --> 00:00:33.240
And how do we figure out what
that same denominator is?

00:00:33.599 --> 00:00:37.479
Well, it's going to be a common
multiple of three and five,

00:00:37.479 --> 00:00:39.937
and, ideally, it's going to be
the least common multiple of

00:00:39.937 --> 00:00:41.502
three and five.

00:00:41.502 --> 00:00:42.768
So, how can we calculate that?

00:00:43.128 --> 00:00:45.017
Well, we can start with the
larger of the two numbers,

00:00:45.017 --> 00:00:47.739
say five, and let's go
through it's multiples and

00:00:47.739 --> 00:00:51.239
see when we get to one that's
divisible perfectly by three.

00:00:51.538 --> 00:00:53.996
So, five is not divisible by three,

00:00:53.996 --> 00:00:57.804
10 is not divisible by three,
15 is divisible by three.

00:00:57.810 --> 00:00:59.997
In fact, 15 is three times five.

00:01:00.168 --> 00:01:02.422
So, I can rewrite both
of these fractions as

00:01:02.422 --> 00:01:04.075
something over 15.

00:01:04.580 --> 00:01:08.572
So, what's 4/3 if I were to
rewrite it as something over 15?

00:01:08.572 --> 00:01:11.743
Well, to get from three
to 15 in the denominator,

00:01:11.743 --> 00:01:16.339
we have to multiply by five.

00:01:16.984 --> 00:01:18.630
So, if you multiply the
denominator by five,

00:01:18.630 --> 00:01:20.418
if you don't want to change
the value of the fraction,

00:01:20.418 --> 00:01:22.679
you have to multiply the
numerator by five as well.

00:01:23.425 --> 00:01:25.863
So, you have to multiply the
numerator by five as well,

00:01:25.863 --> 00:01:28.693
four times five is going to be 20.

00:01:28.890 --> 00:01:32.207
So, 4/3 is the same thing as 20/15.

00:01:32.680 --> 00:01:33.856
Alright.

00:01:34.073 --> 00:01:39.037
Now, how would we rewrite
1/5 and something over 15?

00:01:39.037 --> 00:01:41.197
So, we're going to have
15 in the denominator.

00:01:41.618 --> 00:01:45.808
Well, to go from five to 15,
we had to multiply by three.

00:01:46.757 --> 00:01:48.444
So, if we multiply the
denominator by three,

00:01:48.444 --> 00:01:50.584
we have to multiply the
numerator by three as well.

00:01:51.409 --> 00:01:52.859
So, times three.

00:01:53.015 --> 00:01:54.667
One times three is just three.

00:01:55.087 --> 00:01:58.501
So, 4/3 minus 1/5, we can rewrite that as

00:01:58.501 --> 00:02:02.032
20/15 minus 3/15.

00:02:02.392 --> 00:02:04.450
Now, this becomes a lot
more straight forward.

00:02:04.606 --> 00:02:05.689
What is this going to be?

00:02:05.927 --> 00:02:07.857
Well, this is going to be a
certain number of fifteenths.

00:02:07.857 --> 00:02:09.342
We have 20/15 and we're

00:02:09.342 --> 00:02:11.762
taking away three of those fifteenths.

00:02:11.762 --> 00:02:13.688
So, we are going to have,

00:02:13.688 --> 00:02:16.024
if you have 20 of something and
you take away three of them,

00:02:16.024 --> 00:02:18.692
you're going to have 17 of those things.

00:02:18.727 --> 00:02:20.317
In this case, we're
talking about fifteenths.

00:02:20.616 --> 00:02:23.771
So, this is going to be 17/15.

00:02:24.476 --> 00:02:26.244
And if we wanted to write
it as a mixed number,

00:02:26.244 --> 00:02:28.946
we could say 15 goes into 17 one time with

00:02:28.946 --> 00:02:30.212
a remainder of two.

00:02:30.429 --> 00:02:35.265
So, it's one and 2/15.
Let's do another example.

00:02:35.265 --> 00:02:40.265
Let's see if we can figure
out what 7/10 minus 5/8 is.

00:02:48.223 --> 00:02:49.579
Five over eight.

00:02:49.579 --> 00:02:51.677
And I encourage you to
pause this video and

00:02:51.687 --> 00:02:53.378
see if you can calculate it yourself.

00:02:53.442 --> 00:02:55.231
Well, just like we saw before,

00:02:55.231 --> 00:02:56.531
we have different denominators but

00:02:56.531 --> 00:02:59.090
we need to rewrite them so
they have a common denominator,

00:02:59.090 --> 00:03:00.296
that way, we can subtract.

00:03:00.757 --> 00:03:05.734
And so, what's a common
multiple of 10 and eight and,

00:03:05.734 --> 00:03:07.122
ideally, the least common multiple.

00:03:07.217 --> 00:03:08.275
It doesn't have to be but

00:03:08.275 --> 00:03:09.702
it keeps things a little bit simpler.

00:03:09.899 --> 00:03:12.967
Well, let's start with the
larger of the two numbers and

00:03:12.967 --> 00:03:15.385
then keep finding in their multiples and

00:03:15.385 --> 00:03:18.575
find one that is perfectly
divisible by the other one,

00:03:18.575 --> 00:03:19.530
by eight.

00:03:19.530 --> 00:03:22.536
So, ten isn't perfectly
divisible by eight, 20 isn't,

00:03:22.536 --> 00:03:25.347
30 isn't, 40 is.

00:03:26.011 --> 00:03:29.383
40 is a multiple of 10 and
it's a multiple of eight.

00:03:29.383 --> 00:03:31.644
In fact, it's the least common
multiple of 10 and eight.

00:03:31.964 --> 00:03:34.665
So, we can rewrite both
of these fractions as

00:03:34.665 --> 00:03:35.871
something over 40.

00:03:36.393 --> 00:03:40.010
So, that's going to be
something over 40 minus

00:03:40.010 --> 00:03:41.681
something over 40.

00:03:42.245 --> 00:03:46.700
Minus something over 40
is equal to something.

00:03:47.303 --> 00:03:49.930
So, 7/10 is what over 40?

00:03:50.002 --> 00:03:54.492
Well, to go from 10 to
40 in the denominator,

00:03:54.492 --> 00:03:55.798
we multiplied by four.

00:03:56.198 --> 00:03:57.783
So, we have to do the same
thing in the numerator,

00:03:57.783 --> 00:04:01.407
multiply the numerator by
four. Seven times four is 28.

00:04:01.765 --> 00:04:04.494
So, 7/10 is the same thing as 28/40.

00:04:05.361 --> 00:04:07.400
Now, let's do the same thing
with the other fraction.

00:04:07.759 --> 00:04:10.034
To go from eight to 40 in the denominator,

00:04:10.034 --> 00:04:11.646
we had to multiply the
denominator by five.

00:04:12.433 --> 00:04:14.144
Eight times five is 40.

00:04:14.443 --> 00:04:16.475
So, if we multiply the
denominator by five,

00:04:16.475 --> 00:04:18.960
we have to multiply the
numerator by five as well.

00:04:19.279 --> 00:04:21.499
Five times five is 25.

00:04:22.469 --> 00:04:25.942
So, 7/10 minus 5/8 is
the exact same things as

00:04:25.942 --> 00:04:30.942
28/40 minus 25/40 and now
this makes a lot of sense.

00:04:32.303 --> 00:04:34.197
It's going to be a certain
number of fortieths.

00:04:34.496 --> 00:04:39.231
If I have 28/40 and I take
away 25 of those fortieths,

00:04:39.231 --> 00:04:41.125
how many fortieths am
I going to have left?

00:04:41.730 --> 00:04:43.808
Well, I'm going to 3/40 left.

00:04:43.966 --> 00:04:47.942
28/40 minus 25/40. So,
I'm going to have 3/40.

00:04:47.952 --> 00:04:52.952
28 minus 25 is three and we are done.

00:04:53.782 --> 00:04:58.409
7/10 minus 5/8 is 3/40.


Adding fractions with unlike denominators

WEBVTT
Kind: captions
Language: en

00:00:00.303 --> 00:00:03.046
- [Voiceover] Let's say that
we have the fraction 9/10,

00:00:03.046 --> 00:00:08.046
and I want to add to
that the fraction 1/6.

00:00:09.567 --> 00:00:13.039
What is this, what is this going to equal?

00:00:13.839 --> 00:00:14.827
So when you first look at this, you say,

00:00:14.827 --> 00:00:16.602
"Oh, I have different denominators here.

00:00:16.602 --> 00:00:18.599
It's not obvious how I add these."

00:00:18.599 --> 00:00:21.038
And you'd be right and the way to actually

00:00:21.038 --> 00:00:23.546
move forward is to find
a common denominator,

00:00:23.546 --> 00:00:26.181
to convert both of these fractions into

00:00:26.181 --> 00:00:28.536
fractions that have a common denominator.

00:00:28.536 --> 00:00:30.498
So how do you think about
a common denominator?

00:00:30.498 --> 00:00:32.125
Well, a common denominator's
gonna have to be

00:00:32.125 --> 00:00:36.478
a common multiple of these two
denominators of 10 and six.

00:00:36.478 --> 00:00:38.638
So what's a common multiple of 10 and six?

00:00:38.638 --> 00:00:41.436
And it's usually simplest to
find the least common multiple,

00:00:41.436 --> 00:00:43.502
and a good way of doing that
is start with the larger

00:00:43.502 --> 00:00:47.312
denominator here, 10, and say,
okay is 10 divisible by six?

00:00:47.952 --> 00:00:50.978
No. Okay, now, is 20 divisible by six?

00:00:51.588 --> 00:00:56.005
No. Is 30 divisible by six?
Yes. 30 is divisible by six.

00:00:56.005 --> 00:00:57.722
So I'm just going through
the multiples of 10

00:00:57.722 --> 00:00:59.556
and saying, "Well what is
the smallest multiple of 10

00:00:59.556 --> 00:01:03.644
that is divisible by six?"
And that's going to be 30.

00:01:03.644 --> 00:01:05.639
So I could rewrite both of these fractions

00:01:05.639 --> 00:01:07.605
as something over 30.

00:01:07.605 --> 00:01:10.296
So nine over 10. How would I write that as

00:01:10.296 --> 00:01:11.898
something over 30? Well I multiply

00:01:11.898 --> 00:01:15.984
the denominator, I'm multiplying
the denominator by three.

00:01:17.074 --> 00:01:19.617
So I've just multiplied
the denominator by three.

00:01:19.617 --> 00:01:22.088
So if I don't want to change
the value of the fraction,

00:01:22.088 --> 00:01:23.586
I have to do the same
thing to the numerator.

00:01:23.586 --> 00:01:26.186
I have to multiply that by three as well

00:01:26.996 --> 00:01:29.854
because now I'm just multiplying
the numerator by three

00:01:29.854 --> 00:01:31.433
and the denominator by three,
and that doesn't change

00:01:31.433 --> 00:01:32.954
the value of the fraction.

00:01:32.954 --> 00:01:35.751
So nine times three is 27.

00:01:35.751 --> 00:01:38.549
So once again, 9/10 and 27/30

00:01:38.549 --> 00:01:40.964
represent the same number.

00:01:40.964 --> 00:01:43.564
I've just written it now
with a denominator of 30,

00:01:43.564 --> 00:01:45.631
and that's useful because
I can also write 1/6

00:01:45.631 --> 00:01:49.100
with a denominator of 30. Let's do that.

00:01:49.100 --> 00:01:51.621
So 1/6 is what over 30?

00:01:51.621 --> 00:01:52.724
I encourage you to pause the video

00:01:52.724 --> 00:01:53.850
and try to think about it.

00:01:53.850 --> 00:01:56.149
So what did we do go from six to 30?

00:01:56.149 --> 00:01:59.248
We had to multiply by five.

00:01:59.908 --> 00:02:01.628
So if we multiply the denominator by five,

00:02:01.628 --> 00:02:04.623
we have to multiply the
numerator by five as well,

00:02:04.623 --> 00:02:09.623
so one times five, one times five is five.

00:02:11.008 --> 00:02:13.749
So 9/10 is the same thing as 27/30,

00:02:13.749 --> 00:02:16.453
and 1/6 is the same thing as 5/30.

00:02:16.453 --> 00:02:20.226
And now we can add, now we can add

00:02:20.226 --> 00:02:21.817
and it's fairly straightforward.

00:02:21.817 --> 00:02:23.267
We have a certain number of 30ths,

00:02:23.267 --> 00:02:25.335
added to another number of 30ths,

00:02:25.335 --> 00:02:30.062
so 27/30 + 5/30, well that's going to be

00:02:30.062 --> 00:02:35.062
27, that's going to be 27 plus five,

00:02:35.471 --> 00:02:40.201
plus five, plus 5/30,

00:02:41.181 --> 00:02:43.583
plus 5/30, which of course

00:02:43.583 --> 00:02:47.361
going to be equal to 32/30.

00:02:47.361 --> 00:02:50.781
32 over 30, and

00:02:50.781 --> 00:02:54.321
if we want, we could try
to reduce this fraction.

00:02:54.801 --> 00:02:56.805
We have a common factor of 32 and 30,

00:02:56.805 --> 00:03:00.196
they're both divisible by two.

00:03:00.196 --> 00:03:03.505
So if we divide the numerator
and the denominator by two,

00:03:03.505 --> 00:03:06.118
numerator divided by two is 16,

00:03:06.118 --> 00:03:08.902
denominator divided by two is 15.

00:03:09.452 --> 00:03:12.640
So, this is the same thing
as 16/15, and if I wanted

00:03:12.640 --> 00:03:16.215
to write this as a mixed
number, 15 goes into 16 one time

00:03:16.215 --> 00:03:17.574
with a remainder one.

00:03:17.574 --> 00:03:20.255
So this is the same thing as 1 1/15.

00:03:20.795 --> 00:03:22.534
Let's do another example.

00:03:22.534 --> 00:03:27.018
Let's say that we wanted
to add, we wanted to add

00:03:27.018 --> 00:03:31.881
1/2 to

00:03:31.881 --> 00:03:36.881
to 11/12, to 11 over 12.

00:03:36.893 --> 00:03:38.033
And I encourage you to pause the video

00:03:38.033 --> 00:03:40.864
and see if you could work this out.

00:03:40.864 --> 00:03:42.502
Well like we saw before, we wanna find

00:03:42.502 --> 00:03:43.883
a common denominator.

00:03:43.883 --> 00:03:45.102
If these had the same denominator,

00:03:45.102 --> 00:03:46.264
we could just add them immediately,

00:03:46.264 --> 00:03:48.527
but we wanna find a common denominator

00:03:48.527 --> 00:03:50.222
because right now they're not the same.

00:03:50.902 --> 00:03:53.468
Well what we wanna find is a multiple,

00:03:53.468 --> 00:03:55.794
a common multiple of
two and 12, and ideally

00:03:55.794 --> 00:03:58.164
we'll find the lowest common
multiple of two and 12,

00:03:58.164 --> 00:04:00.264
and just like we did before,
let's start with the larger

00:04:00.264 --> 00:04:01.901
of the two numbers, 12.

00:04:01.901 --> 00:04:05.291
Now we could just say
well 12 times one is 12,

00:04:05.291 --> 00:04:07.949
so that we could view that
as the lowest multiple of 12.

00:04:07.949 --> 00:04:10.632
And is that divisible by two? Yeah, sure.

00:04:10.632 --> 00:04:12.790
12 is divisible by two.

00:04:12.790 --> 00:04:15.855
So 12 is actually the least
common multiple of two and 12,

00:04:15.855 --> 00:04:17.213
so we could write both of these

00:04:17.213 --> 00:04:19.013
fractions as something over 12.

00:04:19.013 --> 00:04:21.625
So 1/2 is what over 12?

00:04:21.625 --> 00:04:24.446
Well to go from two to
12, you multiply by six,

00:04:24.446 --> 00:04:27.104
so we'll also multiply
the numerator by six.

00:04:27.104 --> 00:04:30.588
Now we see 1/2, and 6/12,
these are the same thing.

00:04:30.588 --> 00:04:33.954
One is half of two, six is half of 12.

00:04:34.914 --> 00:04:38.485
And how would we write
11/12 as something over 12?

00:04:38.485 --> 00:04:40.855
Well it's already written
as something over 12,

00:04:40.855 --> 00:04:43.258
11/12 already has 12 in the denominator,

00:04:43.258 --> 00:04:45.029
so we don't have to change that.

00:04:45.615 --> 00:04:48.268
11/12, and now we're ready to add.

00:04:48.600 --> 00:04:51.350
So this is going to be equal to six,

00:04:52.520 --> 00:04:55.820
this is going to be equal to six plus 11,

00:04:56.510 --> 00:05:01.510
six plus 11 over 12.

00:05:02.378 --> 00:05:06.021
Over 12. We have 6/12 plus 11/12,

00:05:06.021 --> 00:05:09.318
it's gonna be six plus 11 over 12,

00:05:10.728 --> 00:05:15.087
which is equal to, six plus 11 is 17/12.

00:05:15.087 --> 00:05:16.504
If we wanted to write
it as a mixed number,

00:05:16.504 --> 00:05:19.487
that is what, 12 goes
into 17 one time with

00:05:19.487 --> 00:05:24.487
a remainder of five, so 1 5/12.

00:05:24.530 --> 00:05:25.710
Let's do one more of these.

00:05:25.710 --> 00:05:29.007
This is strangely fun. Alright.

00:05:29.007 --> 00:05:31.043
Let's say that we wanted to add,

00:05:31.523 --> 00:05:35.894
We're gonna add 3/4 to,

00:05:36.504 --> 00:05:40.584
we're gonna add 3/4 to 1/5.

00:05:41.414 --> 00:05:43.974
To one over five.

00:05:43.974 --> 00:05:44.659
What is this going to be?

00:05:44.659 --> 00:05:46.157
And once again, pause the video and

00:05:46.157 --> 00:05:47.870
see if you could work it out.

00:05:47.870 --> 00:05:49.291
Well we have different denominators here,

00:05:49.291 --> 00:05:52.052
and we wanna find, we wanna rewrite these

00:05:52.052 --> 00:05:53.457
so they have the same denominators,

00:05:53.457 --> 00:05:54.792
so we have to find a common multiple,

00:05:54.792 --> 00:05:57.095
ideally the least common multiple.

00:05:57.095 --> 00:05:59.738
So what's the least common
multiple of four and five?

00:06:00.548 --> 00:06:01.862
Well let's start with the larger number,

00:06:01.862 --> 00:06:04.718
and let's look at its
multiples and keep increasing

00:06:04.718 --> 00:06:07.061
them until we get one
that's divisible by four.

00:06:07.061 --> 00:06:10.064
So five is not divisible by four.

00:06:10.064 --> 00:06:13.622
10 is not divisible by four,
or perfectly divisible by four

00:06:13.622 --> 00:06:14.702
is what we care about.

00:06:14.702 --> 00:06:17.059
15 is not perfectly divisible by four.

00:06:17.059 --> 00:06:20.763
20 is divisible by four, in
fact, that is five times four.

00:06:20.763 --> 00:06:23.514
That is 20. So what we
could do is, we could write

00:06:23.514 --> 00:06:27.460
both of these fractions as
having 20 in the denominator,

00:06:27.460 --> 00:06:28.714
or 20 as the denominator.

00:06:29.454 --> 00:06:32.266
So we could write 3/4
is something over 20.

00:06:32.996 --> 00:06:35.319
So to go from four to
20 in the denominator,

00:06:35.319 --> 00:06:36.949
we multiplied by five.

00:06:36.949 --> 00:06:38.466
So we also do that to the numerator.

00:06:38.466 --> 00:06:41.398
We multiply by three times five to get 15.

00:06:41.398 --> 00:06:44.183
All I did to go from four
to 20, multiplied by five.

00:06:44.183 --> 00:06:45.820
So I have to do the same
thing to the numerator,

00:06:45.820 --> 00:06:47.736
three times five is 15.

00:06:47.736 --> 00:06:52.658
3/4 is the same thing
as 15/20, and over here.

00:06:52.658 --> 00:06:55.004
1/5. What is that over 20?

00:06:55.004 --> 00:06:58.358
Well to go from five to 20,
you have to multiply by four.

00:06:58.358 --> 00:06:59.995
So we have to do the same
thing to the numerator.

00:06:59.995 --> 00:07:03.861
I have to multiply this
numerator times four to get 4/20.

00:07:04.451 --> 00:07:07.181
So now I've rewritten this
instead of 3/4 plus 1/5,

00:07:07.181 --> 00:07:10.815
it's now written as 15/20 plus 4/20.

00:07:10.815 --> 00:07:12.973
And what is that going to be?

00:07:12.973 --> 00:07:17.932
Well that's going to be
15 plus four is 19/20.

00:07:17.932 --> 00:07:22.024
19/20, and we're done.


Area of parallelogram on coordinate plane

WEBVTT
Kind: captions
Language: en

00:00:00.270 --> 00:00:01.920
- [Instructor] Let's see
if we can find the area

00:00:01.920 --> 00:00:03.030
of this parallelogram.

00:00:03.030 --> 00:00:04.680
I encourage you to pause the video and see

00:00:04.680 --> 00:00:07.050
if you can figure it out on your own.

00:00:07.050 --> 00:00:08.670
Well, we just have to remind ourselves

00:00:08.670 --> 00:00:11.250
that the area of a parallelogram
is just going to be

00:00:11.250 --> 00:00:13.800
the base, let me do this
in different colors,

00:00:13.800 --> 00:00:16.470
it's going to be the base
of the parallelogram,

00:00:16.470 --> 00:00:18.270
which I wanna do that
in a different color.

00:00:18.270 --> 00:00:23.270
So let me write the base
of the parallelogram

00:00:23.310 --> 00:00:26.280
times the height of the parallelogram.

00:00:26.280 --> 00:00:31.080
Times the height of the parallelogram.

00:00:31.080 --> 00:00:32.460
Area is equal to base times height.

00:00:32.460 --> 00:00:35.010
So what could we consider to be the base

00:00:35.010 --> 00:00:36.540
of this parallelogram?

00:00:36.540 --> 00:00:39.810
Well, we could imagine it
to be one of these sides.

00:00:39.810 --> 00:00:42.060
So we could go from here,

00:00:42.060 --> 00:00:45.540
and so I could say, well,
I could consider this

00:00:45.540 --> 00:00:46.740
to be the base.

00:00:46.740 --> 00:00:48.943
And so what's the length of that base?

00:00:48.943 --> 00:00:51.720
Well, we're just going in
the vertical direction.

00:00:51.720 --> 00:00:55.590
We go from y equals five to
y is equal to negative seven.

00:00:55.590 --> 00:00:57.210
So this has length 12.

00:00:57.210 --> 00:01:00.720
We have five above the x-axis
and seven below the x-axis,

00:01:00.720 --> 00:01:01.710
adding up to 12.

00:01:01.710 --> 00:01:05.310
Or you could count it, one,
two, three, four, five, six,

00:01:05.310 --> 00:01:08.730
seven, eight, nine, 10, 11, 12.

00:01:08.730 --> 00:01:10.170
So this is our base.

00:01:10.170 --> 00:01:12.870
And we could say that base is equal to 12.

00:01:12.870 --> 00:01:15.030
And now what could we view as our height?

00:01:15.030 --> 00:01:18.480
Well, we could view

00:01:18.480 --> 00:01:20.980
this dimension right over here

00:01:21.930 --> 00:01:23.460
as our height.

00:01:23.460 --> 00:01:24.780
And what is that going to be?

00:01:24.780 --> 00:01:26.850
Well, you can see very
clearly that the height

00:01:26.850 --> 00:01:27.810
is equal to four.

00:01:27.810 --> 00:01:29.220
And this might be a
little counterintuitive,

00:01:29.220 --> 00:01:30.480
'cause normally when you're
talking about height,

00:01:30.480 --> 00:01:33.180
you're used to thinking
about how high something is.

00:01:33.180 --> 00:01:35.850
But you can imagine rotating this around

00:01:35.850 --> 00:01:38.093
so that this is laying,

00:01:38.093 --> 00:01:39.930
the base is laying flat
and then the height

00:01:39.930 --> 00:01:42.690
is the height in the
traditional sense of the word.

00:01:42.690 --> 00:01:46.020
But we could say h is equal to four.

00:01:46.020 --> 00:01:47.790
And now it's pretty straightforward.

00:01:47.790 --> 00:01:50.880
Our area is going to be equal

00:01:50.880 --> 00:01:53.880
to 12, the length of our base,

00:01:53.880 --> 00:01:56.760
times our height, times four,

00:01:56.760 --> 00:02:00.540
times four, which is clearly just 48.

00:02:00.540 --> 00:02:05.540
48 whatever, square, 48 square units.


Finding coordinates of missing vertex example

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.790
- [Instructor] You are
graphing Rectangle ABCD

00:00:02.790 --> 00:00:04.380
in the coordinate plane.

00:00:04.380 --> 00:00:07.290
The following are three of
the vertices of the rectangle,

00:00:07.290 --> 00:00:08.700
and they give us the coordinates.

00:00:08.700 --> 00:00:11.190
What are the coordinates of point D?

00:00:11.190 --> 00:00:12.510
All right, well, a good place to start.

00:00:12.510 --> 00:00:15.900
Let's just plot these, the three
vertices that they give us.

00:00:15.900 --> 00:00:17.760
So it looks like they're
all in the first quadrant.

00:00:17.760 --> 00:00:21.900
So I'm gonna focus my coordinate
axes on the first quadrant.

00:00:21.900 --> 00:00:25.620
So that's going to be my y-axis.

00:00:25.620 --> 00:00:27.060
And let me now,

00:00:27.060 --> 00:00:30.780
that's gonna be my x-axis.

00:00:30.780 --> 00:00:31.860
And let's see,

00:00:31.860 --> 00:00:33.870
the highest x value that I have,

00:00:33.870 --> 00:00:35.640
we have a 2, I have a 5, I have a 5.

00:00:35.640 --> 00:00:37.200
It looks like it goes up to 5.

00:00:37.200 --> 00:00:38.550
So I could say

00:00:38.550 --> 00:00:39.930
1,

00:00:39.930 --> 00:00:41.130
2,

00:00:41.130 --> 00:00:42.390
3,

00:00:42.390 --> 00:00:43.290
4,

00:00:43.290 --> 00:00:44.340
and 5.

00:00:44.340 --> 00:00:45.249
So that's 5.

00:00:45.249 --> 00:00:46.500
I can just number them 1,

00:00:46.500 --> 00:00:47.333
2,

00:00:47.333 --> 00:00:48.166
3,

00:00:48.166 --> 00:00:49.140
4, and 5.

00:00:49.140 --> 00:00:52.020
And let's see the highest
y value of 1, 1, and a 6.

00:00:52.020 --> 00:00:53.250
It goes up to 6.

00:00:53.250 --> 00:00:54.660
So 1,

00:00:54.660 --> 00:00:55.770
2,

00:00:55.770 --> 00:00:56.790
3,

00:00:56.790 --> 00:00:58.020
4,

00:00:58.020 --> 00:00:58.890
5,

00:00:58.890 --> 00:01:00.540
and 6.

00:01:00.540 --> 00:01:01.373
I can number them

00:01:01.373 --> 00:01:02.206
1,

00:01:02.206 --> 00:01:03.039
2,

00:01:03.039 --> 00:01:03.872
3,

00:01:03.872 --> 00:01:04.705
4,

00:01:04.705 --> 00:01:05.538
5,

00:01:05.538 --> 00:01:06.450
and 6.

00:01:06.450 --> 00:01:08.010
Now let's plot the points.

00:01:08.010 --> 00:01:10.170
So let's first plot point A,

00:01:10.170 --> 00:01:11.940
which is at 2,1.

00:01:11.940 --> 00:01:12.960
x equals 2,

00:01:12.960 --> 00:01:14.310
y is equal to 1.

00:01:14.310 --> 00:01:16.920
So that's that point right over there.

00:01:16.920 --> 00:01:19.200
Now let's plot point B.

00:01:19.200 --> 00:01:22.800
Point B is at 5,1.

00:01:22.800 --> 00:01:24.300
So x equals 5,

00:01:24.300 --> 00:01:25.530
y is equal to 1.

00:01:25.530 --> 00:01:27.390
That's right over there.

00:01:27.390 --> 00:01:29.550
Now let's plot point C.

00:01:29.550 --> 00:01:30.747
Point C,

00:01:30.747 --> 00:01:32.580
and you can find another color here,

00:01:32.580 --> 00:01:34.680
point C is at x is equal to 5,

00:01:34.680 --> 00:01:36.090
y is equal to 6.

00:01:36.090 --> 00:01:37.620
x is equal to 5,

00:01:37.620 --> 00:01:39.360
y is equal to 6.

00:01:39.360 --> 00:01:43.170
So it sticks us roughly right over there.

00:01:43.170 --> 00:01:46.620
And so we need to figure
out what D is going to be.

00:01:46.620 --> 00:01:48.900
Well, D is going, this is a rectangle,

00:01:48.900 --> 00:01:50.100
the sides are parallel,

00:01:50.100 --> 00:01:51.240
all the vertices,

00:01:51.240 --> 00:01:53.430
we have right angles at all the vertices.

00:01:53.430 --> 00:01:56.280
So point D is going to have the same,

00:01:56.280 --> 00:01:59.220
the same x-coordinate as
this point right over here.

00:01:59.220 --> 00:02:01.227
So it's gonna have the same x-coordinate,

00:02:01.227 --> 00:02:03.720
and so it's going to have an x-coordinate,

00:02:03.720 --> 00:02:04.553
so lemme write this,

00:02:04.553 --> 00:02:06.440
D is gonna have an x-coordinate of 2,

00:02:07.680 --> 00:02:09.480
and it's gonna have the same y-coordinate

00:02:09.480 --> 00:02:10.890
as this point up here.

00:02:10.890 --> 00:02:13.830
So it's gonna have a y-coordinate of 6,

00:02:13.830 --> 00:02:15.300
of 6.

00:02:15.300 --> 00:02:18.540
So D is going to be at the point 2,6.

00:02:18.540 --> 00:02:19.710
And you see when we do that,

00:02:19.710 --> 00:02:21.360
we have set up a nice,

00:02:21.360 --> 00:02:22.470
a nice rectangle here.

00:02:22.470 --> 00:02:24.240
And we can draw the rectangle.

00:02:24.240 --> 00:02:27.120
So that's one side, that's the top side,

00:02:27.120 --> 00:02:28.770
that's another side,

00:02:28.770 --> 00:02:30.030
another side,

00:02:30.030 --> 00:02:32.670
and then we have that just like that.

00:02:32.670 --> 00:02:33.503
There you go.

00:02:33.503 --> 00:02:34.336
We have our rectangle.

00:02:34.336 --> 00:02:35.910
But most importantly, we
answered our question:

00:02:35.910 --> 00:02:38.520
What are the coordinates of point D?

00:02:38.520 --> 00:02:39.353
All right.


Dimensions of rectangle from coordinates example

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:02.160
- [Instructor] Rectangle ABCD is graphed

00:00:02.160 --> 00:00:03.570
in the coordinate plane.

00:00:03.570 --> 00:00:05.760
The following are the
vertices of the rectangle,

00:00:05.760 --> 00:00:08.160
and they give us the labels
for the vertices as well.

00:00:08.160 --> 00:00:09.060
Given these coordinates,

00:00:09.060 --> 00:00:12.450
what is the length of
side AD of this rectangle?

00:00:12.450 --> 00:00:13.350
So let's just plot it.

00:00:13.350 --> 00:00:14.850
That's one way that we could tackle it.

00:00:14.850 --> 00:00:15.900
So let's,

00:00:15.900 --> 00:00:16.733
let me see all of these

00:00:16.733 --> 00:00:17.850
are actually in the first coordinate.

00:00:17.850 --> 00:00:20.340
So I can focus on the first coordinate.

00:00:20.340 --> 00:00:23.940
So let's say that's my y-axis.

00:00:23.940 --> 00:00:24.990
That's my y-axis.

00:00:24.990 --> 00:00:27.033
And let me draw my x-axis,

00:00:28.050 --> 00:00:29.340
my x-axis.

00:00:29.340 --> 00:00:31.170
And actually I wanna really focus in

00:00:31.170 --> 00:00:32.580
on the points A and D

00:00:32.580 --> 00:00:33.810
because we just need to find

00:00:33.810 --> 00:00:37.170
the length of the side
from point A to point D.

00:00:37.170 --> 00:00:39.930
So point A, lemme just use another color,

00:00:39.930 --> 00:00:43.530
so point A is at

00:00:43.530 --> 00:00:45.660
x equals 7, y is equal to 1.

00:00:45.660 --> 00:00:46.590
So

00:00:46.590 --> 00:00:47.670
1,

00:00:47.670 --> 00:00:48.600
2,

00:00:48.600 --> 00:00:49.470
3,

00:00:49.470 --> 00:00:50.430
4,

00:00:50.430 --> 00:00:51.390
5,

00:00:51.390 --> 00:00:52.440
6,

00:00:52.440 --> 00:00:54.810
7, so that's x equals 7.

00:00:54.810 --> 00:00:56.970
y is equal to 1.

00:00:56.970 --> 00:00:57.870
y is equal to 1,

00:00:57.870 --> 00:00:59.910
so that is point A

00:00:59.910 --> 00:01:02.040
right over there, point A,

00:01:02.040 --> 00:01:03.690
let me label it,

00:01:03.690 --> 00:01:04.740
point A.

00:01:04.740 --> 00:01:06.780
And then where's point D?

00:01:06.780 --> 00:01:09.450
Point D has the exact same,

00:01:09.450 --> 00:01:11.790
has the exact same x-coordinate,

00:01:11.790 --> 00:01:13.860
but its y-coordinate
is a little bit higher.

00:01:13.860 --> 00:01:15.030
Its y-coordinate is 6,

00:01:15.030 --> 00:01:16.740
it's actually 5 higher.

00:01:16.740 --> 00:01:20.100
So point D is that x equals 7,

00:01:20.100 --> 00:01:22.710
y is equal to 6.

00:01:22.710 --> 00:01:24.660
So y is equal to 1,

00:01:24.660 --> 00:01:25.950
2,

00:01:25.950 --> 00:01:27.060
3,

00:01:27.060 --> 00:01:28.170
4,

00:01:28.170 --> 00:01:29.070
5,

00:01:29.070 --> 00:01:30.420
and 6.

00:01:30.420 --> 00:01:32.670
So that's y is equal to 6.

00:01:32.670 --> 00:01:34.680
And so we can draw the point.

00:01:34.680 --> 00:01:37.140
It is going to be right over here.

00:01:37.140 --> 00:01:39.150
This is point D,

00:01:39.150 --> 00:01:41.640
and we can actually connect
them with a line if we like

00:01:41.640 --> 00:01:44.280
to show that this is
a side of a rectangle.

00:01:44.280 --> 00:01:45.540
So let me draw that.

00:01:45.540 --> 00:01:47.700
I can just draw this like this.

00:01:47.700 --> 00:01:48.900
And there you have it.

00:01:48.900 --> 00:01:50.670
I haven't even drawn
the whole rectangle yet,

00:01:50.670 --> 00:01:52.290
but just by plotting these two points,

00:01:52.290 --> 00:01:56.010
we can think about how long side AD is.

00:01:56.010 --> 00:01:59.160
We can see, look, the x doesn't
change going from A to D,

00:01:59.160 --> 00:02:02.430
but we do increase in
the y direction by 5.

00:02:02.430 --> 00:02:06.900
We go from y is equal to 1

00:02:06.900 --> 00:02:08.640
to y is equal to 6.

00:02:08.640 --> 00:02:11.760
So our change in y is equal to 5.

00:02:11.760 --> 00:02:13.650
So what's the length of this line?

00:02:13.650 --> 00:02:14.640
Well, it's gonna be 5.

00:02:14.640 --> 00:02:15.960
It's gonna be 5,

00:02:15.960 --> 00:02:17.550
whatever the units are.

00:02:17.550 --> 00:02:19.470
So that's the length of side AD.

00:02:19.470 --> 00:02:21.180
It's going to be equal to 5.

00:02:21.180 --> 00:02:26.180
We went from the point 7,1 to the point,

00:02:26.880 --> 00:02:29.100
to the point 7,6.

00:02:29.100 --> 00:02:30.627
Now they said that this was a rectangle,

00:02:30.627 --> 00:02:31.800
now just for satisfaction,

00:02:31.800 --> 00:02:33.990
we can draw the entire rectangle.

00:02:33.990 --> 00:02:35.160
We have the point B.

00:02:35.160 --> 00:02:37.410
That is at x equals 5,

00:02:37.410 --> 00:02:38.550
y is equal to 1.

00:02:38.550 --> 00:02:39.750
So let me draw that.

00:02:39.750 --> 00:02:41.460
And I'm just doing this just for fun now.

00:02:41.460 --> 00:02:43.980
At this point, we're
done with the problem.

00:02:43.980 --> 00:02:45.150
So x equals 5,

00:02:45.150 --> 00:02:46.170
y equals 1.

00:02:46.170 --> 00:02:47.373
That's right over here.

00:02:50.580 --> 00:02:51.870
That's point B.

00:02:51.870 --> 00:02:53.250
Let me write B,

00:02:53.250 --> 00:02:55.590
that's the point 5,1.

00:02:55.590 --> 00:02:56.880
And then we have the point C.

00:02:56.880 --> 00:02:58.470
I'll use another color.

00:02:58.470 --> 00:03:01.440
The point C is at x equals 5, y equals 6.

00:03:01.440 --> 00:03:02.910
x equals 5,

00:03:02.910 --> 00:03:04.590
y equals 6.

00:03:04.590 --> 00:03:07.410
Point C is right over here.

00:03:07.410 --> 00:03:12.030
Point C is the point x
equals 5, y equals 6.

00:03:12.030 --> 00:03:13.920
And then we can connect
all the dots if we like

00:03:13.920 --> 00:03:16.980
and clearly see that it is a rectangle.

00:03:16.980 --> 00:03:18.810
So I could connect those,

00:03:18.810 --> 00:03:21.090
I could connect these two,

00:03:21.090 --> 00:03:22.770
and then I could connect these two.

00:03:22.770 --> 00:03:24.810
And then we see that we
indeed have a rectangle.

00:03:24.810 --> 00:03:26.370
But we answered it a while ago

00:03:26.370 --> 00:03:28.290
that the length of side AD

00:03:28.290 --> 00:03:29.550
is equal to 5.

00:03:29.550 --> 00:03:32.430
If we cared about the other
sides, the length of BA,

00:03:32.430 --> 00:03:34.200
well, this is 2,

00:03:34.200 --> 00:03:36.780
a difference of 2 along the x direction,

00:03:36.780 --> 00:03:37.613
the horizontal direction,

00:03:37.613 --> 00:03:39.900
this difference of 2
along the x direction,

00:03:39.900 --> 00:03:42.390
difference of 5 in the y direction,

00:03:42.390 --> 00:03:45.270
we go from y equals 1 to y equals 6.

00:03:45.270 --> 00:03:46.103
So there you have it.

00:03:46.103 --> 00:03:46.950
We are actually able to figure out

00:03:46.950 --> 00:03:49.233
all of the dimensions of this rectangle.


Coordinates of rectangle example

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:02.040
- [Tutor] So I have copy
and pasted this question

00:00:02.040 --> 00:00:03.660
from the exercises on Khan Academy.

00:00:03.660 --> 00:00:05.490
It says, "Sort the following expressions

00:00:05.490 --> 00:00:07.620
from least to greatest
based on the coordinates

00:00:07.620 --> 00:00:09.360
of the rectangle below."

00:00:09.360 --> 00:00:11.250
So we need to figure
out what Y, Z, and X are

00:00:11.250 --> 00:00:12.750
and then we need to sort them.

00:00:15.060 --> 00:00:16.380
Let's look at Y.

00:00:16.380 --> 00:00:17.790
And this is a little bit counterintuitive

00:00:17.790 --> 00:00:19.216
because they're using the variable Y

00:00:19.216 --> 00:00:23.850
to represent the
X-coordinate of this point.

00:00:23.850 --> 00:00:24.840
So let's see if we can figure out

00:00:24.840 --> 00:00:26.850
what the X-coordinate of this point is.

00:00:26.850 --> 00:00:28.020
Well, it's a rectangle.

00:00:28.020 --> 00:00:30.930
The X-coordinate of this
point right over here

00:00:30.930 --> 00:00:34.350
is gonna be the same as the
X-coordinate of this point.

00:00:34.350 --> 00:00:38.070
They sit on the same vertical
line, the way that it's drawn,

00:00:38.070 --> 00:00:41.400
so if this has an
X-coordinate of X equals four,

00:00:41.400 --> 00:00:43.650
then this is going to have
an X-coordinate of four,

00:00:43.650 --> 00:00:46.320
and now we use the variable
Y for the X-coordinate,

00:00:46.320 --> 00:00:47.460
which is a little strange,

00:00:47.460 --> 00:00:49.277
but this is going to be the point (4,7).

00:00:50.850 --> 00:00:53.970
So Y is equal to four.

00:00:53.970 --> 00:00:56.880
Now let's see what we
could figure out about Z.

00:00:56.880 --> 00:01:00.780
Well, Z is the Y-coordinate for,

00:01:00.780 --> 00:01:02.370
let me use another color here.

00:01:02.370 --> 00:01:05.580
Z is the Y-coordinate for this point.

00:01:05.580 --> 00:01:08.160
Now what other point has
the same Y-coordinate?

00:01:08.160 --> 00:01:10.080
Well, it's gonna be at the same level,

00:01:10.080 --> 00:01:12.570
I guess you could say in
the vertical direction.

00:01:12.570 --> 00:01:14.730
It's gonna sit on the
same horizontal line.

00:01:14.730 --> 00:01:17.370
So it's gonna have the same
Y-coordinate as this point.

00:01:17.370 --> 00:01:19.260
Now this point has a
Y-coordinate of three,

00:01:19.260 --> 00:01:21.270
so this point's Y-coordinate
needs to be three

00:01:21.270 --> 00:01:25.140
or we could say that Z is equal to three.

00:01:25.140 --> 00:01:27.300
So this is going to be equal to three.

00:01:27.300 --> 00:01:29.580
And then last but not least,

00:01:29.580 --> 00:01:31.890
we need to figure out what X is.

00:01:31.890 --> 00:01:34.080
And X is the X-coordinate.

00:01:34.080 --> 00:01:35.100
Let me just in another color.

00:01:35.100 --> 00:01:37.830
X is the X-coordinate for this point.

00:01:37.830 --> 00:01:40.110
Well, this X-coordinate
is gonna be the same thing

00:01:40.110 --> 00:01:42.360
as the X-coordinate
for this point up here,

00:01:42.360 --> 00:01:43.620
which is equal to two.

00:01:43.620 --> 00:01:45.300
They sit on the same vertical line,

00:01:45.300 --> 00:01:48.300
they both sit on the line X equals two.

00:01:48.300 --> 00:01:51.630
And so X is going to be two.

00:01:51.630 --> 00:01:54.810
It is the X-coordinate and
it is also the variable X,

00:01:54.810 --> 00:01:56.919
so this is going to be two.

00:01:56.919 --> 00:02:00.540
If we wanna sort them
from least to greatest,

00:02:00.540 --> 00:02:03.690
X is the least, then we would go,

00:02:03.690 --> 00:02:05.820
it's actually the exact
opposite of what we have here.

00:02:05.820 --> 00:02:09.000
It'd be X, then Z, then Y.

00:02:09.000 --> 00:02:10.680
Two, then three, then four.

00:02:10.680 --> 00:02:12.000
So actually lemme get the exercise out

00:02:12.000 --> 00:02:13.470
and just check our answer.

00:02:13.470 --> 00:02:16.200
So X is the least, then Z is in the middle

00:02:16.200 --> 00:02:17.970
and then Y is the largest.

00:02:17.970 --> 00:02:20.270
We could check our answer
and we got it right.


Drawing polygons exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.390 --> 00:00:03.000
- "Click on the graph to
draw an isosceles trapezoid

00:00:03.000 --> 00:00:05.147
"with vertices at negative two comma eight

00:00:05.147 --> 00:00:07.481
"and negative four comma negative one.

00:00:07.481 --> 00:00:09.663
"Another vertex at three comma eight

00:00:09.663 --> 00:00:11.883
"and a base nine units long

00:00:11.883 --> 00:00:14.633
"that is parallel to the x-axis."

00:00:14.633 --> 00:00:16.850
All right, let's see if we can do this.

00:00:16.850 --> 00:00:20.052
So we could just start
plotting these vertices.

00:00:20.052 --> 00:00:21.259
So negative two comma eight.

00:00:21.259 --> 00:00:23.337
X is negative two, y is eight.

00:00:23.337 --> 00:00:25.357
That's that point right over there.

00:00:25.357 --> 00:00:29.355
The next one is x is negative
four, y is negative one.

00:00:29.355 --> 00:00:31.252
x is negative four, y is negative one

00:00:31.252 --> 00:00:31.863
and that's nice.

00:00:31.863 --> 00:00:33.693
They connect the dots for us.

00:00:34.033 --> 00:00:36.564
Then they say three comma eight.

00:00:36.564 --> 00:00:40.102
So three comma eight.

00:00:40.102 --> 00:00:41.809
It's gonna be right up here.

00:00:41.809 --> 00:00:43.202
Now I think when I click it,

00:00:43.202 --> 00:00:45.106
it's going to try to draw a line

00:00:45.106 --> 00:00:46.429
between this point and this point

00:00:46.429 --> 00:00:47.869
which isn't what we want.

00:00:47.869 --> 00:00:50.527
This is a trapezoid,
an isosceles trapezoid.

00:00:50.527 --> 00:00:52.710
I actually want to
connect three comma eight

00:00:52.710 --> 00:00:55.589
to this point right over here
to negative two comma eight.

00:00:55.589 --> 00:00:56.587
So let me see what happens.

00:00:56.587 --> 00:00:58.746
Yep, yeah, that's not
what I wanted to happen.

00:00:58.746 --> 00:01:01.382
So let me actually delete this.

00:01:01.382 --> 00:01:02.101
Let me delete here.

00:01:02.101 --> 00:01:03.866
I'm actually gonna plot this one first.

00:01:03.866 --> 00:01:05.387
Then I'm going to plot this one,

00:01:05.387 --> 00:01:07.581
then I'm going to plot this
one and then I'm going to see

00:01:07.581 --> 00:01:09.009
where I need to put this other one.

00:01:09.009 --> 00:01:10.866
So let me delete all of these.

00:01:10.866 --> 00:01:13.225
We move these all to the trash.

00:01:13.385 --> 00:01:16.439
So move them all to the trash.

00:01:16.439 --> 00:01:18.168
And so let me start over.

00:01:18.168 --> 00:01:19.603
So I'm going to start,

00:01:19.603 --> 00:01:21.326
that looks like some type
of a bug but let's see.

00:01:21.326 --> 00:01:22.963
Let's hopefully it still works.

00:01:22.963 --> 00:01:24.518
Let's plot three comma eight first.

00:01:24.518 --> 00:01:26.968
Three comma eight.

00:01:26.968 --> 00:01:28.230
That's that point.

00:01:28.230 --> 00:01:31.008
Then we're going to plot
negative two comma eight.

00:01:31.008 --> 00:01:32.517
Negative two comma eight.

00:01:32.517 --> 00:01:36.685
This is the top of our
isosceles trapezoid.

00:01:36.685 --> 00:01:38.147
It's nice to give us the length

00:01:38.147 --> 00:01:40.778
that has, the length of this top is five.

00:01:40.778 --> 00:01:44.182
And then, now you can do
negative four comma negative one.

00:01:44.182 --> 00:01:46.613
Negative four comma negative one is

00:01:46.613 --> 00:01:47.599
this point right over here.

00:01:47.599 --> 00:01:49.564
All right, now this is
looking like a trapezoid.

00:01:49.564 --> 00:01:51.713
And now a base nine units long

00:01:51.713 --> 00:01:54.413
that is parallel to the x-axis.

00:01:54.413 --> 00:01:56.561
So base, it's nine units long

00:01:56.561 --> 00:01:57.706
and it's parallel to the x-axis.

00:01:57.706 --> 00:02:00.051
We just have to move to
the right, nine units.

00:02:00.051 --> 00:02:03.172
One, two, three, four,

00:02:03.172 --> 00:02:08.172
five, six, seven, eight, nine units.

00:02:08.173 --> 00:02:10.611
And then we just move back up.

00:02:10.611 --> 00:02:13.409
And it does look like
we have got ourselves

00:02:13.409 --> 00:02:14.930
an isosceles trapezoid.

00:02:14.930 --> 00:02:16.868
What is an isosceles trapezoid mean?

00:02:16.868 --> 00:02:18.656
It means that these two sides,

00:02:18.656 --> 00:02:20.757
the non top or non bottom

00:02:20.757 --> 00:02:23.857
or non base sides are going to be equal.

00:02:23.857 --> 00:02:27.177
And we see that this side
over here is equal to that

00:02:27.177 --> 00:02:28.990
and that this top is,

00:02:28.990 --> 00:02:30.590
you can kind of view, it's in the middle.

00:02:30.590 --> 00:02:31.850
That's one way to think about it.

00:02:31.850 --> 00:02:35.008
So I like this isosceles trapezoid.

00:02:35.008 --> 00:02:36.250
So let's check our answer.

00:02:36.250 --> 00:02:38.235
Make sure that we've drawn it right.

00:02:38.235 --> 00:02:40.171
Yeah, we did good.


Surface area of a box using nets

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.010
- [Instructor] In a previous
video, we figured out how

00:00:02.010 --> 00:00:04.710
to find the surface
area of this cereal box

00:00:04.710 --> 00:00:07.710
by figuring out the areas
of each of the six surfaces

00:00:07.710 --> 00:00:09.810
of the box and then adding them all up.

00:00:09.810 --> 00:00:11.700
I'm gonna do that again in this video,

00:00:11.700 --> 00:00:14.880
but I'm gonna do it by
visualizing a net for the box.

00:00:14.880 --> 00:00:17.049
And the way I think about
a net of a box like this

00:00:17.049 --> 00:00:19.710
is what would happen if you
were to, if you were to,

00:00:19.710 --> 00:00:21.180
if you were to cut the cardboard

00:00:21.180 --> 00:00:22.740
and then flatten it all out.

00:00:22.740 --> 00:00:24.270
So what am I talking about?

00:00:24.270 --> 00:00:26.130
Well, what we have here,

00:00:26.130 --> 00:00:29.010
we could imagine making a cut in the box

00:00:29.010 --> 00:00:33.063
and the cut could be, see,
I could make a cut back here

00:00:33.063 --> 00:00:36.270
so I could make a cut right over there.

00:00:36.270 --> 00:00:37.950
I could cut it.

00:00:37.950 --> 00:00:41.190
I could cut it right like this.

00:00:41.190 --> 00:00:44.040
I could cut it like that.

00:00:44.040 --> 00:00:46.860
So if I just did that, this
top flap would flap open.

00:00:46.860 --> 00:00:49.957
So that would be able
to come out like that.

00:00:49.957 --> 00:00:54.957
And then I could also
make a cut for this side

00:00:56.238 --> 00:00:58.980
so I can make a cut back there.

00:00:58.980 --> 00:01:03.780
And I could make a cut right over here.

00:01:03.780 --> 00:01:06.930
And now this side could flap forward,

00:01:06.930 --> 00:01:07.920
and I could do the same thing

00:01:07.920 --> 00:01:09.210
on this other side right over here.

00:01:09.210 --> 00:01:10.800
Then that could flap forward.

00:01:10.800 --> 00:01:15.030
And then the backside, I could draw it.

00:01:15.030 --> 00:01:17.970
So I would also have a cut,
I'll draw it as a dotted line.

00:01:17.970 --> 00:01:19.620
'Cause you're not be able to,
you're not supposed to be able

00:01:19.620 --> 00:01:22.110
to see this cut, but the corresponding cut

00:01:22.110 --> 00:01:24.390
to this one on this
side that we can't see.

00:01:24.390 --> 00:01:26.340
Lemme draw it a little
bit neater than that.

00:01:26.340 --> 00:01:29.582
The corresponding cut
would be right back here,

00:01:29.582 --> 00:01:31.320
right back there.

00:01:31.320 --> 00:01:32.940
And then a cut right over here.

00:01:32.940 --> 00:01:34.260
And so what would happen if we were

00:01:34.260 --> 00:01:36.300
to flatten all of this out?

00:01:36.300 --> 00:01:39.000
Well, we would have the front of the box.

00:01:39.000 --> 00:01:41.460
I'll try to draw this as neatly as I can.

00:01:41.460 --> 00:01:44.430
So the front of the box looks like this.

00:01:44.430 --> 00:01:49.430
We would have this top
flap, which looks like this.

00:01:49.500 --> 00:01:51.090
If we were to flatten it all out,

00:01:51.090 --> 00:01:52.890
we have these two side flaps.

00:01:52.890 --> 00:01:56.787
So that's a side flap. That's a side flap.

00:01:56.787 --> 00:01:59.760
And this is another, this
side flap right over here.

00:01:59.760 --> 00:02:00.903
That's a side flap.

00:02:01.740 --> 00:02:04.380
Then we would have the bottom of the box.

00:02:04.380 --> 00:02:07.593
So the bottom of the box
is gonna look like this,

00:02:08.520 --> 00:02:09.750
bottom of the box.

00:02:09.750 --> 00:02:11.280
And then we have the back of the box

00:02:11.280 --> 00:02:12.113
that the bottom is going

00:02:12.113 --> 00:02:14.550
to be connected to, we didn't cut that.

00:02:14.550 --> 00:02:16.440
So that we have the back of the box.

00:02:16.440 --> 00:02:20.220
The back of the box looks like this.

00:02:20.220 --> 00:02:21.150
And there we have it.

00:02:21.150 --> 00:02:22.050
We've made the net.

00:02:22.050 --> 00:02:22.920
This is what would happen

00:02:22.920 --> 00:02:24.660
if you made the cuts that I talked about

00:02:24.660 --> 00:02:26.070
and then flatten the box out.

00:02:26.070 --> 00:02:27.420
It would look like this.

00:02:27.420 --> 00:02:28.710
Now how could we use this net

00:02:28.710 --> 00:02:30.840
to find the surface area?

00:02:30.840 --> 00:02:32.487
Well, we just need to
figure out the surface area

00:02:32.487 --> 00:02:34.860
of this shape now.

00:02:34.860 --> 00:02:36.420
So how do we do that?

00:02:36.420 --> 00:02:38.400
Well, we know a lot about the dimensions.

00:02:38.400 --> 00:02:40.200
We know that this width right over here,

00:02:40.200 --> 00:02:41.643
that this is 10 centimeters.

00:02:43.362 --> 00:02:45.583
10 centimeters from there to there.

00:02:45.583 --> 00:02:48.990
We know the height
actually going all the way

00:02:48.990 --> 00:02:50.790
from here all the way up,

00:02:50.790 --> 00:02:53.781
because the height of the
box is 20 centimeters.

00:02:53.781 --> 00:02:57.630
So this is going to be 20
centimeters, right over here,

00:02:57.630 --> 00:02:59.940
then you have another 20 centimeters,

00:02:59.940 --> 00:03:03.580
you have another 20
centimeters right over here

00:03:04.620 --> 00:03:06.450
and right over here if you like.

00:03:06.450 --> 00:03:08.580
And then you have, see the depth

00:03:08.580 --> 00:03:09.900
of the box is three centimeters.

00:03:09.900 --> 00:03:13.200
So this is three centimeters.
Three centimeters.

00:03:13.200 --> 00:03:16.800
And then this is three centimeters.

00:03:16.800 --> 00:03:18.570
And so what is the area, actually,

00:03:18.570 --> 00:03:20.280
let me just do one region first.

00:03:20.280 --> 00:03:24.264
What is the area of this
entire region that I am,

00:03:24.264 --> 00:03:27.533
that I am shading in with this blue color?

00:03:27.533 --> 00:03:29.569
Well, it's 10 centimeters.

00:03:29.569 --> 00:03:31.530
That is, I'll do a color

00:03:31.530 --> 00:03:33.540
that you can see a little bit more easily.

00:03:33.540 --> 00:03:36.660
It is 10 centimeters wide.

00:03:36.660 --> 00:03:40.050
10 centimeters, times, what's the height?

00:03:40.050 --> 00:03:44.490
20 plus three plus 20 plus three.

00:03:44.490 --> 00:03:46.890
So that's going to be 40 plus six.

00:03:46.890 --> 00:03:49.710
So times 46 centimeters.

00:03:49.710 --> 00:03:50.820
That's this blue area.

00:03:50.820 --> 00:03:55.820
So that's gonna be 460 square centimeters,

00:03:56.520 --> 00:03:57.690
460 square centimeters.

00:03:57.690 --> 00:03:59.730
And now we just have
to figure out the area

00:03:59.730 --> 00:04:00.990
of the two flaps.

00:04:00.990 --> 00:04:04.440
So this flap right over
here is 20 centimeters

00:04:04.440 --> 00:04:08.520
by three centimeters, so
that's 60 centimeters squared.

00:04:08.520 --> 00:04:11.610
So 60 centimeters squared,

00:04:11.610 --> 00:04:13.650
or 60 square centimeters I should say.

00:04:13.650 --> 00:04:16.350
And then this flap is gonna
have the exact same area,

00:04:16.350 --> 00:04:19.980
another 60 square centimeters,
60 square centimeters.

00:04:19.980 --> 00:04:22.080
And you add everything together.

00:04:22.080 --> 00:04:24.000
We deserve a little bit of a drum roll.

00:04:24.000 --> 00:04:26.040
We get, well this is gonna add up

00:04:26.040 --> 00:04:28.189
to 580 square centimeters,

00:04:28.189 --> 00:04:30.360
which is the exact thing
we got in the other video

00:04:30.360 --> 00:04:32.400
where we didn't use a net.

00:04:32.400 --> 00:04:34.080
And you should just, it's nice to be able

00:04:34.080 --> 00:04:35.880
to do it either way to be
able to visualize the net

00:04:35.880 --> 00:04:37.691
or be able to look at this

00:04:37.691 --> 00:04:39.090
and think about the different
sides, even the sides

00:04:39.090 --> 00:04:41.613
that you might not necessarily see.


Surface area of a box

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:01.590
- [Instructor] Let's
see if we can figure out

00:00:01.590 --> 00:00:04.530
the surface area of this cereal box.

00:00:04.530 --> 00:00:06.810
And there's a couple of ways to tackle it.

00:00:06.810 --> 00:00:09.060
The first way is, well, let's
figure out the surface area

00:00:09.060 --> 00:00:10.680
of the sides that we can see

00:00:10.680 --> 00:00:12.810
and then think about what
the surface area of the sides

00:00:12.810 --> 00:00:14.940
that we can't see are and
how they might relate.

00:00:14.940 --> 00:00:17.580
And then add them all
together. So let's do that.

00:00:17.580 --> 00:00:20.340
So the front of the box
is 20 centimeters tall

00:00:20.340 --> 00:00:21.870
and 10 centimeters wide.

00:00:21.870 --> 00:00:22.980
It's a rectangle.

00:00:22.980 --> 00:00:24.150
So to figure out its area,

00:00:24.150 --> 00:00:26.970
we can just multiply 20
centimeters times 10 centimeters,

00:00:26.970 --> 00:00:29.354
and that's going to
give us 200 centimeters,

00:00:29.354 --> 00:00:33.480
200 centimeters or 200 square centimeters,

00:00:33.480 --> 00:00:35.400
I should say, 200 square centimeters.

00:00:35.400 --> 00:00:36.911
That's the area of the front.

00:00:36.911 --> 00:00:38.760
And lemme write it over here as well, 200.

00:00:38.760 --> 00:00:40.410
Now we also know there's another side

00:00:40.410 --> 00:00:42.660
that has the exact same area
as the front of the box,

00:00:42.660 --> 00:00:44.150
and that's the back of the box.

00:00:44.150 --> 00:00:47.250
And so let's write another
200 square centimeters

00:00:47.250 --> 00:00:48.527
for the back of the box.

00:00:48.527 --> 00:00:51.346
Now let's figure out the
area of the top of the box.

00:00:51.346 --> 00:00:54.060
The top of the box is, we see it's three.

00:00:54.060 --> 00:00:55.530
The box is three centimeters deep.

00:00:55.530 --> 00:00:58.082
So this right over here
is three centimeters.

00:00:58.082 --> 00:01:00.930
Three, it's three centimeters deep

00:01:00.930 --> 00:01:02.460
and it's 10 centimeters wide.

00:01:02.460 --> 00:01:04.440
We see that the box is
10 centimeters wide.

00:01:04.440 --> 00:01:06.870
So the top of the box is gonna
be three centimeters times 10

00:01:06.870 --> 00:01:10.423
centimeters, which is 30
square centimeters of area.

00:01:10.423 --> 00:01:13.280
So that's the top of the
box. 30 square centimeters.

00:01:13.280 --> 00:01:14.790
Well, the bottom of the box is

00:01:14.790 --> 00:01:16.080
gonna have the exact same area.

00:01:16.080 --> 00:01:17.700
We just can't see it right now.

00:01:17.700 --> 00:01:19.770
So that's gonna be another 30.

00:01:19.770 --> 00:01:20.910
And then we have two more sides

00:01:20.910 --> 00:01:22.530
'cause this box has six sides.

00:01:22.530 --> 00:01:25.743
We have this side panel
that is 20 centimeters tall.

00:01:25.743 --> 00:01:28.380
We see that the height of
the box is 20 centimeters

00:01:28.380 --> 00:01:29.819
and three centimeters deep.

00:01:29.819 --> 00:01:32.781
So three times, three times

00:01:32.781 --> 00:01:35.400
20, 3 times.

00:01:35.400 --> 00:01:37.380
Let me write that a little bit neater.

00:01:37.380 --> 00:01:40.763
Three times 20,

00:01:40.763 --> 00:01:43.407
that's 20 centimeters right there.

00:01:43.407 --> 00:01:45.930
Three centimeters times 20
centimeters is gonna give us

00:01:45.930 --> 00:01:50.930
60 square centimeters,
60 square centimeters.

00:01:51.000 --> 00:01:52.110
Now that's this side panel,

00:01:52.110 --> 00:01:54.210
but there's another side panel
that has the exact same air

00:01:54.210 --> 00:01:56.130
that's on the other side of the box.

00:01:56.130 --> 00:01:57.990
So it's 60 centimeters squared

00:01:57.990 --> 00:01:59.955
or squared centimeters for this side.

00:01:59.955 --> 00:02:01.680
And then another 60 for
the corresponding side

00:02:01.680 --> 00:02:03.480
opposite to it that we can't see.

00:02:03.480 --> 00:02:06.150
And now we can just add
up all of these together.

00:02:06.150 --> 00:02:08.520
And so we get zero.

00:02:08.520 --> 00:02:11.820
Let's see, this is going to be, let's see,

00:02:11.820 --> 00:02:13.560
carry the one or regroup the one.

00:02:13.560 --> 00:02:16.140
It's a 100 and then we have 500.

00:02:16.140 --> 00:02:20.658
So we get 580 square
centimeters is the surface area

00:02:20.658 --> 00:02:23.163
of this box.


Finding surface area using net

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:01.620
- [Instructor] What I
want to do in this video

00:00:01.620 --> 00:00:04.830
is get some practice finding
surface areas of figures

00:00:04.830 --> 00:00:08.040
by opening them up into
what's called nets,

00:00:08.040 --> 00:00:10.410
and one way to think about
it is if you had a figure

00:00:10.410 --> 00:00:12.780
like this, and if it was
made out of cardboard

00:00:12.780 --> 00:00:15.870
and if you were to cut it,
if you were to cut it right

00:00:15.870 --> 00:00:19.440
where I'm drawing this red,
and also right over here

00:00:19.440 --> 00:00:21.510
and right over there and right over there

00:00:21.510 --> 00:00:24.480
and also in the back where
you can't see it just now,

00:00:24.480 --> 00:00:26.640
it would open up into something like this.

00:00:26.640 --> 00:00:28.410
So if you were to open it up,

00:00:28.410 --> 00:00:30.480
it would open up into something like this,

00:00:30.480 --> 00:00:32.730
and when you open it up, it's much easier

00:00:32.730 --> 00:00:34.860
to figure out the surface area.

00:00:34.860 --> 00:00:37.140
So the surface area of this figure,

00:00:37.140 --> 00:00:39.450
when we open it up, we can just
figure out the surface area

00:00:39.450 --> 00:00:41.250
of each of these regions.

00:00:41.250 --> 00:00:42.480
So let's think about it.

00:00:42.480 --> 00:00:45.120
So, what's, first of
all, the surface area?

00:00:45.120 --> 00:00:49.110
What's the surface area
of this right over here?

00:00:49.110 --> 00:00:52.590
Well, in the net that
corresponds to this area.

00:00:52.590 --> 00:00:53.730
It's a triangle.

00:00:53.730 --> 00:00:57.270
It has a base of 12 and a height of eight.

00:00:57.270 --> 00:00:59.680
So this area right over
here is going to be 1/2

00:01:00.660 --> 00:01:05.660
times the base, so times 12,
times the height, times eight.

00:01:06.270 --> 00:01:09.120
So this is the same thing as
six times eight, which is equal

00:01:09.120 --> 00:01:12.630
to 48 whatever units, square units.

00:01:12.630 --> 00:01:14.010
This is gonna be units of area.

00:01:14.010 --> 00:01:15.780
So that's gonna be 48 square units.

00:01:15.780 --> 00:01:18.420
And up here is the exact same thing.

00:01:18.420 --> 00:01:19.440
That's the exact same thing.

00:01:19.440 --> 00:01:20.460
You can't see it in this figure.

00:01:20.460 --> 00:01:22.980
If it was transparent,
if it was transparent,

00:01:22.980 --> 00:01:26.160
it would be this back
side right over here.

00:01:26.160 --> 00:01:30.540
But that's also going to
be 48, 48 square units.

00:01:30.540 --> 00:01:33.390
Now we can think about the areas of,

00:01:33.390 --> 00:01:35.880
I guess you could consider
them to be the side panels.

00:01:35.880 --> 00:01:38.250
So that's a side panel right over there.

00:01:38.250 --> 00:01:40.500
It's 14 high and 10 wide.

00:01:40.500 --> 00:01:42.180
This is the other side panel.

00:01:42.180 --> 00:01:44.760
It's also, this length right over here

00:01:44.760 --> 00:01:45.750
is the same as this length,

00:01:45.750 --> 00:01:48.270
so it's also 14 high and 10 wide.

00:01:48.270 --> 00:01:51.810
So this side panel is
this one right over here.

00:01:51.810 --> 00:01:53.610
And then you have one on the other side.

00:01:53.610 --> 00:01:56.760
And so the area of each
of these, 14 times 10,

00:01:56.760 --> 00:01:59.400
they are 140 square units.

00:01:59.400 --> 00:02:02.670
This one is also 140 square units.

00:02:02.670 --> 00:02:06.000
And then finally we just have
to figure out the area of,

00:02:06.000 --> 00:02:08.130
I guess you could say
the base of this figure.

00:02:08.130 --> 00:02:12.660
So this whole region right
over here, which is this area,

00:02:12.660 --> 00:02:14.670
which is that area right over there.

00:02:14.670 --> 00:02:17.550
And that's going to be 12 by 14.

00:02:17.550 --> 00:02:22.080
So this area is 12 times 14,
which is equal to, let's see,

00:02:22.080 --> 00:02:27.080
12 times 12 is 144, plus
another 24, so it's 168.

00:02:29.520 --> 00:02:33.000
So the total area is
going to be, let's see,

00:02:33.000 --> 00:02:36.990
if you add this one and
that one, you get 96,

00:02:36.990 --> 00:02:38.580
96 square units.

00:02:38.580 --> 00:02:40.080
The two magenta, I guess you could say,

00:02:40.080 --> 00:02:43.710
side panels, 140 plus 140, that's 280.

00:02:43.710 --> 00:02:45.480
280.

00:02:45.480 --> 00:02:49.110
And then you have this
base that comes in at 168.

00:02:49.110 --> 00:02:50.790
Want me do that same color?

00:02:50.790 --> 00:02:53.733
168, 168.

00:02:55.050 --> 00:02:57.120
Add them all together and
we get the surface area

00:02:57.120 --> 00:02:57.960
for the entire figure,

00:02:57.960 --> 00:03:01.950
and it was super valuable
to open it up into the net

00:03:01.950 --> 00:03:03.540
'cause we could make sure
we got all of the sides.

00:03:03.540 --> 00:03:05.520
We didn't have to kind of
rotate it in our brains,

00:03:05.520 --> 00:03:07.110
although you could do that as well.

00:03:07.110 --> 00:03:10.110
So six plus zero plus eight is 14.

00:03:10.110 --> 00:03:13.560
Regroup the one 10 to the tens place.

00:03:13.560 --> 00:03:14.970
So it's now one 10.

00:03:14.970 --> 00:03:16.920
So one plus nine is 10,

00:03:16.920 --> 00:03:21.333
plus eight is 18, plus six is 24.

00:03:22.230 --> 00:03:23.940
And then you have, oops.

00:03:23.940 --> 00:03:26.250
And then you have two
plus two plus one is five.

00:03:26.250 --> 00:03:27.330
So the surface area

00:03:27.330 --> 00:03:32.223
of this figure is 544, 544 square units.


Area of triangles intuition

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:01.650
- [Instructor] We know that we can find

00:00:01.650 --> 00:00:02.880
the area of a rectangle

00:00:02.880 --> 00:00:05.070
by multiplying the base times the height.

00:00:05.070 --> 00:00:08.220
The area of a rectangle is
equal to base times height.

00:00:08.220 --> 00:00:10.950
In another video we saw that
if we're looking at the area

00:00:10.950 --> 00:00:14.370
of a parallelogram and we
also know the length of a base

00:00:14.370 --> 00:00:16.950
and we know its height,
that the area is still going

00:00:16.950 --> 00:00:18.720
to be base times height.

00:00:18.720 --> 00:00:21.180
Now it's not as obvious when
you look at the parallelogram,

00:00:21.180 --> 00:00:25.110
but in that video we did a
little manipulation of the area.

00:00:25.110 --> 00:00:26.670
We said, "Hey, let's
take this little section

00:00:26.670 --> 00:00:28.110
right over here."

00:00:28.110 --> 00:00:31.200
So we took that little
section right over there

00:00:31.200 --> 00:00:35.730
and then we move it over
to the right-hand side.

00:00:35.730 --> 00:00:38.700
And just like that, you see
that as long as the base

00:00:38.700 --> 00:00:41.070
and the height is the same
as this rectangle here,

00:00:41.070 --> 00:00:42.660
I'm able to construct the same rectangle

00:00:42.660 --> 00:00:44.280
by moving that area over.

00:00:44.280 --> 00:00:46.440
And that's why the area
of this parallelogram

00:00:46.440 --> 00:00:47.820
is base times height.

00:00:47.820 --> 00:00:49.560
I didn't add or take away area,

00:00:49.560 --> 00:00:51.960
I just shifted area
from the left-hand side

00:00:51.960 --> 00:00:54.750
to the right-hand side
to show you that the area

00:00:54.750 --> 00:00:56.520
of that parallelogram was the same

00:00:56.520 --> 00:00:58.230
as this area of the rectangle.

00:00:58.230 --> 00:01:01.260
It's still going to be base times height.

00:01:01.260 --> 00:01:03.810
So hopefully that convinces
you, that convinces you

00:01:03.810 --> 00:01:07.590
that the area of a parallelogram
is base times height

00:01:07.590 --> 00:01:10.380
because we're now going to
use that to get the intuition

00:01:10.380 --> 00:01:13.083
for the area of a triangle.

00:01:14.603 --> 00:01:16.680
So let's look at some triangles here.

00:01:16.680 --> 00:01:17.880
So that is a triangle

00:01:17.880 --> 00:01:20.730
and we're given the base and the height.

00:01:20.730 --> 00:01:21.780
And we're gonna try to think about,

00:01:21.780 --> 00:01:23.460
well what's, what's this area?

00:01:23.460 --> 00:01:25.260
What's the area of this
triangle going to be?

00:01:25.260 --> 00:01:26.280
And you can imagine it's going

00:01:26.280 --> 00:01:28.470
to be dependent on base and height.

00:01:28.470 --> 00:01:32.010
Well, to think about that, let me copy

00:01:32.010 --> 00:01:33.780
and paste this triangle.

00:01:33.780 --> 00:01:38.640
So let me copy and then let me paste it.

00:01:38.640 --> 00:01:39.870
And what I'm gonna do is,

00:01:39.870 --> 00:01:41.370
so now I have two of the triangles.

00:01:41.370 --> 00:01:43.230
So this is now going to be twice the area.

00:01:43.230 --> 00:01:45.900
And I'm gonna rotate it around.

00:01:45.900 --> 00:01:49.890
I'm gonna rotate it around like that,

00:01:49.890 --> 00:01:51.960
rotate it around like that,

00:01:51.960 --> 00:01:54.930
and then add it to the original area.

00:01:54.930 --> 00:01:57.960
And you see something very
interesting is happening.

00:01:57.960 --> 00:02:01.170
I have now constructed a parallelogram.

00:02:01.170 --> 00:02:03.270
I have now constructed a parallelogram

00:02:03.270 --> 00:02:06.893
that has twice the area
of our original triangle

00:02:06.893 --> 00:02:08.730
'cause I have two of our original
triangles right over here.

00:02:08.730 --> 00:02:09.810
You saw me do it.

00:02:09.810 --> 00:02:11.040
I copied and pasted it

00:02:11.040 --> 00:02:12.460
and then I flipped it over

00:02:14.006 --> 00:02:15.600
and I constructed the parallelogram.

00:02:15.600 --> 00:02:17.460
Now why is this interesting?

00:02:17.460 --> 00:02:20.610
Well, the area of the
entire parallelogram,

00:02:20.610 --> 00:02:23.490
the area of the entire parallelogram

00:02:23.490 --> 00:02:25.953
is going to be the length of this base,

00:02:27.314 --> 00:02:28.410
base times this height.

00:02:28.410 --> 00:02:29.490
You also have the height written

00:02:29.490 --> 00:02:31.620
on the h upside down over here

00:02:31.620 --> 00:02:33.000
was going to be base times height.

00:02:33.000 --> 00:02:35.820
That's gonna be for the
parallelogram, for the entire,

00:02:35.820 --> 00:02:37.920
lemme draw a parallelogram
right over here.

00:02:37.920 --> 00:02:40.620
That's gonna be the area of
the entire parallelogram.

00:02:40.620 --> 00:02:43.170
So what would be the area
of our original triangle?

00:02:44.370 --> 00:02:48.450
What would be the area
of our original triangle?

00:02:48.450 --> 00:02:51.570
Well, we already saw that this
area of the parallelogram,

00:02:51.570 --> 00:02:53.760
it's twice the area of
our original triangle.

00:02:53.760 --> 00:02:54.780
So our original triangle

00:02:54.780 --> 00:02:56.730
is just going to have half the area.

00:02:56.730 --> 00:02:58.770
So this area right over here

00:02:58.770 --> 00:03:02.520
is going to be one half the
area of the parallelogram,

00:03:02.520 --> 00:03:06.480
one half base, lemme do those same colors.

00:03:06.480 --> 00:03:10.770
One half base times height,

00:03:10.770 --> 00:03:13.732
one half base times height.

00:03:13.732 --> 00:03:16.186
And you might say, "Okay, maybe
it worked for this triangle,

00:03:16.186 --> 00:03:18.180
but I wanna see it work
for more triangles."

00:03:18.180 --> 00:03:19.800
And so to help you there,

00:03:19.800 --> 00:03:21.840
I've added another
triangle right over here.

00:03:21.840 --> 00:03:23.940
You could view this as an obtuse triangle.

00:03:23.940 --> 00:03:26.700
This angle right over here
is greater than 90 degrees.

00:03:26.700 --> 00:03:27.720
But I'm gonna do the same trick.

00:03:27.720 --> 00:03:30.540
We have the base and then
we have the height here.

00:03:30.540 --> 00:03:34.324
You can think of if you start
at this point right over here

00:03:34.324 --> 00:03:36.330
and if you drop a ball,

00:03:36.330 --> 00:03:37.680
the length that the ball goes.

00:03:37.680 --> 00:03:39.480
Or if you had a string here to kinda get

00:03:39.480 --> 00:03:40.950
to the ground level, you could view this

00:03:40.950 --> 00:03:42.600
as the ground level right over there.

00:03:42.600 --> 00:03:44.580
That that's going to be the height.

00:03:44.580 --> 00:03:46.920
It's not sitting in the
triangle like we saw last time,

00:03:46.920 --> 00:03:49.740
but it's still the height of the triangle.

00:03:49.740 --> 00:03:51.337
If this was a building
of some kind, you say,

00:03:51.337 --> 00:03:52.170
"Well, this is the height.

00:03:52.170 --> 00:03:54.210
How far off the ground is it?"

00:03:54.210 --> 00:03:55.770
Well, what's the area of this going to be?

00:03:55.770 --> 00:03:56.603
Well you can imagine,

00:03:56.603 --> 00:03:59.010
it's gonna be one half base times height.

00:03:59.010 --> 00:04:00.930
How do we feel good about that?

00:04:00.930 --> 00:04:05.343
Well, let's do the same magic here.

00:04:06.531 --> 00:04:09.213
So let me copy and paste this.

00:04:10.302 --> 00:04:12.330
So I'm gonna copy and then paste it.

00:04:12.330 --> 00:04:14.460
Whoops, that didn't work.

00:04:14.460 --> 00:04:19.460
Let me, so let me copy and then paste it.

00:04:20.310 --> 00:04:23.130
And so I have two of these triangles now,

00:04:23.130 --> 00:04:25.500
but I'm gonna flip this one over

00:04:25.500 --> 00:04:28.500
so that I can construct a parallelogram.

00:04:28.500 --> 00:04:31.860
So I'm gonna flip it over
and move it over here.

00:04:31.860 --> 00:04:34.200
I'm gonna have to rotate
it a little bit more.

00:04:34.200 --> 00:04:37.680
So I think you get the general idea.

00:04:37.680 --> 00:04:39.720
So now I've constructed a parallelogram

00:04:39.720 --> 00:04:42.663
that has twice the area
of our original triangle.

00:04:45.743 --> 00:04:47.790
It has twice the area of
our original triangle.

00:04:47.790 --> 00:04:49.560
And so if I talked about the area

00:04:49.560 --> 00:04:52.440
of the entire parallelogram, it would be,

00:04:52.440 --> 00:04:57.440
it would be base times the
height of the parallelogram,

00:04:57.570 --> 00:04:59.340
base times the height
of the parallelogram.

00:04:59.340 --> 00:05:02.070
But if we're only talking
about the area of,

00:05:02.070 --> 00:05:04.560
if we're only, if we're only talking

00:05:04.560 --> 00:05:07.290
about this area right over here,

00:05:07.290 --> 00:05:09.030
which is our original triangle,

00:05:09.030 --> 00:05:11.070
it's gonna be half the
area of the parallelogram.

00:05:11.070 --> 00:05:12.840
So it's gonna be one half of that.

00:05:12.840 --> 00:05:15.150
So our area of our original triangle

00:05:15.150 --> 00:05:17.400
is one half base times height.

00:05:17.400 --> 00:05:19.140
So hopefully that makes
you feel pretty good

00:05:19.140 --> 00:05:21.210
about this formula that
you will see in geometry.

00:05:21.210 --> 00:05:23.850
That area of a triangle is
one half base times height

00:05:23.850 --> 00:05:26.520
while the area of a rectangle
or a parallelogram is going

00:05:26.520 --> 00:05:28.233
to be base times height.


Area of parallelograms intuition

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:03.870
- [Narrator] If we have a
rectangle with base length h

00:00:03.870 --> 00:00:07.950
and height length h, we know
how to figure out its area.

00:00:07.950 --> 00:00:12.090
Its area is just going to be the base,

00:00:12.090 --> 00:00:15.510
is going to be the base times the height,

00:00:15.510 --> 00:00:16.680
the base times the height.

00:00:16.680 --> 00:00:19.590
This is just a review of
the area of a rectangle.

00:00:19.590 --> 00:00:22.440
You just multiply the
base times the height.

00:00:22.440 --> 00:00:24.810
Now, let's look at a parallelogram.

00:00:24.810 --> 00:00:28.560
And in this parallelogram,
our base still has length b

00:00:28.560 --> 00:00:31.200
and we still have a height h.

00:00:31.200 --> 00:00:32.400
So when we talk about the height,

00:00:32.400 --> 00:00:34.920
we're not talking about
the length of these sides

00:00:34.920 --> 00:00:37.380
that, at least, the way I've
drawn them, moved diagonally.

00:00:37.380 --> 00:00:38.970
We're talking about if you go from,

00:00:38.970 --> 00:00:41.790
that's from this side up here

00:00:41.790 --> 00:00:43.380
and you were to go straight down,

00:00:43.380 --> 00:00:45.210
if you were to go at a 90 degree angle,

00:00:45.210 --> 00:00:47.610
if you were to go
perpendicularly straight down,

00:00:47.610 --> 00:00:48.570
you get to this side,

00:00:48.570 --> 00:00:52.050
that's going to be, that's
going to be our height.

00:00:52.050 --> 00:00:54.090
So in a situation like this,
when you have a parallelogram,

00:00:54.090 --> 00:00:55.800
you know it's base and its height,

00:00:55.800 --> 00:00:59.610
what do we think its area is going to be?

00:00:59.610 --> 00:01:01.079
So at first, it might
seem, well, you know,

00:01:01.079 --> 00:01:04.650
this isn't as obvious as if
we're dealing with a rectangle,

00:01:04.650 --> 00:01:08.370
but we can do a little visualization
that I think will help.

00:01:08.370 --> 00:01:11.190
So what I'm going to do
is I'm gonna take a chunk

00:01:11.190 --> 00:01:13.320
of area from the left hand side,

00:01:13.320 --> 00:01:15.210
actually this triangle
on the left hand side

00:01:15.210 --> 00:01:17.040
that helps make up the parallelogram

00:01:17.040 --> 00:01:19.230
and then move it to the right

00:01:19.230 --> 00:01:21.723
and then we will see
something somewhat amazing.

00:01:22.590 --> 00:01:24.660
So I'm gonna take this,

00:01:24.660 --> 00:01:26.970
I'm gonna take this
little chunk right there.

00:01:26.970 --> 00:01:28.890
Actually, let me copy it, let
me do it a little bit better.

00:01:28.890 --> 00:01:31.090
So this, I'm gonna take

00:01:32.850 --> 00:01:36.180
that chunk right there and let me cut

00:01:36.180 --> 00:01:39.420
and paste it, so it's still
the same parallelogram,

00:01:39.420 --> 00:01:41.580
but I'm just gonna move
this section of area.

00:01:41.580 --> 00:01:43.290
Remember we're just
thinking about how much,

00:01:43.290 --> 00:01:46.470
how much is space is inside
of the parallelogram.

00:01:46.470 --> 00:01:49.260
And I'm gonna take this
area right over here

00:01:49.260 --> 00:01:52.590
and I'm gonna move it
to the right hand side.

00:01:52.590 --> 00:01:54.420
And what just happened?

00:01:54.420 --> 00:01:55.770
What just happened?

00:01:55.770 --> 00:01:57.600
Let me see if I can move
it a little bit better.

00:01:57.600 --> 00:01:59.583
What just happened when I did that?

00:02:00.630 --> 00:02:05.190
Well, notice it now looks just
like my previous rectangle.

00:02:05.190 --> 00:02:06.900
That just by taking some of the area,

00:02:06.900 --> 00:02:08.130
by taking some of the area on the left

00:02:08.130 --> 00:02:09.420
and moving it to the right,

00:02:09.420 --> 00:02:11.670
I have reconstructed this rectangle.

00:02:11.670 --> 00:02:13.380
So they actually have the same area.

00:02:13.380 --> 00:02:15.180
The area of this parallelogram

00:02:15.180 --> 00:02:16.980
or what used to be the parallelogram

00:02:16.980 --> 00:02:19.140
before I moved that triangle from the left

00:02:19.140 --> 00:02:23.610
to the right is also going to
be the base times the height.

00:02:23.610 --> 00:02:28.590
So the area here is also,
the area here is also

00:02:28.590 --> 00:02:29.820
base times height.

00:02:29.820 --> 00:02:31.560
'Cause once again, I just
took this chunk of area

00:02:31.560 --> 00:02:34.350
that was over there and
I moved it to the right.

00:02:34.350 --> 00:02:36.690
So the area of a parallelogram, the area,

00:02:36.690 --> 00:02:39.840
let me make this look even more
like a parallelogram again.

00:02:39.840 --> 00:02:43.470
The area of a parallelogram
is just going to be,

00:02:43.470 --> 00:02:45.120
if you have the base and the height,

00:02:45.120 --> 00:02:48.630
it's just going to be the
base times the height.

00:02:48.630 --> 00:02:51.630
So the area for both of
these, the area for both

00:02:51.630 --> 00:02:54.573
of these are just base times height.


Graphs of systems of inequalities word problem

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:01.746
- [Voiceover] "Ksenia
wants to chop broccoli

00:00:01.746 --> 00:00:03.471
"and carrots for a competition.

00:00:03.471 --> 00:00:05.294
"It takes her the same number of seconds

00:00:05.294 --> 00:00:08.231
"to chop each carrot, and
it takes her the same number

00:00:08.231 --> 00:00:11.216
"of seconds to chop each broccoli head.

00:00:11.216 --> 00:00:14.430
"Her goal is to chop
at least 20 vegetables

00:00:14.430 --> 00:00:19.430
"with a time limit of
540 seconds," all right.

00:00:20.061 --> 00:00:21.821
"The graph below represents the set

00:00:21.821 --> 00:00:25.814
"of all combinations of
carrots and broccoli.

00:00:25.814 --> 00:00:29.436
"Inequality A," let's see,
"Inequality A represents

00:00:29.436 --> 00:00:33.094
"the range of all combinations
Ksenia wants to chop."

00:00:33.094 --> 00:00:36.363
Because she wants to chop
at least 20 vegetables.

00:00:36.363 --> 00:00:38.466
So, that's what Inequality
A is representing,

00:00:38.466 --> 00:00:40.718
that she wants to chop
at least 20 vegetables.

00:00:40.718 --> 00:00:43.133
So, all this blue shaded
area and even the line,

00:00:43.133 --> 00:00:45.926
is a solid line so it
includes point on the line.

00:00:45.926 --> 00:00:47.169
These are all of the scenarios

00:00:47.169 --> 00:00:49.830
where she's chopping
at least 20 vegetables,

00:00:49.830 --> 00:00:53.697
all this blue area
including the blue line.

00:00:53.697 --> 00:00:55.716
And it says, "Inequality
B represents the range

00:00:55.716 --> 00:00:59.478
of all combinations she can
chop with her time limit."

00:00:59.478 --> 00:01:03.634
So, Inequality B, this is
all of the combinations

00:01:03.634 --> 00:01:05.968
where she is within her time limit,

00:01:05.968 --> 00:01:10.968
where she is not spending
any more than 540 seconds.

00:01:11.519 --> 00:01:13.582
"What is the least number of carrots

00:01:13.582 --> 00:01:17.202
Ksenia can chop while achieving her goal?"

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Well, her goal, remember she wants to chop

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at least 20 vegetables.

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So, you want to be in the blue area.

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You want to be in the solution set

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for Inequality A, which
would be the blue area

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or on the blue line.

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And she wants to achieve her goal

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of meeting the time limit.

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So, she needs to also
be in the solution set

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for Inequality B so she also has to be in

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the green area or on the green line.

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And so the overlap of the two,

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if she's meeting both constraints,

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it's going to be all of this area.

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This is the overlap

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of the two solution sets.

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So, in this overlap where is
the least number of carrots.

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"What is the least
number of carrots Ksenia

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can chop while achieving her goal."

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So, if we see here, the
least number of carrots,

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you might tempted to
say, "Okay, 20 carrots,

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that is in the solution set."

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That would be 20 carrots
and zero broccoli heads

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but you can actually find a combination

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that has even fewer carrots.

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You can go all the way to this point

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because remember the points on the lines

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are also included in the solution sets,

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because they are solid
lines not dash lines.

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So, this point right over here,

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10 carrots and 10 broccoli heads

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actually meets her goal.

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So, let me write that down.

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10 carrots and

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10 broccoli, 10 broccoli heads.

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Let me just write that, 10 broccoli heads.

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So, that's the least amount.

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If you wanted to somehow figure out

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less than 10 carrots, in
any of those scenarios

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there's no overlap.

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You know, if you say,
"Oh, is there any way

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"to do nine carrots?"

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If you look over here there's no overlap

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at c equals nine between
the two solution sets.

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So, the minimum right over here

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is actually the point of intersection

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of these two lines.

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10 carrots, 10 broccoli heads

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that's the combination
that has her chopping

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the minimum number of carrots

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while achieving, frankly, her goals,

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both of her goals.

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Being under time and chopping
at least 20 vegetables.