Comparing fractions visually with pies

WEBVTT
Kind: captions
Language: en

00:00:00.250 --> 00:00:01.570
- What I want to do in this video

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is compare the fraction
7/10 to the fraction 8/9.

00:00:05.850 --> 00:00:07.930
And, like always, I encourage
you to pause the video

00:00:07.930 --> 00:00:10.470
and see if you can figure
out where these things--

00:00:10.470 --> 00:00:12.010
one of these is larger than the other,

00:00:12.010 --> 00:00:14.330
or whether they are equal.

00:00:14.330 --> 00:00:17.570
So, for me in this video, I
want to think about it visually.

00:00:17.570 --> 00:00:21.750
And I'm going to do that
using wholes of the same size

00:00:21.750 --> 00:00:23.390
that are circles.

00:00:23.390 --> 00:00:26.770
So let me draw them or
let me get them, here.

00:00:26.770 --> 00:00:29.250
So there you go, so these
are wholes of the same size.

00:00:29.250 --> 00:00:32.530
I'm gonna compare 7/10 of
this whole of the circle

00:00:32.530 --> 00:00:34.430
to 8/9 or this whole.

00:00:34.430 --> 00:00:36.890
Which is a circle of the exact same size.

00:00:36.890 --> 00:00:40.030
If you're comparing 7/10 of a small circle

00:00:40.030 --> 00:00:43.350
to 8/9 of a bigger circle
or 7/10 of a big circle

00:00:43.350 --> 00:00:46.010
to 8/9 of a smaller circle
or a different shape,

00:00:46.010 --> 00:00:47.770
then you really can't make the comparison.

00:00:47.770 --> 00:00:49.970
But we're gonna compare
7/10 of the same whole

00:00:49.970 --> 00:00:51.950
to 8/9 of the same whole.

00:00:51.950 --> 00:00:54.750
Now, you can see the way
that I've pre-drawn it.

00:00:54.750 --> 00:00:56.750
The circles are the same size,

00:00:56.750 --> 00:01:01.010
but I have divided them into a
different number of sections.

00:01:01.010 --> 00:01:04.710
Here, since I have
10ths, I've divided into,

00:01:04.710 --> 00:01:07.930
you see that I've divided it
into, one, two, three, four,

00:01:07.930 --> 00:01:11.590
five, six, seven, eight,
nine, ten sections.

00:01:11.590 --> 00:01:13.690
Over here, since we're dealing with 9ths,

00:01:13.690 --> 00:01:15.370
you can see I've divided
it into one, two, three,

00:01:15.370 --> 00:01:19.710
four, five, six, seven,
eight, nine sections.

00:01:19.710 --> 00:01:22.570
But let's think about
what 7/10 represents.

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It represents 7 of these 10 sections.

00:01:25.450 --> 00:01:27.590
So let me color them in.

00:01:27.590 --> 00:01:31.270
Let me get my coloring in tool.

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So that represents one, two, three four,

00:01:35.190 --> 00:01:39.850
five, six, seven out of the ten sections.

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Now what about 8/9?

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8/9 is going to represent 8
of these 9 equal sections.

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One, two, three, four, five, six,

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seven, eight of those sections.

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So which one of these is larger?

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Which one is larger?

00:01:57.810 --> 00:01:58.810
Well you can see very clearly,

00:01:58.810 --> 00:02:01.470
remember we're using 8/9 of the same whole

00:02:01.470 --> 00:02:03.830
and 7/10 of that exact same whole.

00:02:03.830 --> 00:02:06.870
You see that we have
colored in more in magenta,

00:02:06.870 --> 00:02:09.330
or this pinkish color
than we have in blue.

00:02:09.330 --> 00:02:11.670
So 8/9 is the larger of these two.

00:02:11.670 --> 00:02:16.010
Or we could say that
7/10 is less than 8/9.

00:02:16.010 --> 00:02:18.890
And once again, the way I
remember what symbol to use,

00:02:18.890 --> 00:02:20.770
we always want it opening to the

00:02:20.770 --> 00:02:22.950
larger of the two number or the little,

00:02:22.950 --> 00:02:25.010
the tip is going to be pointing to the

00:02:25.010 --> 00:02:26.530
smaller of the two number.

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So, 7/10 is less than 8/9.


Comparing fractions on a number line

WEBVTT
Kind: captions
Language: en

00:00:00.170 --> 00:00:03.770
- Let's see if we can compare
the fraction 5/3 to 10/7

00:00:03.770 --> 00:00:05.510
or which-- if we can figure out which one

00:00:05.510 --> 00:00:07.950
of these fractions is larger.

00:00:07.950 --> 00:00:11.170
And you might notice both of
these are larger than a whole.

00:00:11.170 --> 00:00:14.310
A whole would be 3/3, this is 5/3.

00:00:14.310 --> 00:00:17.790
And a whole here would
be 7/7, this is 10/7.

00:00:17.790 --> 00:00:19.550
So which of these is going to be larger?

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And to help us with
that, I'm going to plot

00:00:21.650 --> 00:00:24.170
each of these on a number line

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and I encourage you to pause this video

00:00:25.770 --> 00:00:29.590
and try to do the same
before I work it out.

00:00:29.590 --> 00:00:31.590
Alright, so I have a number line, here.

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We have zero, one, two and, first,

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I divide the number line into thirds.

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You see right over here, this is 1/3,

00:00:40.730 --> 00:00:45.730
this is 2/3, the thirds
are being marked off

00:00:47.110 --> 00:00:48.370
in blue right over here.

00:00:48.370 --> 00:00:51.030
You see that each from
the space from zero to one

00:00:51.030 --> 00:00:54.910
is split into three equal
sections, one, two, three.

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And then the space from one to two

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is split into three equal
sections, one, two, and three,

00:01:01.150 --> 00:01:02.430
you see that right over here.

00:01:02.430 --> 00:01:06.320
So I'm marking off all of the--

00:01:06.320 --> 00:01:08.640
I'm marking off all of the thirds.

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So this is 1/3, this is
2/3, this is 3/3, which is,

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of course, the same thing as one.

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This is 4/3, and then this right over here

00:01:17.180 --> 00:01:18.820
is going to be 5/3.

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And if we were to go over here,

00:01:19.720 --> 00:01:22.380
two would be the same thing as 6/3.

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But what we care about
is 5/3, so that's that.

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Right over there, I don't
want to fill it in so much.

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So 5/3 is that right over there.

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Now let's think about sevenths.

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Now to do sevenths I have
to split the part of the

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number line between zero and one

00:01:40.110 --> 00:01:44.170
or between each whole number
into seven equal spaces.

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So you see that here.

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One, two, three, four, five, six, seven.

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You have seven equal sections.

00:01:49.470 --> 00:01:54.470
So this is 1/7, this is
2/7, 3/7, 4/7, 5/7, 6/7,

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this is 7/7, I could write that down,

00:01:59.370 --> 00:02:02.370
this is, one is the same thing as 7/7,

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8/7, 9/7, 10/7 right over here.

00:02:07.630 --> 00:02:11.530
This, right over here is ten over seven.

00:02:11.530 --> 00:02:15.150
So we see that both 10/7 and 5/3

00:02:15.150 --> 00:02:17.390
are between one and two, but which one

00:02:17.390 --> 00:02:19.970
of these is actually larger?

00:02:19.970 --> 00:02:22.690
Well we see 5/3 is further to the right

00:02:22.690 --> 00:02:25.010
on the number line than 10/7.

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I'm gonna make this a
little bit easier to see.

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So 10/7 and that is right over there.

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So 5/3 is to the right of 10/7,
so 5/3 is greater than 10/7.

00:02:36.150 --> 00:02:37.710
So how do we write the symbol?

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Well we always want to open
it up to the larger number.

00:02:40.270 --> 00:02:42.790
5/3 is the larger number
so we want the larger side

00:02:42.790 --> 00:02:45.150
or the opening on the larger number.

00:02:45.150 --> 00:02:46.840
Or the smaller side, or the point,

00:02:46.840 --> 00:02:48.180
pointing to the smaller number.

00:02:48.180 --> 00:02:52.810
So we have 5/3 is greater than 10/7.


Comparing fractions visually with a bar

WEBVTT
Kind: captions
Language: en

00:00:00.150 --> 00:00:01.650
- [Voiceover] What I
wanna do in this video

00:00:01.650 --> 00:00:05.590
is compare the fractions 3/4

00:00:05.590 --> 00:00:08.220
and 4/5,

00:00:10.050 --> 00:00:12.220
and I wanna do this visually.

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So what I'm gonna do is
I'm gonna have two copies

00:00:14.350 --> 00:00:18.150
of the same whole, so
let me just draw that,

00:00:18.150 --> 00:00:20.720
but I'm gonna divide the first one,

00:00:21.950 --> 00:00:25.190
so this is one whole right
over here, this rectangle,

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when we draw the whole thing.

00:00:27.120 --> 00:00:29.220
So this is a whole,

00:00:29.220 --> 00:00:32.450
and right below that,
we have the same whole.

00:00:32.450 --> 00:00:35.690
We have a rectangle of
exactly the same size.

00:00:39.850 --> 00:00:42.320
Now you might notice
that I've divided them

00:00:42.320 --> 00:00:45.550
into a different number of equal sections.

00:00:45.550 --> 00:00:50.550
In the top one, I've divided
it into four equal sections

00:00:50.620 --> 00:00:54.290
because I am concerned with fourths

00:00:54.290 --> 00:00:58.850
so I've divided this
top whole into fourths

00:00:58.850 --> 00:01:02.382
and I've divided this bottom
whole, or this bottom bar

00:01:02.382 --> 00:01:07.080
or this bottom rectangle, into fifths,

00:01:07.080 --> 00:01:11.600
or five equal sections.

00:01:12.720 --> 00:01:15.890
So let's think about what 3/4 represent.

00:01:15.890 --> 00:01:20.240
So that's gonna be one of
the fourths, right over here,

00:01:22.090 --> 00:01:24.500
two of the fourths,

00:01:26.380 --> 00:01:28.890
and then three of the fourths.

00:01:28.890 --> 00:01:31.350
And what is 4/5 going to be?

00:01:31.350 --> 00:01:35.070
Well, 4/5 is going to be one fifth,

00:01:36.120 --> 00:01:37.660
two fifths,

00:01:38.820 --> 00:01:43.590
three fifths, and four fifths.

00:01:43.590 --> 00:01:46.190
So when you look at
them visually, remember,

00:01:46.190 --> 00:01:48.290
we're taking fractions of the same whole.

00:01:48.290 --> 00:01:50.050
This is 3/4 of that rectangle,

00:01:50.050 --> 00:01:53.120
this is 4/5 of a same-sized rectangle.

00:01:53.120 --> 00:01:54.550
It wouldn't make any
sense if you're doing it

00:01:54.550 --> 00:01:57.190
for different shapes or
different sized rectangles.

00:01:57.190 --> 00:01:58.850
We just divided them
into different sections

00:01:58.850 --> 00:02:01.310
and you see that if you
have four of the fifths,

00:02:01.310 --> 00:02:04.710
that that is going to be more

00:02:04.710 --> 00:02:07.550
than three of the fourths,

00:02:07.550 --> 00:02:11.620
and so 4/5 is greater than 3/4

00:02:11.620 --> 00:02:15.590
or you could say 3/4 is less than 4/5,

00:02:15.590 --> 00:02:17.250
or any way you wanna think about it.

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The symbol you wanna use always
opens to the larger number.

00:02:21.890 --> 00:02:23.750
4/5 is larger than 3/4,

00:02:23.750 --> 00:02:28.690
so the large end of our
symbol is facing the 4/5,

00:02:28.690 --> 00:02:33.140
so we would say 3/4 is less than 4/5.


Equivalent fractions and different wholes

WEBVTT
Kind: captions
Language: en

00:00:00.818 --> 00:00:03.673
- [Voiceover] If I were to
give you the fraction 1/3,

00:00:03.673 --> 00:00:05.090
I could rewrite it.

00:00:05.090 --> 00:00:07.876
If I multiply both the numerator
and the denominator by two,

00:00:07.876 --> 00:00:12.089
one times two is two, and
three times two is six.

00:00:12.089 --> 00:00:15.446
So we know that 1/3 is
the same thing as 2/6.

00:00:15.446 --> 00:00:19.781
This is true, 1/3 is just
another way of saying 2/6.

00:00:19.781 --> 00:00:21.599
Now given that we know that,

00:00:21.599 --> 00:00:24.191
which of these, I guess you could say,

00:00:24.191 --> 00:00:27.844
picture equations actually
show us that that is true?

00:00:27.844 --> 00:00:29.947
Show that to us visually?

00:00:29.947 --> 00:00:31.443
We could start with this first one.

00:00:31.443 --> 00:00:33.046
We have three equal sections,

00:00:33.046 --> 00:00:35.344
so each of these sections are 1/3.

00:00:35.344 --> 00:00:38.665
And we have one of those 1/3 shaded in.

00:00:38.665 --> 00:00:41.822
If we go on the right side,
we have six equal sections,

00:00:41.822 --> 00:00:46.373
so it's divided into 1/6, and
we have two of them shaded in.

00:00:46.373 --> 00:00:48.347
One, two.

00:00:48.347 --> 00:00:52.178
And the important thing
to realize, is over here,

00:00:52.178 --> 00:00:54.593
we divided the same rectangle to the 1/3

00:00:54.593 --> 00:00:56.579
as we divided into 1/6 over here.

00:00:56.579 --> 00:01:00.224
We are comparing fractions
of the same whole.

00:01:00.224 --> 00:01:03.963
So you see that if you were to
take these two magenta boxes

00:01:03.963 --> 00:01:07.327
and put them together, they
actually do have the same area

00:01:07.327 --> 00:01:11.973
as this one larger rectangle.

00:01:11.973 --> 00:01:15.820
So this one looks like a
good picture explanation

00:01:15.820 --> 00:01:19.930
or a visual explanation for
why 1/3 is equal to 2/6.

00:01:19.930 --> 00:01:23.390
You take the same rectangle,
divide it into 1/3

00:01:23.390 --> 00:01:26.640
and take one of them, and
then you take that same exact

00:01:26.640 --> 00:01:29.311
rectangle, divide it into
1/6 and pick two of them,

00:01:29.311 --> 00:01:31.119
you have shaded in the same amount.

00:01:31.119 --> 00:01:33.597
1/3 over here is the same thing

00:01:33.597 --> 00:01:36.357
as 2/6 over here.

00:01:36.357 --> 00:01:39.388
The important thing to
realize is that you're taking

00:01:39.388 --> 00:01:43.196
1/3 and 2/6 of the exact same rectangle.

00:01:43.196 --> 00:01:45.495
When you look over at these figures,

00:01:45.495 --> 00:01:49.442
you see that this is 1/3 of
this figure right over here.

00:01:49.442 --> 00:01:51.470
It's split into three equal sections,

00:01:51.470 --> 00:01:53.536
and this is one of them that's shaded in.

00:01:53.536 --> 00:01:58.293
And this is, right over here, 2/6.

00:01:58.293 --> 00:02:02.415
You have six equal sections and
we've shaded in two of them.

00:02:03.445 --> 00:02:05.518
And we know that 1/3 is equal to 2/6,

00:02:05.518 --> 00:02:08.025
but this picture isn't true.

00:02:08.025 --> 00:02:10.440
This shaded area right over here

00:02:10.440 --> 00:02:13.412
is less than this shaded area over here.

00:02:13.412 --> 00:02:15.154
So you say, "Why doesn't it work?

00:02:15.154 --> 00:02:18.381
"Why isn't this 1/3
the same thing as 2/6?"

00:02:18.381 --> 00:02:22.700
Because you're taking 2/6
of a larger hexagon here.

00:02:22.700 --> 00:02:25.161
Here, you're taking 1/3
of a smaller figure.

00:02:25.161 --> 00:02:27.785
Here, you're taking
2/6 of a larger figure.

00:02:27.785 --> 00:02:31.569
You can't compare fractions
of different wholes.

00:02:31.569 --> 00:02:35.192
2/6 of a larger whole
is going to be larger

00:02:35.192 --> 00:02:37.281
than 1/3 of a smaller whole.

00:02:37.281 --> 00:02:40.363
The only time that 1/3 and
2/6 are going to be equivalent

00:02:40.363 --> 00:02:44.131
is if they are of the same
whole, like we saw up here.

00:02:44.131 --> 00:02:46.314
So this is not true.

00:02:46.314 --> 00:02:50.400
And for the same reason,
this right over here is 1/3

00:02:50.400 --> 00:02:52.629
and this over here is 2/6.

00:02:52.629 --> 00:02:57.064
This is 1/3 of a smaller
circle than this 2/6 is.

00:02:57.064 --> 00:02:58.945
So this is not true.

00:02:58.945 --> 00:03:03.078
By that same logic, we have
these other pictures over here.

00:03:03.078 --> 00:03:05.732
And over here, we're not
talking about 1/3 and 2/6.

00:03:05.732 --> 00:03:07.907
Here, we're talking about,
let's see, splitting something

00:03:07.907 --> 00:03:11.227
into one, two, three, four,
five, six, seven, eight.

00:03:11.227 --> 00:03:12.992
So we're talking about
1/8, and in this case,

00:03:12.992 --> 00:03:16.638
it's one, two, three,
four, five, six, 6/8,

00:03:16.638 --> 00:03:18.216
and we're comparing it,
let's see over here,

00:03:18.216 --> 00:03:21.277
they're dividing into four
and they've shaded in three.

00:03:21.277 --> 00:03:25.453
Now it is true that 6/8 is equal to 3/4.

00:03:25.453 --> 00:03:27.636
Six divided by two is three.

00:03:27.636 --> 00:03:30.213
Eight divided by two is four.

00:03:30.213 --> 00:03:33.951
So if you multiply or divide
the numerator by the exact

00:03:33.951 --> 00:03:38.107
same thing, then you're going
to get an equivalent fraction.

00:03:38.107 --> 00:03:42.565
Now 6/8 is equal to 3/4,
but this picture isn't true

00:03:42.565 --> 00:03:44.794
because 6/8 would be equal to 3/4

00:03:44.794 --> 00:03:47.616
if you're talking about taking 6/8 and 3/4

00:03:47.616 --> 00:03:50.459
of the same size, in this case, diamond.

00:03:50.459 --> 00:03:52.247
These things are not the same size,

00:03:52.247 --> 00:03:54.523
so you can't make this picture statement.

00:03:54.523 --> 00:03:56.404
Same thing for this one over here.

00:03:56.404 --> 00:03:58.099
These circles are of different sizes.

00:03:58.099 --> 00:04:01.674
So taking the same fraction
of different sizes,

00:04:01.674 --> 00:04:04.049
you can't say that
they're going to be equal.

00:04:04.049 --> 00:04:06.225
This last one, you're taking

00:04:06.225 --> 00:04:08.222
the fractions of the same whole.

00:04:08.222 --> 00:04:12.401
It's kind of this weird
left-pointing arrow-looking shape,

00:04:12.401 --> 00:04:14.282
but they are the same shape.

00:04:14.282 --> 00:04:17.533
And you see here, we've
divided that shape into 1/8,

00:04:17.533 --> 00:04:19.638
and we have shaded in six of them.

00:04:19.638 --> 00:04:22.786
One, two, let me do that in
a yellow so you can see it.

00:04:22.786 --> 00:04:26.070
One, two, three, four, five, six.

00:04:26.070 --> 00:04:30.691
Over here, we have divided it into 1/4,

00:04:30.691 --> 00:04:34.506
and we have shaded in three of them.

00:04:35.576 --> 00:04:37.197
So this statement is true.

00:04:37.197 --> 00:04:41.585
We already knew that 6/8
and 3/4 are the same thing,

00:04:41.585 --> 00:04:46.114
but this picture right over
here visually shows us that,

00:04:46.114 --> 00:04:48.485
because we're taking 6/8 of the same whole

00:04:48.485 --> 00:04:50.937
as we're taking 3/4 of
it, and you see that here.

00:04:50.937 --> 00:04:55.327
This area, let me shade
it in a different color.

00:04:55.327 --> 00:04:57.997
So the area that I'm shading in,

00:04:57.997 --> 00:05:02.218
six of these triangles

00:05:02.218 --> 00:05:06.757
with equal area is the same thing as three

00:05:06.757 --> 00:05:09.516
of these parallelograms,
or three of the four.

00:05:09.516 --> 00:05:12.154
These have the same
area, you could see that.

00:05:12.154 --> 00:05:16.618
One of these, like that
one right over there,

00:05:16.618 --> 00:05:20.611
could be equivalent to this over here,

00:05:20.611 --> 00:05:23.112
equivalent to that over there.

00:05:23.112 --> 00:05:24.880
Actually, let me do that for all of them.

00:05:24.880 --> 00:05:28.235
This one over here

00:05:28.235 --> 00:05:30.990
could be equivalent to, let's see,

00:05:30.990 --> 00:05:33.219
if you were to flip it
over, it would be equivalent

00:05:33.219 --> 00:05:35.727
to that over there.

00:05:35.727 --> 00:05:39.465
And finally, I think you
see where this is going,

00:05:39.465 --> 00:05:43.734
this one over here is equivalent

00:05:43.734 --> 00:05:45.820
to this one over here.

00:05:45.820 --> 00:05:48.606
So we have the same fraction shaded in.

00:05:48.606 --> 00:05:51.918
We just divided the one on the
left into more equal pieces

00:05:51.918 --> 00:05:53.928
than we did on the right but
these are equivalent fractions.

00:05:53.928 --> 00:05:58.085
And this picture shows us that
6/8 is, indeed, equal to 3/4.

00:05:58.085 --> 00:06:00.453
Once again, 6/8 of the same whole

00:06:00.453 --> 00:06:04.159
is equal to 3/4 of that same whole.


More on equivalent fractions

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.800
What I want to do in this
video is really digest the idea

00:00:03.800 --> 00:00:07.140
that if we have some fraction,
as long as we multiply

00:00:07.140 --> 00:00:09.740
the numerator and the
denominator of the fraction

00:00:09.740 --> 00:00:12.020
by the same number,
then we're going

00:00:12.020 --> 00:00:14.390
to have an equivalent fraction.

00:00:14.390 --> 00:00:15.850
So let's think about that.

00:00:15.850 --> 00:00:18.625
Let's say we multiply the
denominator here by 2.

00:00:22.490 --> 00:00:25.950
I'm claiming that as long as
we multiply the numerator by 2,

00:00:25.950 --> 00:00:28.260
we are going to get an
equivalent fraction.

00:00:28.260 --> 00:00:30.830
So here, the denominator was 6.

00:00:30.830 --> 00:00:33.105
So here, our
denominator will be 12.

00:00:33.105 --> 00:00:34.990
If our numerator here
is 4, well, we've

00:00:34.990 --> 00:00:39.320
got to multiply by 2 again,
multiply our numerator by 2,

00:00:39.320 --> 00:00:40.430
to get 8.

00:00:40.430 --> 00:00:45.990
So I'm claiming that 8/12
is the same fraction as 4/6.

00:00:45.990 --> 00:00:52.210
And to visualize that,
let me redraw this whole.

00:00:52.210 --> 00:00:55.160
But instead of having
6 equal sections,

00:00:55.160 --> 00:00:58.360
we now have 12 equal sections.

00:00:58.360 --> 00:01:02.580
So each of the six
we can turn into 2.

00:01:02.580 --> 00:01:05.670
That's essentially what
multiplying by 2 does.

00:01:05.670 --> 00:01:09.010
We now have twice as
many equal sections.

00:01:09.010 --> 00:01:11.593
Now that we have twice as many
equal sections-- literally one,

00:01:11.593 --> 00:01:15.860
two, three, four, five, six,
seven, eight, nine, 10, 11 12--

00:01:15.860 --> 00:01:18.370
how many of them are
actually shaded in yellow?

00:01:18.370 --> 00:01:22.760
Well, one, two three, four
five, six, seven, eight-- 8/12.

00:01:22.760 --> 00:01:24.220
And there's no magic here.

00:01:24.220 --> 00:01:27.540
If we have twice
as many sections,

00:01:27.540 --> 00:01:29.890
we're going to have to shade
in twice as many of them

00:01:29.890 --> 00:01:33.660
in order to have the same
fraction of the whole.

00:01:33.660 --> 00:01:35.080
And it goes the other way, too.

00:01:35.080 --> 00:01:37.140
This isn't just true
with multiplication.

00:01:37.140 --> 00:01:40.020
It's also true that if
we divide the numerator

00:01:40.020 --> 00:01:42.470
and the denominator
by the same quantity,

00:01:42.470 --> 00:01:44.967
we are going to have
an equivalent fraction.

00:01:44.967 --> 00:01:46.550
So that's another
way of saying, well,

00:01:46.550 --> 00:01:48.910
what happens if I
were to divide by 2?

00:01:48.910 --> 00:01:53.470
So if I were to divide by
2-- so let me divide by 2--

00:01:53.470 --> 00:01:57.400
I'm going to have 1/2 the
number of equal sections.

00:01:57.400 --> 00:02:00.500
Or I will only have
three equal sections.

00:02:00.500 --> 00:02:03.360
And I'm claiming if I do the
same thing in the numerator,

00:02:03.360 --> 00:02:06.560
that this is going to
represent the same fraction.

00:02:06.560 --> 00:02:09.289
So 4 divided by 2 is 2.

00:02:09.289 --> 00:02:12.940
So I'm claiming that
2/3 is the same fraction

00:02:12.940 --> 00:02:16.060
as 4/6 is the same
fraction as 8/12.

00:02:16.060 --> 00:02:18.180
Well, let's visualize that.

00:02:18.180 --> 00:02:20.130
So here, this is
6 equal sections.

00:02:20.130 --> 00:02:23.000
But now, we're going to have
only three equal sections.

00:02:23.000 --> 00:02:26.600
So we can merge some of
these equal sections.

00:02:26.600 --> 00:02:29.890
So we can merge these
two right over here.

00:02:29.890 --> 00:02:33.760
And we can merge these
two right over here.

00:02:33.760 --> 00:02:39.150
And then, we can merge
these two right over here.

00:02:39.150 --> 00:02:42.310
So our whole is
still the same whole.

00:02:42.310 --> 00:02:45.060
But now, we only have
three equal sections.

00:02:45.060 --> 00:02:47.850
And two of them are
actually shaded in.

00:02:47.850 --> 00:02:49.990
So these are all
equivalent fractions.

00:02:49.990 --> 00:02:53.450
So the big takeaway here
is start with a fraction.

00:02:53.450 --> 00:02:58.190
If you multiply the
numerator and the denominator

00:02:58.190 --> 00:02:59.660
by the same quantity,
you're going

00:02:59.660 --> 00:03:01.150
to have an equivalent fraction.

00:03:01.150 --> 00:03:03.410
If you divide the numerator
and the denominator

00:03:03.410 --> 00:03:05.500
by the same
quantity, you're also

00:03:05.500 --> 00:03:07.730
going to have an
equivalent fraction.

00:03:07.730 --> 00:03:10.810
So with that in our brains,
let's tackle a little bit

00:03:10.810 --> 00:03:13.420
of an equivalent
fractions problem.

00:03:13.420 --> 00:03:20.300
Let's think about-- if
someone says, OK, I have 5/25,

00:03:20.300 --> 00:03:28.740
and I want to write that as some
value, let's call that value t,

00:03:28.740 --> 00:03:35.410
over 100, what would t be?

00:03:35.410 --> 00:03:37.560
Well, we can see
in the denominator

00:03:37.560 --> 00:03:42.387
to go from 25 to 100,
you had to multiply by 4.

00:03:42.387 --> 00:03:43.970
So if you want an
equivalent fraction,

00:03:43.970 --> 00:03:47.830
you have to multiply the
numerator by 4 as well.

00:03:47.830 --> 00:03:51.110
So t will need to
be equal to 20.

00:03:51.110 --> 00:03:53.130
So t is equal to 20.

00:03:53.130 --> 00:04:03.480
5/25 is the same
thing as 20/100.

00:04:03.480 --> 00:04:10.250
But what if someone says, well,
5/25 is equivalent to blank,

00:04:10.250 --> 00:04:13.860
let's say question mark, over 5?

00:04:13.860 --> 00:04:15.020
Well now what would you do?

00:04:15.020 --> 00:04:16.561
Actually, let's do
it the other way--

00:04:16.561 --> 00:04:21.870
is equal to 1 over
question mark.

00:04:21.870 --> 00:04:25.460
Well, you could say, look, to
get our numerator from 5 to 1,

00:04:25.460 --> 00:04:27.830
we have to divide by 5.

00:04:27.830 --> 00:04:31.530
We have to divide by
5 to go from 5 to 1.

00:04:31.530 --> 00:04:35.970
And so similarly, we have to
divide the denominator by 5.

00:04:35.970 --> 00:04:40.220
So if you divide the denominator
by 5, 25 divided by 5

00:04:40.220 --> 00:04:43.390
is going to get you 5.

00:04:46.540 --> 00:04:48.530
So these are all
equivalent fractions.

00:04:48.530 --> 00:04:52.604
1/5 is equivalent to 5/25,
which is equal to 20/100.


Intro to equivalent fractions

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:02.950
So I've got a whole
pizza here, and let's say

00:00:02.950 --> 00:00:05.590
that I were to cut it
into two equal pieces.

00:00:05.590 --> 00:00:11.040
Let me cut it right over
here into 2 equal pieces.

00:00:11.040 --> 00:00:14.940
And let's say that I ate
one of those 2 equal pieces.

00:00:14.940 --> 00:00:17.840
So let's say I ate all
of this right over here.

00:00:17.840 --> 00:00:20.910
What fraction of the
pizza have I eaten?

00:00:20.910 --> 00:00:23.360
Well, I took the
whole and I divide it

00:00:23.360 --> 00:00:29.430
into two equal pieces, and
then I ate one of those pieces.

00:00:33.660 --> 00:00:37.630
So I ate 1/2 of the pizza.

00:00:37.630 --> 00:00:43.150
Now, let's imagine that instead
of cutting that pizza into only

00:00:43.150 --> 00:00:45.730
2 equal pieces,
let's imagine instead

00:00:45.730 --> 00:00:48.590
that decide to cut it
into 4 equal pieces.

00:00:48.590 --> 00:00:50.290
So let's draw that.

00:00:50.290 --> 00:00:52.820
So 4 equal pieces.

00:00:52.820 --> 00:00:56.520
So I could cut once
this way and then

00:00:56.520 --> 00:00:59.950
I could cut it once this way.

00:00:59.950 --> 00:01:02.860
And so here I have
4 equal pieces.

00:01:05.440 --> 00:01:09.060
But let's say that I want to
eat the same amount of pizza.

00:01:09.060 --> 00:01:12.180
How many of these 4 equal
pieces would I have to eat.

00:01:12.180 --> 00:01:15.520
I encourage you to pause the
video and think about that.

00:01:15.520 --> 00:01:18.110
Well, I would eat this piece.

00:01:18.110 --> 00:01:20.770
You could imagine me eating
this piece and this piece right

00:01:20.770 --> 00:01:21.560
over here.

00:01:21.560 --> 00:01:25.140
I've eaten the same
amount of the pizza.

00:01:25.140 --> 00:01:28.840
Each of these pieces you could
imagine got cut into 2 pieces

00:01:28.840 --> 00:01:31.010
when I cut the whole
pizza this way.

00:01:31.010 --> 00:01:34.350
And so now I have to
eat 2 slices of the 4,

00:01:34.350 --> 00:01:36.850
as opposed to 1 slice of the 2.

00:01:36.850 --> 00:01:40.250
So I just ate 2 out
of the 4 slices.

00:01:42.836 --> 00:01:44.210
I'm using different
numbers here.

00:01:44.210 --> 00:01:47.014
Here I'm using a
1 in the numerator

00:01:47.014 --> 00:01:48.055
and 2 in the denominator.

00:01:48.055 --> 00:01:49.554
Here, I'm using a
2 in the numerator

00:01:49.554 --> 00:01:51.340
and a 4 in the denominator.

00:01:51.340 --> 00:01:54.800
These two fractions
represent the same quantity.

00:01:54.800 --> 00:01:56.410
I ate the same amount of pizza.

00:01:56.410 --> 00:02:00.530
If I eat 2/4 of a pizza, if I
eat 2 out of 4 equal pieces,

00:02:00.530 --> 00:02:02.590
that's the same
fraction of the pizza

00:02:02.590 --> 00:02:05.780
as if I eat 1 out
of 2 equal pieces.

00:02:05.780 --> 00:02:07.930
So we would say that
these two things

00:02:07.930 --> 00:02:10.930
are equivalent fractions.

00:02:10.930 --> 00:02:12.960
Now let's do another
one like this.

00:02:12.960 --> 00:02:15.190
Instead of just dividing
it into 4 equal pieces,

00:02:15.190 --> 00:02:17.735
let's divide it
into 8 equal pieces.

00:02:23.040 --> 00:02:26.780
So now we could
cut once like this.

00:02:26.780 --> 00:02:29.040
So now we have 2 equal pieces.

00:02:29.040 --> 00:02:30.880
Cut once like this.

00:02:30.880 --> 00:02:32.510
Now we have 4 equal pieces.

00:02:32.510 --> 00:02:35.300
And then divide each of
those 4 pieces into 2 pieces.

00:02:35.300 --> 00:02:39.320
So I'll cut those
in-- So let's see.

00:02:39.320 --> 00:02:40.695
I want to make
them equal pieces.

00:02:40.695 --> 00:02:43.410
Those don't look as
equal as I would like.

00:02:43.410 --> 00:02:51.480
So that looks more equal, and
that looks reasonably equal.

00:02:51.480 --> 00:02:53.500
So now how many equal
pieces do I have?

00:02:53.500 --> 00:02:55.090
I have 8 equal pieces.

00:02:57.880 --> 00:03:01.180
But let's say I wanted to eat
the same fraction of the pizza.

00:03:01.180 --> 00:03:04.410
So I could eat all of these
pieces right over here.

00:03:04.410 --> 00:03:07.090
Well, how many of those 8
equal pieces have I eaten?

00:03:07.090 --> 00:03:11.110
Well, I've eaten 1, 2, 3,
4 of those 8 equal pieces.

00:03:11.110 --> 00:03:15.400
And so once again, this
fraction, 4 of 8, or 4/8,

00:03:15.400 --> 00:03:19.230
is equivalent to 2/4,
which is equivalent to 1/2.

00:03:19.230 --> 00:03:21.780
And you might see a little
bit of a pattern here.

00:03:21.780 --> 00:03:26.580
Going from this scenario
to this scenario,

00:03:26.580 --> 00:03:29.400
I got twice as
many equal slices.

00:03:29.400 --> 00:03:32.060
Because I had twice
as many equal slices,

00:03:32.060 --> 00:03:35.810
I needed to eat two times
the number of slices.

00:03:35.810 --> 00:03:38.970
So I multiply the
denominator by 2,

00:03:38.970 --> 00:03:41.370
and I multiply the
numerator by 2.

00:03:41.370 --> 00:03:43.820
If I multiply the numerator
and the denominator

00:03:43.820 --> 00:03:46.510
by the same number,
then I'm not changing

00:03:46.510 --> 00:03:48.860
what that fraction represents.

00:03:48.860 --> 00:03:50.640
And you see that over here.

00:03:50.640 --> 00:03:54.170
Going from 4 slices to 8
slices, I cut every slice,

00:03:54.170 --> 00:03:57.140
I turned every slice
into 2 more slices,

00:03:57.140 --> 00:03:59.401
so I had twice as many slices.

00:03:59.401 --> 00:04:01.150
And then if I want to
eat the same amount,

00:04:01.150 --> 00:04:07.020
I have to eat twice
as many pieces.

00:04:07.020 --> 00:04:10.590
So all of these, 1/2, 2/4, four
4/8, and I could keep going.

00:04:10.590 --> 00:04:12.520
I could do 8/16.

00:04:12.520 --> 00:04:14.170
I could do 16/32.

00:04:14.170 --> 00:04:17.262
All of these would be
equivalent fractions.


Dividing numbers: long division example

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.500
Let's tackle a slightly more
interesting division problem.

00:00:03.500 --> 00:00:06.606
So we're going to take
280 and divide it by 5.

00:00:06.606 --> 00:00:08.230
And I encourage you
to pause this video

00:00:08.230 --> 00:00:09.760
and try to apply the
technique that we've

00:00:09.760 --> 00:00:11.634
seen in a previous video
to try to figure out

00:00:11.634 --> 00:00:14.630
what 280 divided by 5 is.

00:00:14.630 --> 00:00:15.890
So let's have our go at it.

00:00:15.890 --> 00:00:16.970
So we could rewrite this.

00:00:16.970 --> 00:00:24.390
We've already seen this
as 280 divided by 5.

00:00:24.390 --> 00:00:26.200
So the first question
we can ask ourselves

00:00:26.200 --> 00:00:28.300
is, how many times
does 5 go into 2?

00:00:28.300 --> 00:00:30.360
Well, 5 doesn't
go into 2 at all.

00:00:30.360 --> 00:00:31.845
It goes 0 times.

00:00:31.845 --> 00:00:35.590
0 times 5 is 0.

00:00:35.590 --> 00:00:36.940
And then we subtract.

00:00:36.940 --> 00:00:39.220
2 minus 0 is 2.

00:00:39.220 --> 00:00:41.390
And now, we can bring
down the next digit,

00:00:41.390 --> 00:00:44.139
which, in this case, is an 8.

00:00:44.139 --> 00:00:45.930
And so we can ask
ourselves, how many times

00:00:45.930 --> 00:00:49.570
does 5 go into 28
without going over it?

00:00:49.570 --> 00:00:54.450
So we could think, well,
let's see 5 times 5 is 25.

00:00:54.450 --> 00:00:58.770
5 times 6 is 30.

00:00:58.770 --> 00:01:01.150
So 5 times 6 would go over 28.

00:01:01.150 --> 00:01:03.760
So we want to go
back to 5 times 5.

00:01:03.760 --> 00:01:06.630
So 5 goes into 28 5 times.

00:01:06.630 --> 00:01:09.280
5 times 5 is 25.

00:01:09.280 --> 00:01:11.340
And you subtract.

00:01:11.340 --> 00:01:13.490
28 minus 25 is 3.

00:01:13.490 --> 00:01:15.710
And now, let's bring
down the next digit.

00:01:15.710 --> 00:01:18.360
In this case, the digit is a 0.

00:01:18.360 --> 00:01:20.290
So how many times
does 5 go into 30?

00:01:20.290 --> 00:01:21.830
Well, it goes exactly 6 times.

00:01:21.830 --> 00:01:23.520
We've already talked about that.

00:01:23.520 --> 00:01:26.110
6 times 5 is 30.

00:01:26.110 --> 00:01:27.350
We subtract.

00:01:27.350 --> 00:01:30.260
And we have nothing left over.

00:01:30.260 --> 00:01:37.830
So we just figured out that 280
divided by 5 is equal to 56.

00:01:37.830 --> 00:01:39.989
Now, why did this work?

00:01:39.989 --> 00:01:41.780
Well, the first way to
think about it, this

00:01:41.780 --> 00:01:42.606
was actually 200.

00:01:42.606 --> 00:01:43.980
And we were thinking
about, do we

00:01:43.980 --> 00:01:46.080
put anything in the
hundreds place here?

00:01:46.080 --> 00:01:49.020
So one way of thinking about
it, how many hundreds of times

00:01:49.020 --> 00:01:52.175
does 5 go into 200?

00:01:52.175 --> 00:01:53.800
And it doesn't go
any multiples of 100.

00:01:53.800 --> 00:01:54.930
And if that's a
little confusing,

00:01:54.930 --> 00:01:56.140
don't worry too much about it.

00:01:56.140 --> 00:01:57.681
But I want you to
really think deeply

00:01:57.681 --> 00:01:59.190
about the place notation.

00:01:59.190 --> 00:02:00.860
But then we were able
to, again, extend

00:02:00.860 --> 00:02:04.170
it to instead of just
200 but then 280.

00:02:04.170 --> 00:02:06.150
This 28 right over here,
if you think about it,

00:02:06.150 --> 00:02:08.020
this 2 is in the hundreds place.

00:02:08.020 --> 00:02:09.710
This 8 is in the tens place.

00:02:09.710 --> 00:02:12.030
So this is really
representing 280.

00:02:12.030 --> 00:02:15.340
And we said, how many tens
of times does 5 go into 280?

00:02:15.340 --> 00:02:19.220
Well, it goes 5 tens
times, or it goes 50 times.

00:02:19.220 --> 00:02:22.220
50 times 5 is 250.

00:02:22.220 --> 00:02:26.340
When you subtract 250
from 280, you've got 30.

00:02:26.340 --> 00:02:28.590
And there were no other
things in the ones place here.

00:02:28.590 --> 00:02:30.423
So we just have to
figure out how many times

00:02:30.423 --> 00:02:31.990
does 5 go into 30.

00:02:31.990 --> 00:02:33.590
So hopefully, that
gives you a sense

00:02:33.590 --> 00:02:36.860
of what's actually happening
behind this little process.

00:02:36.860 --> 00:02:38.070
It's not just magic.

00:02:38.070 --> 00:02:41.194
We're really just keeping
track of place value here.

00:02:41.194 --> 00:02:42.610
The other thing I
want to show you

00:02:42.610 --> 00:02:45.629
is that we didn't necessarily
have to write this 0 here.

00:02:45.629 --> 00:02:47.170
Another way of
computing it, we could

00:02:47.170 --> 00:02:52.465
have said 280 divided by 5.

00:02:52.465 --> 00:02:55.080
And we could've said, how
many times does 5 go into 2?

00:02:55.080 --> 00:02:56.330
Well, it doesn't go any times.

00:02:56.330 --> 00:03:00.140
So let's just think
about 5 going into 28.

00:03:00.140 --> 00:03:03.330
Well, 5 goes into 28 5 times.

00:03:03.330 --> 00:03:05.820
5 times 5 is 25.

00:03:05.820 --> 00:03:07.020
Subtract.

00:03:07.020 --> 00:03:09.310
28 minus 25 is 3.

00:03:09.310 --> 00:03:11.190
Bring down the 0.

00:03:11.190 --> 00:03:13.690
5 goes into 30 six times.

00:03:13.690 --> 00:03:15.630
6 times 5 is 30.

00:03:15.630 --> 00:03:17.330
And we have nothing left over.

00:03:17.330 --> 00:03:19.580
And again, what
we're really saying

00:03:19.580 --> 00:03:24.282
is that 5 doesn't go into
200 any hundreds of times.

00:03:24.282 --> 00:03:25.865
If this was a 500,
you could say, hey,

00:03:25.865 --> 00:03:27.660
it's going to go
in it 100 times.

00:03:27.660 --> 00:03:30.600
But instead, we go, well,
what about 5 going into 280?

00:03:30.600 --> 00:03:32.080
Well, it goes 50 times.

00:03:32.080 --> 00:03:34.400
50 times 5 is 250.

00:03:34.400 --> 00:03:36.880
280 minus 250 is 30.

00:03:36.880 --> 00:03:38.870
5 goes into 30 six times.

00:03:38.870 --> 00:03:41.644
So hopefully, that makes
a little bit of sense.


Dividing numbers: intro to long division

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:02.060
In this video,
I'll introduce you

00:00:02.060 --> 00:00:04.730
to a new way of
computing division,

00:00:04.730 --> 00:00:06.552
especially for larger numbers.

00:00:06.552 --> 00:00:08.760
And then we'll think a little
bit about why it works.

00:00:08.760 --> 00:00:13.336
So we're going to try to
compute what 96 divided by 4 is.

00:00:13.336 --> 00:00:15.460
And I'm going to write it
a little bit differently.

00:00:15.460 --> 00:00:20.746
I'm going to write 96
divided by-- so I'm

00:00:20.746 --> 00:00:22.780
going to write this
strange-looking symbol

00:00:22.780 --> 00:00:25.950
right over here, this
thing that covers the 96.

00:00:25.950 --> 00:00:30.319
But you could view this
as 96 divided by 4.

00:00:30.319 --> 00:00:32.610
And I'll show you in a second
why we write it this way.

00:00:32.610 --> 00:00:35.772
This is actually a very useful
way of actually computing it.

00:00:35.772 --> 00:00:37.730
So the first thing we'll
do is we'll say, well,

00:00:37.730 --> 00:00:42.330
how many times does 4 go into 9?

00:00:42.330 --> 00:00:47.970
Well, we know that 4 times 2 is
equal to 8 and that 4 times 3

00:00:47.970 --> 00:00:50.170
is equal to 12.

00:00:50.170 --> 00:00:51.540
So 3 would be too much.

00:00:51.540 --> 00:00:52.590
We would go above 9.

00:00:52.590 --> 00:00:56.040
So we want to be below 9 but
not have too much left over.

00:00:56.040 --> 00:00:58.170
We want the largest
number that gets us

00:00:58.170 --> 00:01:01.060
into 9 without going over 9.

00:01:01.060 --> 00:01:03.160
So we'll say it goes two times.

00:01:03.160 --> 00:01:05.710
4 goes into 9 two times.

00:01:05.710 --> 00:01:07.350
And then we say,
what's 2 times 4?

00:01:07.350 --> 00:01:10.450
Well, 2 times 4 is 8.

00:01:10.450 --> 00:01:11.480
4 times 2 is 8.

00:01:11.480 --> 00:01:13.180
Or 2 times 4 is 8.

00:01:13.180 --> 00:01:15.000
And now, we subtract.

00:01:15.000 --> 00:01:17.710
We subtract the 8 from the 9.

00:01:17.710 --> 00:01:20.720
And we get 1.

00:01:20.720 --> 00:01:24.102
And now we bring down the
next digit, which is the 6.

00:01:24.102 --> 00:01:26.060
And then we ask ourselves,
well, how many times

00:01:26.060 --> 00:01:27.910
does 4 go into 16?

00:01:27.910 --> 00:01:31.440
Well, in this case, we know
that 4 goes into 16 exactly

00:01:31.440 --> 00:01:32.330
four times.

00:01:32.330 --> 00:01:34.200
4 times 4 is 16.

00:01:34.200 --> 00:01:37.000
So we say 4 goes
into 16 four times.

00:01:37.000 --> 00:01:40.280
Then we multiply
4 times 4 is 16.

00:01:40.280 --> 00:01:41.600
We subtract.

00:01:41.600 --> 00:01:45.682
And 16 minus 16, we have
absolutely nothing left over.

00:01:45.682 --> 00:01:46.890
And there we have our answer.

00:01:46.890 --> 00:01:48.770
I know it seems kind of
magical at this point.

00:01:48.770 --> 00:01:50.144
But in a few
seconds, we're going

00:01:50.144 --> 00:01:52.200
to think about why
this actually worked.

00:01:52.200 --> 00:01:58.048
We got that 96 divided
by 4 is equal to-- I

00:01:58.048 --> 00:02:06.100
want to do that in a
different color-- 24.

00:02:06.100 --> 00:02:08.880
Now, what I want you to do
right now is pause this video

00:02:08.880 --> 00:02:11.180
and think about why
did this actually work.

00:02:11.180 --> 00:02:13.140
How did we magically get
the right answer here?

00:02:13.140 --> 00:02:14.139
And you can verify this.

00:02:14.139 --> 00:02:18.406
Multiply 4 times 24,
and you will get 96.

00:02:18.406 --> 00:02:20.490
Well, I'm assuming
you gave a go at it.

00:02:20.490 --> 00:02:22.210
And the important
thing always is

00:02:22.210 --> 00:02:23.660
to keep track of
the place value.

00:02:23.660 --> 00:02:25.160
And it really tells
you what's going

00:02:25.160 --> 00:02:27.930
on when we do this process.

00:02:27.930 --> 00:02:29.870
When we looked at this
9 right over here,

00:02:29.870 --> 00:02:31.080
this 9 is in the tens place.

00:02:31.080 --> 00:02:32.525
This is actually
representing 90.

00:02:32.525 --> 00:02:34.499
It represents 9 tens.

00:02:34.499 --> 00:02:36.040
So we're saying,
well, how many times

00:02:36.040 --> 00:02:44.420
does 4 go into 90 if we're
thinking about multiples of 10?

00:02:44.420 --> 00:02:46.560
Well, it goes 20 times.

00:02:46.560 --> 00:02:49.390
4 times 20 is 80.

00:02:49.390 --> 00:02:51.710
And so we said, well,
4 times 20 is 80.

00:02:51.710 --> 00:02:53.950
But we still have 16 left over.

00:02:53.950 --> 00:02:55.910
You do 96 minus 80.

00:02:55.910 --> 00:02:58.890
You have 16 left over
to divide 4 into.

00:02:58.890 --> 00:03:02.274
And then 4 goes
into 16 four times.

00:03:02.274 --> 00:03:04.190
So really, a lot of this
is just saying, well,

00:03:04.190 --> 00:03:06.234
we first figured out that
we could go 20 times.

00:03:06.234 --> 00:03:08.025
And then we said, well,
that doesn't get us

00:03:08.025 --> 00:03:08.880
all the way to 96.

00:03:08.880 --> 00:03:11.690
We have to go
another four times.

00:03:11.690 --> 00:03:13.362
Hopefully, that helps.


Comparing with multiplication: money and distance

WEBVTT
Kind: captions
Language: en

00:00:01.290 --> 00:00:05.630
So we're told that Minli
has $5.00 more than James.

00:00:05.630 --> 00:00:07.730
Minli has $33.00.

00:00:07.730 --> 00:00:10.160
How much does James have?

00:00:10.160 --> 00:00:13.150
So let's use some letters to
represent the amount of money

00:00:13.150 --> 00:00:16.020
Minli has and the amount
of money James has.

00:00:16.020 --> 00:00:20.300
So let's say that m
represents the amount of money

00:00:20.300 --> 00:00:22.010
that Minli has.

00:00:22.010 --> 00:00:25.390
And j represents the
amount that James has.

00:00:25.390 --> 00:00:28.760
So we're told that Minli
has $5.00 more than James.

00:00:28.760 --> 00:00:31.770
So Minli's money, or the
amount of money Minli has,

00:00:31.770 --> 00:00:34.270
is going to be equal
to the amount of money

00:00:34.270 --> 00:00:36.648
James has plus 5.

00:00:41.060 --> 00:00:42.630
Start with the amount James has.

00:00:42.630 --> 00:00:44.310
Add 5, you have Minli's.

00:00:44.310 --> 00:00:48.440
Minli has $5.00 more than James.

00:00:48.440 --> 00:00:52.400
Now they tell us,
Minli has $33.00.

00:00:52.400 --> 00:00:53.790
So let me write this down.

00:00:53.790 --> 00:00:58.800
So we know that m is equal to--
so this the sentence right here

00:00:58.800 --> 00:01:05.197
tells us, this right here tells
us, that m is equal to $33.00.

00:01:05.197 --> 00:01:06.780
So instead of an m
here, we could say,

00:01:06.780 --> 00:01:15.205
well $33.00 is equal to the
amount of money James has plus

00:01:15.205 --> 00:01:15.967
5.

00:01:15.967 --> 00:01:17.550
And so now we just
need to think about

00:01:17.550 --> 00:01:19.110
how much money does James have.

00:01:19.110 --> 00:01:20.693
And I encourage you
to pause the video

00:01:20.693 --> 00:01:22.280
and think about it on your own.

00:01:22.280 --> 00:01:25.860
Well, one way to visualize
this is maybe on a number line.

00:01:25.860 --> 00:01:29.020
So let me draw a number
line right over here.

00:01:29.020 --> 00:01:32.200
And let's say this is 0.

00:01:32.200 --> 00:01:35.460
This is the amount that
Minli has right up here.

00:01:35.460 --> 00:01:38.290
So Minli has $33.00.

00:01:38.290 --> 00:01:40.930
That represents that
point right over here.

00:01:40.930 --> 00:01:43.600
And j is the amount
the James has.

00:01:43.600 --> 00:01:46.700
So let's say that this
is j right over here.

00:01:46.700 --> 00:01:51.470
And we know that if we add 5
to j-- if you take j, add 5,

00:01:51.470 --> 00:01:52.150
you get to 33.

00:01:59.210 --> 00:02:01.410
Well how can I go to 33?

00:02:01.410 --> 00:02:03.110
How can I start
with m or how can I

00:02:03.110 --> 00:02:05.100
start with the amount
Minli has and then end up

00:02:05.100 --> 00:02:06.429
with the amount James has?

00:02:06.429 --> 00:02:08.220
Well, I could just go
the other way around.

00:02:08.220 --> 00:02:13.370
I could start with m,
and I could subtract 5.

00:02:13.370 --> 00:02:17.070
So we could say that 33, which
is m, the amount of money

00:02:17.070 --> 00:02:23.170
Minli has, minus 5, is
equal to the amount of money

00:02:23.170 --> 00:02:24.840
that James has.

00:02:24.840 --> 00:02:27.030
So how much money
does James have?

00:02:27.030 --> 00:02:31.110
Well, 33 minus 5
is going to be 28.

00:02:31.110 --> 00:02:36.500
So we can say that James
has, 28 is equal to j,

00:02:36.500 --> 00:02:38.860
or James has $28.00.

00:02:38.860 --> 00:02:40.742
So this right over here is 28.

00:02:40.742 --> 00:02:41.450
And you see that.

00:02:41.450 --> 00:02:44.630
28 plus 5 is 33.

00:02:44.630 --> 00:02:47.360
33 is 5 more than 28.

00:02:47.360 --> 00:02:48.955
So this all works out.

00:02:48.955 --> 00:02:51.210
Now let's think about
this next question.

00:02:51.210 --> 00:02:54.500
Jessica's house
is 5 times as far

00:02:54.500 --> 00:02:56.790
from school as Paulette's house.

00:02:56.790 --> 00:02:59.110
Jessica's house is
15 miles from school.

00:02:59.110 --> 00:03:01.750
How far is Paulette's
house from the school?

00:03:01.750 --> 00:03:04.420
So just like we did, let's just
look at each of these sentences

00:03:04.420 --> 00:03:05.753
and see what they're telling us.

00:03:05.753 --> 00:03:08.220
And I encourage you to
define use of letters

00:03:08.220 --> 00:03:11.080
to represent the distance
Jessica's house from school

00:03:11.080 --> 00:03:12.550
and Paulette's
house from school.

00:03:12.550 --> 00:03:13.900
And try to figure
this out on your own.

00:03:13.900 --> 00:03:15.316
Pause the video
right now, and try

00:03:15.316 --> 00:03:16.930
to figure it out on your own.

00:03:16.930 --> 00:03:20.250
So you could imagine j might
be a good letter for Jessica's,

00:03:20.250 --> 00:03:22.770
the distance from
Jessica's house to school.

00:03:22.770 --> 00:03:26.170
And let's say that p we'll
use for Paulette's house

00:03:26.170 --> 00:03:27.530
from school.

00:03:27.530 --> 00:03:30.940
So we're told that, however
far Paulette's house

00:03:30.940 --> 00:03:33.260
is from school, you
take 5 times that

00:03:33.260 --> 00:03:36.220
to see how far Jessica's
house is from school.

00:03:36.220 --> 00:03:40.680
So we could write that p, which
is how far Paulette's house is

00:03:40.680 --> 00:03:49.770
from school, times 5, is equal
to how far Jessica's house is

00:03:49.770 --> 00:03:50.650
from school.

00:03:50.650 --> 00:03:54.240
Jessica's house is 5
times as far from school

00:03:54.240 --> 00:03:57.790
as Paulette's house. p is
Paulette's house's distance

00:03:57.790 --> 00:03:58.390
from school.

00:03:58.390 --> 00:04:01.670
j is Jessica's house's
distance from school.

00:04:01.670 --> 00:04:05.350
They then tell us that Jessica's
house is 15 miles from school.

00:04:05.350 --> 00:04:07.700
So they're essentially telling
us, this sentence right

00:04:07.700 --> 00:04:10.710
over here, they're telling
us that j is equal to 15.

00:04:14.400 --> 00:04:17.919
So we can rewrite this as
the distance Paulette's house

00:04:17.919 --> 00:04:20.860
is from school times
5, which we know

00:04:20.860 --> 00:04:22.930
to be Jessica's
distance from school,

00:04:22.930 --> 00:04:24.910
or Jessica's house's
distance from school,

00:04:24.910 --> 00:04:30.830
which we now know to be 15,
is going to be equal to 15.

00:04:30.830 --> 00:04:32.630
So what would p be?

00:04:32.630 --> 00:04:37.320
Some number times 5 is
going to be equal to 15.

00:04:37.320 --> 00:04:41.524
Some number of miles times
5 is equal to 15 miles.

00:04:41.524 --> 00:04:43.190
Well, you might already
be able to think

00:04:43.190 --> 00:04:46.190
about this in your head, but
we could also visualize it.

00:04:46.190 --> 00:04:48.110
So let's draw a
number line again.

00:04:51.190 --> 00:04:53.040
And if you start
with that number,

00:04:53.040 --> 00:04:57.430
p-- so this is Paulette's
distance from school so, p--

00:04:57.430 --> 00:04:59.770
and you multiply it by 5.

00:04:59.770 --> 00:05:07.510
So that's times 1, times 2,
times 3, times 4, times 5.

00:05:07.510 --> 00:05:11.070
So notice this is p, this
distance is p right over here,

00:05:11.070 --> 00:05:13.790
this is another p
right over here,

00:05:13.790 --> 00:05:16.880
this is another p
right over here,

00:05:16.880 --> 00:05:18.810
that's another p
right over there,

00:05:18.810 --> 00:05:20.860
and another p right over there.

00:05:20.860 --> 00:05:22.360
Or another way of
thinking about it,

00:05:22.360 --> 00:05:23.818
this whole distance
right over here

00:05:23.818 --> 00:05:29.045
is going to be p plus p plus
p plus p plus p, or p times 5.

00:05:32.080 --> 00:05:34.710
And that's equal to how
far Jessica is from school,

00:05:34.710 --> 00:05:36.550
or Jessica's house
is from school.

00:05:36.550 --> 00:05:39.900
So this right over
here is equal to 15,

00:05:39.900 --> 00:05:43.480
which is the same thing as j.

00:05:43.480 --> 00:05:44.960
So how would you figure it out?

00:05:44.960 --> 00:05:47.360
Well, if 5 times
p is equal to 15--

00:05:47.360 --> 00:05:48.800
if I want to figure
out what p is,

00:05:48.800 --> 00:05:53.890
I could just divide 15
into 5 equal groups.

00:05:53.890 --> 00:06:01.080
So we could say that
p is equal to 15

00:06:01.080 --> 00:06:03.100
divided into 5 equal groups.

00:06:03.100 --> 00:06:07.300
If you take 15 miles and
divide it into 5 equal groups,

00:06:07.300 --> 00:06:09.130
you're going to end up with p.

00:06:09.130 --> 00:06:10.400
So what is this?

00:06:10.400 --> 00:06:15.800
Well, we get p, 15
divided by 5 is 3.

00:06:15.800 --> 00:06:18.722
So Paulette lives 3
miles from school.

00:06:18.722 --> 00:06:19.430
And you see that.

00:06:19.430 --> 00:06:22.610
This is a 3, another 3,
another 3, plus another 3,

00:06:22.610 --> 00:06:23.425
and another 3.

00:06:23.425 --> 00:06:25.460
3 time 5 is 15.

00:06:25.460 --> 00:06:27.730
Jessica's house, which
is 15 miles from school,

00:06:27.730 --> 00:06:30.080
is 5 times as far from
school as Paulette's house,

00:06:30.080 --> 00:06:33.960
which is 3 miles from school
so it all makes sense.


Comparing with multiplication: ages and heights

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:04.600
We're told that Imran is
two years older than Diya.

00:00:04.600 --> 00:00:05.800
Diya is 2.

00:00:05.800 --> 00:00:07.170
How old is Imran?

00:00:07.170 --> 00:00:08.789
So I encourage you
to pause this video

00:00:08.789 --> 00:00:10.330
and try to think
about this yourself.

00:00:10.330 --> 00:00:14.290
How old is Imran given
the information here?

00:00:14.290 --> 00:00:17.520
So let's use some letters to
represent Imran and Diya's age.

00:00:17.520 --> 00:00:21.290
Let's say that I is
equal to Imran's age.

00:00:24.280 --> 00:00:27.526
And let's say that D
is equal to Diya's age.

00:00:32.390 --> 00:00:35.320
So given that, what does
this first sentence tell us?

00:00:35.320 --> 00:00:39.460
It says that Imran's age, Imran
is two years older than Diya.

00:00:39.460 --> 00:00:47.890
So Imran's age is going to be
equal to Diya's age plus 2.

00:00:50.382 --> 00:00:52.090
That's what this first
sentence tells us.

00:00:52.090 --> 00:00:54.920
Imran's age is
Diya's age plus 2.

00:00:54.920 --> 00:00:58.139
Imran is two years
older than Diya.

00:00:58.139 --> 00:00:59.680
Another way you
could think about it,

00:00:59.680 --> 00:01:02.480
if we were to imagine
their ages on a number line

00:01:02.480 --> 00:01:05.960
right over here,
so if this is 0,

00:01:05.960 --> 00:01:10.870
and then this is Diya's
age right over here--

00:01:10.870 --> 00:01:12.710
let me do it in that same color.

00:01:12.710 --> 00:01:15.170
This is Diya's age
right over here.

00:01:15.170 --> 00:01:17.910
If you were to add
2 to Diya's age,

00:01:17.910 --> 00:01:22.950
if you were to go
to more, so plus 2,

00:01:22.950 --> 00:01:24.595
you would get to Imran's age.

00:01:27.210 --> 00:01:29.340
So that's what this
first sentence tells us.

00:01:29.340 --> 00:01:31.830
Then they tell us
that Diya is 2.

00:01:31.830 --> 00:01:34.380
So that's like
saying that D, which

00:01:34.380 --> 00:01:36.570
we're using to represent
Diya's age, that's

00:01:36.570 --> 00:01:38.180
going to be equal to 2.

00:01:38.180 --> 00:01:43.360
So this thing is equal to 2.

00:01:43.360 --> 00:01:47.230
So we can rewrite
this as Imran's age

00:01:47.230 --> 00:01:54.320
is equal to Diya's age,
which we know to be 2 plus 2.

00:01:54.320 --> 00:01:55.690
So what's 2 plus 2?

00:01:55.690 --> 00:01:56.570
Well, you know that.

00:01:56.570 --> 00:01:58.160
That's going to be 4.

00:01:58.160 --> 00:02:02.360
So Imran's age is
going to be equal to 4.

00:02:02.360 --> 00:02:04.640
If this is 2 right
over here, and then you

00:02:04.640 --> 00:02:08.889
were to go up 2 more,
you're going to get to 4.

00:02:08.889 --> 00:02:11.100
Imran's age is 4.

00:02:11.100 --> 00:02:13.590
Let's try one more of these.

00:02:13.590 --> 00:02:17.330
So here we're told that the
giraffe at the zoo is 3 times

00:02:17.330 --> 00:02:19.240
taller than Shantanu.

00:02:19.240 --> 00:02:21.160
Shantanu is 6 feet tall.

00:02:21.160 --> 00:02:22.617
How tall is the giraffe?

00:02:22.617 --> 00:02:24.700
So once again, I encourage
you to pause this video

00:02:24.700 --> 00:02:26.735
and try to think
about it on your own.

00:02:26.735 --> 00:02:28.700
Well, let's do the same idea.

00:02:28.700 --> 00:02:30.600
Let's use some
letters to represent

00:02:30.600 --> 00:02:37.880
the different heights, so G
for giraffe and S for Shantanu.

00:02:37.880 --> 00:02:40.670
I'll use G for the
height of the giraffe.

00:02:40.670 --> 00:02:43.680
The height of the
giraffe is 3 times

00:02:43.680 --> 00:02:50.470
taller-- not 3 more
feet-- 3 times taller

00:02:50.470 --> 00:02:51.862
than Shantanu's height.

00:02:51.862 --> 00:02:53.570
So we could say that
the giraffe's height

00:02:53.570 --> 00:02:59.180
is equal to 3 times
Shantanu's height.

00:03:02.510 --> 00:03:04.285
So let's be clear.

00:03:04.285 --> 00:03:06.600
G represents the
height of the giraffe,

00:03:06.600 --> 00:03:08.910
S represents the
height of Shantanu.

00:03:08.910 --> 00:03:10.750
So another way of
thinking about it--

00:03:10.750 --> 00:03:13.640
let me see if I can draw
a giraffe reasonably here.

00:03:13.640 --> 00:03:17.780
So if this is the
giraffe-- actually,

00:03:17.780 --> 00:03:20.010
this isn't a bad job
at drawing a giraffe.

00:03:20.010 --> 00:03:23.050
So this is the giraffe
right over here.

00:03:23.050 --> 00:03:25.640
They're saying that if you
take Shantanu's height times 3,

00:03:25.640 --> 00:03:27.380
you get to the giraffe's height.

00:03:27.380 --> 00:03:30.080
Actually, a giraffe might
have longer legs than that.

00:03:30.080 --> 00:03:37.460
So this is G. The height
of the giraffe is G.

00:03:37.460 --> 00:03:39.540
So let's say this is
Shantanu's height.

00:03:39.540 --> 00:03:42.970
If I multiply that by 3, I
get the giraffe's height.

00:03:42.970 --> 00:03:45.760
Let me see-- one, two.

00:03:45.760 --> 00:03:50.390
So Shantanu might look
something like this.

00:03:50.390 --> 00:03:52.040
And if you took 3
times his height,

00:03:52.040 --> 00:03:55.180
so this is 1 time his height,
2 times his height, 3 times

00:03:55.180 --> 00:03:58.470
his height, you would get to
the height of the giraffe.

00:03:58.470 --> 00:04:00.830
So that's what this
first sentence tells us.

00:04:00.830 --> 00:04:02.570
Shantanu is 6 feet tall.

00:04:02.570 --> 00:04:08.140
So they tell us that this height
right over here is 6 feet.

00:04:08.140 --> 00:04:12.680
Or another way of saying
it is S is equal to 6 feet.

00:04:12.680 --> 00:04:14.520
How tall is the giraffe?

00:04:14.520 --> 00:04:16.769
Well, once again, S
is equal to 6 feet.

00:04:16.769 --> 00:04:22.800
So the giraffe's height is equal
to 3 times Shantanu's height,

00:04:22.800 --> 00:04:29.280
which we now know to be 6
feet, so 3 times 6 feet.

00:04:29.280 --> 00:04:32.290
So in feet, the giraffe's
height is 3 times 6.

00:04:32.290 --> 00:04:36.530
Well, we know that
3 times 6 is 18.

00:04:36.530 --> 00:04:37.900
And we're dealing with feet.

00:04:37.900 --> 00:04:40.115
So the giraffe's
height is 18 feet.

00:04:40.115 --> 00:04:41.490
And you see that
right over here.

00:04:41.490 --> 00:04:49.230
6 feet plus another 6
feet plus another 6 feet,

00:04:49.230 --> 00:04:56.740
or 6 feet times 3 is going
to get you to 18 feet.


Multiplying: 2 digit numbers

WEBVTT
Kind: captions
Language: en

00:00:01.000 --> 00:00:04.390
In this video, we're going
to multiply 36 times 27.

00:00:04.390 --> 00:00:06.470
So we're multiplying
a two-digit number

00:00:06.470 --> 00:00:08.152
times another two-digit number.

00:00:08.152 --> 00:00:09.735
And the way that
we're going to tackle

00:00:09.735 --> 00:00:13.130
it is we're going to
first multiply 36 times 7,

00:00:13.130 --> 00:00:15.190
figure out what that is.

00:00:15.190 --> 00:00:18.190
Then we're going to
multiply 36 times 20,

00:00:18.190 --> 00:00:19.850
figure out what that
is, and then add

00:00:19.850 --> 00:00:21.530
those two numbers together.

00:00:21.530 --> 00:00:24.440
And what I want to do is
first do this, just show you

00:00:24.440 --> 00:00:26.835
the process for how to
multiply these two numbers.

00:00:26.835 --> 00:00:28.210
And then I'm going
to do it again

00:00:28.210 --> 00:00:30.043
where we're going to
think a little bit more

00:00:30.043 --> 00:00:32.340
about what the different
numbers represent.

00:00:32.340 --> 00:00:34.070
So first let's start
with the process.

00:00:34.070 --> 00:00:38.140
So I'm going to multiply
36 times 7 or 7 times 36.

00:00:38.140 --> 00:00:40.010
So I can start in
the ones place.

00:00:40.010 --> 00:00:43.090
7 times 6 is 42.

00:00:43.090 --> 00:00:44.760
Write the 2 down here.

00:00:44.760 --> 00:00:46.790
And then the 4,
which represents 40,

00:00:46.790 --> 00:00:48.770
I can put in the tens place.

00:00:48.770 --> 00:00:55.680
7 times 3 is 21, plus 4 is 25.

00:00:55.680 --> 00:00:57.980
So I could write the
25 right over here.

00:00:57.980 --> 00:00:59.710
There's no place
to carry this 2,

00:00:59.710 --> 00:01:03.450
so I just wrote the 2 right
over here in the hundreds place.

00:01:03.450 --> 00:01:05.290
Now let's move over.

00:01:05.290 --> 00:01:07.430
So let me clean this up
so we don't get confused.

00:01:07.430 --> 00:01:10.600
So we just figured out
what 7 times 36 is.

00:01:10.600 --> 00:01:12.760
It is 252.

00:01:12.760 --> 00:01:16.647
Now let's worry about
what 36 times 20 is.

00:01:16.647 --> 00:01:18.230
And so what we're
going to do is we're

00:01:18.230 --> 00:01:20.290
going to throw a
0 right over here,

00:01:20.290 --> 00:01:21.760
because we're now
going to multiply

00:01:21.760 --> 00:01:23.880
36 times something
in the tens place.

00:01:23.880 --> 00:01:25.250
This isn't just a 2.

00:01:25.250 --> 00:01:27.470
This is a 20.

00:01:27.470 --> 00:01:31.090
So let's just go with
the digits right now

00:01:31.090 --> 00:01:34.660
and then we'll think about
it in terms of place value

00:01:34.660 --> 00:01:36.020
the second time around.

00:01:36.020 --> 00:01:39.200
2 times 6 is 12.

00:01:39.200 --> 00:01:41.160
Write the 2 right over here.

00:01:41.160 --> 00:01:42.490
Carry the 1.

00:01:42.490 --> 00:01:47.750
2 times 3 is 6, plus 1 is 7.

00:01:47.750 --> 00:01:53.059
So we just figured out
that 36 times 20 is 720.

00:01:53.059 --> 00:01:54.850
And just think about
what would've happened

00:01:54.850 --> 00:01:56.230
if we didn't put the 0 here.

00:01:56.230 --> 00:02:00.130
Then we would have figured
out that 36 times 2 is 72,

00:02:00.130 --> 00:02:01.510
but this 2 isn't just a 2.

00:02:01.510 --> 00:02:03.520
This is a 20.

00:02:03.520 --> 00:02:07.010
So 36 times 20 is 720.

00:02:07.010 --> 00:02:11.030
And now we can add these two
things because 36 times 27

00:02:11.030 --> 00:02:16.280
is the same thing as 36
times 20 plus 36 times 7.

00:02:16.280 --> 00:02:18.480
So let's add these
two numbers together.

00:02:18.480 --> 00:02:22.450
2 plus 0 is 2.

00:02:22.450 --> 00:02:24.860
5 plus 2 is 7.

00:02:24.860 --> 00:02:26.970
2 plus 7 is 9.

00:02:26.970 --> 00:02:29.680
And we get 972.

00:02:29.680 --> 00:02:32.600
Now I'm going to do this
exact same problem again.

00:02:32.600 --> 00:02:36.070
But this time, I'm
going to really talk

00:02:36.070 --> 00:02:38.200
about what these
digits represent.

00:02:38.200 --> 00:02:40.640
Hopefully the first pass,
you kind of saw the process

00:02:40.640 --> 00:02:41.515
for doing it.

00:02:41.515 --> 00:02:43.890
Now we're going to think about
what these digits actually

00:02:43.890 --> 00:02:44.990
represent.

00:02:44.990 --> 00:02:49.090
So we're going to
multiply 36 times 7.

00:02:49.090 --> 00:02:52.650
So 7 times 6 is 42.

00:02:52.650 --> 00:02:56.170
We would write the 2 in the
ones place, and then the 40

00:02:56.170 --> 00:02:57.950
we can write in the tens place.

00:02:57.950 --> 00:02:59.900
This 4 represents 40.

00:02:59.900 --> 00:03:07.990
7 times 30 is 210,
plus 40 is 250.

00:03:07.990 --> 00:03:11.750
And we already had that 2 in
the ones place, so we get 252.

00:03:11.750 --> 00:03:14.460
36 times 7 is 252.

00:03:14.460 --> 00:03:16.210
Now let's clean this up.

00:03:16.210 --> 00:03:26.050
Now let's multiply-- 20 times
6 is going to give us 120.

00:03:26.050 --> 00:03:29.340
So let's write the
20 in the tens place,

00:03:29.340 --> 00:03:34.520
and then carry the 100, or carry
the 1, which represents 100.

00:03:34.520 --> 00:03:38.060
Now, 2 times-- or I
should say 20 times

00:03:38.060 --> 00:03:42.970
30 is going to give you
600, plus one more 100

00:03:42.970 --> 00:03:47.770
is going to give you 700.

00:03:47.770 --> 00:03:53.912
So we just figured out
that 36 times 20 is 720.

00:03:53.912 --> 00:03:57.350
The 7 is in the hundreds
place, 2 in the tens place.

00:03:57.350 --> 00:03:59.150
And then we can add again.

00:03:59.150 --> 00:04:04.510
And just like we did the
last time, we got 972.


Number patterns: visualizing sequence relationships

WEBVTT
Kind: captions
Language: en

00:00:00.794 --> 00:00:02.976
Voiceover: You are given
the following starting

00:00:02.976 --> 00:00:06.589
numbers and rules for
two sequences of numbers.

00:00:06.897 --> 00:00:09.159
The first sequence, Sequence x,

00:00:09.159 --> 00:00:12.155
starting number should be one,

00:00:12.155 --> 00:00:14.702
and then the rule is add one.

00:00:14.837 --> 00:00:17.311
Sequence y, starting
number should be five,

00:00:17.311 --> 00:00:19.672
and then the rule should be add five.

00:00:19.880 --> 00:00:23.505
Fill in the table with the
first three terms of x and y.

00:00:23.505 --> 00:00:26.227
Then plot the ordered pairs (x,y)

00:00:26.227 --> 00:00:27.234
on the graph below.

00:00:27.234 --> 00:00:29.143
So let's see, Sequence x.

00:00:29.143 --> 00:00:30.980
They say, the starting number,

00:00:30.980 --> 00:00:33.828
the starting number should be one.

00:00:33.828 --> 00:00:35.848
So the starting number is one,

00:00:35.848 --> 00:00:37.388
and then the rule, to
get to the next number,

00:00:37.388 --> 00:00:38.788
you just add one.

00:00:38.838 --> 00:00:41.025
So, one plus one is two.

00:00:41.025 --> 00:00:43.069
Two plus one is three.

00:00:43.069 --> 00:00:44.798
Fairly straight forward.

00:00:44.798 --> 00:00:46.099
Now, let's look at Sequence y.

00:00:46.099 --> 00:00:48.356
They're saying the starting
number should be five.

00:00:49.064 --> 00:00:51.988
Starting number five,
and then the rule is,

00:00:51.988 --> 00:00:53.778
to get the next term, we just add five.

00:00:53.778 --> 00:00:55.488
So, five plus five is ten,

00:00:55.934 --> 00:00:58.879
ten plus five is fifteen.

00:00:58.879 --> 00:01:00.869
Now they want us to plot these things.

00:01:01.515 --> 00:01:04.885
Let's see, we plot them as ordered pairs,

00:01:04.885 --> 00:01:07.037
so we're going to have the point (1,5).

00:01:07.037 --> 00:01:11.339
When x is one, y is five.

00:01:11.339 --> 00:01:14.464
We see that there, x is one, y is five.

00:01:14.627 --> 00:01:16.610
When x is two, y is ten.

00:01:17.010 --> 00:01:20.940
When x is two, y is ten,

00:01:21.111 --> 00:01:24.129
and then when x is three, y is fifteen.

00:01:24.544 --> 00:01:31.109
When x is three, y is fifteen,

00:01:31.109 --> 00:01:32.213
and wee see that.

00:01:32.213 --> 00:01:35.711
For every one we move to the right,

00:01:35.711 --> 00:01:38.752
for every one we increase
in the horizontal direction,

00:01:38.752 --> 00:01:44.025
every one we increase in
x, we increase five for y.

00:01:44.025 --> 00:01:48.044
We increase one for x,
we increase five for y.

00:01:48.114 --> 00:01:51.293
So now we just have one
last thing to answer.

00:01:51.429 --> 00:01:53.537
The terms in Sequence y are blank,

00:01:53.537 --> 00:01:56.150
times the terms in Sequence x.

00:01:56.352 --> 00:01:59.044
So you immediately see, this term, five,

00:01:59.044 --> 00:02:00.575
is five times one.

00:02:00.575 --> 00:02:02.308
Ten is five times two.

00:02:02.308 --> 00:02:04.870
Fifteen is five times
three, and it makes sense.

00:02:04.870 --> 00:02:06.866
You started five times higher,

00:02:07.096 --> 00:02:09.892
and here you added one
each time, and we see that

00:02:09.892 --> 00:02:10.999
visually right over here,

00:02:10.999 --> 00:02:14.442
we add one each time, while here we add

00:02:14.442 --> 00:02:15.869
five times as much each time.

00:02:15.900 --> 00:02:17.946
We add five each time.

00:02:18.750 --> 00:02:25.179
The terms in Sequence y are five
times the terms in Sequence x.

00:02:25.570 --> 00:02:26.846
We got it right.


Quadrilaterals: classifying shapes

WEBVTT
Kind: captions
Language: en

00:00:01.420 --> 00:00:07.250
A parallelogram is a blank with
two sets of parallel lines.

00:00:07.250 --> 00:00:09.280
So let's see what
the options are.

00:00:09.280 --> 00:00:10.880
So one option is
a quadrilateral.

00:00:10.880 --> 00:00:13.405
And a parallelogram is
definitely a quadrilateral.

00:00:13.405 --> 00:00:16.120
A quadrilateral is
a four-sided figure,

00:00:16.120 --> 00:00:18.350
and it is definitely
a four-sided figure.

00:00:18.350 --> 00:00:21.970
A parallelogram is
not always a rhombus.

00:00:21.970 --> 00:00:25.830
A rhombus is a special case of a
parallelogram where not only do

00:00:25.830 --> 00:00:27.680
you have to sets
of parallel lines

00:00:27.680 --> 00:00:31.560
as your sides, two
sets of parallel sides,

00:00:31.560 --> 00:00:34.410
but all of the sides are the
same length in a rhombus.

00:00:34.410 --> 00:00:37.610
And a square is a special
case of a rhombus where

00:00:37.610 --> 00:00:40.400
all of the angles
are 90 degrees.

00:00:40.400 --> 00:00:42.450
So here, all we can say
is that a parallelogram

00:00:42.450 --> 00:00:44.080
is a quadrilateral.

00:00:44.080 --> 00:00:45.330
And so let's check our answer.

00:00:45.330 --> 00:00:47.244
And it's always a good
idea to look at hints.

00:00:47.244 --> 00:00:49.660
And so it'll kind of say the
same thing that we just said,

00:00:49.660 --> 00:00:51.576
but it would say it for
the particular problem

00:00:51.576 --> 00:00:53.240
that you're actually looking at.

00:00:53.240 --> 00:00:55.120
Let's do a few more of these.

00:00:55.120 --> 00:00:58.200
Suzanne is on an expedition
to save the universe.

00:00:58.200 --> 00:01:01.320
Sounds like a reasonable
expedition to go on.

00:01:01.320 --> 00:01:03.130
For her final challenge,
she has to play

00:01:03.130 --> 00:01:05.800
a game called Find
the Rhombuses.

00:01:05.800 --> 00:01:10.310
A wizard tells her that she
has a square, a quadrilateral,

00:01:10.310 --> 00:01:12.410
and a parallelogram,
and she must

00:01:12.410 --> 00:01:16.220
identify which of the
shapes are also rhombuses.

00:01:16.220 --> 00:01:20.580
Which of these shapes should
she pick to save the universe?

00:01:20.580 --> 00:01:23.850
So a square is a special
case of a rhombus.

00:01:23.850 --> 00:01:28.626
Just to remind ourselves, a
rhombus, the opposite sides

00:01:28.626 --> 00:01:29.750
are parallel to each other.

00:01:29.750 --> 00:01:32.650
You have two sets
of parallel sides.

00:01:32.650 --> 00:01:35.140
A square has two sets
of parallel sides,

00:01:35.140 --> 00:01:37.650
and it has the extra condition
that all of the angles

00:01:37.650 --> 00:01:38.900
are right angles.

00:01:38.900 --> 00:01:43.120
So a square is definitely
going to be a rhombus.

00:01:43.120 --> 00:01:46.240
Now, all rhombuses
have four sides.

00:01:46.240 --> 00:01:48.930
So all rhombuses
are quadrilaterals.

00:01:48.930 --> 00:01:51.830
But not all quadrilaterals
are rhombuses.

00:01:51.830 --> 00:01:54.080
You could have a quadrilateral
where none of the sides

00:01:54.080 --> 00:01:55.770
are parallel to each other.

00:01:55.770 --> 00:01:58.380
So we won't click this one.

00:01:58.380 --> 00:02:00.600
Once again, a parallelogram.

00:02:00.600 --> 00:02:03.370
So all rhombuses
are parallelograms.

00:02:03.370 --> 00:02:06.000
They have two sets
of parallel sides,

00:02:06.000 --> 00:02:07.620
two sets of parallel
line segments

00:02:07.620 --> 00:02:09.020
representing their sides.

00:02:09.020 --> 00:02:12.911
But all parallelograms
are not rhombuses.

00:02:12.911 --> 00:02:14.910
So I would say that if
someone gives you square,

00:02:14.910 --> 00:02:18.400
you can say, look, a square is
always going to be a rhombus.

00:02:18.400 --> 00:02:21.170
A quadrilateral isn't always
going to be a rhombus,

00:02:21.170 --> 00:02:25.210
nor is a parallelogram
always going to be a rhombus.

00:02:25.210 --> 00:02:27.108
We got it right.


Coordinate plane: graphing points word problem

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:04.250
We're told that Naomi
lives at 2nd Avenue and 3rd

00:00:04.250 --> 00:00:08.600
Street representing 2,
3 on the graph below.

00:00:08.600 --> 00:00:12.080
Her school is it 4th
Avenue and 10th street,

00:00:12.080 --> 00:00:15.730
representing 4, 10 on the graph.

00:00:15.730 --> 00:00:19.410
She walks over to 4th Avenue
and up to 10th Street.

00:00:19.410 --> 00:00:21.970
Plot her home and
school on the map.

00:00:21.970 --> 00:00:24.770
How many blocks does
Naomi walk to school?

00:00:24.770 --> 00:00:28.520
So let's plot her home first,
2nd Avenue and 3rd Street,

00:00:28.520 --> 00:00:31.180
which they are giving
us the coordinates 2, 3.

00:00:31.180 --> 00:00:33.760
So the x-axis right
over here, this

00:00:33.760 --> 00:00:35.820
is representing the avenues.

00:00:35.820 --> 00:00:38.150
And then the y-axis,
or the vertical axis,

00:00:38.150 --> 00:00:40.290
this represents the street.

00:00:40.290 --> 00:00:41.880
So 2, 3.

00:00:41.880 --> 00:00:46.870
So this right over here is
2nd Avenue and 3rd Street.

00:00:46.870 --> 00:00:48.634
So you see here, we
went to 2nd Avenue

00:00:48.634 --> 00:00:50.050
and then we went
up to 3rd Street.

00:00:50.050 --> 00:00:51.383
This is where she lives at home.

00:00:51.383 --> 00:00:55.150
We could have also said
3rd Street and 2nd Avenue.

00:00:55.150 --> 00:00:59.537
Now her school is a 4th
Avenue and 10th street.

00:00:59.537 --> 00:01:01.870
And now we're going to go,
so if we start at the origin,

00:01:01.870 --> 00:01:06.040
we go to 4th Avenue and then
go up 10 to 10th Street.

00:01:06.040 --> 00:01:09.150
So this is the coordinate 4, 10.

00:01:09.150 --> 00:01:10.860
So this is where our home is.

00:01:10.860 --> 00:01:12.740
This is where a school is.

00:01:12.740 --> 00:01:16.350
They say she walks over to 4th
Avenue and up to 10th Street.

00:01:16.350 --> 00:01:19.730
So she walks from 2nd
Avenue to 4th Avenue

00:01:19.730 --> 00:01:21.830
and then up to 10th Street.

00:01:21.830 --> 00:01:24.840
How many blocks does
Naomi walk to school?

00:01:24.840 --> 00:01:30.220
Well as she walks from
2nd Avenue to 4th Avenue,

00:01:30.220 --> 00:01:31.960
she's going to walk 2 blocks.

00:01:31.960 --> 00:01:33.780
1 block and 2 blocks.

00:01:33.780 --> 00:01:36.500
You shouldn't get confused by
they're actually marking off

00:01:36.500 --> 00:01:38.600
with these lines
every half a block.

00:01:38.600 --> 00:01:42.590
So that's 1 block to 3rd
Avenue, 2 blocks to 4th Avenue.

00:01:42.590 --> 00:01:44.250
So that's 2 blocks so far.

00:01:44.250 --> 00:01:46.982
And then she goes from
3rd Street to 10th Street.

00:01:46.982 --> 00:01:48.940
Well, if you go from 3rd
Street to 10th Street,

00:01:48.940 --> 00:01:50.730
that's going to
be 7 more blocks.

00:01:50.730 --> 00:01:56.830
1, 2, 3, 4, 5, 6, 7.

00:01:56.830 --> 00:01:59.820
So 2 in the horizontal
direction and then 7

00:01:59.820 --> 00:02:01.430
in the vertical direction.

00:02:01.430 --> 00:02:04.770
So that's 9 total blocks.


Redistributing trail mix

WEBVTT
Kind: captions
Language: en

00:00:00.472 --> 00:00:02.307
- [Voiceover] The plot
below shows the amount

00:00:02.307 --> 00:00:05.952
of trail mix in each
of four camper's bags.

00:00:05.952 --> 00:00:06.812
So let's look at this.

00:00:06.812 --> 00:00:09.110
We see there's a dot
for each camper's bag,

00:00:09.110 --> 00:00:11.688
and it tells us that there's one bag

00:00:11.688 --> 00:00:13.290
that has this much trail mix,

00:00:13.290 --> 00:00:15.252
one bag that has this much trail mix,

00:00:15.252 --> 00:00:17.806
two bags that have this much trail mix.

00:00:17.806 --> 00:00:19.175
If the total amount of trail mix

00:00:19.175 --> 00:00:21.415
in all of the bags were redistributed

00:00:21.415 --> 00:00:23.969
so that each bag contained an equal amount

00:00:23.969 --> 00:00:26.408
of trail mix, how much trail mix

00:00:26.408 --> 00:00:28.531
would be in each bag?

00:00:28.531 --> 00:00:29.926
All measurements are rounded

00:00:29.926 --> 00:00:32.584
to the nearest eighth of an ounce.

00:00:32.584 --> 00:00:34.848
So right now the bags,
there's different amounts

00:00:34.848 --> 00:00:37.065
depending on which bag you look at.

00:00:37.065 --> 00:00:38.853
But they want us to redistribute it.

00:00:38.853 --> 00:00:40.165
So what we really wanna do is think

00:00:40.165 --> 00:00:42.011
about let's take all of the trail mix

00:00:42.011 --> 00:00:44.553
in the four bags together,
and then divide it

00:00:44.553 --> 00:00:47.211
by four so that they
all have an even amount.

00:00:47.211 --> 00:00:48.628
So let's just think
about how much trail mix

00:00:48.628 --> 00:00:51.366
we have in these four bags together.

00:00:51.366 --> 00:00:53.329
So we have this bag right over here,

00:00:53.329 --> 00:00:55.121
and it has this amount.

00:00:55.121 --> 00:00:56.860
And let's see, it says
rounded to the nearest

00:00:56.860 --> 00:00:57.549
eighth of an ounce.

00:00:57.549 --> 00:00:58.675
Are these eighths?

00:00:58.675 --> 00:01:01.883
Let's see, this is one, two, three, four,

00:01:01.883 --> 00:01:03.324
five, six, seven, eight.

00:01:03.324 --> 00:01:06.747
Yep, so between two and three is divided

00:01:06.747 --> 00:01:08.304
into eight sections.

00:01:08.304 --> 00:01:10.404
So each of these hash marks are an eighth.

00:01:10.404 --> 00:01:13.145
So this is 2 1/8, 2 2/8.

00:01:13.145 --> 00:01:15.664
So this is 2 2/8.

00:01:15.664 --> 00:01:19.681
Now 2/8 is the same thing as 1/4.

00:01:19.681 --> 00:01:22.339
You can divide two by
two and eight by two,

00:01:22.339 --> 00:01:23.499
and you would get 1/4.

00:01:23.499 --> 00:01:27.853
So 2 2/8 is the same thing as 2 1/4.

00:01:28.224 --> 00:01:33.032
Now you have this bag right over here,

00:01:33.032 --> 00:01:34.529
and how much is in this bag?

00:01:34.529 --> 00:01:38.431
Well let's see, that is 1/8, 2/8, 3/8,

00:01:38.431 --> 00:01:40.810
4/8, 5/8, 6/8.

00:01:40.810 --> 00:01:45.184
So that's going to be 2 6/8.

00:01:45.593 --> 00:01:48.252
Now 6/8, six and eight are both divisible

00:01:48.252 --> 00:01:50.445
by two, so if you divide them both

00:01:50.445 --> 00:01:52.523
by two, that's the same thing as 3/4.

00:01:52.523 --> 00:01:56.056
So that's the same thing as 2 3/4.

00:01:56.056 --> 00:01:58.461
2 3/4.

00:01:58.850 --> 00:02:00.727
And now you have two dots right over

00:02:00.727 --> 00:02:01.667
at this point over here.

00:02:01.667 --> 00:02:03.281
So there's two dots that have this many

00:02:03.281 --> 00:02:04.244
ounces of trail mix.

00:02:04.244 --> 00:02:05.823
Well how many ounces is this?

00:02:05.823 --> 00:02:10.656
3 1/8, 2/8, 3/8, 4/8.

00:02:10.656 --> 00:02:13.176
So that's 3 4/8.

00:02:13.176 --> 00:02:14.894
Well 4/8 is the same thing,

00:02:14.894 --> 00:02:16.601
they're both divisible by four.

00:02:16.601 --> 00:02:18.214
Four divided by four is one,

00:02:18.214 --> 00:02:19.515
eight divided by four is two.

00:02:19.515 --> 00:02:22.464
This is the same thing as 3 1/2.

00:02:22.464 --> 00:02:23.311
3 1/2.

00:02:23.311 --> 00:02:25.003
And you see that this dot right over here,

00:02:25.003 --> 00:02:28.577
it is halfway between three and four.

00:02:28.577 --> 00:02:30.122
And so let's think about,

00:02:30.122 --> 00:02:32.176
let's list the amount of ounces

00:02:32.176 --> 00:02:33.952
in each of the bags.

00:02:33.952 --> 00:02:37.837
So you have 2 1/4 ounces.

00:02:37.837 --> 00:02:39.171
That's one of the bags.

00:02:39.171 --> 00:02:44.171
You have another bag that's 2 3/4 ounces.

00:02:44.211 --> 00:02:45.649
Then you have two bags,

00:02:45.649 --> 00:02:46.798
remember there's two bags here,

00:02:46.798 --> 00:02:48.959
that's what the dot plot does for us.

00:02:48.959 --> 00:02:52.557
We have two bags here
that are 3 1/2 ounces.

00:02:52.557 --> 00:02:56.689
So 3 1/2 and 3 1/2.

00:02:56.689 --> 00:02:58.117
Now let me make sure that doesn't look

00:02:58.117 --> 00:02:59.452
like an exponent to you.

00:02:59.452 --> 00:03:01.797
3 1/2.

00:03:01.797 --> 00:03:03.353
Now what was our strategy?

00:03:03.353 --> 00:03:05.396
What we wanted to do is redistribute all

00:03:05.396 --> 00:03:06.709
of the trail mix so that we have

00:03:06.709 --> 00:03:08.589
the same amount in each bag.

00:03:08.589 --> 00:03:10.238
So one way that you could imagine this

00:03:10.238 --> 00:03:12.037
is imagine everyone
pouring their trail mix

00:03:12.037 --> 00:03:13.581
into a big pile and then dividing

00:03:13.581 --> 00:03:16.576
that pile into four,
and then putting exactly

00:03:16.576 --> 00:03:18.294
1/4 of the total into each bag.

00:03:18.294 --> 00:03:20.083
Then you would redistribute
it so that you have

00:03:20.083 --> 00:03:22.821
the same amount in each bag.

00:03:22.821 --> 00:03:24.261
So let's add it all together.

00:03:24.261 --> 00:03:25.758
Let's create our pile.

00:03:25.758 --> 00:03:30.737
Let's add 2 1/4 plus 2
3/4 plus 3 1/2 plus 3 1/2.

00:03:31.201 --> 00:03:32.899
Well what is,

00:03:34.545 --> 00:03:38.533
let's see, two plus two is four.

00:03:38.533 --> 00:03:43.013
1/4 plus 3/4 is going to be 4/4.

00:03:43.013 --> 00:03:47.791
So this right over here
is going to equal to five.

00:03:47.791 --> 00:03:49.243
Another way to think about it,

00:03:49.243 --> 00:03:51.042
you could rewrite these.

00:03:51.042 --> 00:03:53.166
You could say 2 1/4, two is the same thing

00:03:53.166 --> 00:03:58.032
as 8/4, so this would be 9/4

00:04:01.959 --> 00:04:04.593
plus, let's see, four times two is eight

00:04:04.593 --> 00:04:06.416
plus three is 11.

00:04:06.416 --> 00:04:08.958
Plus 11/4.

00:04:08.958 --> 00:04:13.958
Then you would have 3 1/2 plus 3 1/2,

00:04:14.097 --> 00:04:15.686
you might be able to say
three plus three is six,

00:04:15.686 --> 00:04:18.241
1/2 plus 1/2 is one, so
it's going to be seven.

00:04:18.241 --> 00:04:19.656
But if you wanted to write it like this,

00:04:19.656 --> 00:04:23.737
you could write 3 1/2, that's
two times three is six,

00:04:23.737 --> 00:04:25.699
plus one is seven.

00:04:25.699 --> 00:04:28.660
Plus 7/2 plus 7/2.

00:04:28.660 --> 00:04:30.877
So these are sometimes
called improper fractions,

00:04:30.877 --> 00:04:31.956
although some people disagree with calling

00:04:31.956 --> 00:04:33.304
them improper fractions.

00:04:33.304 --> 00:04:35.102
These are just fractions
where the numerator

00:04:35.102 --> 00:04:37.552
is greater than the denominator.

00:04:37.552 --> 00:04:40.442
But anyway, let's add all of this up.

00:04:40.442 --> 00:04:45.061
So 9/11, let me see what
color I haven't used yet.

00:04:46.165 --> 00:04:49.574
Sorry, not 9/11, 9/4 plus 11/4

00:04:49.574 --> 00:04:51.268
is going to be equal to what?

00:04:51.268 --> 00:04:53.950
This part is going to be equal to 20/4,

00:04:53.950 --> 00:04:56.353
which is the same thing as five.

00:04:56.353 --> 00:04:58.525
20 divided by four is five,

00:04:58.525 --> 00:05:01.485
or you could say 20/4 is
going to be five wholes.

00:05:01.485 --> 00:05:05.809
Then 7/2 plus 7/2,
that's going to be 14/2,

00:05:05.809 --> 00:05:08.503
plus 14/2, that's this
part right over here.

00:05:08.503 --> 00:05:10.883
14/2 is the same thing as seven.

00:05:10.883 --> 00:05:12.844
Five plus seven.

00:05:12.844 --> 00:05:14.889
So in total, you have 12 ounces.

00:05:14.889 --> 00:05:17.140
Once again, I just did this
as another way to do it,

00:05:17.140 --> 00:05:18.127
but you could have just said, look,

00:05:18.127 --> 00:05:20.119
two plus two is four,

00:05:20.119 --> 00:05:22.754
1/4 plus 3/4 is another whole.

00:05:22.754 --> 00:05:24.775
So four plus one is five.

00:05:24.775 --> 00:05:27.061
Then 3 1/2 plus 3 1/2,
three plus three is six,

00:05:27.061 --> 00:05:30.277
1/2 plus 1/2 is one,
six plus one is seven.

00:05:30.277 --> 00:05:32.414
So in total, you have 12 ounces.

00:05:32.414 --> 00:05:34.457
Now that's if you combine
everything together,

00:05:34.457 --> 00:05:36.025
but now we wanna redistribute it

00:05:36.025 --> 00:05:37.823
into the four bags.

00:05:37.823 --> 00:05:40.593
We wanna divide it equally into four bags.

00:05:40.593 --> 00:05:42.636
So you just take your 12 ounces,

00:05:42.636 --> 00:05:44.459
let me do this in a new color,

00:05:44.459 --> 00:05:47.155
you just wanna take your 12 ounces

00:05:48.917 --> 00:05:50.407
and divide it by four.

00:05:50.407 --> 00:05:52.502
Divide it into four equal groups.

00:05:52.502 --> 00:05:56.368
Then you're gonna get
three ounces per bag.

00:05:56.368 --> 00:05:57.535
So that's our answer.

00:05:58.410 --> 00:06:01.042
How much trail mix would be in each bag?

00:06:01.042 --> 00:06:03.641
Three ounces.

00:06:03.641 --> 00:06:06.181
And we are done.


Volume through decomposition

WEBVTT
Kind: captions
Language: en

00:00:00.497 --> 00:00:01.449
- [Voiceover] Let's see
if we can figure out

00:00:01.449 --> 00:00:05.650
the volume of this figure over here.

00:00:05.650 --> 00:00:07.508
They've given us some of the dimensions.

00:00:07.508 --> 00:00:09.655
We see this side over
here is two centimeters,

00:00:09.655 --> 00:00:12.583
this is seven centimeters,
this is 12 centimeters,

00:00:12.583 --> 00:00:15.483
this is five centimeters,
this is three centimeters.

00:00:15.483 --> 00:00:17.956
And so like always, pause this video

00:00:17.956 --> 00:00:20.463
and see if you can figure it out.

00:00:20.463 --> 00:00:22.240
Well there's a bunch of ways to do this,

00:00:22.240 --> 00:00:23.598
but the way I'd like to do it is

00:00:23.598 --> 00:00:26.802
just to break it up into
two rectangular prisms.

00:00:26.802 --> 00:00:28.462
So what I'm gonna do is, in fact most

00:00:28.462 --> 00:00:30.156
of the reasonable ways to do this

00:00:30.156 --> 00:00:32.363
would be to break it up
into two rectangular prisms,

00:00:32.363 --> 00:00:34.081
and the ones that jump out at me

00:00:34.081 --> 00:00:39.027
is one prism like this that is

00:00:39.027 --> 00:00:41.230
three centimeters wide,

00:00:41.230 --> 00:00:43.519
five centimeters high,

00:00:43.519 --> 00:00:46.479
and then it is seven centimeters long,

00:00:46.479 --> 00:00:48.163
or seven centimeters deep.

00:00:48.163 --> 00:00:49.637
So this one right over here.

00:00:49.637 --> 00:00:50.798
And if this part right over here

00:00:50.798 --> 00:00:55.105
was transparent you would
see it look just like this.

00:00:55.105 --> 00:00:57.343
You would see it look just like this.

00:00:57.343 --> 00:00:59.789
And so this one once again,

00:00:59.789 --> 00:01:04.158
it is three centimeters wide,

00:01:06.143 --> 00:01:10.545
seven centimeters long.

00:01:10.545 --> 00:01:12.228
So this distance right over here

00:01:12.228 --> 00:01:15.107
is going to be the same as
this distance right over here.

00:01:15.107 --> 00:01:17.397
So seven centimeters long.

00:01:17.397 --> 00:01:20.513
So the width times the length

00:01:20.513 --> 00:01:23.762
times the height is five centimeters.

00:01:23.762 --> 00:01:25.722
Gets us to, let's see.

00:01:25.722 --> 00:01:28.875
Three times seven is 21,

00:01:28.875 --> 00:01:31.321
times five is equal to,

00:01:31.321 --> 00:01:33.892
20 times five is 100,
one times five is five.

00:01:33.892 --> 00:01:36.128
So it's going to be 105.

00:01:36.128 --> 00:01:39.071
We can say 105 cubic
centimeters, cause you have

00:01:39.071 --> 00:01:41.428
centimeters times centimeters
times centimeters.

00:01:41.428 --> 00:01:43.030
So this blue part right over here,

00:01:43.030 --> 00:01:44.852
this blue rectangular prism,

00:01:44.852 --> 00:01:49.571
has a volume of 105 cubic centimeters.

00:01:50.110 --> 00:01:52.050
So now we can separately
figure out the volume

00:01:52.050 --> 00:01:55.924
of what I'm now highlighting
in this magenta color.

00:01:55.924 --> 00:01:58.134
What I'm highlighting
in this magenta color.

00:01:58.134 --> 00:01:59.595
If this was transparent, you would see

00:01:59.595 --> 00:02:02.474
this part back over here
and right over here.

00:02:02.474 --> 00:02:04.378
So what are its dimensions?

00:02:04.378 --> 00:02:07.675
Well, we know its height
is two centimeters,

00:02:07.675 --> 00:02:11.878
we know that this
dimension right over here,

00:02:11.878 --> 00:02:14.119
I guess you could say its depth,

00:02:14.119 --> 00:02:16.463
we could call it that,
is seven centimeters.

00:02:16.463 --> 00:02:17.686
But what is this right over here?

00:02:17.686 --> 00:02:20.062
If we want to consider
this, maybe it's length,

00:02:20.062 --> 00:02:21.828
or maybe it's width, depending on

00:02:21.828 --> 00:02:23.603
what we want to call it.

00:02:23.603 --> 00:02:26.633
Well, let's see, this whole
thing is 12 centimeters,

00:02:26.633 --> 00:02:29.374
from here to here is 12 centimeters,

00:02:29.374 --> 00:02:33.587
and we know that from here
to here is three centimeters,

00:02:33.587 --> 00:02:38.242
so this piece right over here
must be nine centimeters.

00:02:38.242 --> 00:02:40.947
So that must be nine centimeters,

00:02:40.947 --> 00:02:43.687
is this distance right over here.

00:02:43.687 --> 00:02:46.369
So the volume of this magenta part

00:02:46.369 --> 00:02:50.607
is going to be nine centimeters
times seven centimers

00:02:50.607 --> 00:02:53.322
times the height, times two centimeters.

00:02:53.322 --> 00:02:56.121
Which is going to get us, let's see,

00:02:56.121 --> 00:03:01.107
nine times seven is 63,
63 times two is equal to,

00:03:01.107 --> 00:03:04.422
60 times two is 120,
three times two is six,

00:03:04.422 --> 00:03:08.903
so it's 126 cubic centimeters.

00:03:08.903 --> 00:03:11.016
So the total volume of the entire thing

00:03:11.016 --> 00:03:13.558
is going to be the volume
of the magenta stuff,

00:03:13.558 --> 00:03:17.796
which is 126 cubic centimeters,

00:03:17.796 --> 00:03:20.013
plus the volume of the blue stuff,

00:03:20.013 --> 00:03:24.529
plus 105 cubic centimeters.

00:03:24.529 --> 00:03:27.554
And that's going to give
us, for the entire figure,

00:03:27.554 --> 00:03:30.241
six plus five is 11,

00:03:30.241 --> 00:03:32.853
so one plus two is three, that's really

00:03:32.853 --> 00:03:35.534
one ten plus two tens is three tens.

00:03:35.534 --> 00:03:38.554
And then 100 plus 100 is 200,

00:03:38.554 --> 00:03:43.323
so we get 231 cubic centimeters

00:03:43.538 --> 00:03:46.041
is the volume of the entire thing.

00:03:46.041 --> 00:03:48.089
Fascinating.


Volume in unit cubes by decomposing shape

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.775
- [Voiceover] So these are two
pictures of the same figure.

00:00:02.775 --> 00:00:06.374
This is a front view of the object,

00:00:06.374 --> 00:00:09.810
and this is the back view of the object.

00:00:09.810 --> 00:00:13.490
And if a unit cube looks like this,

00:00:13.490 --> 00:00:15.673
what I want to do is I want
to figure out the volume

00:00:15.673 --> 00:00:18.749
of this figure in terms of unit cubes,

00:00:18.749 --> 00:00:20.572
or in terms of cubic units.

00:00:20.572 --> 00:00:22.058
I encourage you to pause the video

00:00:22.058 --> 00:00:26.189
and think of it on your own
before we try it together.

00:00:26.248 --> 00:00:28.048
All right, well, there's
a bunch of ways that

00:00:28.048 --> 00:00:29.429
we could tackle this,

00:00:29.429 --> 00:00:31.287
all of them kind of breaking this

00:00:31.287 --> 00:00:32.947
figure up in different ways.

00:00:32.947 --> 00:00:37.853
One way we could do it, we
could break it up into this.

00:00:37.853 --> 00:00:41.213
I guess we could call
it a rectangular prism.

00:00:41.213 --> 00:00:43.426
So, if you could see through there,

00:00:43.426 --> 00:00:46.467
it would be like this, so
this piece right over here,

00:00:46.467 --> 00:00:48.673
this piece right over here.

00:00:48.673 --> 00:00:51.250
And I'll redraw it here,
so you can visualize it.

00:00:51.250 --> 00:00:54.884
So, if I were to redraw
it, it looks like this.

00:00:54.884 --> 00:00:58.111
It looks like this.

00:00:58.111 --> 00:01:01.687
And what are its dimensions?

00:01:01.687 --> 00:01:05.483
Well, its four units wide.

00:01:05.483 --> 00:01:08.025
It's two units high.

00:01:08.025 --> 00:01:10.347
And then it's four
units, we could say long,

00:01:10.347 --> 00:01:12.448
or four units deep.

00:01:12.448 --> 00:01:13.922
So just like that.

00:01:13.922 --> 00:01:16.372
So what's the volume of this yellow part?

00:01:16.372 --> 00:01:17.893
Well, the volume is just you multiply

00:01:17.893 --> 00:01:20.563
these three dimensions,
the length times the width,

00:01:20.563 --> 00:01:22.165
times the height.

00:01:22.165 --> 00:01:24.680
So the volume is going to be our length

00:01:24.680 --> 00:01:28.151
times our width times our height.

00:01:28.151 --> 00:01:33.151
Four times four is 16 times two is 32.

00:01:33.572 --> 00:01:34.397
But we're not done.

00:01:34.397 --> 00:01:36.800
That's just the volume
of this yellow part.

00:01:36.800 --> 00:01:39.458
We still have to take into
consideration the volume

00:01:39.458 --> 00:01:40.932
of this piece right over here that

00:01:40.932 --> 00:01:44.611
we haven't figured out yet,
this piece right over there.

00:01:44.611 --> 00:01:46.478
Now, this one might just jump out at you.

00:01:46.478 --> 00:01:48.579
You could just count the unit cubes,

00:01:48.579 --> 00:01:50.181
but I'll redraw it here,

00:01:50.181 --> 00:01:52.700
just to show you what's going on.

00:01:52.700 --> 00:01:54.732
All right. So it looks like this.

00:01:54.732 --> 00:01:56.257
It looks like this.

00:01:57.460 --> 00:01:59.352
And what are its dimensions?

00:01:59.352 --> 00:02:02.080
Well, it's two wide, two high,

00:02:02.080 --> 00:02:04.692
and we could say one deep or one long.

00:02:04.692 --> 00:02:07.699
So its volume is going to be equal to

00:02:07.699 --> 00:02:12.699
your length times your
width times your height,

00:02:14.374 --> 00:02:15.616
which is equal to four.

00:02:15.616 --> 00:02:16.453
And you just see that.

00:02:16.453 --> 00:02:20.318
There's one, two, three, four
unit cubes in this object.

00:02:20.318 --> 00:02:22.965
So the total volume is going to be

00:02:22.965 --> 00:02:27.965
32 plus 32 plus four is equal to 36.

00:02:29.884 --> 00:02:31.939
Now, of course, there's
other ways to tackle it.

00:02:31.939 --> 00:02:33.343
You could just say, hey,
let's figure out the

00:02:33.343 --> 00:02:35.828
volume of the blue layer
and then just double it

00:02:35.828 --> 00:02:37.546
because that red layer has the same

00:02:37.546 --> 00:02:39.403
volume as the blue layer.

00:02:39.403 --> 00:02:41.173
And the blue layer, you
could, say, well, look,

00:02:41.173 --> 00:02:42.171
it's only one deep.

00:02:42.171 --> 00:02:44.830
So we just have to
count the cubes up here,

00:02:44.830 --> 00:02:46.966
and then we'll know how
many unit cubes fit into it.

00:02:46.966 --> 00:02:49.882
So you literally could
just count one, two...

00:02:49.882 --> 00:02:51.843
Let me do it a color you can actually see.

00:02:51.843 --> 00:02:54.896
You just have to go one,
two, three, four, five,

00:02:54.896 --> 00:02:57.858
six, seven, eight, nine, 10, 11,

00:02:57.858 --> 00:03:01.716
12, 13, 14, 15, 16, 17, 18.

00:03:01.716 --> 00:03:05.240
So you see there are 18
cubes in the blue layer,

00:03:05.240 --> 00:03:09.024
and there are going to be
another 18 in the red layer,

00:03:09.024 --> 00:03:12.844
plus 18, that also gets
you to 36 unit cubes,

00:03:12.844 --> 00:03:16.318
or a volume of 36 cubic units.