Creating a fraction through division of whole numbers

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.950
My wife and I have recently
purchased an assortment pack

00:00:03.950 --> 00:00:08.870
of soap, and we both want
to experience all 3 bars.

00:00:08.870 --> 00:00:11.960
And we are not willing to
share soap with each other.

00:00:11.960 --> 00:00:13.760
So we have a little
bit of a conundrum.

00:00:13.760 --> 00:00:16.090
How do we share
these 3 bars of soap

00:00:16.090 --> 00:00:18.810
so that we each get to
experience all of the smells?

00:00:18.810 --> 00:00:20.650
So this might be
a nice rose smell.

00:00:20.650 --> 00:00:22.932
This might be some
type of ivory smell.

00:00:22.932 --> 00:00:24.890
I don't even know if
that's a legitimate smell.

00:00:24.890 --> 00:00:26.473
This might be some
type of sandalwood,

00:00:26.473 --> 00:00:28.160
which is always very nice.

00:00:28.160 --> 00:00:29.730
And my wife has an idea.

00:00:29.730 --> 00:00:32.460
She says, look I'm going
to take these 3 bars,

00:00:32.460 --> 00:00:36.100
so we're starting
with 3 bars of soap.

00:00:36.100 --> 00:00:38.692
And I'm going to divide
it into 2 equal groups.

00:00:38.692 --> 00:00:40.400
And I said, how are
you going to do that?

00:00:40.400 --> 00:00:42.316
And she says, well, I'm
just going to take out

00:00:42.316 --> 00:00:44.299
some type of carving
saw or carving knife,

00:00:44.299 --> 00:00:45.840
or who knows what
it is, and I'm just

00:00:45.840 --> 00:00:49.330
going to cut it right down the
middle, so right down this.

00:00:49.330 --> 00:00:52.800
This is my wife cutting the soap
right over here with her saw.

00:00:52.800 --> 00:00:56.720
So she's cutting the
soap right over here.

00:00:56.720 --> 00:01:01.650
So she has divided the
soap into 2 equal groups.

00:01:01.650 --> 00:01:05.850
So she's taken 3 bars, and she's
divided into 2 equal groups.

00:01:05.850 --> 00:01:09.160
So the interesting question
here is, how many bars of soap

00:01:09.160 --> 00:01:10.800
do we each have now?

00:01:10.800 --> 00:01:12.634
And I encourage you
to pause the video

00:01:12.634 --> 00:01:14.050
and think about
that for a second.

00:01:14.050 --> 00:01:18.110
How many bars of soap
do we each have now?

00:01:18.110 --> 00:01:19.760
Well, let's just
visualize my share.

00:01:19.760 --> 00:01:22.540
Let's say I take this
bottom half right over here.

00:01:22.540 --> 00:01:25.910
So this is Sal's
share of the soap.

00:01:25.910 --> 00:01:27.430
Let me write this out.

00:01:27.430 --> 00:01:31.410
So this is Sal's share.

00:01:31.410 --> 00:01:33.290
My wife took this top half.

00:01:33.290 --> 00:01:35.470
Well, what do I have?

00:01:35.470 --> 00:01:40.730
Well, I have 1/2, I have another
1/2, and I have another 1/2

00:01:40.730 --> 00:01:41.780
of a bar.

00:01:41.780 --> 00:01:45.770
So I have 3/2 bars of soap.

00:01:45.770 --> 00:01:53.540
Or you could say that I have
3 times 1/2 bars of soap.

00:01:53.540 --> 00:01:55.790
Notice, something very
interesting happened here.

00:01:55.790 --> 00:01:59.699
3 divided by 2 is
equal to 3 times 1/2.

00:01:59.699 --> 00:02:01.490
And we could make it
even more interesting,

00:02:01.490 --> 00:02:05.810
because we know 2 is the
exact same thing as 2/1.

00:02:05.810 --> 00:02:10.850
So 3 divided by 2/1 is
equal to 3 times 1/2.

00:02:10.850 --> 00:02:13.520
Notice we went from a
division to a multiplication,

00:02:13.520 --> 00:02:14.790
and we took the reciprocal.

00:02:14.790 --> 00:02:16.040
And that makes complete sense.

00:02:16.040 --> 00:02:18.390
We have 3/2 bars now.

00:02:18.390 --> 00:02:20.600
But what's 3 times 1/2?

00:02:20.600 --> 00:02:26.175
Well, 3 times 1/2
is equal to 3/2.

00:02:29.010 --> 00:02:32.830
So just doing this little
simple, smelly soap example,

00:02:32.830 --> 00:02:34.770
we've got a very
interesting result.

00:02:34.770 --> 00:02:39.300
3 divided by 2 is the
same thing as 3 times 1/2,

00:02:39.300 --> 00:02:43.245
which is the same thing as 3/2.


Decimals: writing it out in words 2

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:01.990
So I'm going to
write out a number

00:00:01.990 --> 00:00:04.489
that we're going to think about
how we could say or actually

00:00:04.489 --> 00:00:05.830
write that number.

00:00:05.830 --> 00:00:07.350
So I'm just going
to write it out.

00:00:07.350 --> 00:00:10.030
I'm going to resist the
temptation to actually speak it

00:00:10.030 --> 00:00:11.780
out because that's
normally how I operate.

00:00:11.780 --> 00:00:13.920
But I'm not going to
do that right now.

00:00:17.070 --> 00:00:19.480
So there's several ways
that we can pronounce.

00:00:19.480 --> 00:00:21.050
So I encourage you to pause
it and try to pronounce it,

00:00:21.050 --> 00:00:21.550
yourself.

00:00:21.550 --> 00:00:23.649
You might not even
need to pause it.

00:00:23.649 --> 00:00:25.190
Well, the first
thing that jumps out,

00:00:25.190 --> 00:00:27.990
well we've got
20,000 and then some.

00:00:27.990 --> 00:00:30.050
So maybe we should
write it that way.

00:00:30.050 --> 00:00:33.315
So we've got 20,000.

00:00:33.315 --> 00:00:36.880
20-- actually let
me write it out

00:00:36.880 --> 00:00:39.520
as numbers first to
really decompose it.

00:00:39.520 --> 00:00:41.460
So we have 20,000.

00:00:41.460 --> 00:00:43.720
And then what do we
have on top of that?

00:00:43.720 --> 00:00:47.300
Well we have 5/10.

00:00:47.300 --> 00:00:49.250
This is the tenths place.

00:00:49.250 --> 00:00:53.910
So we can literally
write that as 5/10.

00:00:53.910 --> 00:00:55.200
5/10.

00:00:55.200 --> 00:00:59.606
Then we have 0, 0/100.

00:00:59.606 --> 00:01:01.480
I'll write that as a
hundredths place just so

00:01:01.480 --> 00:01:03.170
that we can keep track of it.

00:01:03.170 --> 00:01:07.170
And then finally,
we have 7/1000,

00:01:07.170 --> 00:01:09.740
that's the 1000th place.

00:01:09.740 --> 00:01:14.100
So we could write
that, plus 7/1000.

00:01:14.100 --> 00:01:16.060
So if we would write
down everything

00:01:16.060 --> 00:01:18.710
that I just spoke
out loud, we would

00:01:18.710 --> 00:01:22.730
say that this is 20-- let me
write that a little bit neater.

00:01:22.730 --> 00:01:29.130
This is 20,000.

00:01:29.130 --> 00:01:32.660
20,000 and 5/10.

00:01:32.660 --> 00:01:46.780
And 5-- let me write out the
word-- and 5/10 and 7/1000.

00:01:46.780 --> 00:01:49.880
Now, this isn't the
only way to say this.

00:01:49.880 --> 00:01:52.320
Another way of
thinking about it is

00:01:52.320 --> 00:01:56.690
to try to merge the 5/10 and the
7/1000 in terms of thousandths.

00:01:56.690 --> 00:01:57.890
So let's think about this.

00:01:57.890 --> 00:02:00.380
So we could write this
as-- so once again, we

00:02:00.380 --> 00:02:03.590
would have our 20,000.

00:02:03.590 --> 00:02:05.560
But instead of 5/10
and 7/1000, let's

00:02:05.560 --> 00:02:07.846
write our 5/10 in
terms of thousandths.

00:02:07.846 --> 00:02:10.220
And the easiest way to do it
is to multiply the numerator

00:02:10.220 --> 00:02:12.580
and denominator,
both here, by 100.

00:02:12.580 --> 00:02:14.990
So then we will
have-- so this 5/10

00:02:14.990 --> 00:02:20.930
is the same thing
as 500 over 1,000.

00:02:20.930 --> 00:02:25.900
And the 7/1000 is still 7/1000.

00:02:25.900 --> 00:02:32.150
And these two
combined are 507/1000.

00:02:32.150 --> 00:02:37.560
So we could just call
this 20,000 and 507/1000.

00:02:37.560 --> 00:02:39.450
so let's write that down.

00:02:39.450 --> 00:03:06.140
So we could just say this
is 20,000 and 507/1000.

00:03:06.140 --> 00:03:11.000
This is 1/1000, while this right
over here, 1,000, of course,

00:03:11.000 --> 00:03:13.280
actually represents 1,000.

00:03:13.280 --> 00:03:16.710
So we got 20 thousands,
that's that right over there,

00:03:16.710 --> 00:03:18.525
and 507/1000.


Decimals: expanding out place value

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:15.390
Let's say I have
the number 905.074.

00:00:15.390 --> 00:00:16.970
So how could I expand this out?

00:00:16.970 --> 00:00:19.390
And what does this
actually represent?

00:00:19.390 --> 00:00:22.790
So let's just think about
each of the place values here.

00:00:22.790 --> 00:00:26.550
The 9 right over here, this
is in the hundreds place.

00:00:26.550 --> 00:00:29.050
This literally
represents nine hundreds.

00:00:29.050 --> 00:00:32.939
So we could rewrite
that 9 as nine hundreds.

00:00:32.939 --> 00:00:33.980
Let me write it two ways.

00:00:33.980 --> 00:00:35.560
We could write it
as 900, which is

00:00:35.560 --> 00:00:40.230
the same thing as 9 times 100.

00:00:40.230 --> 00:00:41.300
Now, there's a 0.

00:00:41.300 --> 00:00:43.240
That's just going to
represent zero tens.

00:00:43.240 --> 00:00:45.637
But zero tens is still just 0.

00:00:45.637 --> 00:00:47.470
So we don't have to
really worry about that.

00:00:47.470 --> 00:00:51.660
It's not adding any value to
our expression or to our number.

00:00:51.660 --> 00:00:53.290
Now we have this 5.

00:00:53.290 --> 00:00:54.910
This 5 is in the ones place.

00:00:54.910 --> 00:00:56.990
It literally
represents five ones,

00:00:56.990 --> 00:00:58.859
or you could just
say it represents 5.

00:00:58.859 --> 00:01:00.650
Now, if we wanted to
write it as five ones,

00:01:00.650 --> 00:01:05.060
we could say well, that's
going to be 5 times 1.

00:01:05.060 --> 00:01:09.540
So far, we've represented
905, 900 plus 5 or 9 times

00:01:09.540 --> 00:01:11.601
100 plus 5 times 1.

00:01:11.601 --> 00:01:13.100
And you might say
hey, how do I know

00:01:13.100 --> 00:01:15.074
whether I should
multiply or add first?

00:01:15.074 --> 00:01:17.490
Should I do this addition
before I do this multiplication?

00:01:17.490 --> 00:01:20.217
And I'll always remind
you, order of operations.

00:01:20.217 --> 00:01:22.300
In this scenario, you would
do your multiplication

00:01:22.300 --> 00:01:23.470
before you do your addition.

00:01:23.470 --> 00:01:26.730
So you would multiply your 5
times 1 and your 9 times 100

00:01:26.730 --> 00:01:29.450
before adding these
two things together.

00:01:29.450 --> 00:01:30.310
But let's move on.

00:01:30.310 --> 00:01:31.450
You have another 0.

00:01:31.450 --> 00:01:34.480
This 0 is in the tenths place.

00:01:34.480 --> 00:01:38.250
This is telling us the number
of tenths we're going to have.

00:01:38.250 --> 00:01:41.990
This is zero tenths, so
it's really not adding much,

00:01:41.990 --> 00:01:43.340
or it's not adding anything.

00:01:43.340 --> 00:01:44.756
Now we go to the
hundredths place.

00:01:48.890 --> 00:01:51.970
So this literally
represents seven hundredths.

00:01:51.970 --> 00:02:03.230
So we could write this as
7/100, or 7 times 1/100.

00:02:03.230 --> 00:02:06.320
And then finally, we go
to the thousandths place.

00:02:06.320 --> 00:02:10.520
So we go to the
thousandths place.

00:02:10.520 --> 00:02:12.430
And we have four thousandths.

00:02:12.430 --> 00:02:21.370
So that literally represents 4
over 1,000, or 4 times 1/1000.

00:02:21.370 --> 00:02:26.160
Notice this is coming
from the hundreds place.

00:02:26.160 --> 00:02:28.569
You have zero tens, but I'll
write the tens place there

00:02:28.569 --> 00:02:29.360
just so you see it.

00:02:29.360 --> 00:02:32.400
So it's zero tens, so I didn't
even bother to write that down.

00:02:32.400 --> 00:02:34.170
Then you have your ones place.

00:02:34.170 --> 00:02:35.650
You have five ones.

00:02:35.650 --> 00:02:36.970
Then you have zero tenths.

00:02:36.970 --> 00:02:38.470
So I didn't write that down.

00:02:38.470 --> 00:02:42.100
Then you have seven
hundredths and then

00:02:42.100 --> 00:02:43.183
you have four thousandths.

00:02:46.170 --> 00:02:46.850
And we are done.

00:02:46.850 --> 00:02:49.610
We've written this out,
really just understanding

00:02:49.610 --> 00:02:51.535
what this number represents.


Multiplying fractions word problem: bike to a friend

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:05.660
You can ride your bike
1/5 of a mile per minute.

00:00:05.660 --> 00:00:07.910
If it takes you
3 and 1/3 minutes

00:00:07.910 --> 00:00:09.890
to get to your friend's
house, how many miles

00:00:09.890 --> 00:00:12.020
away does your friend live?

00:00:12.020 --> 00:00:14.845
And this here is pictures
of these guys on bicycles.

00:00:14.845 --> 00:00:16.720
It's pretty clear they're
not riding to work,

00:00:16.720 --> 00:00:18.845
or some of these guys aren't
even riding a bicycle.

00:00:18.845 --> 00:00:20.490
But let's focus on the question.

00:00:20.490 --> 00:00:24.120
So you can ride your bike
1/5 of a mile per minute.

00:00:24.120 --> 00:00:26.200
And you're going to
do this for 3 and 1/3

00:00:26.200 --> 00:00:29.611
minutes-- times 3 and 1/3.

00:00:29.611 --> 00:00:31.110
So we really have
to figure out, how

00:00:31.110 --> 00:00:36.260
do we multiply 1/5
times 3 and 1/3?

00:00:36.260 --> 00:00:38.630
So there's a couple of
ways to think about it.

00:00:38.630 --> 00:00:42.310
You could literally view a 3 and
1/3 as this is the same thing

00:00:42.310 --> 00:00:49.730
as 1/5 times 3 plus 1/3.

00:00:49.730 --> 00:00:52.920
That's exactly
what 3 and 1/3 is.

00:00:52.920 --> 00:00:55.350
And then we can just apply
the distributive property.

00:00:55.350 --> 00:01:00.106
This would be 1/5
times 3-- I'm going

00:01:00.106 --> 00:01:06.770
to keep the colors the
same-- plus 1/5 times 1/3.

00:01:11.380 --> 00:01:13.860
And this is going to
be equal to-- well,

00:01:13.860 --> 00:01:20.440
we could rewrite 1/5
times 3 as 1/5 times 3/1.

00:01:20.440 --> 00:01:23.550
That's what 3 really is if
we wrote it as a fraction.

00:01:23.550 --> 00:01:28.210
And then, of course, we're going
to have plus 1/5 times 1/3.

00:01:31.430 --> 00:01:34.082
And let's just think about
what each of these evaluate to.

00:01:34.082 --> 00:01:35.540
Here you multiplied
the numerators,

00:01:35.540 --> 00:01:37.040
and you multiplied
the denominators.

00:01:37.040 --> 00:01:42.070
So this is going to be equal
to 1 times 3 over 5 times 1.

00:01:42.070 --> 00:01:44.426
And this business
right over here

00:01:44.426 --> 00:01:46.550
is going to be-- and
remember, order of operations.

00:01:46.550 --> 00:01:49.030
We want to do our
multiplication first.

00:01:49.030 --> 00:01:56.500
So this is going to be 1
times 1 over 5 times 3.

00:01:56.500 --> 00:02:05.387
And so that's going to be
equal to 3/5 plus 1/15.

00:02:05.387 --> 00:02:07.220
And now we have different
denominators here.

00:02:07.220 --> 00:02:09.520
But lucky for us,
3/5, if we multiplied

00:02:09.520 --> 00:02:12.390
the numerator and
the denominator by 3,

00:02:12.390 --> 00:02:14.940
we're going to get
a denominator of 15.

00:02:14.940 --> 00:02:22.490
And so that's equal to 9/15
plus 1/15, which equals 10/15.

00:02:22.490 --> 00:02:25.380
And if you divide the numerator
and the denominator both by 5,

00:02:25.380 --> 00:02:27.750
you're going to get 2/3.

00:02:27.750 --> 00:02:30.956
So your friend lives 2/3
miles away from your house.

00:02:30.956 --> 00:02:32.330
Well, that's kind
of interesting.

00:02:32.330 --> 00:02:34.770
And this was kind of
a long way to do it.

00:02:34.770 --> 00:02:37.410
Let's think about if there's
a simpler way to do it.

00:02:37.410 --> 00:02:40.570
So this is the same
thing as 1/5 times--

00:02:40.570 --> 00:02:44.930
and I'm just going to write
3 and 1/3 as a mixed number.

00:02:44.930 --> 00:02:50.890
So it's 1/5 times 3 and 1/3 can
be rewritten as 9/3-- sorry,

00:02:50.890 --> 00:02:53.680
I'm going to rewrite 3 and
1/3 as an improper fraction.

00:02:53.680 --> 00:02:55.150
So this is the
same thing as 9/3--

00:02:55.150 --> 00:03:01.795
that's 3-- plus 1/3, which is
the same thing as 1/5-- well, I

00:03:01.795 --> 00:03:03.370
switched colors
arbitrarily-- which

00:03:03.370 --> 00:03:09.710
is the same thing-- I'm still
on the same color-- as 1/5

00:03:09.710 --> 00:03:16.180
times 9/3 plus 1/3 is 10/3.

00:03:16.180 --> 00:03:17.950
And now we can just
multiply the numerator

00:03:17.950 --> 00:03:21.700
and multiply the denominator--
or multiply the numerators.

00:03:21.700 --> 00:03:26.010
So this is 1 times
10-- I'm trying

00:03:26.010 --> 00:03:30.010
to stay good with the
color coding-- over 5 times

00:03:30.010 --> 00:03:37.820
3, which is exactly equal
to what we just got.

00:03:37.820 --> 00:03:40.410
1 times 10 is equal to 10.

00:03:40.410 --> 00:03:42.312
5 times 3 is 15.

00:03:42.312 --> 00:03:46.790
10/15, we already established,
is the same thing as 2/3.

00:03:46.790 --> 00:03:51.020
So your friend lives 2/3
of a mile away from you.


Multiplying fractions word problem: laundry emergency

WEBVTT
Kind: captions
Language: en

00:00:00.950 --> 00:00:05.240
Gina had 2/3 cups of
laundry detergent.

00:00:05.240 --> 00:00:07.180
She used half of the
detergent on Friday

00:00:07.180 --> 00:00:08.690
to wash all of her sheets.

00:00:08.690 --> 00:00:10.910
How much detergent
does she have left?

00:00:10.910 --> 00:00:14.630
So we're essentially going to
take half of 2/3, or 1/2 times

00:00:14.630 --> 00:00:15.860
2/3.

00:00:15.860 --> 00:00:17.970
So let's see which of
these choices match up.

00:00:17.970 --> 00:00:20.820
So we should be
taking 1/2 times 2/3.

00:00:20.820 --> 00:00:23.550
Well, here they're
taking 2/3 minus 1/2,

00:00:23.550 --> 00:00:25.290
so this isn't going to be right.

00:00:25.290 --> 00:00:28.530
Here they're going to
be taking-- let's see,

00:00:28.530 --> 00:00:32.290
1/3, because 2/3 times 1/2, this
is exactly what we want to do.

00:00:32.290 --> 00:00:34.280
We want to take half of 2/3.

00:00:34.280 --> 00:00:36.870
And we multiply the
numerators, 2 times 1 is 2.

00:00:36.870 --> 00:00:39.810
Multiply the denominators,
3 times 2 is 6.

00:00:39.810 --> 00:00:41.470
You have 2/6.

00:00:41.470 --> 00:00:45.340
You could say the 2/6 has either
been used up or 2/6 is left.

00:00:45.340 --> 00:00:47.740
And 2/6 is the exact
same thing as 1/3.

00:00:47.740 --> 00:00:50.640
Divide the numerator and
the denominator by 2,

00:00:50.640 --> 00:00:52.430
so this looks right.

00:00:52.430 --> 00:00:56.320
And so here 1/6 cup, well,
we know it's not 1/6 cup.

00:00:56.320 --> 00:00:57.860
We know it's 1/3 cup.

00:00:57.860 --> 00:01:01.950
So here, 1/3 cup, because
2/3 is equal to 1/3 plus 1/3,

00:01:01.950 --> 00:01:02.770
that's right.

00:01:02.770 --> 00:01:04.840
So half of 2/3 must be 1/3.

00:01:04.840 --> 00:01:06.610
This is exactly right as well.

00:01:06.610 --> 00:01:08.470
These are both completely
reasonable ways

00:01:08.470 --> 00:01:10.200
to getting to the
right answer that Gina

00:01:10.200 --> 00:01:14.340
has 1/3-- she used 1/3 of
a cup of laundry detergent,

00:01:14.340 --> 00:01:17.350
and she has 1/3 left.


Multiplying fractions word problem: movie marathon

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:02.860
You're having a movie marathon.

00:00:02.860 --> 00:00:06.960
Each movie lasts
about 1 and 3/4 hours.

00:00:06.960 --> 00:00:09.420
If you want to watch
3 movies, how long

00:00:09.420 --> 00:00:11.660
will your movie marathon last?

00:00:11.660 --> 00:00:13.490
You can use the
number line below

00:00:13.490 --> 00:00:15.970
to help you find your answer.

00:00:15.970 --> 00:00:20.020
So each movie lasts
1 and 3/4 hours long.

00:00:20.020 --> 00:00:23.330
So let's say the
first movie happens

00:00:23.330 --> 00:00:24.710
to be Star Wars,
which I probably

00:00:24.710 --> 00:00:27.250
would include in
any movie marathon.

00:00:27.250 --> 00:00:29.480
So it's going to be
1 and 3/4 hours long.

00:00:29.480 --> 00:00:32.388
So this is 1,.

00:00:32.388 --> 00:00:35.150
1, and then we're
going to have 3/4 more.

00:00:35.150 --> 00:00:36.940
So each of these are
a fourth because we

00:00:36.940 --> 00:00:40.770
have 4 sections between
1 and 2-- 1, 2, 3, 4.

00:00:40.770 --> 00:00:42.070
So we're going to watch 3/4.

00:00:42.070 --> 00:00:44.340
1, 2, 3.

00:00:44.340 --> 00:00:46.960
So that's how long
my first movie is.

00:00:46.960 --> 00:00:50.920
And then, my second movie is
going to be another 1 and 3/4.

00:00:50.920 --> 00:00:53.150
So 1 is 4/4.

00:00:53.150 --> 00:00:55.842
So 1, 2, 3, 4.

00:00:55.842 --> 00:00:57.550
And then we're going
to have another 3/4.

00:00:57.550 --> 00:00:59.940
1, 2, 3.

00:00:59.940 --> 00:01:03.370
And just as the way I said
that, 4/4 and 3/4 is 7/4.

00:01:03.370 --> 00:01:06.606
So 1 and 3/4, that's
the same thing as 7/4.

00:01:06.606 --> 00:01:07.730
And you see that over here.

00:01:07.730 --> 00:01:09.450
When I did the 1
and 3/4 in blue,

00:01:09.450 --> 00:01:11.580
I actually marked off 7/4.

00:01:11.580 --> 00:01:15.300
Same thing here,
here in magenta.

00:01:15.300 --> 00:01:18.700
And now, let's do it in green.

00:01:18.700 --> 00:01:21.770
So Gandhi is actually a good
bit longer than 1 and 3/4,

00:01:21.770 --> 00:01:23.660
but it's a good movie,
so I put it here

00:01:23.660 --> 00:01:25.570
just because it's
one of my favorites.

00:01:25.570 --> 00:01:29.320
And so let's do another
1 and 3/4 hours.

00:01:29.320 --> 00:01:33.180
Or we could do that as 4/4
plus 3/4 hours, or 7/4.

00:01:33.180 --> 00:01:38.390
1, 2, 3, 4, 5, 6, 7.

00:01:38.390 --> 00:01:42.610
So that gets us to 5.

00:01:42.610 --> 00:01:48.650
So this point right over
here is 5 and 1/4 hours.

00:01:48.650 --> 00:01:51.870
Now, let's see if we
get that same result

00:01:51.870 --> 00:01:54.200
when we do it mathematically.

00:01:54.200 --> 00:01:56.270
I guess this is
mathematical as well.

00:01:56.270 --> 00:01:58.240
But every movie lasts
1 and 3/4 hours.

00:01:58.240 --> 00:02:00.700
We already said
that 1 and 3/4 is

00:02:00.700 --> 00:02:04.810
the same thing as 1 plus 3/4.

00:02:04.810 --> 00:02:07.010
1 is the same thing as 4/4.

00:02:07.010 --> 00:02:11.870
So it's 4/4 plus 3/4, which
is the same thing as 7/4.

00:02:11.870 --> 00:02:13.810
And another way to
convert this mixed number

00:02:13.810 --> 00:02:16.000
into an improper fraction,
you say, hey, look.

00:02:16.000 --> 00:02:18.140
4 times 1 is 4--
and that's really

00:02:18.140 --> 00:02:21.120
where you're getting the
4/4 from-- plus 3 is 7.

00:02:21.120 --> 00:02:23.330
7/4.

00:02:23.330 --> 00:02:26.600
Now, we're going to do 7/4
of an hour three times.

00:02:26.600 --> 00:02:33.000
So we're going to
multiply 7/4 times 3.

00:02:33.000 --> 00:02:35.610
Now, there are several ways
that you could think about this.

00:02:35.610 --> 00:02:39.410
You could literally view
this as having three 7/4

00:02:39.410 --> 00:02:41.170
and adding them together.

00:02:41.170 --> 00:02:50.180
So 7/4 plus 7/4 plus 7/4,
which would be equal to 21/4.

00:02:50.180 --> 00:02:52.430
And if you wanted to convert
this into a mixed number,

00:02:52.430 --> 00:02:56.620
21/4 is the same thing as--
well, 4 goes into 20 evenly.

00:02:56.620 --> 00:03:00.240
So you could call
this 20/4 plus 1/4.

00:03:00.240 --> 00:03:03.030
And 20/4 is the same thing as 5.

00:03:03.030 --> 00:03:08.220
So it's 5 and 1/4, which is
exactly what we saw here.

00:03:08.220 --> 00:03:12.090
Another way of thinking
about it, you could say 7/4.

00:03:12.090 --> 00:03:18.192
And then we can multiply
it by 3/1, times 3/1,

00:03:18.192 --> 00:03:20.650
just to make it clear that
we're multiplying the numerators

00:03:20.650 --> 00:03:22.820
and multiplying
the denominators.

00:03:22.820 --> 00:03:26.230
So in the numerator, I'm
to have 7 plus 7 plus 7.

00:03:26.230 --> 00:03:27.205
That's 7 times 3.

00:03:27.205 --> 00:03:28.580
And in the
denominator, I'm going

00:03:28.580 --> 00:03:30.900
to have 4 times 1,
which is just 4.

00:03:30.900 --> 00:03:34.560
Well, this is just
7 times 3 over 4,

00:03:34.560 --> 00:03:40.808
or 21/4, which we've
already seen is 5 and 1/4.


Multiplying fractions word problem: milk love

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:02.440
If there are 6 people
in your family,

00:00:02.440 --> 00:00:06.650
and each person drinks 4/5 of
a gallon of milk each month,

00:00:06.650 --> 00:00:09.390
between what two whole numbers
is the number of gallons

00:00:09.390 --> 00:00:13.080
of milk your family
will drink each month?

00:00:13.080 --> 00:00:15.920
So let's think about
this a little bit.

00:00:15.920 --> 00:00:19.990
So I've got my milk gallon
here for inspiration.

00:00:19.990 --> 00:00:21.657
So there's 6 people
in my family,

00:00:21.657 --> 00:00:23.990
or in your family, whoever's
family we're talking about.

00:00:23.990 --> 00:00:27.960
And each of them drink
4/5 of a gallon per month.

00:00:27.960 --> 00:00:32.910
So we're going to go 6
times 4/5, or 4/5 times 6.

00:00:32.910 --> 00:00:34.930
Now, what is this
going to be equal to?

00:00:34.930 --> 00:00:37.420
Well, there are several ways
we could think about this.

00:00:37.420 --> 00:00:40.320
One, we can express
6 as a fraction.

00:00:40.320 --> 00:00:45.110
We could say that this is the
same thing as 6/1 times 4/5.

00:00:48.100 --> 00:00:49.510
And you multiply the numerators.

00:00:49.510 --> 00:00:51.320
You multiply the denominators.

00:00:51.320 --> 00:00:53.590
6 times 4 is 24.

00:00:53.590 --> 00:00:55.570
And 1 times 5 is 5.

00:00:55.570 --> 00:01:00.185
So 24/5 gallons are going
to be drunk every month.

00:01:00.185 --> 00:01:01.810
And this is the same
thing-- let's see,

00:01:01.810 --> 00:01:04.980
5 goes into 20 evenly
and has a remainder of 4.

00:01:04.980 --> 00:01:08.861
So we could write this
as 20/5 plus 4/5 5--

00:01:08.861 --> 00:01:10.860
let me move over a little
bit so I don't run out

00:01:10.860 --> 00:01:17.430
of space, which
is the same thing.

00:01:17.430 --> 00:01:18.300
This is 4.

00:01:18.300 --> 00:01:20.670
So this is 4 and 4/5.

00:01:20.670 --> 00:01:22.470
So that's one way
to think about it.

00:01:22.470 --> 00:01:27.280
The other way, you could
think of this as 4/5 times 6,

00:01:27.280 --> 00:01:31.730
which is the same thing
as 4/5 plus 4/5 plus 4/5--

00:01:31.730 --> 00:01:36.720
I think you get the idea--
plus 4/5 plus 4/5 plus 4/5.

00:01:36.720 --> 00:01:39.599
We took literally six 4/5's
and added them together.

00:01:39.599 --> 00:01:41.140
And what's this
going to be equal to?

00:01:41.140 --> 00:01:44.810
Well, this is going to
be 4 plus 4 plus 4 plus 4

00:01:44.810 --> 00:01:49.400
plus 4 plus 4 over 5.

00:01:49.400 --> 00:01:51.210
This is how many 1/5's we have.

00:01:51.210 --> 00:01:53.530
Well, this is 6
times 4, or you could

00:01:53.530 --> 00:01:59.740
say 4 times 6, over 5,
which is equal to 24 over 5,

00:01:59.740 --> 00:02:02.610
which we've already
figured out is 4 and 4/5.

00:02:02.610 --> 00:02:04.450
So either way,
we're going to drink

00:02:04.450 --> 00:02:05.900
4 and 4/5 gallons per month.

00:02:05.900 --> 00:02:08.139
Now let's try to answer
the actual question.

00:02:08.139 --> 00:02:09.539
Between what two whole numbers?

00:02:09.539 --> 00:02:12.650
Well, between 4 and
5 gallons a month.

00:02:12.650 --> 00:02:15.950
4 and 4/5 is right
in between 4 and 5.

00:02:15.950 --> 00:02:18.390
We got it right.


Multiplying fractions word problem: pigging out on pumpkin pie

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.340
Ken and Isaiah are eating
pumpkin pie with their friends.

00:00:04.340 --> 00:00:08.029
They want to figure out how
much pie they have eaten so far.

00:00:08.029 --> 00:00:10.120
There are 7 of them in
total and each of them

00:00:10.120 --> 00:00:12.870
has eaten 2/5 of a pie.

00:00:12.870 --> 00:00:14.720
Ken said you can
solve this problem

00:00:14.720 --> 00:00:16.455
by multiplying 2/5 times 7.

00:00:16.455 --> 00:00:17.330
And that makes sense.

00:00:17.330 --> 00:00:19.310
Each of them ate 2/5,
and there are 7 folks.

00:00:19.310 --> 00:00:21.340
I would multiply 2/5 times 7.

00:00:21.340 --> 00:00:24.480
So he multiplied and got 14/35.

00:00:24.480 --> 00:00:26.387
Now this is a little
bit suspicious.

00:00:26.387 --> 00:00:27.720
And here's the work that he did.

00:00:27.720 --> 00:00:32.540
He said 2/5 times 7 is equal
to 2 times 7 over 5 times 7,

00:00:32.540 --> 00:00:34.410
which equals 14/35.

00:00:34.410 --> 00:00:37.650
So this is starting to
smell real fishy right now.

00:00:37.650 --> 00:00:42.250
Isaiah says that 14/35 cannot
be correct because 14/35 is less

00:00:42.250 --> 00:00:43.330
than one whole pie.

00:00:43.330 --> 00:00:45.680
So this is definitely
true, that 14/35

00:00:45.680 --> 00:00:48.350
is less than one whole pie.

00:00:48.350 --> 00:00:52.820
Which of the following best
explains this situation?

00:00:52.820 --> 00:00:54.620
So before even looking
at the explanations,

00:00:54.620 --> 00:00:56.150
let's see what's
fishy about this.

00:00:56.150 --> 00:00:59.150
So let me get my little
notepad and write it out.

00:00:59.150 --> 00:01:04.069
So he literally took 2/5, and he
attempted to multiply it by 7.

00:01:04.069 --> 00:01:06.360
And he said that this is the
same thing as taking the 2

00:01:06.360 --> 00:01:10.160
times the 7, over
the 5 times the 7.

00:01:10.160 --> 00:01:12.880
And that's how he got the 14/35.

00:01:12.880 --> 00:01:15.300
And I encourage you to pause
this and try to figure out

00:01:15.300 --> 00:01:17.599
why this doesn't
make sense yourself.

00:01:17.599 --> 00:01:19.640
Now let's think about what
actually went on here.

00:01:23.620 --> 00:01:26.620
If you multiply the
numerator times 7

00:01:26.620 --> 00:01:28.460
and the denominator
by 7, you're actually

00:01:28.460 --> 00:01:30.450
not changing the
value of the fraction.

00:01:30.450 --> 00:01:36.820
This is equivalent to saying 2/5
times 7/7, which is equivalent

00:01:36.820 --> 00:01:41.640
to 2/5 times 1, which
is equivalent to 2/5.

00:01:41.640 --> 00:01:45.390
14/35 is just another way, it's
an equivalent representation

00:01:45.390 --> 00:01:45.990
of 2/5.

00:01:45.990 --> 00:01:49.970
This is just how much pie only
one person should have eaten.

00:01:49.970 --> 00:01:51.732
So how should he have
thought about this?

00:01:51.732 --> 00:01:53.940
Well, there's a couple of
ways you could think of it.

00:01:53.940 --> 00:01:57.800
2/5 times 7 could
literally mean seven 2/5.

00:01:57.800 --> 00:02:00.520
It literally could
mean, so 2/5 times 7,

00:02:00.520 --> 00:02:05.370
literally means one 2/5, plus
another 2/5, plus another 2/5.

00:02:05.370 --> 00:02:07.330
So we do this seven times.

00:02:07.330 --> 00:02:09.080
So that's four 2/5.

00:02:09.080 --> 00:02:10.990
That's five 2/5.

00:02:10.990 --> 00:02:12.710
That's six 2/5.

00:02:12.710 --> 00:02:14.640
And that's seven 2/5.

00:02:14.640 --> 00:02:16.430
Sorry, my brain isn't working.

00:02:16.430 --> 00:02:19.012
And that's seven 2/5.

00:02:19.012 --> 00:02:20.470
And if you were to
add all of these

00:02:20.470 --> 00:02:22.820
together, how many
fifths do you have?

00:02:22.820 --> 00:02:24.750
How many fifths do you have?

00:02:24.750 --> 00:02:29.802
Well, you have 2 plus
2 plus 2 plus 2 plus 2.

00:02:29.802 --> 00:02:30.760
Let's see, that's five.

00:02:30.760 --> 00:02:32.900
Plus 2, plus 2.

00:02:32.900 --> 00:02:36.220
You have 2, seven times, fifths.

00:02:36.220 --> 00:02:39.650
Or this is another way of saying
you have 7 times-- let me write

00:02:39.650 --> 00:02:44.420
it this way-- you
have 2 times 7 fifths.

00:02:44.420 --> 00:02:47.335
Or another way of saying it
is this is equal to 14/5.

00:02:53.930 --> 00:02:55.830
This is well over more than 1.

00:02:55.830 --> 00:02:59.460
5 goes into 14 two times, and
you have a remainder of 4.

00:02:59.460 --> 00:03:01.290
So it's 2 and 4/5 pie.

00:03:01.290 --> 00:03:03.590
So this is what he
should have done.

00:03:03.590 --> 00:03:06.730
But you might be saying, well,
how would I just multiply this

00:03:06.730 --> 00:03:09.030
if I didn't even have
to think it through,

00:03:09.030 --> 00:03:10.874
adding all these 2/5 together?

00:03:10.874 --> 00:03:12.290
Well, one way to
think about it is

00:03:12.290 --> 00:03:15.900
that 7 is the same thing as 7/1.

00:03:15.900 --> 00:03:22.160
So he could have just
said 2/5 times 7/1.

00:03:22.160 --> 00:03:24.410
7/1 is the exact
same thing as 7.

00:03:24.410 --> 00:03:26.190
And that would be
equal to 2 times 7

00:03:26.190 --> 00:03:28.250
in the numerator, which is 14.

00:03:28.250 --> 00:03:33.240
And the denominator would be 5
times 1, which is equal to 5.

00:03:33.240 --> 00:03:35.360
And you would get
the same answer.

00:03:35.360 --> 00:03:38.440
So now let's actually go back
and select the right choice.

00:03:38.440 --> 00:03:41.570
I forgot that we actually
had to say which explanation

00:03:41.570 --> 00:03:42.660
is the right explanation.

00:03:42.660 --> 00:03:45.440
So explanation 1 is that Ken
didn't multiply correctly.

00:03:45.440 --> 00:03:49.254
Multiplying 2/5 times 7 is the
same as adding 2/5 seven times.

00:03:49.254 --> 00:03:50.170
This is exactly right.

00:03:50.170 --> 00:03:52.100
This is exactly what
we just went through.

00:03:52.100 --> 00:03:55.460
The correct answer
is 14/5 or 2 and 4/5.

00:03:55.460 --> 00:03:57.570
So explanation A seems
to be the right one,

00:03:57.570 --> 00:03:59.230
but we'll just read
the other ones just

00:03:59.230 --> 00:04:01.810
to see if there's
some flaws in them.

00:04:01.810 --> 00:04:04.820
Explanation B, Ken multiplied
correctly but forgot

00:04:04.820 --> 00:04:07.760
to cancel out the
7's in the fractions.

00:04:07.760 --> 00:04:11.040
Since 7/7 is equal
to 1, 2/5 times 7

00:04:11.040 --> 00:04:13.610
is equal to 2/5 times 1.

00:04:13.610 --> 00:04:15.340
Well, obviously,
2/5 times 7 is not

00:04:15.340 --> 00:04:16.750
the same thing as 2/5 times 1.

00:04:16.750 --> 00:04:18.700
So this is kind of nutty.

00:04:18.700 --> 00:04:22.060
Explanation C, since Ken
must add up all the pie

00:04:22.060 --> 00:04:24.930
for 7 of them, he should
have added 7 and 2/5

00:04:24.930 --> 00:04:26.450
instead of multiplying.

00:04:26.450 --> 00:04:27.510
No, that makes no sense.

00:04:27.510 --> 00:04:28.551
He should be multiplying.

00:04:28.551 --> 00:04:29.989
Ken is correct.

00:04:29.989 --> 00:04:31.780
His friends just didn't
each that much pie.

00:04:31.780 --> 00:04:33.600
No, that doesn't
make sense either.

00:04:33.600 --> 00:04:35.710
It's explanation A.


Word problem: What fraction of an hour should the piano still be practiced?

WEBVTT
Kind: captions
Language: en

00:00:00.980 --> 00:00:03.050
Pedro is supposed
to practice piano

00:00:03.050 --> 00:00:05.780
for 3/4 of an hour every day.

00:00:05.780 --> 00:00:09.980
Today, he has practiced
for 1/4 of an hour.

00:00:09.980 --> 00:00:13.900
What fraction of an hour
does he need to practice?

00:00:13.900 --> 00:00:16.239
So let's visualize 3/4.

00:00:16.239 --> 00:00:18.480
So he needs to practice
3/4 of an hour.

00:00:18.480 --> 00:00:24.300
So if this represents
an entire hour,

00:00:24.300 --> 00:00:26.270
so let's divide it into fourths.

00:00:26.270 --> 00:00:27.547
So now, divide it into halves.

00:00:27.547 --> 00:00:29.380
Now, let's make it into
four equal sections.

00:00:29.380 --> 00:00:30.870
Now, it's into fourths.

00:00:30.870 --> 00:00:35.730
So he needs to practice
3/4 of an hour.

00:00:38.450 --> 00:00:41.110
That was my attempt
at drawing an arrow.

00:00:41.110 --> 00:00:43.040
So let me shade it in.

00:00:43.040 --> 00:00:47.930
So he needs to practice
3/4 of an hour.

00:00:47.930 --> 00:00:52.180
This gets us to 3/4
right over here.

00:00:52.180 --> 00:00:55.640
So he needs to practice 3 out
of the fourths of an hour.

00:00:55.640 --> 00:00:58.060
So this is 3/4.

00:00:58.060 --> 00:01:01.460
Now, it says he has already
practiced for 1/4 of an hour.

00:01:01.460 --> 00:01:03.490
So he's already
practiced for 1/4.

00:01:06.270 --> 00:01:08.965
So how much more does
he need to practice?

00:01:11.720 --> 00:01:15.650
Well, he needs to
practice this much more.

00:01:15.650 --> 00:01:17.630
And you might already
see the answer visually.

00:01:17.630 --> 00:01:19.710
But let's think about
how to represent it

00:01:19.710 --> 00:01:23.190
as a fraction expression.

00:01:23.190 --> 00:01:25.550
So let me write it like this.

00:01:25.550 --> 00:01:27.130
So this is how much
he needs to go.

00:01:27.130 --> 00:01:31.160
How much does he
need to practice?

00:01:31.160 --> 00:01:34.510
So he needs to do
a total of 3/4.

00:01:34.510 --> 00:01:36.110
He's already done 1/4.

00:01:39.080 --> 00:01:41.331
So if you subtract
1/4 from the 3/4,

00:01:41.331 --> 00:01:43.330
you're going to get this
amount right over here.

00:01:43.330 --> 00:01:47.260
You're going to get the amount
that he needs to practice.

00:01:47.260 --> 00:01:49.740
Now, this already has
the same denominator.

00:01:49.740 --> 00:01:55.850
So this is going to be
equal to 3 minus 1 over 4,

00:01:55.850 --> 00:01:58.814
which is equal to 2/4.

00:01:58.814 --> 00:02:00.230
And you see that
right over there.

00:02:00.230 --> 00:02:06.580
He still needs to practice 1
and 2 out of the fourths, 2/4.

00:02:06.580 --> 00:02:09.759
Now, 2/4, we know, is
the same thing as 1/2.

00:02:09.759 --> 00:02:11.595
There's a couple of
ways to think about it.

00:02:11.595 --> 00:02:12.970
You could say,
hey, look, this is

00:02:12.970 --> 00:02:14.820
half of the length
of this entire thing.

00:02:14.820 --> 00:02:17.350
It has a little bit on
this end and that end.

00:02:17.350 --> 00:02:27.890
Or if you make 4 equal
blocks, and if you

00:02:27.890 --> 00:02:30.640
were to shade in
2 of them, you see

00:02:30.640 --> 00:02:33.160
that you have shaded
in 1/2 of the blocks.

00:02:33.160 --> 00:02:35.990
This is the exact same
amount as if I just divide it

00:02:35.990 --> 00:02:39.140
into 2 sections and I
shaded in only 1 of them.

00:02:39.140 --> 00:02:42.160
2/4 is the same thing as 1/2.

00:02:42.160 --> 00:02:44.160
And if you wanted to work
it out mathematically,

00:02:44.160 --> 00:02:45.180
you just have to
do the same thing

00:02:45.180 --> 00:02:46.790
to the numerator
and the denominator.

00:02:46.790 --> 00:02:49.540
So let's divide the
numerator and the denominator

00:02:49.540 --> 00:02:52.576
by 2, because they are
both divisible by 2.

00:02:52.576 --> 00:02:54.450
That's actually their
greatest common factor.

00:02:54.450 --> 00:02:56.560
So 2 divided by 2 is 1.

00:02:56.560 --> 00:02:58.472
4 divided by 2 is 2.

00:02:58.472 --> 00:03:00.680
So what fraction of an hour
does he need to practice?

00:03:00.680 --> 00:03:03.440
He's got to practice
1/2 an hour.


Word problem: How long is this lizard?

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.710
James and Karlee caught a
lizard in their backyard.

00:00:03.710 --> 00:00:07.920
Its body is 17/8 inches
long, and its tail

00:00:07.920 --> 00:00:09.930
is 10/8 inches long.

00:00:09.930 --> 00:00:12.230
What is the length of the
lizard's body and tail

00:00:12.230 --> 00:00:12.740
together?

00:00:12.740 --> 00:00:15.880
So good thing we have
the lizard obediently

00:00:15.880 --> 00:00:19.400
standing next to our
measuring stick or our ruler

00:00:19.400 --> 00:00:21.250
or whatever this happens to be.

00:00:21.250 --> 00:00:23.014
And we have it
marked off in inches.

00:00:23.014 --> 00:00:24.680
And not only that,
we have it marked off

00:00:24.680 --> 00:00:25.620
in eighths of inches.

00:00:25.620 --> 00:00:30.850
Notice-- this is an inch, and
we have 1, 2, 3, 4, 5, 6, 7,

00:00:30.850 --> 00:00:34.220
8 sections of that actual inch.

00:00:34.220 --> 00:00:37.290
So, let's see, 17/8 inches long.

00:00:37.290 --> 00:00:46.013
This is going to be 1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13,

00:00:46.013 --> 00:00:49.710
14, 15, 16, 17.

00:00:49.710 --> 00:00:54.820
This right over
here is 17/8, which

00:00:54.820 --> 00:00:58.440
you might notice is
also equal to 2 and 1/8.

00:00:58.440 --> 00:01:01.040
8 goes into 17 two times
with a remainder of 1.

00:01:01.040 --> 00:01:02.900
But we won't talk
about that just yet.

00:01:02.900 --> 00:01:09.110
But we see that's the length
of the lizard's body-- 17/8.

00:01:09.110 --> 00:01:13.960
I just counted off 17 of these
8's right over here-- 17/8.

00:01:13.960 --> 00:01:19.130
And then they say that the
tail is 10/8 inches long.

00:01:19.130 --> 00:01:28.970
So it's another 1, 2,
3, 4, 5, 6, 7, 8, 9, 10.

00:01:28.970 --> 00:01:33.040
So this distance right over
here, I just counted off 10/8.

00:01:33.040 --> 00:01:35.840
So that's another
10/8 inches long,

00:01:35.840 --> 00:01:39.540
and it gets us all the way to
the end of the lizard's tail.

00:01:39.540 --> 00:01:42.100
And they say what is the length
of the lizard's body and tail

00:01:42.100 --> 00:01:42.460
together?

00:01:42.460 --> 00:01:44.470
Well, that's going to be
the length of the body

00:01:44.470 --> 00:01:46.370
plus the length of the tail.

00:01:46.370 --> 00:01:49.560
So you have 17/8 plus 10/8.

00:01:49.560 --> 00:01:58.010
Well, that's going to
get you to 17 plus 10/8,

00:01:58.010 --> 00:02:04.870
which is equal to 27-- 27/8.

00:02:04.870 --> 00:02:08.100
And so this mark right
over here is 27/8.

00:02:08.100 --> 00:02:10.289
And if you wanted to write
this as a mixed number,

00:02:10.289 --> 00:02:13.330
you could just say, well,
8 goes into 27 three times,

00:02:13.330 --> 00:02:15.140
and you have a remainder of 3.

00:02:15.140 --> 00:02:17.974
So this is equal to 3 and 3/8.

00:02:17.974 --> 00:02:19.890
And you see that on the
ruler right over here.

00:02:19.890 --> 00:02:23.210
It's equal to 1, 2, 3.

00:02:23.210 --> 00:02:26.140
And then we have
another 1, 2, 3/8.

00:02:26.140 --> 00:02:28.650
These two things are equivalent.


Word problem: What fraction of spider eyes are looking at me?

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.150
A common house
spider has 8 eyes.

00:00:04.150 --> 00:00:09.070
If a spider is looking at
you with its 2 front eyes

00:00:09.070 --> 00:00:15.849
and 3 of its other eyes,
what fraction of its eyes

00:00:15.849 --> 00:00:16.640
are looking at you?

00:00:16.640 --> 00:00:18.848
And just to show that we're
not making this stuff up,

00:00:18.848 --> 00:00:20.540
this is actually
pictures of spiders.

00:00:20.540 --> 00:00:22.290
And you see in every
one of these pictures

00:00:22.290 --> 00:00:24.456
they have eight eyes, even
this one's got other eyes

00:00:24.456 --> 00:00:27.980
right over here-- 1,
2, 3, 4, 5, 6, 7, 8.

00:00:27.980 --> 00:00:29.140
I never even realized this.

00:00:29.140 --> 00:00:31.450
This is actually quite creepy.

00:00:31.450 --> 00:00:34.070
But anyway, let's actually
focus on the problem.

00:00:34.070 --> 00:00:38.430
So it's looking at you
with 2 of its 8 eyes.

00:00:38.430 --> 00:00:42.420
So the 2 front eyes represent
what fraction of the eyes?

00:00:42.420 --> 00:00:48.089
Well, it represents
2 of the 8 eyes.

00:00:48.089 --> 00:00:49.630
You could look at
it right over here.

00:00:49.630 --> 00:00:50.449
You have 8 eyes.

00:00:50.449 --> 00:00:51.490
We have the 2 front eyes.

00:00:51.490 --> 00:00:54.110
That's 1 and 2 of the 8.

00:00:54.110 --> 00:00:57.270
And it says it's also looking
at us with 3 of its other eyes.

00:00:57.270 --> 00:01:02.700
So it's also looking at
us with 3 of the 8 eyes.

00:01:02.700 --> 00:01:04.269
And so you can
imagine that might

00:01:04.269 --> 00:01:07.527
be that one, that
one, and that one.

00:01:07.527 --> 00:01:09.235
So this is really
about adding fractions.

00:01:09.235 --> 00:01:12.460
This is about adding 2/8 of
something to 3/8 of something.

00:01:12.460 --> 00:01:16.380
So what fraction of its
eyes are looking at us?

00:01:16.380 --> 00:01:18.770
Well, it's still going
to be a fraction of 8.

00:01:18.770 --> 00:01:28.120
It's going to be still out
of 8-- 2 plus 3 over 8,

00:01:28.120 --> 00:01:32.280
which is just going to be
equal to 5/8 of the eyes.

00:01:32.280 --> 00:01:34.890
5 out of the 8 eyes
are looking at us.

00:01:34.890 --> 00:01:38.250
And we see it right over
here-- 1, 2, 3, 4, 5.

00:01:38.250 --> 00:01:41.770
5 of the total of 8
eyes are looking at us

00:01:41.770 --> 00:01:44.128
in a fairly creepy way.


Decomposing a fraction visually

WEBVTT
Kind: captions
Language: en

00:00:00.654 --> 00:00:02.570
So let's think about all
of the different ways

00:00:02.570 --> 00:00:04.105
that we can represent 7/9.

00:00:07.510 --> 00:00:10.130
So let's just visualize 7/9.

00:00:10.130 --> 00:00:14.280
So here I have 9 equal sections.

00:00:14.280 --> 00:00:18.820
And 7/9 you could represent
as 7 of those equal sections.

00:00:18.820 --> 00:00:22.710
So let me get myself a
bigger thing to draw with,

00:00:22.710 --> 00:00:25.237
so that I can fill this in fast.

00:00:25.237 --> 00:00:26.820
Actually, I don't
like how that looks.

00:00:26.820 --> 00:00:29.520
I'm going to use
the paint brush.

00:00:29.520 --> 00:00:30.530
So here we go.

00:00:30.530 --> 00:00:39.020
So that's 1, 2, 3,
4-- you know where

00:00:39.020 --> 00:00:45.470
this is going-- 5, 6, and 7.

00:00:45.470 --> 00:00:47.366
So that's one way
of representing 7/9.

00:00:47.366 --> 00:00:48.240
We already know that.

00:00:48.240 --> 00:00:49.364
That's not too interesting.

00:00:49.364 --> 00:00:51.150
But let's see if we
can represent 7/9

00:00:51.150 --> 00:00:53.290
as the sum of other fractions.

00:00:53.290 --> 00:00:56.350
So let's imagine maybe
we can represent it

00:00:56.350 --> 00:01:00.040
as-- let's do it as 2/9.

00:01:00.040 --> 00:01:02.470
Let me use a
different brush here.

00:01:02.470 --> 00:01:06.285
So let's represent it as 2/9.

00:01:06.285 --> 00:01:18.350
2/9 plus-- I don't know,
let's see, maybe 3/9.

00:01:18.350 --> 00:01:21.140
But that doesn't quite
get us to 7/9 yet.

00:01:21.140 --> 00:01:24.960
2/9 plus 3/9 is going
to get us to 5/9.

00:01:24.960 --> 00:01:26.210
So we're going to need 2 more.

00:01:26.210 --> 00:01:28.140
So it's going to be
plus another 2/9.

00:01:35.200 --> 00:01:37.270
So what would this look like?

00:01:37.270 --> 00:01:40.964
So let's just draw
another grid here.

00:01:40.964 --> 00:01:42.630
So this is going to
look like-- and I'll

00:01:42.630 --> 00:01:44.254
try to do it right
below it, so that we

00:01:44.254 --> 00:01:46.940
can see how they match up.

00:01:46.940 --> 00:01:51.000
So we have 2/9, this
2/9 right over here.

00:01:51.000 --> 00:01:53.000
Well, we have each
of these as a ninth.

00:01:53.000 --> 00:01:56.270
We have 9 equal sections.

00:01:56.270 --> 00:02:01.880
So we're going to get 1 and 2.

00:02:01.880 --> 00:02:05.110
And then we're going
to add 3 more ninths.

00:02:05.110 --> 00:02:13.100
So 1, 2, 3.

00:02:13.100 --> 00:02:18.380
So we add 3/9 right over
there and then 2 more ninths,

00:02:18.380 --> 00:02:23.300
1 and 2.

00:02:23.300 --> 00:02:31.249
So notice, when I added 2/9 to
3/9 to 2/9, this equals 7/9.

00:02:31.249 --> 00:02:33.290
And we know that when we
add a bunch of fractions

00:02:33.290 --> 00:02:35.181
like this that have
the same denominator,

00:02:35.181 --> 00:02:36.430
we can just add the numerator.

00:02:36.430 --> 00:02:37.100
And this is why.

00:02:37.100 --> 00:02:41.694
This is 2/9 plus 3/9 times
2/9 is going to give me 7/9.

00:02:41.694 --> 00:02:42.860
Let's do this one more time.

00:02:42.860 --> 00:02:44.600
This is actually a lot of fun.

00:02:44.600 --> 00:02:46.065
So let me draw my grid again.

00:02:49.170 --> 00:02:52.890
And then let's see
what we can do.

00:02:52.890 --> 00:02:56.430
So let me get my pen tool out.

00:02:56.430 --> 00:03:00.420
Let me make sure my
ink isn't too thick.

00:03:00.420 --> 00:03:01.600
Well, this is fine.

00:03:01.600 --> 00:03:03.360
And let's add a
couple of ninths here.

00:03:03.360 --> 00:03:06.700
So let's add first 1/9.

00:03:06.700 --> 00:03:08.810
And I'm going to draw
out all the 9's in blue.

00:03:08.810 --> 00:03:10.980
And let's add 2/9.

00:03:10.980 --> 00:03:14.915
And then we could add-- I don't
know, maybe we could add-- let

00:03:14.915 --> 00:03:17.050
me give some space here
so we can add more.

00:03:17.050 --> 00:03:18.880
And maybe we could add 3/9.

00:03:21.640 --> 00:03:26.135
And then we could add, let's
see-- actually, let me just

00:03:26.135 --> 00:03:26.760
write this out.

00:03:26.760 --> 00:03:29.980
I'm going to try to add
four fractions here.

00:03:29.980 --> 00:03:33.320
So let's say add first 1/9
and see where that gets us.

00:03:33.320 --> 00:03:37.640
So 1/9 is going to get
us right over here.

00:03:37.640 --> 00:03:39.740
So that's 1/9.

00:03:39.740 --> 00:03:41.940
So let's say we add 2/9 to that.

00:03:46.140 --> 00:03:47.990
I've got my little
paint brush going on.

00:03:47.990 --> 00:03:54.070
So that's 1 and 2 more, 2/9.

00:03:54.070 --> 00:03:55.540
So that still
doesn't get us there.

00:03:55.540 --> 00:03:57.310
This gives us a total of 3/9.

00:03:57.310 --> 00:04:00.370
1 plus 2 is 3-- 3/9.

00:04:00.370 --> 00:04:06.100
So let's add 4/9.

00:04:06.100 --> 00:04:08.780
And I'll do that
in this blue color.

00:04:08.780 --> 00:04:10.760
So 4/9.

00:04:10.760 --> 00:04:13.890
That's different enough.

00:04:13.890 --> 00:04:15.760
So let's see where this gets us.

00:04:15.760 --> 00:04:17.200
Actually, well, why not?

00:04:17.200 --> 00:04:19.370
So 4/9.

00:04:19.370 --> 00:04:28.390
And so that's going
to get us 1, 2, 3, 4.

00:04:28.390 --> 00:04:30.250
So that looks like it
got us all the way,

00:04:30.250 --> 00:04:34.057
because 1 plus 2 plus 4 is
going to give us 7-- 7/9.

00:04:34.057 --> 00:04:35.140
So what could we put here?

00:04:35.140 --> 00:04:37.090
Well, we could say 0/9.

00:04:37.090 --> 00:04:38.290
Why not?

00:04:38.290 --> 00:04:44.017
So we could call this
one right over here 0/9.

00:04:44.017 --> 00:04:45.350
And how would we visualize that?

00:04:45.350 --> 00:04:47.110
Well, we're saying
none of these.

00:04:47.110 --> 00:04:49.360
No ninths right over here.

00:04:49.360 --> 00:04:53.770
So this is 1/9 plus 2/9
plus 4/9 is equal to 7/9.

00:04:53.770 --> 00:04:55.840
So these are all different
ways to decompose

00:04:55.840 --> 00:04:58.130
the exact same fraction.


Decomposing a mixed number

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:01.970
Let's now think
about different ways

00:00:01.970 --> 00:00:03.220
to represent a mixed number.

00:00:03.220 --> 00:00:11.361
And let's say that our
mixed number is 2 and 1/8.

00:00:11.361 --> 00:00:13.610
Actually, let's make it a
little bit more interesting.

00:00:13.610 --> 00:00:16.210
Let's make it 2 and 1/4.

00:00:18.720 --> 00:00:21.890
So let's first think about
the whole number part, the 2.

00:00:21.890 --> 00:00:27.260
Well, the 2 is
literally two holes.

00:00:27.260 --> 00:00:29.010
You could literally
view that if you want.

00:00:29.010 --> 00:00:30.550
Right here we've
drawn each hole.

00:00:30.550 --> 00:00:35.400
We've cut it up into sections
of 8, so it literally is 8/8.

00:00:35.400 --> 00:00:36.970
So let me just do it like this.

00:00:36.970 --> 00:00:41.120
So the 2 is this whole region
right over here, that's 1.

00:00:41.120 --> 00:00:45.080
So this right over here is 1.

00:00:45.080 --> 00:00:49.460
And then this right
over here is 2, 2 holes,

00:00:49.460 --> 00:00:52.130
so let me paint that in.

00:00:52.130 --> 00:00:55.180
So that is 2 holes.

00:00:55.180 --> 00:00:56.410
And then I have 1/4.

00:00:56.410 --> 00:00:58.770
So this last piece,
this last hole,

00:00:58.770 --> 00:01:00.900
is divided into 8 sections.

00:01:00.900 --> 00:01:03.680
So let me divide it
into fourths first.

00:01:03.680 --> 00:01:11.310
So that's one 1/4, 2/4, and 3/4.

00:01:11.310 --> 00:01:13.230
So we want one of
those four to be

00:01:13.230 --> 00:01:17.180
filled in-- one of
those four in orange.

00:01:17.180 --> 00:01:20.952
So one of those four to be
filled in, just like that.

00:01:20.952 --> 00:01:23.160
You might notice that I
filled in two of the eighths,

00:01:23.160 --> 00:01:26.690
and that's because 1/4
and 2/8 is the same thing.

00:01:26.690 --> 00:01:29.780
So there I've represented
this mixed number, 2 and 1/4.

00:01:29.780 --> 00:01:31.400
Let's see how we
can decompose this.

00:01:31.400 --> 00:01:33.950
So let's get our grids back.

00:01:33.950 --> 00:01:35.540
So how else could we do it?

00:01:35.540 --> 00:01:38.100
And I'm just going to throw
a bunch of fractions up there

00:01:38.100 --> 00:01:41.220
and see what I get.

00:01:41.220 --> 00:01:45.570
So the first thing I'm
going to throw out is 1/2.

00:01:45.570 --> 00:01:51.004
So how would I
represent 1/2 here?

00:01:51.004 --> 00:01:52.420
Well, if I take
one of these holes

00:01:52.420 --> 00:01:54.860
and I put it into two
sections right over here,

00:01:54.860 --> 00:01:57.920
1/2 would be this
section right over there.

00:01:57.920 --> 00:02:00.070
So let me color that in.

00:02:00.070 --> 00:02:02.480
So we have 1/2.

00:02:02.480 --> 00:02:09.039
So I'm first going to add 1/2,
which is the same thing as 4/8.

00:02:09.039 --> 00:02:11.330
And you see that I just filled
in four out of the eight

00:02:11.330 --> 00:02:15.670
sections, which is exactly
half of this first hole.

00:02:15.670 --> 00:02:17.720
So we're making some progress.

00:02:17.720 --> 00:02:24.000
Now let's throw in 3/8.

00:02:24.000 --> 00:02:26.140
So what would 3/8 look like?

00:02:29.660 --> 00:02:32.134
Each of these boxes
are literally an 1/8

00:02:32.134 --> 00:02:34.050
and I could fill it in
however I want, but let

00:02:34.050 --> 00:02:42.630
me just put this as 1, 2, and 3.

00:02:42.630 --> 00:02:52.192
And then let's fill
in plus another 8/8.

00:02:52.192 --> 00:02:53.400
Now, what's 8/8?

00:02:53.400 --> 00:02:56.115
Well, 8/8 is a whole, and
I'll do that over here.

00:02:56.115 --> 00:02:57.740
I still haven't filled
this one in yet,

00:02:57.740 --> 00:02:59.970
but I'll fill in this
one right over here.

00:02:59.970 --> 00:03:01.100
So let's do that.

00:03:01.100 --> 00:03:14.150
So 8/8-- so that's 1/8, 2/8,
3/8, 4/8, 5/8, 6/8, 7/8,

00:03:14.150 --> 00:03:16.100
and 8/8, and it's a whole.

00:03:16.100 --> 00:03:18.971
So I have a whole hole
here, so that's 8/8.

00:03:18.971 --> 00:03:21.470
I want to make this one a whole,
because I want to get to 2,

00:03:21.470 --> 00:03:23.830
so let me put in a 1/8 there.

00:03:23.830 --> 00:03:30.360
So plus 1/8, well, that's going
to be this one right over here,

00:03:30.360 --> 00:03:32.850
so that's my 1/8.

00:03:32.850 --> 00:03:41.165
And then let's add another
2/8, plus another 2/8.

00:03:45.080 --> 00:03:46.940
Well, this is in
eighths right over here,

00:03:46.940 --> 00:03:50.580
so 2/8 is going to
be two of these.

00:03:50.580 --> 00:03:55.010
And notice, you see that the
2/8 is the same thing as 1/4.

00:03:57.890 --> 00:04:00.900
If you took this 1/4
and split it into two,

00:04:00.900 --> 00:04:03.910
so you have two times as
many pieces, it becomes 2/8.

00:04:03.910 --> 00:04:07.850
And you see that if 1 times
2 is 2, 4 times 2 is 8.

00:04:07.850 --> 00:04:09.720
So that 1/4 is the
same thing as 2/8.

00:04:09.720 --> 00:04:12.390
You see the 8/8 is the
same thing as a whole.

00:04:12.390 --> 00:04:15.490
Now, you see, you could make
another whole out of 1/2,

00:04:15.490 --> 00:04:20.390
plus 3/8, plus 1/8, and
they add up to a whole.

00:04:20.390 --> 00:04:22.770
And just to make sense
of why that worked,

00:04:22.770 --> 00:04:27.460
1/2 is the same thing
as 4/8-- because you

00:04:27.460 --> 00:04:32.360
see that, we filled in the
4/8-- then you have 3/8,

00:04:32.360 --> 00:04:35.930
and then you have 1/8.

00:04:35.930 --> 00:04:39.160
And if you add all of these
together, 4/8 plus 3/8 plus

00:04:39.160 --> 00:04:42.520
1/8, you are going
to get, in terms

00:04:42.520 --> 00:04:47.540
of eighths, 4/8 plus 3/8
plus 1/8 is going to be 8/8.

00:04:47.540 --> 00:04:51.890
4 plus 3 plus 1 is 8, so you get
8/8 which is this entire whole.

00:04:51.890 --> 00:04:55.300
So hopefully that helps give
you a visual understanding

00:04:55.300 --> 00:04:57.570
of what we're doing when
we're adding and decomposing

00:04:57.570 --> 00:05:00.230
these fractions a
little bit more.


Adding up fractions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.900
We're asked to select which
fractions add together

00:00:02.900 --> 00:00:07.350
to make 25 over 22, or 25/22.

00:00:07.350 --> 00:00:09.100
You can use as many
fractions as you need.

00:00:09.100 --> 00:00:11.615
Put all unused fractions
into the trash can.

00:00:11.615 --> 00:00:13.760
So let's think about
how we could do this.

00:00:13.760 --> 00:00:16.409
So I actually want to use
the largest fraction first,

00:00:16.409 --> 00:00:17.700
just so I can get pretty close.

00:00:17.700 --> 00:00:20.300
So I'm going to make 16/22.

00:00:20.300 --> 00:00:23.620
And let's see, if
I add 8/22 to that,

00:00:23.620 --> 00:00:27.450
then that's going to get me
to--let's see, 16 plus 8,

00:00:27.450 --> 00:00:31.010
that gets me to 24/22.

00:00:31.010 --> 00:00:35.490
This is 16/22 plus 8/22
is going to get me 24/22.

00:00:35.490 --> 00:00:40.170
And if I get one more,
that gets me to 25/22.

00:00:40.170 --> 00:00:43.240
And then I could put these
unused ones down here

00:00:43.240 --> 00:00:43.957
in the trash can.

00:00:43.957 --> 00:00:46.890
So I'm going to put these
unused ones down here.

00:00:46.890 --> 00:00:48.760
And let's actually
check our answer.

00:00:48.760 --> 00:00:49.360
Got it right.

00:00:49.360 --> 00:00:51.901
And there's multiple
ways that we

00:00:51.901 --> 00:00:53.150
could have actually done this.

00:00:53.150 --> 00:00:54.820
Actually, it's not clear
that there's multiple ways

00:00:54.820 --> 00:00:55.623
that we could have done this.

00:00:55.623 --> 00:00:57.335
Let's see-- is
there any other way?

00:00:57.335 --> 00:01:05.530
Yeah, because even if we
did the 2 and the 4 here,

00:01:05.530 --> 00:01:08.100
we would have to get to-- let's
see, 8 would get us to-- this

00:01:08.100 --> 00:01:13.280
is 2/22 plus 4/22
is 6/22, plus 8/22

00:01:13.280 --> 00:01:15.005
is going to be 14/22, yeah.

00:01:15.005 --> 00:01:17.510
And then if you put 16/22,
there's going to be too many.

00:01:17.510 --> 00:01:19.400
So actually, the way
we did it was the way

00:01:19.400 --> 00:01:20.441
that you've got to do it.

00:01:20.441 --> 00:01:23.720
So the 2 and the 4 I'm going
to put in the trash can.

00:01:23.720 --> 00:01:28.180
And 16 of something plus 1 of
something plus 8 of something

00:01:28.180 --> 00:01:30.170
is going to be 25
of that something.

00:01:30.170 --> 00:01:32.503
And in this case, the something
that we're talking about

00:01:32.503 --> 00:01:33.620
are 22nds.


Equivalent fraction word problem example 3

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:02.970
You and your friends
are making name cards

00:00:02.970 --> 00:00:04.970
to assign seats for dinner.

00:00:04.970 --> 00:00:08.440
You each are given an
identical strip of blank paper

00:00:08.440 --> 00:00:11.540
and begin coloring parts
of it with crayons.

00:00:11.540 --> 00:00:14.510
You start with blue,
or you start with blue.

00:00:14.510 --> 00:00:18.150
Your friend Micah starts off
with green, Joelle starts

00:00:18.150 --> 00:00:21.260
with the red, Holden
starts with pink,

00:00:21.260 --> 00:00:22.980
And Kate starts with purple.

00:00:26.080 --> 00:00:28.030
After one color is
used, whose strip

00:00:28.030 --> 00:00:30.120
has the same part of its
area colored as yours?

00:00:30.120 --> 00:00:32.409
So let's look at my
strip right over here.

00:00:32.409 --> 00:00:34.400
It has how many equal sections?

00:00:34.400 --> 00:00:38.960
1, 2, 3, 4, 5, 6 equal sections.

00:00:38.960 --> 00:00:41.404
And how many of them
have been colored blue?

00:00:41.404 --> 00:00:42.070
Well, let's see.

00:00:42.070 --> 00:00:46.750
1, 2, 3, 4 of my equal sections
have been colored blue.

00:00:46.750 --> 00:00:49.130
So I need to figure out
which of these other folks

00:00:49.130 --> 00:00:52.905
have colored exactly
4/6 of their strip blue.

00:00:52.905 --> 00:00:55.030
Or another way of thinking
about it, I want to say,

00:00:55.030 --> 00:00:58.910
whoever has an equivalent
fraction colored, not blue,

00:00:58.910 --> 00:01:01.750
colored whatever
their color might be.

00:01:01.750 --> 00:01:04.530
So let's first look at
Micah, or "Mee-kah."

00:01:04.530 --> 00:01:07.130
So he's got 3 equal sections.

00:01:07.130 --> 00:01:10.130
He's got 3 equal sections,
and he's colored,

00:01:10.130 --> 00:01:13.930
he's colored 2 of
them, 2 of them green.

00:01:13.930 --> 00:01:18.550
Now is 4/6 and 2/3
the same thing?

00:01:18.550 --> 00:01:20.420
Well, he has 3 equal sections.

00:01:20.420 --> 00:01:21.760
I had 6.

00:01:21.760 --> 00:01:25.600
If I were to group each
of these into groups of 2,

00:01:25.600 --> 00:01:28.930
so if I were to divide
by 2, let me do that.

00:01:28.930 --> 00:01:31.380
If I were divide by 2, so
I'm going to group this

00:01:31.380 --> 00:01:33.680
into groups of 2.

00:01:33.680 --> 00:01:40.850
Then I would have 1, 2-- so I
have 1, 2, 3 equal sections.

00:01:40.850 --> 00:01:46.660
And then if I were to group the
blue strips into groups of 2,

00:01:46.660 --> 00:01:50.680
then I would have
1, 2 of them that

00:01:50.680 --> 00:01:52.512
have actually been colored blue.

00:01:52.512 --> 00:01:54.720
So if I divide both the
numerator and the denominator

00:01:54.720 --> 00:01:59.210
by 2, I get 2/3, which is the
exact same fraction as Micah.

00:01:59.210 --> 00:02:00.720
Another way to
think about it, is

00:02:00.720 --> 00:02:03.740
if you divide each of
Micah's sections into 2.

00:02:03.740 --> 00:02:09.889
So if you divide this into 2,
this into 2, and this into 2.

00:02:09.889 --> 00:02:12.660
Well, then how many total
sections does he have?

00:02:12.660 --> 00:02:17.060
Well, he had 3,
now each of those 3

00:02:17.060 --> 00:02:18.330
have been divided into 2.

00:02:18.330 --> 00:02:22.160
So we can multiply
by 2 to get 6.

00:02:22.160 --> 00:02:24.230
So this is multiplying by 2.

00:02:24.230 --> 00:02:26.600
And each of the 2
in green, there now

00:02:26.600 --> 00:02:29.670
each of those sections have
now been divided into 2.

00:02:29.670 --> 00:02:37.050
So you can multiply by 2
as well, and you get 4.

00:02:37.050 --> 00:02:39.390
So 2/3 and 4/6 are
the same thing.

00:02:39.390 --> 00:02:42.267
So Micah has colored in the
same amount that I have.

00:02:42.267 --> 00:02:44.350
Now let's see whether any
of these other fractions

00:02:44.350 --> 00:02:45.690
are the same.

00:02:45.690 --> 00:02:49.600
So Joelle has colored
in 4, has colored

00:02:49.600 --> 00:02:53.942
in 4 out of 1, 2, 3, 4, 5, 6, 7.

00:02:53.942 --> 00:02:55.940
4 out of 7.

00:02:55.940 --> 00:03:02.884
So it's not clear to me that we
can somehow multiply or divide

00:03:02.884 --> 00:03:04.550
the numerator and the
denominator here--

00:03:04.550 --> 00:03:06.080
remember it has to
be by the same number

00:03:06.080 --> 00:03:08.163
in order to not change the
value of the fraction--

00:03:08.163 --> 00:03:11.410
to somehow make
that equal to 4/6.

00:03:11.410 --> 00:03:13.690
She's colored in
the same number,

00:03:13.690 --> 00:03:17.040
but the size of each of
her strips is different.

00:03:17.040 --> 00:03:20.110
So 4/7 is definitely
different than 4/6.

00:03:20.110 --> 00:03:21.910
So we can rule Joelle out.

00:03:21.910 --> 00:03:30.940
Holden has colored in 3 out of
5 sections, 3 out of 5 sections.

00:03:30.940 --> 00:03:33.410
Once again, not obvious
that I can multiply

00:03:33.410 --> 00:03:37.970
both the numerator and the
denominator 3/5 times something

00:03:37.970 --> 00:03:41.150
to get you to 4/6, so
these aren't equivalent.

00:03:41.150 --> 00:03:45.092
And then Kate has colored
in, Kate has colored in 1, 2,

00:03:45.092 --> 00:03:48.249
3, 4, 5, 6, 7, 8.

00:03:48.249 --> 00:03:50.230
8 out of 11.

00:03:50.230 --> 00:03:54.100
And once again, I could try
to get a numerator up here, 4,

00:03:54.100 --> 00:03:56.020
I could divide by 2.

00:03:56.020 --> 00:03:58.092
But then if I divide by
2 in the denominator,

00:03:58.092 --> 00:03:59.550
I'm going to get
something strange.

00:03:59.550 --> 00:04:03.530
I'm going to get 4
over 11 divided by 2,

00:04:03.530 --> 00:04:05.990
which 11 divided
by 2 is like 5.5.

00:04:05.990 --> 00:04:08.886
So this is clearly not
the same thing as 4/6.

00:04:08.886 --> 00:04:10.260
We'll have rule
this out as well.

00:04:10.260 --> 00:04:14.490
So Micah and I have
colored in the same amount.


Equivalent fraction word problem example 2

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:04.370
Mary wants the fraction
of pink blocks in model B

00:00:04.370 --> 00:00:07.530
to be equivalent to the fraction
of pink blocks in model A.

00:00:07.530 --> 00:00:10.090
How many blocks in
model B need to be pink

00:00:10.090 --> 00:00:11.420
in order to make that happen?

00:00:11.420 --> 00:00:12.930
So let's look at model A.

00:00:12.930 --> 00:00:14.640
So how many equal
sections are there?

00:00:14.640 --> 00:00:18.390
There's 1, 2, 3, 4,
5 equal sections.

00:00:18.390 --> 00:00:20.390
There are 5 equal sections.

00:00:20.390 --> 00:00:22.730
And what fraction
of them are pink?

00:00:22.730 --> 00:00:26.440
Well, you have one
of them is pink.

00:00:26.440 --> 00:00:28.070
One of the blocks is pink.

00:00:28.070 --> 00:00:34.175
So 1/5 of the blocks over
here in model A is pink.

00:00:34.175 --> 00:00:35.800
Now, we have to think
about how to make

00:00:35.800 --> 00:00:38.420
1/5 pink right over here.

00:00:38.420 --> 00:00:39.420
So let's think about it.

00:00:39.420 --> 00:00:41.540
So how many, first of all,
total blocks are there?

00:00:41.540 --> 00:00:43.673
There's 1, 2, 3--
let me actually

00:00:43.673 --> 00:00:44.960
do this in another color.

00:00:44.960 --> 00:00:52.824
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17,

00:00:52.824 --> 00:00:54.890
18, 19, 20.

00:00:54.890 --> 00:00:59.042
There are 20 total blocks here.

00:00:59.042 --> 00:01:01.000
And you see that this is
the exact same length.

00:01:01.000 --> 00:01:04.900
Model B is the exact
same length as model A.

00:01:04.900 --> 00:01:06.540
And so for every
block that you have

00:01:06.540 --> 00:01:09.040
in model A, for
every block here,

00:01:09.040 --> 00:01:13.655
you must have 4 in model B
because we went from 5 sections

00:01:13.655 --> 00:01:15.280
to 20.

00:01:15.280 --> 00:01:17.050
So we're essentially
multiplying by 4.

00:01:17.050 --> 00:01:18.091
And you even see it here.

00:01:18.091 --> 00:01:20.810
If we were to just
draw a line right down

00:01:20.810 --> 00:01:25.050
here you see, you see for
every block in model A,

00:01:25.050 --> 00:01:28.940
you have 1, 2, 3, 4
blocks in model B.

00:01:28.940 --> 00:01:31.980
So one block in
model A that is pink

00:01:31.980 --> 00:01:35.700
would be the same thing
as 4 blocks in model B.

00:01:35.700 --> 00:01:38.890
We would literally just
multiply this times 4.

00:01:38.890 --> 00:01:47.850
So 4 of the 20 will need to
be made pink, so 1, 2, 3, 4.

00:01:47.850 --> 00:01:50.380
And you see that it
exactly matches up.

00:01:50.380 --> 00:01:55.069
This is the exact same fraction
of the entire model over here.

00:01:55.069 --> 00:01:56.860
So when they say how
many blocks in model B

00:01:56.860 --> 00:01:58.950
need to be pink to
make that happen?

00:01:58.950 --> 00:02:00.940
It would be 4.


Visualizing equivalent fractions

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:03.180
Let's think about what
fraction of this grid

00:00:03.180 --> 00:00:04.852
is actually shaded in pink.

00:00:04.852 --> 00:00:06.560
So the first thing we
want to think about

00:00:06.560 --> 00:00:09.790
is how many equal
sections do we have here?

00:00:09.790 --> 00:00:15.080
Well, this is a 1, 2,
3, 4, 5 by 1, 2, 3 grid.

00:00:15.080 --> 00:00:16.400
So there's 15 sections here.

00:00:16.400 --> 00:00:22.000
You could also count it-- 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11,

00:00:22.000 --> 00:00:24.200
12, 13, 14, 15.

00:00:24.200 --> 00:00:28.900
So there are 15
equal sections here.

00:00:28.900 --> 00:00:30.510
And how many of
those equal sections

00:00:30.510 --> 00:00:33.139
are actually shaded in
this kind of pinkish color?

00:00:33.139 --> 00:00:37.610
Well, We have 1, 2, 3, 4, 5, 6.

00:00:37.610 --> 00:00:40.480
So it's 6/15 is shaded in.

00:00:40.480 --> 00:00:42.100
But I want to
simplify this more.

00:00:42.100 --> 00:00:44.600
I have a feeling that there's
some equivalent fractions that

00:00:44.600 --> 00:00:47.310
represent the exact
same thing as 6/15.

00:00:47.310 --> 00:00:50.100
And to get a sense of that, let
me redraw this a little bit,

00:00:50.100 --> 00:00:52.260
where I still shade in
six of these rectangles,

00:00:52.260 --> 00:00:55.230
but I'll shade them a
little bit in one chunk.

00:00:55.230 --> 00:00:59.570
So let me throw in another
grid right over here,

00:00:59.570 --> 00:01:05.459
and let me attempt to
shade in the rectangles

00:01:05.459 --> 00:01:06.530
as fast as possible.

00:01:06.530 --> 00:01:09.055
So that is 1-- 1 rectangle.

00:01:09.055 --> 00:01:12.950
I'll even make my
thing even bigger.

00:01:12.950 --> 00:01:18.470
All right, 1 rectangle,
2 rectangles,

00:01:18.470 --> 00:01:28.090
3 rectangles-- halfway
there-- 4 rectangles,

00:01:28.090 --> 00:01:35.120
5 rectangles shaded in and
now 6 rectangles shaded in.

00:01:35.120 --> 00:01:36.810
So this right over
here, what I just

00:01:36.810 --> 00:01:43.090
did, this is still 6
rectangles of the 15 rectangles

00:01:43.090 --> 00:01:43.700
are shaded in.

00:01:43.700 --> 00:01:45.210
So this is still 6/15.

00:01:45.210 --> 00:01:47.230
These are representing
the same thing.

00:01:47.230 --> 00:01:50.175
But how can I simplify
this even more?

00:01:50.175 --> 00:01:52.290
Well, when you look
at it numerically,

00:01:52.290 --> 00:01:55.430
you see that both 6 and
15 are divisible by 3.

00:01:55.430 --> 00:01:58.120
In fact, their greatest
common factor is 3.

00:01:58.120 --> 00:02:01.289
So what happens if we divide the
numerator and denominator by 3?

00:02:01.289 --> 00:02:03.830
If we do the same thing to the
numerator and the denominator,

00:02:03.830 --> 00:02:06.890
we're not going to be changing
the value of the fraction.

00:02:06.890 --> 00:02:10.060
So let's divide
the numerator by 3

00:02:10.060 --> 00:02:12.800
and divide the denominator by 3.

00:02:12.800 --> 00:02:14.136
And what do we get?

00:02:14.136 --> 00:02:18.950
We get 2 over 5.

00:02:18.950 --> 00:02:23.700
Now how does this make sense
in the context of this diagram

00:02:23.700 --> 00:02:24.900
right here?

00:02:24.900 --> 00:02:27.030
Well, we started off
with 6 shaded in.

00:02:27.030 --> 00:02:29.134
You divide by 3, you
have 2 shaded in.

00:02:29.134 --> 00:02:31.300
So you're essentially saying,
hey, let's group these

00:02:31.300 --> 00:02:32.960
into sections of 3.

00:02:32.960 --> 00:02:38.330
So let's say that this right
over here is one section of 3.

00:02:38.330 --> 00:02:41.520
This is one section
of 3 right over here.

00:02:41.520 --> 00:02:43.460
So that's one section of 3.

00:02:43.460 --> 00:02:48.100
And then this is another
section of 3 right over here.

00:02:48.100 --> 00:02:51.590
And so you have
two sections of 3.

00:02:51.590 --> 00:02:54.740
And actually let me color
it in a little bit better.

00:02:54.740 --> 00:02:56.580
So you have two sections of 3.

00:02:56.580 --> 00:03:01.120
And if you were to combine
them, it looks just like this.

00:03:01.120 --> 00:03:03.560
Notice this is covering
the exact same area

00:03:03.560 --> 00:03:06.430
as your 6 smaller ones did.

00:03:06.430 --> 00:03:08.630
And then how many equal
sections of this size

00:03:08.630 --> 00:03:10.960
do you have on
this entire thing?

00:03:10.960 --> 00:03:14.190
Well, you have 5 equal sections.

00:03:14.190 --> 00:03:18.500
Because this is one section
of 3 right over here,

00:03:18.500 --> 00:03:21.430
this is another section of 3.

00:03:21.430 --> 00:03:25.510
And then this is
another section of 3.

00:03:25.510 --> 00:03:28.220
So notice, you're covering
the exact same area

00:03:28.220 --> 00:03:29.930
of the original thing.

00:03:29.930 --> 00:03:33.680
You're covering 2 out
of the 5 equal sections.

00:03:33.680 --> 00:03:37.669
So 2/5 and 6/15 are
equivalent fractions.

00:03:37.669 --> 00:03:39.460
So if you want to say
what fraction of this

00:03:39.460 --> 00:03:43.735
is covered in the simplest
form, you would say 2/5.


Equivalent fraction word problem example

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:03.640
You are filling up
trays to make ice cubes.

00:00:03.640 --> 00:00:08.250
You notice that each tray holds
the exact same amount of water

00:00:08.250 --> 00:00:11.580
but has a different number
of ice cubes that it makes.

00:00:11.580 --> 00:00:15.660
The blue tray makes 8
equally sized ice cubes.

00:00:15.660 --> 00:00:20.250
The pink tray makes 16
equally sized ice cubes.

00:00:20.250 --> 00:00:24.580
So let me draw the
blue tray here.

00:00:24.580 --> 00:00:26.960
The blue tray, I'll
draw it like this.

00:00:26.960 --> 00:00:28.140
That is not blue.

00:00:28.140 --> 00:00:30.690
Let me draw it in blue.

00:00:30.690 --> 00:00:33.450
So the blue tray makes 8.

00:00:33.450 --> 00:00:35.830
So this is the blue
tray right over here.

00:00:35.830 --> 00:00:37.920
And let me draw it and
divide it into 8 sections

00:00:37.920 --> 00:00:40.250
to represent the 8
equally sized ice cubes.

00:00:40.250 --> 00:00:41.357
And I'll try to do it.

00:00:41.357 --> 00:00:41.940
So, let's see.

00:00:41.940 --> 00:00:44.470
That's about in
half, and then I'm

00:00:44.470 --> 00:00:47.630
going to put each of those
in half, and then each

00:00:47.630 --> 00:00:48.660
of those in half.

00:00:48.660 --> 00:00:51.380
So there we go.

00:00:51.380 --> 00:00:56.887
This is pretty close to
8 equally sized cubes,

00:00:56.887 --> 00:00:57.970
except for this one there.

00:00:57.970 --> 00:01:00.260
Let me actually make it
a little bit cleaner.

00:01:00.260 --> 00:01:08.267
So there's the half,
half, half, half, half.

00:01:08.267 --> 00:01:10.350
All right, so this is
looking a little bit better,

00:01:10.350 --> 00:01:12.390
so 8 equally sized ice cubes.

00:01:12.390 --> 00:01:13.880
This is the blue tray.

00:01:13.880 --> 00:01:15.970
Now, the pink tray takes
the same amount of water,

00:01:15.970 --> 00:01:18.430
so I'm going to make it
the exact same length.

00:01:18.430 --> 00:01:21.320
So the pink tray is
the exact same length,

00:01:21.320 --> 00:01:23.791
but it has 16 equally
sized ice cubes.

00:01:23.791 --> 00:01:25.290
So what I'm going
to do is I'm going

00:01:25.290 --> 00:01:27.970
to make these same
sections here,

00:01:27.970 --> 00:01:30.960
but then I'm going to cut
these sections in two.

00:01:30.960 --> 00:01:35.760
So that's 8, and to 16, I
got to divide these into two.

00:01:35.760 --> 00:01:38.430
So almost done.

00:01:38.430 --> 00:01:39.876
This is the hard part.

00:01:39.876 --> 00:01:42.730
That last one wasn't that neat.

00:01:42.730 --> 00:01:43.800
Almost done.

00:01:43.800 --> 00:01:44.910
All right.

00:01:44.910 --> 00:01:45.450
There we go.

00:01:45.450 --> 00:01:46.850
Set up the problem.

00:01:46.850 --> 00:01:50.970
Now, you put 3 ice cubes from
the blue tray into your drink.

00:01:50.970 --> 00:01:52.340
So, let's do that.

00:01:52.340 --> 00:02:00.430
So, 1, 2, 3 ice cubes from
the blue tray into your drink.

00:02:00.430 --> 00:02:02.230
How many ice cubes
from the pink tray

00:02:02.230 --> 00:02:07.039
would you need to equal the
same exact amount of ice?

00:02:07.039 --> 00:02:08.539
So there's a bunch
of ways to do it.

00:02:08.539 --> 00:02:09.940
We can think about
it with numbers,

00:02:09.940 --> 00:02:11.009
or we can think
about it visually.

00:02:11.009 --> 00:02:12.800
Let's first think
about it with numbers.

00:02:12.800 --> 00:02:16.150
So how much of this
tray have I pulled out?

00:02:16.150 --> 00:02:21.350
Well, I have 8 equally sized
cubes and I took 3 of them out,

00:02:21.350 --> 00:02:23.210
so this literally
represents 3/8.

00:02:27.200 --> 00:02:30.890
So the question
is 3 over 8, if I

00:02:30.890 --> 00:02:34.490
take 3 out of 8 equally
sized cubes, that's

00:02:34.490 --> 00:02:38.220
the same quantity
as taking what?

00:02:38.220 --> 00:02:39.710
I want to do that in white.

00:02:39.710 --> 00:02:41.870
That's the same
quantity as taking

00:02:41.870 --> 00:02:48.440
how many cubes out of 16, out
of 16 equally sized cubes?

00:02:48.440 --> 00:02:51.110
Well, let's look at
it over here visually.

00:02:51.110 --> 00:02:54.950
So if we want the exact
same amount of ice,

00:02:54.950 --> 00:03:01.200
so we're gonna have
1, 2, 3, 4, 5, 6.

00:03:01.200 --> 00:03:04.135
We have 6/16, so this
is equal to 6/16.

00:03:07.870 --> 00:03:10.180
Now, does that
make actual sense?

00:03:10.180 --> 00:03:11.590
Well, sure.

00:03:11.590 --> 00:03:17.660
To go from 3/8 to 6/16, you
multiply the numerator by 2,

00:03:17.660 --> 00:03:19.770
and you multiply the
denominator by 2.

00:03:19.770 --> 00:03:21.910
Now, does that
actually make sense?

00:03:21.910 --> 00:03:22.720
Well, sure it does.

00:03:22.720 --> 00:03:26.420
Because for the
pink ice tray, you

00:03:26.420 --> 00:03:31.050
have 2 ice cubes for every 1
that you have in the blue ice

00:03:31.050 --> 00:03:31.960
tray.

00:03:31.960 --> 00:03:34.709
So the blue ice tray, you
have 8 equally sized cubes.

00:03:34.709 --> 00:03:36.250
Well, for each of
those, you're going

00:03:36.250 --> 00:03:38.580
to have 2 in the pink
ice tray, so you multiply

00:03:38.580 --> 00:03:41.730
by 2 to have 16
equally sized cubes.

00:03:41.730 --> 00:03:44.360
And out of the blue tray,
if you take 3, well,

00:03:44.360 --> 00:03:46.650
that equivalent amount
for each of those cubes,

00:03:46.650 --> 00:03:48.600
you would get 2 from
the pink ice tray.

00:03:48.600 --> 00:03:51.015
So you're multiplying
by 2 right over there.

00:03:51.015 --> 00:03:52.640
So the answer to the
question, how many

00:03:52.640 --> 00:03:54.223
ice cubes from the
pink tray would you

00:03:54.223 --> 00:03:55.780
need to equal the
same amount of ice?

00:03:55.780 --> 00:04:00.080
Well, that is you
would need 6 cubes.


Common Core and the Khan Academy

WEBVTT
Kind: captions
Language: en

00:00:01.801 --> 00:00:02.279
Voiceover: Few people realize,

00:00:02.279 --> 00:00:05.216
that before I made even the
first video for Khan Academy,

00:00:05.216 --> 00:00:08.917
the whole point of Khan Academy
was to be a exercise, practice,

00:00:08.917 --> 00:00:13.418
feedback, reporting engine for
students, for parents and for teachers.

00:00:13.418 --> 00:00:16.135
The first videos were really
just a way to compliment that.

00:00:16.873 --> 00:00:19.602
Even if we go back as far back as 2005,

00:00:19.602 --> 00:00:21.200
we had a lot of exercises on Khan Academy,

00:00:21.200 --> 00:00:23.855
and we got our first real funding in 2010.

00:00:23.855 --> 00:00:27.053
That's where we invested most heavily.

00:00:27.053 --> 00:00:28.943
Now that the Common Core has come out,

00:00:28.943 --> 00:00:31.278
and not only has it come out, not only
have the standards been published,

00:00:31.278 --> 00:00:34.755
but now that there's a
sense of the adoptions,

00:00:34.755 --> 00:00:37.520
there's a sense of how it
is going to be assessed.

00:00:37.520 --> 00:00:41.349
We've decided to do a serious
push to intentionally make sure,

00:00:41.349 --> 00:00:43.951
that our exercises match the Common Core.

00:00:43.951 --> 00:00:45.403
Now, I want to be very clear here.

00:00:45.403 --> 00:00:48.341
Our intention isn't to just
map to the Common Core,

00:00:48.341 --> 00:00:51.403
because a lot of states
have happened to adopt it.

00:00:51.403 --> 00:00:54.945
Our intention is not to map to just
superficially take some content,

00:00:54.945 --> 00:00:59.607
and do a superficial
mapping to the Common Core.

00:00:59.607 --> 00:01:03.124
The whole purpose of why we're doing
this is, when we looked at the standards,

00:01:03.124 --> 00:01:06.327
when we looked at the items coming
out of the Assessment Consortia,

00:01:06.342 --> 00:01:07.825
from Smarter Balanced from park,

00:01:07.825 --> 00:01:10.403
we realized that this is a
very, very, very good standard.

00:01:10.403 --> 00:01:11.465
It's a rigorous standard.

00:01:11.465 --> 00:01:15.581
It's going to measure real
deep student understanding.

00:01:15.581 --> 00:01:19.263
And there's going to be a huge
need for students and teachers,

00:01:19.263 --> 00:01:24.000
and parents to have resources
where they can get practice,

00:01:24.000 --> 00:01:28.553
get feedback, get explanations
around Common Core subjects.

00:01:28.553 --> 00:01:32.545
What you see right over here is
a screen of our Common Core site.

00:01:32.545 --> 00:01:35.599
This is a place, this
is really a dashboard,

00:01:35.599 --> 00:01:38.959
where teachers or parents or even students
can look at Common Core standards,

00:01:38.959 --> 00:01:42.798
and then think about which Khan
Academy content maps to that standard.

00:01:42.798 --> 00:01:47.408
It's available at
kahnacademy.org/commoncore.

00:01:47.408 --> 00:01:49.041
You could go by different grade levels.

00:01:49.041 --> 00:01:53.139
Grade 7, grade 8, and you could see
the different Khan Academy exercises,

00:01:53.139 --> 00:01:56.810
that cover that standard.

00:01:56.810 --> 00:02:00.074
We're actually in process
on doing this right now.

00:02:00.735 --> 00:02:05.599
We have a team of teachers, of
professors, of graduate students,

00:02:05.599 --> 00:02:08.131
who are working closely with us.

00:02:08.131 --> 00:02:12.238
Working very closely with many of the
authors of the Common Core itself,

00:02:12.238 --> 00:02:16.784
to make sure that we
intentionally create content,

00:02:16.784 --> 00:02:21.686
that really hits at the conceptual
spirit of the Common Core.

00:02:21.686 --> 00:02:23.923
Just as a sense of the type
of team we're looking at,

00:02:23.923 --> 00:02:27.308
of the type of folks who
are working on this with us,

00:02:27.308 --> 00:02:28.920
let me show you our team page.

00:02:29.668 --> 00:02:31.543
If you go to the bottom
of any Khan Academy page,

00:02:31.543 --> 00:02:34.661
and you go to our content specialists
you'll go to this page right over here.

00:02:34.661 --> 00:02:38.320
This is a sample of the folks
who are working with us.

00:02:38.320 --> 00:02:42.632
Many of them are experts in developmental
mathematics, in early learning.

00:02:42.632 --> 00:02:45.217
Some of them are graduate students,
some of them are professors,

00:02:45.217 --> 00:02:47.871
of mathematics at the University level.

00:02:47.871 --> 00:02:52.134
It's a really high quality team
of folks who have been helping us,

00:02:52.134 --> 00:02:55.808
make sure that we have really deep
and really peer vetted content,

00:02:55.808 --> 00:02:58.858
intentionally created for the Common Core.

00:02:58.858 --> 00:03:02.058
Once again it's not just because 40
something states have adopted it.

00:03:02.058 --> 00:03:04.725
We like the Common Core because we think,

00:03:04.725 --> 00:03:07.556
when properly implemented, and
that's a very important thing to say.

00:03:07.556 --> 00:03:10.725
When properly implemented, it
is really trying to measure,

00:03:10.725 --> 00:03:13.193
deep conceptual understanding.

00:03:13.193 --> 00:03:16.332
I encourage you to look at the
bios of all of these folks here,

00:03:16.332 --> 00:03:18.141
and we're constantly adding
more and more people,

00:03:18.141 --> 00:03:20.142
to collaborate on this effort.

00:03:20.142 --> 00:03:22.144
To see the type of content
that they're creating,

00:03:22.144 --> 00:03:25.874
it's anything from fairly
simple exercises like this:

00:03:25.874 --> 00:03:28.945
"Rewrite the expression 75
plus 20 in the following form."

00:03:28.945 --> 00:03:31.860
The whole point of this
is to measure or to test,

00:03:31.860 --> 00:03:34.547
does someone really understand
the distributive property.

00:03:34.547 --> 00:03:37.945
We've built tools and widgets so that we
can do free expression entry around it.

00:03:37.945 --> 00:03:43.195
For example, 75 plus 20 we would
want to factor out, let's see.

00:03:43.195 --> 00:03:44.724
We could factor out a 5.

00:03:44.724 --> 00:03:49.108
That's 5 times 5 times.

00:03:49.108 --> 00:03:56.466
We could write that as 15 plus 4.

00:03:56.466 --> 00:04:01.062
I should write this down as a
multiplication symbol just like this.

00:04:01.062 --> 00:04:02.203
There you have it.

00:04:02.203 --> 00:04:04.267
We've written this in the
form that they wanted,

00:04:04.267 --> 00:04:05.463
and we can check our answer.

00:04:05.463 --> 00:04:07.201
This is just a very simple
widget to understand,

00:04:07.201 --> 00:04:10.799
does someone understand factoring and
undoing the distributive property.

00:04:10.799 --> 00:04:12.679
Then there's more complicated problems.

00:04:12.679 --> 00:04:14.937
This is really the spirit
of the Common Core.

00:04:14.937 --> 00:04:17.203
A lot of people, a lot of
times in an algebra class,

00:04:17.203 --> 00:04:20.339
you'll have people talking about
functions, evaluating functions,

00:04:20.339 --> 00:04:22.197
but the spirit of the Common
Core's to really understand,

00:04:22.197 --> 00:04:24.867
what a function is saying and
really constructing functions.

00:04:24.867 --> 00:04:28.975
This is an example of a word
problem, an application problem,

00:04:28.975 --> 00:04:31.059
where the student really has
to interpret the function,

00:04:31.059 --> 00:04:35.588
to make some fairly nuance conclusions
about what's going to happen,

00:04:35.588 --> 00:04:38.934
given the data that's going on.

00:04:38.934 --> 00:04:42.217
We also have things like
this, that are widgets,

00:04:42.217 --> 00:04:46.744
that actually kind of are, for lack of
a better word, virtual manipulatives,

00:04:46.744 --> 00:04:50.535
to help build these content items.

00:04:50.535 --> 00:04:53.402
For example here, "Move each
expression to the right category,

00:04:53.402 --> 00:04:57.362
"to indicate if each one is less
than, equal to, or greater than 6/7."

00:04:57.362 --> 00:04:58.470
A student will say, "Okay, let's see."

00:04:58.470 --> 00:05:01.546
5/7 plus 4/7, that's 9/7.

00:05:01.546 --> 00:05:04.953
That's greater than 6/7.

00:05:04.953 --> 00:05:06.472
Let's see 2 plus ...

00:05:06.472 --> 00:05:08.028
Let's see, this is going to be 5/7.

00:05:08.028 --> 00:05:09.472
That's less than.

00:05:09.472 --> 00:05:11.549
This right over here is
going to be, let's see,

00:05:11.549 --> 00:05:15.782
this is going to be exactly equal
to 6/7 and you get the general idea.

00:05:15.782 --> 00:05:17.078
Over here.

00:05:17.078 --> 00:05:18.357
Then we have things like this,

00:05:18.357 --> 00:05:20.694
and these are all written
by experts in their field.

00:05:20.694 --> 00:05:23.477
These are written by people who are
looking very closely at the standards.

00:05:23.477 --> 00:05:25.778
They're working with specialists,

00:05:25.778 --> 00:05:28.058
they're working with some of the folks
who've even written the standards,

00:05:28.058 --> 00:05:30.261
and being trained by the folks who
have been written the standards,

00:05:30.261 --> 00:05:31.464
to come up with these questions.

00:05:31.464 --> 00:05:34.409
To once again, really go at
the conceptual understanding,

00:05:34.409 --> 00:05:38.316
that the standard is trying to call out.

00:05:38.316 --> 00:05:42.855
Over here, "In the number 47,339
how many times less is the value,

00:05:42.855 --> 00:05:46.746
"of the second 3 than the
value of the first 3?"

00:05:46.746 --> 00:05:49.265
So the second 3 represents 30.

00:05:49.265 --> 00:05:51.335
It's in the tens place, it's 3 tens.

00:05:51.335 --> 00:05:53.417
This 3 is in the hundreds place,

00:05:53.417 --> 00:05:55.260
it represents 3 hundreds.

00:05:55.260 --> 00:05:58.388
30 is 10 times less than 300.

00:05:58.388 --> 00:06:00.465
You would type in 10.

00:06:00.465 --> 00:06:02.476
This is just an example.

00:06:02.476 --> 00:06:07.480
Just coming this fall we have
on the order of 100 exercises,

00:06:07.480 --> 00:06:10.067
that have been intentionally
created for the Common Core.

00:06:10.067 --> 00:06:11.382
It's not just the exercises.

00:06:11.382 --> 00:06:14.459
On top of that we have a
whole series of videos,

00:06:14.459 --> 00:06:17.612
that have been intentionally
created for the Common Core as well.

00:06:17.612 --> 00:06:18.677
For example this one.

00:06:18.677 --> 00:06:20.415
This interpreting features of functions.

00:06:20.415 --> 00:06:24.880
We have a whole series of videos
that explain the underlying idea,

00:06:24.880 --> 00:06:27.075
behind what's going on.

00:06:27.075 --> 00:06:30.338
As we go over the next few months
and definitely over the next year,

00:06:30.338 --> 00:06:34.040
especially because Fall of 2014 is when
you're going to have major adoption,

00:06:34.040 --> 00:06:36.362
of the Common Core in most of the country,

00:06:36.362 --> 00:06:39.868
the goal is to have all of
the major standards covered,

00:06:39.868 --> 00:06:43.439
from K through high school
and then even beyond.

00:06:43.439 --> 00:06:46.959
For us to really be a super
set of the Common Core.

00:06:46.959 --> 00:06:48.332
That's the goal by next Fall.

00:06:48.332 --> 00:06:50.926
Even this Fall we want to cover
most of the major standards,

00:06:50.926 --> 00:06:54.114
starting this Fall going
through middle school.

00:06:54.114 --> 00:06:57.965
We're going to be aggressively
adding content above and beyond that.

00:06:57.965 --> 00:07:00.617
For regions, whether
it's states or countries,

00:07:00.617 --> 00:07:02.460
that are not using the Common Core,

00:07:02.460 --> 00:07:05.054
I want to emphasize that we're
not doing this just to cater,

00:07:05.054 --> 00:07:06.410
to the Common Core states.

00:07:06.410 --> 00:07:10.587
We want to do this because we want
to have rigorous math content,

00:07:10.587 --> 00:07:14.387
and exercises on the site that
would hopefully allow students,

00:07:14.387 --> 00:07:17.273
to succeed in any
environment that they're in.

00:07:17.273 --> 00:07:19.150
We're really intending to be a super set.

00:07:19.150 --> 00:07:23.150
Not a exact covering of the Common Core.


Multiplying: understanding by using area models

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:03.890
I'm going to multiply 78
times 65 in a little less

00:00:03.890 --> 00:00:06.340
than standard way, but
hopefully it'll make some sense,

00:00:06.340 --> 00:00:08.180
and you realize that
there's multiple ways

00:00:08.180 --> 00:00:09.370
that you can multiply.

00:00:09.370 --> 00:00:11.536
And this is actually the
way that I multiply numbers

00:00:11.536 --> 00:00:12.160
in my head.

00:00:12.160 --> 00:00:13.577
So 78 times 65.

00:00:13.577 --> 00:00:15.160
And then we're
actually going to think

00:00:15.160 --> 00:00:17.076
about what the different
parts of this process

00:00:17.076 --> 00:00:19.320
represent on this area model.

00:00:19.320 --> 00:00:23.600
So 78 times 65.

00:00:23.600 --> 00:00:25.892
So I'm going to start
just the way we normally

00:00:25.892 --> 00:00:26.850
start when we multiply.

00:00:26.850 --> 00:00:29.360
I'm going to start with
this 5 in this ones place,

00:00:29.360 --> 00:00:32.850
and I'm going to
say 5 times 8 is 40.

00:00:32.850 --> 00:00:35.060
And instead of just
writing a 0 and carrying

00:00:35.060 --> 00:00:38.090
a 4 right over here, I'm just
going to write the number 40.

00:00:38.090 --> 00:00:40.630
So this was the 8 times the 8.

00:00:40.630 --> 00:00:44.100
Now, I'm going to multiply
the 5 times the 7.

00:00:44.100 --> 00:00:46.420
And we have to be a
little bit careful here

00:00:46.420 --> 00:00:49.970
because 5 times-- this isn't
just any 7, this is a 70.

00:00:49.970 --> 00:00:52.910
So what is 5 times 70?

00:00:52.910 --> 00:00:58.460
Well, 5 times 7 would be
35, so five times 70 is 350.

00:00:58.460 --> 00:01:02.440
So I'll write that down, 350.

00:01:02.440 --> 00:01:08.080
So just as a review, 5 times
8 is 40, 5 times 7 is 350.

00:01:08.080 --> 00:01:13.040
If you add these two together,
this is going to be 5 times 78.

00:01:13.040 --> 00:01:15.180
Now let's go over to the 6.

00:01:15.180 --> 00:01:17.370
So let's multiply
the 6 times the 8.

00:01:17.370 --> 00:01:18.970
Now we have to be careful again.

00:01:18.970 --> 00:01:21.640
This 6 is not just a regular
6, it's in the tens place.

00:01:21.640 --> 00:01:23.190
This is a 60.

00:01:23.190 --> 00:01:25.070
60 times 8.

00:01:25.070 --> 00:01:32.110
Well, 6 times 8 is 40, so 60
times 8 is going to be 480.

00:01:32.110 --> 00:01:35.920
So it's going to be 480.

00:01:35.920 --> 00:01:38.220
And then 6 times 7.

00:01:38.220 --> 00:01:40.520
Well, that would be 42,
but we have to be careful.

00:01:40.520 --> 00:01:43.390
This is 60 times
70, so we're going

00:01:43.390 --> 00:01:45.040
to have two zeroes at the end.

00:01:45.040 --> 00:01:48.060
This is 4,200, not just 42.

00:01:48.060 --> 00:01:54.320
So 6 times 7 is 4,200.

00:01:54.320 --> 00:01:56.340
And now we can add
everything together.

00:01:56.340 --> 00:01:57.830
And this is a very
similar process

00:01:57.830 --> 00:01:59.910
to what we do when we do
the traditional method

00:01:59.910 --> 00:02:00.772
of multiplying.

00:02:00.772 --> 00:02:02.480
I just made it a little
bit more explicit

00:02:02.480 --> 00:02:06.240
what parts are from
multiplying which digits.

00:02:06.240 --> 00:02:08.080
But we can add
everything together.

00:02:08.080 --> 00:02:10.449
In the ones place, we have a 0.

00:02:10.449 --> 00:02:14.324
In the tens place, we
have 4 plus 5 is 9.

00:02:14.324 --> 00:02:17.450
9 plus 8 is 17.

00:02:17.450 --> 00:02:19.340
And now we can carry a 1.

00:02:19.340 --> 00:02:21.400
1 plus 3 is 4.

00:02:21.400 --> 00:02:23.382
4 plus 4 is 8.

00:02:23.382 --> 00:02:25.590
8 plus 2 is 10.

00:02:25.590 --> 00:02:28.440
Carry a 1, regroup
of 1 even, and then

00:02:28.440 --> 00:02:29.710
you have a 5 right over there.

00:02:29.710 --> 00:02:31.735
So you get 5,070.

00:02:31.735 --> 00:02:33.280
Now, I want to
think about-- I want

00:02:33.280 --> 00:02:36.440
to visualize what was going
on here using this area model.

00:02:36.440 --> 00:02:38.570
So once again, we had 78.

00:02:38.570 --> 00:02:41.310
So I'm going to make this
vertical length represent 78.

00:02:41.310 --> 00:02:47.160
So this distance right over
here represents the 70.

00:02:47.160 --> 00:02:50.220
That's the 70, and then we'll
make this distance right over

00:02:50.220 --> 00:02:53.409
here represent the 8.

00:02:53.409 --> 00:02:54.950
Let me make that a
little bit cleaner

00:02:54.950 --> 00:02:56.390
so you see what
I'm talking about.

00:02:56.390 --> 00:02:59.482
So this distance right
over here represents the 8

00:02:59.482 --> 00:03:01.440
and then we're going to
multiply that times 65.

00:03:01.440 --> 00:03:04.640
So this distance
right over here--

00:03:04.640 --> 00:03:06.170
it's not drawn
perfectly to scale,

00:03:06.170 --> 00:03:10.000
but it gives the
idea-- this is 60.

00:03:10.000 --> 00:03:16.100
And then this distance
right over here is the 5.

00:03:16.100 --> 00:03:18.150
So this whole distance is 65.

00:03:18.150 --> 00:03:20.380
This entire distance is 78.

00:03:20.380 --> 00:03:24.470
So when you multiply-- if
you had a rectangle that's

00:03:24.470 --> 00:03:27.210
65 units wide and
you multiplied it,

00:03:27.210 --> 00:03:30.520
and it had a height of
78 units, then its area

00:03:30.520 --> 00:03:33.110
is going to be 78 times 65.

00:03:33.110 --> 00:03:35.480
It's area is going to be 5,070.

00:03:35.480 --> 00:03:37.480
Now, each of these
parts we can map

00:03:37.480 --> 00:03:41.010
to one part of this area
model right over here.

00:03:41.010 --> 00:03:43.020
When we multiplied
the 5 times the 8

00:03:43.020 --> 00:03:48.050
and got the 40, that was
this section right over here,

00:03:48.050 --> 00:03:53.130
5 times 8 is equal to 40.

00:03:53.130 --> 00:03:58.204
When we multiplied 5
times 70, and got 350,

00:03:58.204 --> 00:03:59.370
that's this right over here.

00:03:59.370 --> 00:04:04.560
5 times 70, and we got 350.

00:04:04.560 --> 00:04:09.680
When we multiplied 60-- the 6
in the tens place-- 60 times 8

00:04:09.680 --> 00:04:12.830
and got 480-- that's
this right over here--

00:04:12.830 --> 00:04:17.680
this is 60 times
8 is equal to 480.

00:04:17.680 --> 00:04:22.060
And then finally when we
multiplied 60 times 70

00:04:22.060 --> 00:04:25.040
and got 4,200-- that's
this area right over here--

00:04:25.040 --> 00:04:27.900
this area is 60 wide, 70 tall.

00:04:27.900 --> 00:04:33.390
So this is 60 times 70,
which is equal to 4,200.

00:04:33.390 --> 00:04:36.074
And then when we added
everything up to get 5,070,

00:04:36.074 --> 00:04:37.490
we were essentially
just adding up

00:04:37.490 --> 00:04:40.030
the areas of each
of these tiles.

00:04:40.030 --> 00:04:43.130
This big one is 4,200, that
makes up most of the area,

00:04:43.130 --> 00:04:46.280
and we get 480 from this
magenta one, then 350

00:04:46.280 --> 00:04:48.890
from this greenish-yellow
one, and then

00:04:48.890 --> 00:04:52.140
40 from this greenish-blue
one to get 5,070.

00:04:52.140 --> 00:04:57.280
So the area of this entire
thing is 5,070 square units.