Division resulting in decimals example 2

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:05.470
Let's say that we wanted to
compute 78 divided by 12.

00:00:05.780 --> 00:00:07.170
What would this be?

00:00:07.170 --> 00:00:09.850
Pause this video and
try to figure that out.

00:00:09.850 --> 00:00:11.400
Well, one way to think about it is,

00:00:11.400 --> 00:00:15.330
this is the same thing as 78

00:00:16.930 --> 00:00:18.710
divided by 12, heh.

00:00:18.710 --> 00:00:21.430
I know when I said it in
English, it sounds the same,

00:00:21.430 --> 00:00:24.810
but here, on the right,
I've written it as 78/12,

00:00:24.810 --> 00:00:27.500
or 78 divided by 12.

00:00:27.500 --> 00:00:30.260
Now, how can we re-express this?

00:00:30.260 --> 00:00:31.350
Well, let's think about it.

00:00:31.350 --> 00:00:34.280
Is there a multiple of 12 that's near 78

00:00:34.280 --> 00:00:35.350
without going above it?

00:00:35.350 --> 00:00:38.210
Well, we know that 12 times five is 60,

00:00:38.210 --> 00:00:41.040
12 times six is 72,

00:00:41.040 --> 00:00:44.320
12 times seven is 84, so
12 times seven is too high,

00:00:44.320 --> 00:00:48.000
but we can write this
numerator as, instead of 78,

00:00:48.000 --> 00:00:50.370
I can write it as a multiple of 12

00:00:50.370 --> 00:00:51.670
plus whatever's left over.

00:00:51.670 --> 00:00:55.410
So, 72 plus six, that's
the same thing as 78,

00:00:55.410 --> 00:00:57.930
so it's that divided by 12,

00:00:57.930 --> 00:01:00.500
and, so, I can write this as,

00:01:00.500 --> 00:01:03.290
I can write the 72/12,

00:01:03.290 --> 00:01:07.600
so this is equal to 72 over 12

00:01:07.600 --> 00:01:09.580
plus six over 12,

00:01:09.580 --> 00:01:13.767
plus six over 12 or plus 6/12.

00:01:13.767 --> 00:01:16.520
Now, what's 72 over 12?

00:01:16.520 --> 00:01:20.680
We know that 72 is the
same thing as six times 12.

00:01:20.680 --> 00:01:24.436
So, this is going to be
six times 12 divided by 12,

00:01:24.436 --> 00:01:27.220
well, that's just going
to simplify to six,

00:01:27.220 --> 00:01:29.880
and then what is six 12ths?

00:01:29.880 --> 00:01:32.860
Well, you might recognize
six as half of 12

00:01:32.860 --> 00:01:34.590
or you could divide the numerator

00:01:34.590 --> 00:01:36.660
and the denominator both by six.

00:01:36.660 --> 00:01:39.050
Either way, you are going to get 1/2.

00:01:39.050 --> 00:01:41.880
So, you can view this as six plus 1/2

00:01:43.440 --> 00:01:45.540
or you could view this as six and a half,

00:01:45.540 --> 00:01:47.970
and a lot of times, or as
you get more used to this,

00:01:47.970 --> 00:01:50.100
you won't go do all of these steps,

00:01:50.100 --> 00:01:51.261
but I want to make sure
you really understand

00:01:51.261 --> 00:01:52.930
what's going on.

00:01:52.930 --> 00:01:56.200
So, this is the same
thing as six and a half,

00:01:56.200 --> 00:01:58.720
and if I wanted to
express that as a decimal,

00:01:58.720 --> 00:02:03.720
that's the same thing as six
ones and then how many tenths?

00:02:04.261 --> 00:02:08.900
Well, 1/2 is the same thing as 5/10,

00:02:08.900 --> 00:02:11.240
so six and 5/10.

00:02:11.240 --> 00:02:12.073
So, there you go!

00:02:12.073 --> 00:02:13.890
That's one way of trying to compute

00:02:13.890 --> 00:02:16.300
what 78 divided by 12 is.

00:02:16.300 --> 00:02:17.370
Let's do another example.

00:02:17.370 --> 00:02:22.370
Let's say we wanted to compute
what 20 divided by 80 is.

00:02:22.540 --> 00:02:25.810
Pause this video and see
if you can figure that out.

00:02:25.810 --> 00:02:27.208
Well, we could use a similar technique.

00:02:27.208 --> 00:02:30.510
We could say, hey, this is
the same thing as 20 80ths,

00:02:30.510 --> 00:02:33.430
or we could write 20 the
numerator and 80 the denominator,

00:02:33.430 --> 00:02:36.280
so it's the same thing
as 20 divided by 80,

00:02:41.690 --> 00:02:42.840
and then we could think about, well,

00:02:42.840 --> 00:02:44.960
how can we simplify this fraction,

00:02:44.960 --> 00:02:46.940
or re-express it in some way?

00:02:46.940 --> 00:02:47.773
Well, let's see.

00:02:47.773 --> 00:02:51.460
We can write the numerator as...

00:02:51.460 --> 00:02:55.130
We could write this numerator
as equal to one times 20,

00:02:55.130 --> 00:02:59.660
and then we could write the
denominator as four times 20,

00:02:59.660 --> 00:03:01.810
as four times 20.

00:03:01.810 --> 00:03:06.340
And, so, you could just view
this as being equal to 1/4,

00:03:06.340 --> 00:03:09.276
this is equal to 1/4,

00:03:09.276 --> 00:03:12.295
times 20 over 20.

00:03:12.295 --> 00:03:15.650
Times 20 over 20, well, what's 20 over 20?

00:03:15.650 --> 00:03:16.973
Well, that's just one.

00:03:16.973 --> 00:03:21.130
That's just one, so this
all just becomes 1/4.

00:03:21.130 --> 00:03:24.800
Now, how would we express
that as a decimal?

00:03:24.800 --> 00:03:28.720
Well, let's see, it's hard to
express 1/4 in terms of tenths

00:03:28.720 --> 00:03:31.180
'cause four doesn't divide easily into 10,

00:03:31.180 --> 00:03:34.290
but you can express it
in terms of hundredths.

00:03:34.290 --> 00:03:38.330
So, 1/4 is the same thing
as 25 over a hundred.

00:03:38.330 --> 00:03:40.463
That's the same thing as 25/100.

00:03:41.410 --> 00:03:44.570
Lemme write it down here
so that I get more space.

00:03:44.570 --> 00:03:48.820
So, 1/4 is the same thing as 25 over 100,

00:03:48.820 --> 00:03:51.148
and you could get that by
multiplying the numerator

00:03:51.148 --> 00:03:53.037
and the denominator here by 25,

00:03:53.037 --> 00:03:55.000
and the reason why I care about hundredths

00:03:55.000 --> 00:03:58.150
is we know how to represent
hundredths as decimals.

00:03:58.150 --> 00:04:03.150
This is going to be the same
thing as zero ones and 25/100.

00:04:04.160 --> 00:04:08.630
And, so, 20 divided by
80 is 0.25, or 25/100.

00:04:08.630 --> 00:04:11.131
So, once again, these are
all different strategies

00:04:11.131 --> 00:04:13.890
for thinking about how
we can divide numbers

00:04:13.890 --> 00:04:15.753
that result in decimals.


Example multiplying multi digit numbers

WEBVTT
Kind: captions
Language: en

00:00:00.230 --> 00:00:01.130
- [Instructor] In this video,

00:00:01.130 --> 00:00:04.810
we're going to try to
compute 6742 times 23.

00:00:04.810 --> 00:00:06.980
So like always, pause this video and try

00:00:06.980 --> 00:00:08.473
to compute it for yourself.

00:00:09.350 --> 00:00:11.860
All right, now let's
work on this together.

00:00:11.860 --> 00:00:13.830
And I'm going to do it
using what's often known

00:00:13.830 --> 00:00:16.040
as the standard algorithm.

00:00:16.040 --> 00:00:20.370
Algorithm is just a fancy
word for a series of steps,

00:00:20.370 --> 00:00:22.520
a process for doing something.

00:00:22.520 --> 00:00:26.670
So we have 6742 times 23.

00:00:26.670 --> 00:00:28.350
And so I'm gonna write the 23

00:00:28.350 --> 00:00:29.850
in the same place values.

00:00:29.850 --> 00:00:31.360
So that's two 10s, so I'm gonna write it

00:00:31.360 --> 00:00:33.100
under the four 10s over there.

00:00:33.100 --> 00:00:34.310
And then three ones.

00:00:34.310 --> 00:00:35.990
I'll write it under the two ones.

00:00:35.990 --> 00:00:37.020
And it's important to realize

00:00:37.020 --> 00:00:39.020
this isn't the only way
to multiply numbers.

00:00:39.020 --> 00:00:41.970
In fact, we've studied
other methods for doing it

00:00:41.970 --> 00:00:42.960
in other videos.

00:00:42.960 --> 00:00:45.130
And it's important to realize
what's really going on

00:00:45.130 --> 00:00:46.770
and how these different methods are all,

00:00:46.770 --> 00:00:48.380
on some level, doing the same thing,

00:00:48.380 --> 00:00:49.790
maybe just writing them different

00:00:49.790 --> 00:00:51.850
or doing them in different orders.

00:00:51.850 --> 00:00:53.460
So the way that we would tackle it using

00:00:53.460 --> 00:00:56.180
the standard algorithm,
probably the way that your

00:00:56.180 --> 00:00:57.990
parents first learned to multiply

00:00:57.990 --> 00:00:59.910
multi-digit numbers like this

00:00:59.910 --> 00:01:03.210
is we'll take all of the numbers in 6742,

00:01:03.210 --> 00:01:05.560
all of the various places,
and multiply it by three.

00:01:05.560 --> 00:01:07.470
And then we're gonna
multiply it times two 10s.

00:01:07.470 --> 00:01:09.190
And then we're gonna add everything up.

00:01:09.190 --> 00:01:11.730
So let's first multiply it times three.

00:01:11.730 --> 00:01:15.330
So we have two times three, that is six.

00:01:15.330 --> 00:01:16.950
Then we have four times three.

00:01:16.950 --> 00:01:18.047
And what people often say is,

00:01:18.047 --> 00:01:20.997
"Four times three is 12, write the two,

00:01:20.997 --> 00:01:22.850
"and then carry the one."

00:01:22.850 --> 00:01:24.810
But what really just happened is you said

00:01:24.810 --> 00:01:28.960
four 10s times three is 12 10s.

00:01:28.960 --> 00:01:33.960
12 10s can be written as two 10s plus 100.

00:01:34.430 --> 00:01:37.110
Then we say seven times three is 21.

00:01:37.110 --> 00:01:39.527
And then you'll say, "Oh, I
have to add that other one,

00:01:39.527 --> 00:01:41.167
"so I get 22."

00:01:42.280 --> 00:01:44.250
But once again, what just happened?

00:01:44.250 --> 00:01:47.217
We said seven hundreds times three is 2100

00:01:48.690 --> 00:01:51.330
plus another hundred is 22 hundreds,

00:01:51.330 --> 00:01:53.360
which can be expressed as two hundreds

00:01:53.360 --> 00:01:54.930
and two thousands.

00:01:54.930 --> 00:01:59.290
And then, last but not
least, six times three is 18

00:01:59.290 --> 00:02:01.560
plus two is 20.

00:02:01.560 --> 00:02:03.680
But remember, we're
talking about thousands.

00:02:03.680 --> 00:02:05.653
So this is 20 thousands.

00:02:06.570 --> 00:02:11.570
So then we will move on to
the two 10s right over here.

00:02:11.610 --> 00:02:15.150
So two times two 10s is four 10s.

00:02:15.150 --> 00:02:17.350
Now some folks might be tempted to put

00:02:17.350 --> 00:02:19.500
the four over there,
but that's not four 10s.

00:02:19.500 --> 00:02:21.280
Four 10s would be right over here.

00:02:21.280 --> 00:02:23.110
And so it's common practice as you move

00:02:23.110 --> 00:02:26.120
to the next place value
over, as you get to this two,

00:02:26.120 --> 00:02:28.670
that people will just put a zero here

00:02:28.670 --> 00:02:30.830
just so they don't make that mistake.

00:02:30.830 --> 00:02:33.020
All right, now let's keep going.

00:02:33.020 --> 00:02:35.660
What is four times two?

00:02:35.660 --> 00:02:36.580
Well that's eight.

00:02:36.580 --> 00:02:38.130
We'll just write the
eight right over there.

00:02:38.130 --> 00:02:38.963
Why did that work?

00:02:38.963 --> 00:02:42.150
Well we're having four 10s times two 10s,

00:02:42.150 --> 00:02:42.983
well that's going to be

00:02:42.983 --> 00:02:45.710
eight times 10 times 10, eight hundreds.

00:02:45.710 --> 00:02:48.950
And then we say what is seven times two.

00:02:48.950 --> 00:02:51.820
That is 14, which of course we can write

00:02:51.820 --> 00:02:56.570
the four and then we can
carry the one, so to speak.

00:02:56.570 --> 00:02:58.660
And I'll cross these out
so I don't get confused.

00:02:58.660 --> 00:03:03.660
And then six times two is
going to be equal to 12,

00:03:04.220 --> 00:03:07.990
plus this one that we had carried, is 13.

00:03:07.990 --> 00:03:10.010
So there we go.

00:03:10.010 --> 00:03:13.200
And then we just have
to add everything up.

00:03:13.200 --> 00:03:15.000
And we are going to get

00:03:15.000 --> 00:03:17.430
six plus zero ones is six.

00:03:17.430 --> 00:03:21.060
Two 10s plus four 10s is six 10s.

00:03:21.060 --> 00:03:24.850
Two hundreds plus eight
hundreds is 10 hundreds,

00:03:24.850 --> 00:03:28.510
which you could do as zero
hundreds and one thousand.

00:03:28.510 --> 00:03:31.160
One thousand plus zero
thousands plus four thousands

00:03:31.160 --> 00:03:33.170
is five thousands.

00:03:33.170 --> 00:03:36.660
Two 10 thousands plus three 10 thousands

00:03:36.660 --> 00:03:39.820
is going to be five 10 thousands.

00:03:39.820 --> 00:03:42.230
And then we just have one hundred thousand

00:03:42.230 --> 00:03:43.063
right over there.

00:03:43.063 --> 00:03:46.210
So we've got 155,066.

00:03:46.210 --> 00:03:47.743
And we are done.


Estimating with decimal multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:01.670
- [Instructor] We are asked to estimate

00:00:01.670 --> 00:00:06.670
what is 2.7 times 4 roughly equal to?

00:00:06.730 --> 00:00:09.130
Pause this video and see
if you can answer that.

00:00:10.040 --> 00:00:14.890
Alright, so we can think of 2.7 times 4

00:00:14.890 --> 00:00:17.300
as being roughly equal to,

00:00:17.300 --> 00:00:18.650
or some people might say,

00:00:18.650 --> 00:00:20.528
as approximately equal to.

00:00:20.528 --> 00:00:24.290
Well let's see, 2.7, that's two ones

00:00:24.290 --> 00:00:25.550
and seven tenths.

00:00:25.550 --> 00:00:27.232
So we could think about

00:00:27.232 --> 00:00:29.190
well what's the nearest
whole number to 2.7?

00:00:29.190 --> 00:00:32.080
And so 2.7 is pretty close to 3,

00:00:32.080 --> 00:00:33.947
it's in fact closer to 3 than it is to 2.

00:00:33.947 --> 00:00:35.990
So we could say that this is roughly

00:00:35.990 --> 00:00:38.730
equal to three times four.

00:00:38.730 --> 00:00:40.577
And then three times four is, of course,

00:00:40.577 --> 00:00:43.290
equal to 12.

00:00:43.290 --> 00:00:45.300
And you can see that these other answers

00:00:45.300 --> 00:00:46.598
actually seem quite unreasonable.

00:00:46.598 --> 00:00:49.380
1.2, if I take something

00:00:49.380 --> 00:00:50.810
and I multiply times four,

00:00:50.810 --> 00:00:52.170
I shouldn't get an answer that is

00:00:52.170 --> 00:00:53.960
less than my original something.

00:00:53.960 --> 00:00:55.330
And then to go from,

00:00:55.330 --> 00:00:56.350
and then to take something that's

00:00:56.350 --> 00:00:58.370
roughly equal to three times four

00:00:58.370 --> 00:00:59.714
to get to 120, that doesn't make sense,

00:00:59.714 --> 00:01:02.225
and definitely doesn't
make sense to get to 1,200.

00:01:02.225 --> 00:01:04.470
Let's do another example.

00:01:04.470 --> 00:01:08.440
So here we said what is roughly

00:01:08.440 --> 00:01:12.840
equal to 78 times 19.88.

00:01:12.840 --> 00:01:15.640
So pause this video
and try to answer that.

00:01:15.640 --> 00:01:17.820
Alright, well this is
really the same idea.

00:01:17.820 --> 00:01:19.040
We wanna think about,

00:01:19.040 --> 00:01:22.740
well what are numbers that
these numbers are close to

00:01:22.740 --> 00:01:24.369
that are easy to multiply with.

00:01:24.369 --> 00:01:28.730
So for example, 78, that
is pretty close to 80

00:01:28.730 --> 00:01:33.651
and then 19.88, or 19 and 88 hundredths,

00:01:33.651 --> 00:01:36.070
that's pretty close to 20.

00:01:36.070 --> 00:01:37.930
It's closer to 20 than it is to 19

00:01:37.930 --> 00:01:39.370
and even if it was closer to 19,

00:01:39.370 --> 00:01:40.763
just to estimate I probably
would still go to 20

00:01:40.763 --> 00:01:44.030
because it's easier to multiply with 20.

00:01:44.030 --> 00:01:47.090
So this is going to be pretty close to 20.

00:01:47.090 --> 00:01:49.300
So all of this is going to be roughly

00:01:49.300 --> 00:01:51.790
equal to 80 times 20.

00:01:51.790 --> 00:01:53.460
And you might already recognize

00:01:53.460 --> 00:01:58.460
that this is going to be 8
times 2 times 10 times 10

00:01:58.620 --> 00:02:00.360
and I could write it that way,

00:02:00.360 --> 00:02:02.100
no reason for me to skip steps.

00:02:02.100 --> 00:02:03.950
But you would normally
do this in your head.

00:02:03.950 --> 00:02:05.951
So 80 is the same thing as 8 times 10,

00:02:05.951 --> 00:02:07.980
we do that in that purple color

00:02:07.980 --> 00:02:09.270
so that you can see it.

00:02:09.270 --> 00:02:12.570
So this is the same thing as 8 times 10

00:02:12.570 --> 00:02:17.570
times two times ten
which we could then write

00:02:18.470 --> 00:02:23.470
this is going to be 8 times 2 which is 16,

00:02:24.274 --> 00:02:28.900
so the 8 and the 2, you
get 16 times 10 times 10

00:02:28.900 --> 00:02:33.034
is times 100 so this should be 1,600.

00:02:33.034 --> 00:02:37.720
1,600 which is this
choice right over here.

00:02:37.720 --> 00:02:40.020
And it's always good to
just do reality check

00:02:40.020 --> 00:02:41.363
to make sure it's reasonable.

00:02:41.363 --> 00:02:43.120
It wouldn't make any sense if I

00:02:43.120 --> 00:02:44.780
take a number close to 80

00:02:44.780 --> 00:02:47.970
and I multiply it by some
number that's close to 20

00:02:47.970 --> 00:02:50.129
and get a smaller number than 80.

00:02:50.129 --> 00:02:52.590
It also, if you have a
number that's close to 80,

00:02:52.590 --> 00:02:53.633
to get 160 you'd only have to multiply

00:02:53.633 --> 00:02:55.610
it by roughly 2.

00:02:55.610 --> 00:02:57.510
But here we're multiplying by roughly 20.

00:02:57.510 --> 00:03:02.510
And then to go to 80, or
roughly 80, to 16,000,

00:03:02.828 --> 00:03:04.830
well then you wouldn't have to multiply

00:03:04.830 --> 00:03:05.960
by something that's close to 20,

00:03:05.960 --> 00:03:06.930
you'd have to multiply by something

00:03:06.930 --> 00:03:08.610
that's close to 200.

00:03:08.610 --> 00:03:10.163
So I like our answer.


Multiplying decimals using estimation

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:02.320
- [Instructor] So let's
see if we can come up ways

00:00:02.320 --> 00:00:07.320
to compute what 2.8 times 4.73 is.

00:00:07.360 --> 00:00:09.650
So pause this video,
and try to work it out.

00:00:09.650 --> 00:00:11.010
And, actually, I'll give you a hint.

00:00:11.010 --> 00:00:13.210
Try to figure out, just using the digits,

00:00:13.210 --> 00:00:15.430
not even paying attention to the decimals,

00:00:15.430 --> 00:00:17.610
the digits that the product would have,

00:00:17.610 --> 00:00:19.300
and then use estimation to think about

00:00:19.300 --> 00:00:21.860
where to place the decimal in your product

00:00:21.860 --> 00:00:24.100
so you get a reasonable answer.

00:00:24.100 --> 00:00:25.850
All right, now, let's do this together.

00:00:25.850 --> 00:00:27.590
So let's just imagine
that we were multiplying

00:00:27.590 --> 00:00:29.110
these numbers without decimals.

00:00:29.110 --> 00:00:30.360
So that would be a situation

00:00:30.360 --> 00:00:35.360
where we would have 473 times 28,

00:00:35.760 --> 00:00:37.700
and so we could try to compute that.

00:00:37.700 --> 00:00:40.430
And, so, we could think about
let's multiply everything

00:00:40.430 --> 00:00:44.010
times the eight, so
three times eight is 24,

00:00:44.010 --> 00:00:48.430
seven times eight is 56, plus two is 58.

00:00:48.430 --> 00:00:52.480
And then four times eight
is 32, plus five is 37.

00:00:52.480 --> 00:00:54.590
And then we could multiply
everything times the two.

00:00:54.590 --> 00:00:56.630
I'll cross those out so
I don't get confused.

00:00:56.630 --> 00:00:59.830
Three times two is going to be six,

00:00:59.830 --> 00:01:01.230
so we have to be very careful,

00:01:01.230 --> 00:01:04.100
we are now in the tens place,
so we want a zero here,

00:01:04.100 --> 00:01:08.500
so three times two tens
is going to be six tens.

00:01:08.500 --> 00:01:11.520
Seven times two is 14.

00:01:11.520 --> 00:01:15.530
Four times two is eight, plus one is nine.

00:01:15.530 --> 00:01:18.000
We add everything together,

00:01:18.000 --> 00:01:20.090
and we get four plus zero is four,

00:01:20.090 --> 00:01:22.220
eight plus six is 14,

00:01:22.220 --> 00:01:25.770
and then one plus seven plus four is 12,

00:01:25.770 --> 00:01:29.080
and then we get one plus three is four,

00:01:29.080 --> 00:01:32.150
plus nine is equal to 13.

00:01:32.150 --> 00:01:35.950
So we know that the final
answer has the digits one,

00:01:35.950 --> 00:01:38.660
three, two, four, four in that order.

00:01:38.660 --> 00:01:42.730
One, three, two, four, four.

00:01:42.730 --> 00:01:45.200
So now we have to think about
where would we put a decimal

00:01:45.200 --> 00:01:46.820
for this to be a reasonable answer?

00:01:46.820 --> 00:01:48.720
And here's where estimation is useful.

00:01:48.720 --> 00:01:52.580
We know that 2.8 times 4.73

00:01:52.580 --> 00:01:54.950
is going to be roughly equal to what?

00:01:54.950 --> 00:01:57.770
Well, 2.8 is pretty close to three,

00:01:57.770 --> 00:02:00.950
so I'll estimate 2.8 as being three.

00:02:00.950 --> 00:02:05.060
And 4.73 is, if I had to estimate,

00:02:05.060 --> 00:02:07.540
I would say, "Hey, it'd
be pretty close to five."

00:02:07.540 --> 00:02:09.730
So this should be pretty
close to three times five,

00:02:09.730 --> 00:02:12.090
so it should be close to 15.

00:02:12.090 --> 00:02:14.170
So if I were to put the decimal there,

00:02:14.170 --> 00:02:17.160
that's way more than 15, so
that doesn't seem reasonable.

00:02:17.160 --> 00:02:18.800
Even if I were to put the decimal there,

00:02:18.800 --> 00:02:22.230
1,324.4 is still way more than 15.

00:02:22.230 --> 00:02:23.590
If I were to put the decimal there,

00:02:23.590 --> 00:02:25.410
still way more than 15.

00:02:25.410 --> 00:02:26.870
If I were to put the decimal there,

00:02:26.870 --> 00:02:28.880
hey, that actually feels about right.

00:02:28.880 --> 00:02:33.720
13 and 244 thousandths is,

00:02:33.720 --> 00:02:37.010
is approaching 15, it's in the ballpark.

00:02:37.010 --> 00:02:38.660
And it's actually the closest,

00:02:38.660 --> 00:02:40.070
'cause if we were to
put the decimal there,

00:02:40.070 --> 00:02:44.000
then we go to 1.3244,
so a lot less than 15.

00:02:44.000 --> 00:02:46.820
So if we want this to
be roughly equal to 15,

00:02:46.820 --> 00:02:50.040
we definitely would wanna put
the decimal right over there.

00:02:50.040 --> 00:02:54.630
That is the most reasonable
computation we can do,

00:02:54.630 --> 00:02:55.770
'cause we know the digits

00:02:55.770 --> 00:02:58.230
are going to be one,
three, two, four, four,

00:02:58.230 --> 00:02:59.970
and this helps us put the decimal.

00:02:59.970 --> 00:03:02.060
In the future, we're
gonna come up with ways

00:03:02.060 --> 00:03:03.270
of doing it that where
you don't necessarily

00:03:03.270 --> 00:03:04.930
have to estimate, but I encourage you,

00:03:04.930 --> 00:03:07.010
the estimation is always key.

00:03:07.010 --> 00:03:09.680
If you ever in your life
forget some type of method

00:03:09.680 --> 00:03:11.740
or process for multiplying decimals,

00:03:11.740 --> 00:03:15.010
it's the estimation that
allows you to understand

00:03:15.010 --> 00:03:18.020
whether you're coming up
with a reasonable answer.

00:03:18.020 --> 00:03:19.460
This is really important,

00:03:19.460 --> 00:03:21.030
'cause the decimal, there's a,

00:03:21.030 --> 00:03:23.430
I remember reading a news
story a couple years ago

00:03:23.430 --> 00:03:27.050
where someone put in a stock trade

00:03:27.050 --> 00:03:28.620
where they got the decimal wrong.

00:03:28.620 --> 00:03:31.950
And because of that, they
essentially sold 10 times

00:03:31.950 --> 00:03:33.630
as many shares as they were supposed to,

00:03:33.630 --> 00:03:35.940
and so they lost hundreds
of millions of dollars.

00:03:35.940 --> 00:03:37.723
So, anyway, decimals are important.


Strategies for multiplying multi digit decimals

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.470
- [Instructor] So in this
video, we're gonna try

00:00:01.470 --> 00:00:06.470
to think of ways to compute
what 31.2 times 19 is.

00:00:06.520 --> 00:00:08.580
And there's multiple
ways to approach this,

00:00:08.580 --> 00:00:10.970
but like always, try to pause this video

00:00:10.970 --> 00:00:13.650
and see if you can work
through this on your own.

00:00:13.650 --> 00:00:16.050
All right, now let's do this together.

00:00:16.050 --> 00:00:18.180
Now one way to think about this is

00:00:18.180 --> 00:00:22.880
you could view 31.2 as a
certain number of tenths.

00:00:22.880 --> 00:00:25.160
And how many tenths would this be?

00:00:25.160 --> 00:00:30.080
Well, you could view
this, 31.2, as 312 tenths.

00:00:33.000 --> 00:00:38.000
And so, this is the same
thing as 312 tenths times 19.

00:00:38.420 --> 00:00:40.250
And so, what we could do
is we could figure out

00:00:40.250 --> 00:00:45.250
what is 312 of something times
19, or what's 312 times 19.

00:00:45.631 --> 00:00:47.310
It's gonna be that many tenths,

00:00:47.310 --> 00:00:49.290
and then we could convert
it back to a decimal.

00:00:49.290 --> 00:00:52.110
I'll also show another strategy,
but let's just do that.

00:00:52.110 --> 00:00:57.110
So if we were to just
multiply 312 times 19,

00:00:57.270 --> 00:01:01.040
times 19, this is going to be,

00:01:01.040 --> 00:01:03.250
let's see, two, obviously, another color.

00:01:03.250 --> 00:01:05.510
Two times nine is 18.

00:01:05.510 --> 00:01:07.480
One times nine is nine.

00:01:07.480 --> 00:01:09.790
Plus one is 10.

00:01:09.790 --> 00:01:13.640
Three times nine is 27, plus one is 28.

00:01:13.640 --> 00:01:15.480
If what I just did looks unfamiliar,

00:01:15.480 --> 00:01:19.660
we have videos that explain
how this process works.

00:01:19.660 --> 00:01:22.680
And then, we go to the tens
place, right over here.

00:01:22.680 --> 00:01:25.400
And so, we'd say one times two is two.

00:01:25.400 --> 00:01:26.910
One times one is one.

00:01:26.910 --> 00:01:28.610
One times is three is three.

00:01:28.610 --> 00:01:30.580
And then we add everything together.

00:01:30.580 --> 00:01:32.010
We get eight.

00:01:32.010 --> 00:01:34.440
Zero plus two is two.

00:01:34.440 --> 00:01:36.160
Eight plus one is nine.

00:01:36.160 --> 00:01:37.700
Two plus three is five.

00:01:37.700 --> 00:01:41.070
So we get five, nine, two, eight, or 5928.

00:01:41.070 --> 00:01:43.090
Now that's not going
to be the answer here.

00:01:43.090 --> 00:01:47.510
The answer is going to be 5928 tenths.

00:01:47.510 --> 00:01:51.667
So this is going to be
equal to 5928 tenths.

00:01:53.720 --> 00:01:56.720
Now, how can we express this as a decimal?

00:01:56.720 --> 00:01:58.500
Well, we could think of it this way.

00:01:58.500 --> 00:02:01.940
If that's the decimal,
this is the tenths place,

00:02:01.940 --> 00:02:04.270
this is the ones place, which is the same,

00:02:04.270 --> 00:02:06.280
this is the same thing as 10 tenths place,

00:02:06.280 --> 00:02:07.680
which is the ones place.

00:02:07.680 --> 00:02:12.680
This is the tens place, and
this is the hundreds place.

00:02:13.060 --> 00:02:15.350
Well, you have eight tenths.

00:02:15.350 --> 00:02:17.450
We could put that in the tenths place.

00:02:17.450 --> 00:02:18.980
You have these 20 tenths.

00:02:18.980 --> 00:02:21.270
That's the same thing as two ones.

00:02:21.270 --> 00:02:24.480
You have the 900 tenths.

00:02:24.480 --> 00:02:26.280
Do that in a different color.

00:02:26.280 --> 00:02:29.410
900 tenths is the same thing as nine tens.

00:02:29.410 --> 00:02:34.140
And then, your 5000 tenths

00:02:34.140 --> 00:02:36.780
is the same thing as five hundreds.

00:02:36.780 --> 00:02:38.590
Another way to think about it is

00:02:38.590 --> 00:02:39.860
we wrote all the places out,

00:02:39.860 --> 00:02:41.500
and we wrote it in terms of tenths.

00:02:41.500 --> 00:02:43.460
So the eight went there,
and then every place

00:02:43.460 --> 00:02:46.530
to the left of that went to
the place to the left of that.

00:02:46.530 --> 00:02:49.970
So this is going to be
592 and eight tenths.

00:02:49.970 --> 00:02:51.050
So we could write it like that.

00:02:51.050 --> 00:02:54.440
592 and eight tenths.

00:02:54.440 --> 00:02:56.390
Now another way to approach this

00:02:56.390 --> 00:02:58.610
is to just think about the digits,

00:02:58.610 --> 00:03:00.830
not the actual numbers, to figure out,

00:03:00.830 --> 00:03:03.520
well, the answer will
have what digits in it.

00:03:03.520 --> 00:03:05.690
And then try to estimate to think about

00:03:05.690 --> 00:03:07.480
where the decimal place should go.

00:03:07.480 --> 00:03:10.890
So, for example, you
could do 312 times 19.

00:03:10.890 --> 00:03:14.040
So since you remove the
decimal, do the computation,

00:03:14.040 --> 00:03:17.610
and say, okay, the answer
should have the digits

00:03:17.610 --> 00:03:20.220
five, nine, two, eight in that order.

00:03:20.220 --> 00:03:21.870
Now where should I put the decimal

00:03:21.870 --> 00:03:24.070
in order for that to
be a reasonable answer?

00:03:24.070 --> 00:03:26.100
And that's where estimation comes in.

00:03:26.100 --> 00:03:31.100
You could say, hey, 31.2 times 19,

00:03:31.120 --> 00:03:33.788
that's going to be approximately equal to,

00:03:33.788 --> 00:03:34.870
in fact, if I were to estimate these

00:03:34.870 --> 00:03:36.380
with numbers that are easy to multiply,

00:03:36.380 --> 00:03:37.650
that's going to be roughly equal to

00:03:37.650 --> 00:03:42.110
30 times 20, which is equal to 600.

00:03:42.110 --> 00:03:43.980
So that tells me that my product here

00:03:43.980 --> 00:03:46.150
should be roughly equal to 600.

00:03:46.150 --> 00:03:48.590
And so where would I put the decimal here

00:03:48.590 --> 00:03:50.580
for it to be roughly equal to 600?

00:03:50.580 --> 00:03:54.520
So I know the answer has the
digits five, nine, two, eight.

00:03:54.520 --> 00:03:57.580
Where do I put a decimal for
it to be roughly equal to 600?

00:03:57.580 --> 00:03:58.940
Well, if I were to put the decimal there,

00:03:58.940 --> 00:04:00.960
that's not roughly equal to 600.

00:04:00.960 --> 00:04:02.130
If I were to put the decimal there,

00:04:02.130 --> 00:04:03.550
that's not roughly equal to 600.

00:04:03.550 --> 00:04:04.730
That's close to 60.

00:04:04.730 --> 00:04:07.110
If I put the decimal
there, that's close to six.

00:04:07.110 --> 00:04:08.690
If I wanna be close to 600, I'd have

00:04:08.690 --> 00:04:10.980
to put the decimal right over there.

00:04:10.980 --> 00:04:12.730
And so that's also a good way to test

00:04:12.730 --> 00:04:15.160
the reasonableness of what's going on.

00:04:15.160 --> 00:04:18.100
This should be roughly equal to 600,

00:04:18.100 --> 00:04:19.250
if we were to estimate it.

00:04:19.250 --> 00:04:21.130
And so, we like that our process

00:04:21.130 --> 00:04:24.343
got an answer that is
roughly equal to 600.


Visual representations of decimal multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:02.580
- [Instructor] So what we
have here on this number line

00:00:02.580 --> 00:00:04.890
that we've now marked off with the tenths

00:00:04.890 --> 00:00:07.210
and you can see that this
is three tenths here.

00:00:07.210 --> 00:00:11.720
We can think about this as a
multiplication of a decimal.

00:00:11.720 --> 00:00:13.890
And so what is this representing?

00:00:13.890 --> 00:00:14.820
And I'll give you a hint.

00:00:14.820 --> 00:00:18.280
It's representing something
times three tenths.

00:00:18.280 --> 00:00:20.721
So pause the video and
try to think about that.

00:00:20.721 --> 00:00:21.568
Well, let's see.

00:00:21.568 --> 00:00:25.360
We are going one times three tenths,

00:00:25.360 --> 00:00:29.130
two times three tenths,
three times three tenths,

00:00:29.130 --> 00:00:31.550
and then four times three tenths.

00:00:31.550 --> 00:00:36.550
So what's represented here
is four times three tenths,

00:00:36.910 --> 00:00:40.023
and so what is this going to be equal to?

00:00:40.023 --> 00:00:42.780
Well you can see you go from three tenths,

00:00:42.780 --> 00:00:45.450
to six tenths, to nine tenths,

00:00:45.450 --> 00:00:47.590
and then you could view
this as twelve tenths,

00:00:47.590 --> 00:00:49.470
but twelve tenths is the same thing

00:00:49.470 --> 00:00:52.230
as one, one and two tenths.

00:00:52.230 --> 00:00:55.250
So you could view this as 1.2.

00:00:55.250 --> 00:00:58.130
One and two tenths.

00:00:58.130 --> 00:00:59.840
Let's do another example.

00:00:59.840 --> 00:01:01.670
No, actually I'll do it
on the same number line.

00:01:01.670 --> 00:01:04.160
If we wanted to represent

00:01:04.160 --> 00:01:09.160
three times 0.2

00:01:10.410 --> 00:01:12.580
What would that look
like on this number line?

00:01:12.580 --> 00:01:14.640
And what would this be equal to?

00:01:14.640 --> 00:01:17.192
So I'll put a little equal sign here.

00:01:17.192 --> 00:01:20.100
Pause this video and see
if you an figure that out.

00:01:20.100 --> 00:01:22.470
All right, so let's think
about where two tenths is

00:01:22.470 --> 00:01:25.230
this is one tenth, two
tenths is right over there.

00:01:25.230 --> 00:01:29.620
This is 0.2, and we're gonna
multiply it times three.

00:01:29.620 --> 00:01:32.059
So, we're gonna multiply it times one,

00:01:32.059 --> 00:01:34.444
then we're gonna multiply it times two,

00:01:34.444 --> 00:01:35.460
that takes us to four tenths

00:01:35.460 --> 00:01:37.630
and then we're gonna
multiply it times three

00:01:37.630 --> 00:01:41.243
to get us to six tenths, 0.6.

00:01:42.300 --> 00:01:44.890
So it's six tenths just like that.

00:01:44.890 --> 00:01:47.100
Now you could also visualize two tenths

00:01:47.100 --> 00:01:48.810
as parts of a whole.

00:01:48.810 --> 00:01:51.100
So for example, this
represents two tenths.

00:01:51.100 --> 00:01:53.629
I have this whole, this square is a whole

00:01:53.629 --> 00:01:57.080
it's split into ten equal columns here

00:01:57.080 --> 00:01:58.330
and we have two of them filled in.

00:01:58.330 --> 00:02:00.010
So this represents two tenths.

00:02:00.010 --> 00:02:02.020
So if you have three times two tenths,

00:02:02.020 --> 00:02:04.150
Well this is one times two tenths,

00:02:04.150 --> 00:02:06.000
this is two times two tenths,

00:02:06.000 --> 00:02:08.360
and this is three times two tenths.

00:02:08.360 --> 00:02:10.700
And so how many tenths do we now have?

00:02:10.700 --> 00:02:15.660
Well we have one, two, three,
four, five, six tenths.

00:02:15.660 --> 00:02:19.050
Which is exactly what we
have here, six tenths.

00:02:19.050 --> 00:02:20.660
Let's do one more example,

00:02:20.660 --> 00:02:22.650
that gets a little bit more involved.

00:02:22.650 --> 00:02:24.510
So here we're told to multiply.

00:02:24.510 --> 00:02:26.270
It says you many use the models shown

00:02:26.270 --> 00:02:27.380
to help find the product.

00:02:27.380 --> 00:02:29.850
And this is a screen
shot from the exercise

00:02:29.850 --> 00:02:30.970
on Khan Academy.

00:02:30.970 --> 00:02:31.820
So pause this video

00:02:31.820 --> 00:02:34.560
and see if you can
figure out what this is.

00:02:34.560 --> 00:02:38.110
All right, so they're saying 52 hundredths

00:02:38.110 --> 00:02:40.999
times three, and they have 52 hundredths

00:02:40.999 --> 00:02:42.840
depicted right over here

00:02:42.840 --> 00:02:44.910
and then they have it
depicted three times.

00:02:44.910 --> 00:02:47.500
So the total number of
hundredths depicted here

00:02:47.500 --> 00:02:49.420
that is 52 hundredths times three,

00:02:49.420 --> 00:02:51.010
because we have 52 hundredths here,

00:02:51.010 --> 00:02:54.520
another 52 hundredths, and
then another 52 hundredths.

00:02:54.520 --> 00:02:56.925
So how many hundredths
is that going to be?

00:02:56.925 --> 00:03:01.570
Well, you could view
this as 52 times three

00:03:01.570 --> 00:03:05.290
and that will give you the
number of hundredths we have.

00:03:05.290 --> 00:03:06.720
So let's think about this.

00:03:06.720 --> 00:03:09.950
So if we were to just say 52 times three,

00:03:09.950 --> 00:03:12.130
well this is going to be two times three

00:03:12.130 --> 00:03:15.950
is equal to six and then
five tens times three

00:03:15.950 --> 00:03:18.850
is 15 tens, which is the same thing.

00:03:18.850 --> 00:03:20.960
We either just write it as 15 tens,

00:03:20.960 --> 00:03:22.789
or that's 100 and five tens.

00:03:22.789 --> 00:03:25.830
But either way if I have 52 of something

00:03:25.830 --> 00:03:27.010
and I multiply that by three,

00:03:27.010 --> 00:03:29.010
I now have 156 of that something.

00:03:29.010 --> 00:03:31.990
And here the something is hundredths.

00:03:31.990 --> 00:03:36.990
So if I say 52 hundredths times three

00:03:37.380 --> 00:03:41.069
that's going to be 156 hundredths.

00:03:41.069 --> 00:03:45.960
And how do we represent 156 hundredths.

00:03:45.960 --> 00:03:47.660
Well there is a couple
of ways to think about it

00:03:47.660 --> 00:03:50.880
if this is the ones place,
this is the tenths place,

00:03:50.880 --> 00:03:53.040
this is the hundredths place.

00:03:53.040 --> 00:03:55.890
Well we would write the
six there, the five there,

00:03:55.890 --> 00:03:57.140
and the one there.

00:03:57.140 --> 00:03:59.250
So you could recognize this as hey look,

00:03:59.250 --> 00:04:02.140
a hundred hundredths,
let me color code it,

00:04:02.140 --> 00:04:05.708
a hundred hundredths is
the same thing as a whole

00:04:05.708 --> 00:04:10.680
and I'll circle that in
red, and fifty hundredths

00:04:10.680 --> 00:04:13.340
is the same thing as five tenths,

00:04:13.340 --> 00:04:15.650
and of course six
hundredths is the same thing

00:04:15.650 --> 00:04:17.310
as six hundredths.

00:04:17.310 --> 00:04:20.753
So this is going to be equal to 1.56,

00:04:21.889 --> 00:04:24.190
or you could view this as 156 hundredths,

00:04:24.190 --> 00:04:25.820
or you could view this as a whole,

00:04:25.820 --> 00:04:28.310
which is a hundred
hundredths, and five tenths,

00:04:28.310 --> 00:04:31.563
which is fifty hundredths,
and six hundredths.


Division resulting in decimals

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:01.170
- [Instructor] So in this video,

00:00:01.170 --> 00:00:03.240
we're going to think
about dividing numbers

00:00:03.240 --> 00:00:06.480
where the quotient, the
result of our division,

00:00:06.480 --> 00:00:09.000
might result in a decimal.

00:00:09.000 --> 00:00:10.390
So let's say we wanted to compute

00:00:10.390 --> 00:00:13.370
what five divided by two is.

00:00:13.370 --> 00:00:14.590
In the past, you might've said,

00:00:14.590 --> 00:00:16.470
hey, two times two is four

00:00:16.470 --> 00:00:18.110
and then you'd have one leftover

00:00:18.110 --> 00:00:20.040
and so you might have
said something like this,

00:00:20.040 --> 00:00:23.420
this is two with a remainder of one.

00:00:23.420 --> 00:00:24.940
But now we don't want,
we don't wanna just say,

00:00:24.940 --> 00:00:27.420
hey, how many times can it
go in and what's left over.

00:00:27.420 --> 00:00:31.260
We wanna know exactly how many
times can two go into five

00:00:31.260 --> 00:00:33.780
so we want a decimal
answer right over here.

00:00:33.780 --> 00:00:36.580
So, pause this video and see
if you can figure that out.

00:00:37.440 --> 00:00:39.070
All right, now let's work
through this together.

00:00:39.070 --> 00:00:42.150
And there's many ways that
we could approach this.

00:00:42.150 --> 00:00:45.160
One way is to realize
that five divided by two,

00:00:45.160 --> 00:00:48.480
we could re-express as five divided by two

00:00:48.480 --> 00:00:53.160
or as five halves and five
halves is the same thing

00:00:53.160 --> 00:00:56.930
as four plus one over two

00:00:56.930 --> 00:01:01.660
which is the same thing as
four halves plus one half

00:01:01.660 --> 00:01:04.760
which is the same thing
as four halves is two.

00:01:04.760 --> 00:01:06.190
Let me do that in a different color.

00:01:06.190 --> 00:01:07.840
This part over here is two

00:01:07.840 --> 00:01:09.290
and then you have this one half.

00:01:09.290 --> 00:01:11.950
So it's the same thing as two and a half

00:01:11.950 --> 00:01:14.600
and if we were to
express that as a decimal

00:01:14.600 --> 00:01:17.020
that's going to be equal to two

00:01:17.020 --> 00:01:20.950
and how many tenths is equal to a half?

00:01:20.950 --> 00:01:24.140
Well five tenths is equal to a half.

00:01:24.140 --> 00:01:26.290
So five divided by two,
you could say, 2.5.

00:01:27.810 --> 00:01:30.540
Now another way that you could
approach this is you say,

00:01:30.540 --> 00:01:34.070
five divided by two is the same thing as

00:01:34.070 --> 00:01:36.420
50 tenths divided by two.

00:01:36.420 --> 00:01:38.040
So let me use a different color here.

00:01:38.040 --> 00:01:40.390
So instead of a five, I could say five

00:01:40.390 --> 00:01:43.410
is the same thing as 50 tenths,

00:01:43.410 --> 00:01:46.110
50 tenths divided by,

00:01:46.110 --> 00:01:47.710
divided by two

00:01:47.710 --> 00:01:49.960
and so if I have 50 of something

00:01:49.960 --> 00:01:52.280
and if I were to divide it by two,

00:01:52.280 --> 00:01:55.350
well that's going to result
in 25 of the something.

00:01:55.350 --> 00:01:59.730
So this is going to be equal to 25 tenths.

00:01:59.730 --> 00:02:03.920
And how would we express
25 tenths as a decimal?

00:02:03.920 --> 00:02:05.780
Well, this is going to be equal to,

00:02:05.780 --> 00:02:08.810
if I could put some place values here,

00:02:08.810 --> 00:02:11.400
so we definitely have five tenths.

00:02:11.400 --> 00:02:13.060
I'll put that in the tenths place

00:02:13.060 --> 00:02:18.060
and then 20 tenths is the
same thing as two ones.

00:02:18.520 --> 00:02:20.160
And that's exactly what we had up here.

00:02:20.160 --> 00:02:21.580
Either way, you get 2.5,

00:02:21.580 --> 00:02:23.153
two ones and five tenths.


Example using estimation for decimal products

WEBVTT
Kind: captions
Language: en

00:00:00.310 --> 00:00:04.920
- [Instructor] We are told
that 52 times 762 is equal

00:00:04.920 --> 00:00:09.920
to 39,624, and then we're
told to match each expression

00:00:11.840 --> 00:00:13.410
to its product.

00:00:13.410 --> 00:00:15.290
And these products, this is an exercise

00:00:15.290 --> 00:00:17.270
on Khan Academy, you can move them around

00:00:17.270 --> 00:00:21.000
so the product can be matched
to the appropriate expression.

00:00:21.000 --> 00:00:24.230
So pause this video and see
if you can figure that out.

00:00:24.230 --> 00:00:26.800
All right, now what you
might have realized is all

00:00:26.800 --> 00:00:29.490
of these expressions
deal with the same digits

00:00:29.490 --> 00:00:32.870
as 52 times 762.

00:00:32.870 --> 00:00:35.400
They just have the decimal
in different places.

00:00:35.400 --> 00:00:37.540
And so what we can do
is, we can say, hey look,

00:00:37.540 --> 00:00:40.210
the answer is going to have
the digits three, nine,

00:00:40.210 --> 00:00:41.780
six, two, four, in that order.

00:00:41.780 --> 00:00:44.480
And you could see all of
these have the digits three,

00:00:44.480 --> 00:00:46.250
nine, six, two, four, in that order.

00:00:46.250 --> 00:00:47.800
And then we can estimate what

00:00:47.800 --> 00:00:49.570
these expressions should be equal to,

00:00:49.570 --> 00:00:51.260
what the products should be equal to,

00:00:51.260 --> 00:00:52.810
to think about the decimal.

00:00:52.810 --> 00:00:57.810
So this first expression, 0.52 times 76.2,

00:00:58.480 --> 00:01:02.010
the way I think about
it is 0.52, that's close

00:01:02.010 --> 00:01:04.550
to 50/100, that's close to a half,

00:01:04.550 --> 00:01:08.690
and so 76.2, that's close to 76.

00:01:08.690 --> 00:01:10.850
And so this first expression,

00:01:10.850 --> 00:01:15.410
this first product should
be roughly half of 76.

00:01:15.410 --> 00:01:19.000
Half of 76 would be around 38.

00:01:19.000 --> 00:01:21.520
And so which of these is close to 38?

00:01:21.520 --> 00:01:24.000
Well this first one is 39.624,

00:01:24.000 --> 00:01:25.630
so that's actually the closest to 38.

00:01:25.630 --> 00:01:27.750
The second one is 396.

00:01:27.750 --> 00:01:30.500
And then we have 3,962.

00:01:30.500 --> 00:01:33.731
So I like this first one, the 39.624.

00:01:33.731 --> 00:01:35.840
That feels right.

00:01:35.840 --> 00:01:37.047
Now the second expression,

00:01:37.047 --> 00:01:40.970
0.52 times 762.

00:01:40.970 --> 00:01:45.810
Well once again, 0.52 is
roughly equal to 50/100,

00:01:45.810 --> 00:01:49.930
roughly equal to 5/10,
roughly equal to 1/2,

00:01:49.930 --> 00:01:53.210
and so, and 762, we
could say hey, you know,

00:01:53.210 --> 00:01:55.010
that's, if we wanna be really rough,

00:01:55.010 --> 00:01:57.100
really, really approximate it,

00:01:57.100 --> 00:01:59.200
we could say, hey, it's roughly 800.

00:01:59.200 --> 00:02:01.690
And so this should be about half of 800,

00:02:01.690 --> 00:02:04.120
so it should be around 400.

00:02:04.120 --> 00:02:06.690
And so we actually had
that choice already there.

00:02:06.690 --> 00:02:08.960
So this would be 396.24.

00:02:08.960 --> 00:02:12.030
Definitely wouldn't be the the 3,962.4.

00:02:12.030 --> 00:02:13.860
And so I'm already feeling good

00:02:13.860 --> 00:02:16.290
that this last choice sits down here.

00:02:16.290 --> 00:02:17.234
But I can verify it.

00:02:17.234 --> 00:02:21.100
5.2, well let's just say that's roughly 5.

00:02:21.100 --> 00:02:24.200
762, let's say that's roughly 800.

00:02:24.200 --> 00:02:26.970
So five times 800, that would be around,

00:02:26.970 --> 00:02:28.450
that would be 4000.

00:02:28.450 --> 00:02:31.110
And so we would expect this expression

00:02:31.110 --> 00:02:33.200
to be close to 4000, and indeed,

00:02:33.200 --> 00:02:34.510
that's what this choice is.

00:02:34.510 --> 00:02:37.060
So it turns out that it
was already in the order

00:02:37.060 --> 00:02:38.220
that we needed it to be, but it's good

00:02:38.220 --> 00:02:39.420
that we checked on that.


Multiplying 2-digit by 1-digit with partial products

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.580
- [Instructor] In this
video, we're going to dig

00:00:01.580 --> 00:00:03.840
a little bit deeper and try to understand

00:00:03.840 --> 00:00:07.230
how we might multiply
larger and larger numbers.

00:00:07.230 --> 00:00:08.360
In particular, we're gonna focus on

00:00:08.360 --> 00:00:10.740
multiplying two-digit numbers
times one-digit numbers.

00:00:10.740 --> 00:00:12.560
So, I always encourage
you pause this video,

00:00:12.560 --> 00:00:14.220
and see if you can have a go at this.

00:00:14.220 --> 00:00:17.690
Try to see if you can figure
out what 37 times six is

00:00:17.690 --> 00:00:19.053
with some type of method.

00:00:20.080 --> 00:00:23.010
All right, now let's try to
come up with a method ourselves.

00:00:23.010 --> 00:00:25.290
One way to think about it
is the different places.

00:00:25.290 --> 00:00:26.780
This three is in the tens place,

00:00:26.780 --> 00:00:29.400
so it represents three
tens, and this seven is

00:00:29.400 --> 00:00:31.640
in the ones place, so it
represents seven ones.

00:00:31.640 --> 00:00:36.640
So, you can break up the 37
right over here as three tens.

00:00:36.870 --> 00:00:38.160
I'll write it like this.

00:00:38.160 --> 00:00:40.700
Three tens plus,

00:00:40.700 --> 00:00:45.460
plus, and I'll do this
in yellow, seven ones.

00:00:45.460 --> 00:00:50.460
And then, we are going to
multiply that times six.

00:00:50.480 --> 00:00:52.020
Now, another way you
could think about this,

00:00:52.020 --> 00:00:56.470
this is the same thing as,
I could rewrite this as,

00:00:56.470 --> 00:01:00.360
three tens is the same thing
as 30, and seven ones is,

00:01:00.360 --> 00:01:03.890
of course, just the same
thing as seven times six.

00:01:03.890 --> 00:01:06.100
And, you could also view
this, and you could call

00:01:06.100 --> 00:01:07.660
the distributive property if you like,

00:01:07.660 --> 00:01:10.170
or I really want you to think
about it, why it's intuitive.

00:01:10.170 --> 00:01:14.600
If I have 37 sixes, that's
the same thing as 30 sixes,

00:01:14.600 --> 00:01:19.600
30 sixes plus seven sixes,

00:01:19.935 --> 00:01:24.860
plus seven sixes, and I'll
put the parentheses there

00:01:24.860 --> 00:01:26.390
just so we can keep track.

00:01:26.390 --> 00:01:28.770
Once again, 37 of something is equal to

00:01:28.770 --> 00:01:31.930
30 of that something plus
seven of that something.

00:01:31.930 --> 00:01:36.580
37 times six is 30 times
six plus seven times six.

00:01:36.580 --> 00:01:37.580
Then, we could figure these out.

00:01:37.580 --> 00:01:39.140
What is 30 times six?

00:01:39.140 --> 00:01:43.190
Well, three times six is
18, so three tens times six

00:01:43.190 --> 00:01:47.270
is going to be 18 tens which
is the same thing as 180,

00:01:47.270 --> 00:01:50.704
and seven times six is 42.

00:01:50.704 --> 00:01:53.350
We just then need to
add these two numbers,

00:01:53.350 --> 00:01:54.480
and we could think about them

00:01:54.480 --> 00:01:55.950
in terms of the different places.

00:01:55.950 --> 00:01:59.140
We could look at the ones
place, zero ones, two ones,

00:01:59.140 --> 00:02:01.390
so that'll give us two ones.

00:02:01.390 --> 00:02:04.900
Then, we can think in
terms of the tens place,

00:02:04.900 --> 00:02:07.590
so we have four tens and eight tens,

00:02:07.590 --> 00:02:10.690
so that is going to be 12 tens.

00:02:10.690 --> 00:02:13.260
We could write that down as 120.

00:02:13.260 --> 00:02:16.510
Then, we only have 100 right over here,

00:02:16.510 --> 00:02:18.360
so we don't have a
hundreds place over here,

00:02:18.360 --> 00:02:21.250
so I'll do 100, and if
you add these together,

00:02:21.250 --> 00:02:25.320
100 plus 120, you add the
hundreds, you get 220.

00:02:25.320 --> 00:02:26.440
Then, you add the two ones.

00:02:26.440 --> 00:02:28.360
You get 222.

00:02:28.360 --> 00:02:31.820
So, this is all going to be equal to 222.

00:02:31.820 --> 00:02:33.940
Now, this isn't the
only way to approach it.

00:02:33.940 --> 00:02:36.610
We could actually try to
do it kind of like this,

00:02:36.610 --> 00:02:37.980
but we could write things so it's

00:02:37.980 --> 00:02:40.040
a little bit easier to keep track of.

00:02:40.040 --> 00:02:45.040
So, I could write, let me write
it this way, 37 times six.

00:02:45.380 --> 00:02:47.210
I'll have you, this column
here is where I want

00:02:47.210 --> 00:02:49.700
to write my ones places,
and this column here is

00:02:49.700 --> 00:02:51.030
where I wanna write my tens places,

00:02:51.030 --> 00:02:53.430
and if hundreds places show
up, I'll write it over there.

00:02:53.430 --> 00:02:55.983
That's just really to
keep track of things.

00:02:55.983 --> 00:02:57.810
Then, I can do the same thing.

00:02:57.810 --> 00:03:00.050
I could take my six, and
I could say all right,

00:03:00.050 --> 00:03:02.990
this is the same thing
as three tens times six

00:03:02.990 --> 00:03:04.950
plus seven ones times six.

00:03:04.950 --> 00:03:06.330
And, I could do it in either order.

00:03:06.330 --> 00:03:07.820
Let's do the ones first.

00:03:07.820 --> 00:03:11.950
So if I do the seven ones times
six, that's going to be 42,

00:03:11.950 --> 00:03:14.330
four tens plus two ones.

00:03:14.330 --> 00:03:15.230
How did I get that?

00:03:15.230 --> 00:03:20.060
That is seven ones times
six or seven sixes.

00:03:20.060 --> 00:03:22.180
That's how I got this number here.

00:03:22.180 --> 00:03:26.730
Then, we could multiply our
three tens times the six

00:03:26.730 --> 00:03:31.730
which we did over here,
so three tens times six.

00:03:32.160 --> 00:03:36.480
Well, that's 30 sixes which
we already know is 180.

00:03:36.480 --> 00:03:40.660
One in the hundreds place,
eight tens, zero ones.

00:03:40.660 --> 00:03:42.490
Now, this makes it a little bit easier

00:03:42.490 --> 00:03:43.850
to add everything together.

00:03:43.850 --> 00:03:46.080
Exactly what we did at
this step right over here.

00:03:46.080 --> 00:03:48.140
Everything is in the right place value,

00:03:48.140 --> 00:03:51.120
so it's easy to add them up.

00:03:51.120 --> 00:03:56.120
Two ones plus zero ones
is going to be two ones.

00:03:57.120 --> 00:04:01.820
Four tens plus eight tens is 12 tens

00:04:01.820 --> 00:04:05.540
or 102 tens, so I could
put the two tens here

00:04:05.540 --> 00:04:10.540
and then put that 100 up here,
and then 100 plus 100 is 200.

00:04:10.660 --> 00:04:13.290
So, there you have it, 222.

00:04:13.290 --> 00:04:15.050
The whole point here though is

00:04:15.050 --> 00:04:17.420
to really appreciate what is going on.

00:04:17.420 --> 00:04:19.560
That we're just breaking up the 37.

00:04:19.560 --> 00:04:22.820
We're breaking up this two-digit
number into its places,

00:04:22.820 --> 00:04:25.220
the three tens, the seven ones,
and then we're multiplying

00:04:25.220 --> 00:04:28.090
each of those times the
six in this situation,

00:04:28.090 --> 00:04:29.510
and we did it both ways.

00:04:29.510 --> 00:04:32.100
These are just different
ways to write the same,

00:04:32.100 --> 00:04:33.353
really the same method.


Understanding factor pairs

WEBVTT
Kind: captions
Language: en

00:00:00.230 --> 00:00:02.170
- What we're going to do in this video is

00:00:02.170 --> 00:00:06.130
talk about factors and factor pairs.

00:00:06.130 --> 00:00:07.730
Now when we talk about factors,

00:00:07.730 --> 00:00:10.780
these are really numbers that
can be multiplied together

00:00:10.780 --> 00:00:12.330
to make some number.

00:00:12.330 --> 00:00:16.710
So, for example, if I were
to talk about factors of six,

00:00:16.710 --> 00:00:21.710
I could multiply two
times three to get six

00:00:22.270 --> 00:00:27.190
and so we would say that two
and three are factors of six.

00:00:27.190 --> 00:00:29.480
In fact we would also
say that two and three

00:00:29.480 --> 00:00:32.190
is a factor pair for six
'cause when I multiply

00:00:32.190 --> 00:00:34.600
those two I can get six.

00:00:34.600 --> 00:00:37.160
Now to think about all of
the different factor pairs

00:00:37.160 --> 00:00:40.020
for a number we could think
about it in terms of area.

00:00:40.020 --> 00:00:43.500
How can we make a rectangle with area six?

00:00:43.500 --> 00:00:46.450
Well you could do it if it's
two units by three units.

00:00:46.450 --> 00:00:48.170
So it could look something like this,

00:00:48.170 --> 00:00:49.820
I'll just hand draw it.

00:00:49.820 --> 00:00:53.920
So let's say it has, so
let's say our rectangle

00:00:53.920 --> 00:00:55.460
looks like this.

00:00:55.460 --> 00:00:58.870
So it has two rows and then three columns

00:00:58.870 --> 00:01:00.910
and lets say these all have equal area,

00:01:00.910 --> 00:01:03.410
it's hand drawn but you
can see that the area here

00:01:03.410 --> 00:01:07.190
would be two times three,
which would be equal to

00:01:07.190 --> 00:01:09.160
six square units.

00:01:09.160 --> 00:01:14.160
Now, what if, what are other
ways to get to an area of six?

00:01:14.620 --> 00:01:18.410
Well you could have
something that is one row

00:01:18.410 --> 00:01:20.060
but then it has six columns.

00:01:20.060 --> 00:01:23.080
So maybe it looks something like this.

00:01:23.080 --> 00:01:24.490
It looks like this.

00:01:24.490 --> 00:01:29.490
So it's one row and then
you have one, two, three,

00:01:29.560 --> 00:01:33.570
four, five, and I want them all
to be roughly the same size.

00:01:33.570 --> 00:01:38.070
So it's one times six would
be also an area of six

00:01:38.070 --> 00:01:39.790
so that would be another factor pair.

00:01:39.790 --> 00:01:42.530
We know that two times
three is equal to six

00:01:42.530 --> 00:01:46.690
and we know that one
times six is equal to six

00:01:46.690 --> 00:01:50.150
and these are actually the
two factor pairs for six

00:01:50.150 --> 00:01:52.760
and we can do it with larger numbers.

00:01:52.760 --> 00:01:54.790
We could think about what
about all the factor pairs

00:01:54.790 --> 00:01:57.910
for something like 16?

00:01:57.910 --> 00:02:00.560
Pause this video and see if
you can think about that.

00:02:01.660 --> 00:02:04.910
Well I'll set up a little
table here to think about that

00:02:04.910 --> 00:02:08.900
and so in this column
I'll put the first factor

00:02:09.830 --> 00:02:13.890
and in this column I'll
put the second factor

00:02:13.890 --> 00:02:16.870
and the way I like to
do it is I start at one

00:02:16.870 --> 00:02:19.650
and I keep working up, up to the number,

00:02:19.650 --> 00:02:21.490
to try to figure out all of the factors.

00:02:21.490 --> 00:02:23.900
So let's start with one.

00:02:23.900 --> 00:02:26.300
So one is definitely divisible into 16

00:02:26.300 --> 00:02:28.020
and as long as you put
a whole number here it's

00:02:28.020 --> 00:02:30.030
going to be divisible by one
and we don't have to multiply

00:02:30.030 --> 00:02:34.120
by one to get to 16, we'll
have to multiply it by 16

00:02:34.120 --> 00:02:37.530
so that's a factor pair
right there, one and 16.

00:02:37.530 --> 00:02:38.520
Now what about two?

00:02:38.520 --> 00:02:40.840
Does two go into 16?

00:02:40.840 --> 00:02:43.660
Well sure, two times eight is 16 so

00:02:43.660 --> 00:02:44.770
that's another factor pair.

00:02:44.770 --> 00:02:47.070
Two times eight.

00:02:47.070 --> 00:02:48.800
We found another factor pair.

00:02:48.800 --> 00:02:49.710
Now what about three?

00:02:49.710 --> 00:02:53.140
Does three go evenly into 16?

00:02:53.140 --> 00:02:56.400
Well no, three times five is 15

00:02:56.400 --> 00:02:59.310
and three times six is 18
so three doesn't go into 16

00:02:59.310 --> 00:03:02.580
so three would not be a factor of 16.

00:03:02.580 --> 00:03:04.130
What about four?

00:03:04.130 --> 00:03:08.040
Well four times four is 16 so
that's a factor pair there.

00:03:08.040 --> 00:03:10.710
Four and four.

00:03:10.710 --> 00:03:12.180
What about five?

00:03:12.180 --> 00:03:15.510
Well no, five times two is
10, five times three is 15,

00:03:15.510 --> 00:03:16.970
five times four is 20.

00:03:16.970 --> 00:03:19.370
Five does not go evenly into 16.

00:03:19.370 --> 00:03:21.720
Same thing for six, six times two is 12,

00:03:21.720 --> 00:03:24.480
six times three is 18,
does not go evenly into 16.

00:03:24.480 --> 00:03:25.550
What about seven?

00:03:25.550 --> 00:03:27.610
Seven does not evenly go into 16.

00:03:27.610 --> 00:03:30.400
Seven times two is 14,
seven times three is 21.

00:03:30.400 --> 00:03:31.660
What about eight?

00:03:31.660 --> 00:03:34.180
Well we already know that
eight goes evenly into 16,

00:03:34.180 --> 00:03:36.170
you might be tempted to say
oh there's another factor pair

00:03:36.170 --> 00:03:39.370
of eight times two but we
already wrote that down.

00:03:39.370 --> 00:03:41.380
We just happened to say
that two is the first factor

00:03:41.380 --> 00:03:42.580
and eight is the second
factor but you could

00:03:42.580 --> 00:03:44.420
say it the other way around
so we don't have to then

00:03:44.420 --> 00:03:46.260
go to eight times two.

00:03:46.260 --> 00:03:49.640
And once you've gone half
way you can be confident

00:03:49.640 --> 00:03:51.650
that you've already found
all the factor pairs

00:03:51.650 --> 00:03:53.130
because then you could go to nine,

00:03:53.130 --> 00:03:58.130
nine doesn't go into it,
10, 11, 12, 13, 14, 15

00:03:58.210 --> 00:04:01.210
but then 16, of course, is divisible by 16

00:04:01.210 --> 00:04:03.330
but we've already written
that here in this factor pair.

00:04:03.330 --> 00:04:04.940
So we have these three factor pairs,

00:04:04.940 --> 00:04:08.013
one and 16, two and
eight and four and four.


Multiplying two 2-digit numbers using partial products

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:01.670
- [Instructor] In a previous video,

00:00:01.670 --> 00:00:04.120
we figured out a way to
multiply a two-digit number

00:00:04.120 --> 00:00:05.800
times one-digit number.

00:00:05.800 --> 00:00:07.920
What we did is we broke
up the two-digit numbers

00:00:07.920 --> 00:00:10.410
in terms of its place value,
so the three here in the tenths

00:00:10.410 --> 00:00:13.420
place that's three tens,
this is seven ones.

00:00:13.420 --> 00:00:16.251
So we view 37 sixes as the
same thing as 30 sixes,

00:00:16.251 --> 00:00:21.251
three tens times six plus
seven sixes, seven times six.

00:00:23.530 --> 00:00:27.640
And then we added those
together to get a total of 222.

00:00:27.640 --> 00:00:30.370
What we'll try in this video
is now see what happens if we

00:00:30.370 --> 00:00:33.539
try to do two digits times
two digits, so let's try

00:00:33.539 --> 00:00:38.539
to tackle 37, 37 instead of six
let's multiply that times 26

00:00:41.040 --> 00:00:44.450
so we're trying to figure
out 37 26's is one way

00:00:44.450 --> 00:00:45.530
to think about it.

00:00:45.530 --> 00:00:47.850
So pause this video and
see if you can tackle that

00:00:47.850 --> 00:00:49.630
and see if you can maybe
use a similar method to what

00:00:49.630 --> 00:00:50.603
we used before.

00:00:51.492 --> 00:00:54.450
Well one way to think about
it is you could view this

00:00:54.450 --> 00:00:59.450
as 37 26's or you could view
this as 37 sixes plus 37 20's.

00:01:02.040 --> 00:01:06.130
So first we could do the 37
sixes which is exactly what

00:01:06.130 --> 00:01:09.430
we did over here, we said hey
that's the same thing and we

00:01:09.430 --> 00:01:12.190
could do it in either order
we could say hey let's first

00:01:12.190 --> 00:01:15.810
think about 30 sixes, so we're
going to multiply 30 times

00:01:15.810 --> 00:01:20.810
six 30 sixes is 180, so that
right over there is three tens

00:01:24.129 --> 00:01:25.080
times six.

00:01:25.080 --> 00:01:29.970
And then we could think about
the seven ones times six.

00:01:29.970 --> 00:01:33.010
And so that's going to be 42.

00:01:33.010 --> 00:01:37.037
That's the seven ones times
six or the seven sixes.

00:01:38.520 --> 00:01:41.500
And then we could do the same
thing with the 20's we could

00:01:41.500 --> 00:01:46.500
say hey what are three tens,
what are 30 20's going to be.

00:01:47.850 --> 00:01:49.840
So let's write that over here
or you could say what are

00:01:49.840 --> 00:01:52.260
three tens times two tens.

00:01:52.260 --> 00:01:56.130
Well that would be six times
ten times ten, so let me write

00:01:56.130 --> 00:02:00.510
this down this is three tens
times two tens that's what

00:02:00.510 --> 00:02:02.120
we're going to do now.

00:02:02.120 --> 00:02:06.450
Times two tens and that's
the same thing as which is

00:02:06.450 --> 00:02:10.480
equal to 30 times 20 which
is the same thing as three

00:02:10.480 --> 00:02:12.620
times two times ten times ten.

00:02:12.620 --> 00:02:15.850
Well that's going to be 600,
so we could write that here

00:02:15.850 --> 00:02:18.756
600 and just be very clear,
we've already thought about

00:02:18.756 --> 00:02:21.392
37 sixes, that's these two
numbers up here we have to add

00:02:21.392 --> 00:02:24.480
that we still have to add them.

00:02:24.480 --> 00:02:27.060
But now we're thinking about
37 20's so first we thought

00:02:27.060 --> 00:02:29.889
about 30 20's which is 600
and now we could think about

00:02:29.889 --> 00:02:34.889
seven 20's so seven 20's is
going to be seven times two

00:02:36.810 --> 00:02:40.050
is 14 so seven times 20 is 140.

00:02:40.050 --> 00:02:42.300
So I'll write that right over here.

00:02:42.300 --> 00:02:47.300
140 to be clear this is
seven ones times two tens.

00:02:51.124 --> 00:02:54.917
Or seven 20's and now we
can add it all together

00:02:54.917 --> 00:02:57.150
to get what the total would be.

00:02:57.150 --> 00:03:00.440
So in the total we have,
that's why it's useful to have

00:03:00.440 --> 00:03:03.640
everything stacked by their
place value, we could look

00:03:03.640 --> 00:03:05.790
at the ones place so we say
okay we only have a total

00:03:05.790 --> 00:03:08.010
of two ones here, so I'll put a two there.

00:03:08.010 --> 00:03:10.930
Now let's see tens we have
eight tens plus four tens

00:03:10.930 --> 00:03:15.610
is 12 tens plus another
four tens is 16 tens.

00:03:15.610 --> 00:03:20.610
16 tens we can also break up
as 106 tens, so we could write

00:03:20.997 --> 00:03:23.690
the six tens here and
then put that 100 up here.

00:03:23.690 --> 00:03:28.690
So 100 plus another hundred
is 200 plus 600's is 800's

00:03:29.290 --> 00:03:31.710
plus one more hundred is 900's.

00:03:31.710 --> 00:03:34.920
And so there you have
it, this is equal to 962.

00:03:34.920 --> 00:03:37.630
I really want you to
understand what we just did,

00:03:37.630 --> 00:03:40.316
it might look a little bit
complicated but first we thought

00:03:40.316 --> 00:03:45.316
about what is 37 sixes that's
where we got these numbers

00:03:45.574 --> 00:03:48.120
from and that's what we had
done in a previous video.

00:03:48.120 --> 00:03:51.445
And then we just thought
about well what is 37 20's

00:03:51.445 --> 00:03:55.700
going to be and that's where
these numbers came from

00:03:55.700 --> 00:03:58.054
and actually let me write that down.

00:03:58.054 --> 00:04:01.320
This whole thing that I'm
circling in orange, that is 37

00:04:01.320 --> 00:04:06.320
sixes or I'll write it as 37
times six and then this is

00:04:07.925 --> 00:04:09.004
37 20's so 37 times 20.

00:04:09.004 --> 00:04:14.004
So if it's not obvious, pause
the video or after this video

00:04:15.280 --> 00:04:17.580
reflect on why this works,
try it with other numbers

00:04:17.580 --> 00:04:19.740
'cause if you really
understand this then your

00:04:19.740 --> 00:04:21.900
multiplication life and
actually your mathematical lives

00:04:21.900 --> 00:04:24.700
in your future will only
become more and more intuitive.


Examples identifying multiples

WEBVTT
Kind: captions
Language: en

00:00:00.270 --> 00:00:01.103
- [Instructor] In this video,

00:00:01.103 --> 00:00:03.210
we're gonna start thinking
about what it means

00:00:03.210 --> 00:00:04.980
for something to be a
multiple of a number.

00:00:04.980 --> 00:00:05.813
So, we're asked,

00:00:05.813 --> 00:00:09.480
which of the following
numbers is a multiple of nine?

00:00:09.480 --> 00:00:12.443
So, pause this video and see
if you can figure that out.

00:00:13.460 --> 00:00:15.480
All right, now let's do it together.

00:00:15.480 --> 00:00:17.410
And one way to think about a multiple,

00:00:17.410 --> 00:00:20.680
a multiple is a number that you can get to

00:00:20.680 --> 00:00:24.170
by multiplying the
number, in this case nine,

00:00:24.170 --> 00:00:25.710
by a whole number.

00:00:25.710 --> 00:00:27.730
So we could figure out
the multiples of nine

00:00:27.730 --> 00:00:29.510
by skip counting, that's one way to do it.

00:00:29.510 --> 00:00:31.070
So you could go from nine,

00:00:31.070 --> 00:00:33.520
and then you add nine
to that, you get to 18.

00:00:33.520 --> 00:00:35.530
You add nine to that, you go to 27.

00:00:35.530 --> 00:00:38.710
You add nine to that, you
are going to get to 36.

00:00:38.710 --> 00:00:40.850
You add nine to that, 45.

00:00:40.850 --> 00:00:42.610
Add nine to that, 54.

00:00:42.610 --> 00:00:44.950
Add nine to that, 63.

00:00:44.950 --> 00:00:47.440
Add nine to that, you get 72.

00:00:47.440 --> 00:00:49.880
Add nine to that, you get to 81.

00:00:49.880 --> 00:00:51.560
And we could keep going.

00:00:51.560 --> 00:00:53.670
But to figure out whether
these are multiples,

00:00:53.670 --> 00:00:54.640
you really just have to say, hey,

00:00:54.640 --> 00:00:57.240
are any of these numbers in this list?

00:00:57.240 --> 00:00:59.730
Now, if one of these
numbers are larger than 81

00:00:59.730 --> 00:01:02.100
we would have to keep
going to see it's included.

00:01:02.100 --> 00:01:06.500
But we can see that 46 sits
between two multiples of nine.

00:01:06.500 --> 00:01:08.090
46,

00:01:08.090 --> 00:01:11.010
that sits closer to 45
but it sits in there,

00:01:11.010 --> 00:01:13.080
it sits between two multiples.

00:01:13.080 --> 00:01:15.200
So that's not going to be a multiple.

00:01:15.200 --> 00:01:16.680
Another way to think about it is,

00:01:16.680 --> 00:01:17.800
for something to be a multiple,

00:01:17.800 --> 00:01:19.310
if you divide by nine,

00:01:19.310 --> 00:01:21.050
you're not going to get a remainder.

00:01:21.050 --> 00:01:24.140
But if you divided 46 by nine,

00:01:24.140 --> 00:01:25.380
you are going to get a remainder.

00:01:25.380 --> 00:01:27.750
You're not going to be able
to divide nine into it evenly.

00:01:27.750 --> 00:01:30.550
So I am just going to,

00:01:30.550 --> 00:01:32.910
I'm going to take that
one out of the contention.

00:01:32.910 --> 00:01:35.430
77 is right over here,

00:01:35.430 --> 00:01:37.160
it's between 72 and 81.

00:01:37.160 --> 00:01:40.690
Once again, it's between two
multiples but not a multiple.

00:01:40.690 --> 00:01:43.920
39 is between 36 and 45,

00:01:43.920 --> 00:01:46.210
so not a multiple, between two multiples.

00:01:46.210 --> 00:01:47.080
Rule that out.

00:01:47.080 --> 00:01:50.390
And we can see very clearly
that 18 is a multiple.

00:01:50.390 --> 00:01:52.710
If I was doing this on my own,

00:01:52.710 --> 00:01:54.740
I would just maybe be
skip counting in my head.

00:01:54.740 --> 00:01:57.360
I'd be going, nine, all
right I don't see a nine.

00:01:57.360 --> 00:01:59.940
18, oh, I see an 18, there you go.

00:01:59.940 --> 00:02:02.170
Specially if I'm only
going to pick one choice.

00:02:02.170 --> 00:02:03.633
Let's do another example.

00:02:04.510 --> 00:02:08.560
Which of the following shows
only multiples of eight?

00:02:08.560 --> 00:02:10.903
So, pause this video and think about that.

00:02:11.870 --> 00:02:14.040
All right, well, I could do
it choice by choice here.

00:02:14.040 --> 00:02:14.873
So, let's see.

00:02:14.873 --> 00:02:17.923
This first one, is four
a multiple of eight?

00:02:19.100 --> 00:02:22.380
Well, four, it can divide into eight.

00:02:22.380 --> 00:02:25.280
We could say that eight
is a multiple of four,

00:02:25.280 --> 00:02:27.350
but four is not a multiple of eight.

00:02:27.350 --> 00:02:29.530
What whole number am I
going to multiple eight by

00:02:29.530 --> 00:02:30.840
to get to four?

00:02:30.840 --> 00:02:32.220
So we can rule this out.

00:02:32.220 --> 00:02:34.220
And you can think about
what they're showing here.

00:02:34.220 --> 00:02:35.830
These are actually multiples of four,

00:02:35.830 --> 00:02:36.720
not multiples of eight.

00:02:36.720 --> 00:02:37.570
We can skip count here.

00:02:37.570 --> 00:02:40.870
Four, eight, 12, 16, 20, 24, 28.

00:02:40.870 --> 00:02:43.660
These are multiples of four,
not multiples of eight.

00:02:43.660 --> 00:02:47.010
Some of the multiples of four
are also multiples of eight.

00:02:47.010 --> 00:02:48.420
Eight is a multiple of eight,

00:02:48.420 --> 00:02:50.180
16 is a multiple of eight,

00:02:50.180 --> 00:02:51.840
24 is a multiple of eight,

00:02:51.840 --> 00:02:54.070
but not all of the multiples
of four are multiples of eight.

00:02:54.070 --> 00:02:55.980
And I think you might be
seeing a little pattern here,

00:02:55.980 --> 00:02:58.050
which ones are multiples of eight.

00:02:58.050 --> 00:03:00.400
Now, what about this
choice right over here?

00:03:00.400 --> 00:03:01.890
16 is eight times two.

00:03:01.890 --> 00:03:03.510
24 is eight times three.

00:03:03.510 --> 00:03:05.230
32 is eight times four.

00:03:05.230 --> 00:03:06.480
40 is eight times five.

00:03:06.480 --> 00:03:07.530
In fact, we can skip count.

00:03:07.530 --> 00:03:12.530
Eight then 16, 24, 32, 40, then
48, 56, so on and so forth.

00:03:13.150 --> 00:03:15.320
But these are all multiples of eight,

00:03:15.320 --> 00:03:17.320
so I like this choice.

00:03:17.320 --> 00:03:18.200
And then over here,

00:03:18.200 --> 00:03:20.700
one, two, four, and eight.

00:03:20.700 --> 00:03:22.060
Well, these are showing numbers

00:03:22.060 --> 00:03:24.780
that can be divided into
eight without a remainder.

00:03:24.780 --> 00:03:27.080
You could think of them
as factors of eight.

00:03:27.080 --> 00:03:27.913
You could say, hey,

00:03:27.913 --> 00:03:30.070
I could multiply one
times eight to get eight.

00:03:30.070 --> 00:03:32.950
I could multiple two
times four to get eight.

00:03:32.950 --> 00:03:34.640
But these are not multiples of eight.

00:03:34.640 --> 00:03:38.160
What whole number can I
multiply eight by to get one

00:03:38.160 --> 00:03:40.300
or to get two or to get four?

00:03:40.300 --> 00:03:43.010
In general, your multiples of a number

00:03:43.010 --> 00:03:46.559
are going to be that
number or larger than it.

00:03:46.559 --> 00:03:48.430
I was about to say that
number or multiples of it

00:03:48.430 --> 00:03:49.263
but then I realized

00:03:49.263 --> 00:03:51.010
I can't use multiples to define multiples.

00:03:51.010 --> 00:03:54.580
It would be that number
or larger numbers than it.

00:03:54.580 --> 00:03:57.220
And if you were to skip
count with that number,

00:03:57.220 --> 00:03:59.353
you would hit all of the multiples.


Reasoning about factors and multiples

WEBVTT
Kind: captions
Language: en

00:00:00.270 --> 00:00:01.240
- [Instructor] We're told we know

00:00:01.240 --> 00:00:04.000
that five times three is equal to 15.

00:00:04.000 --> 00:00:05.230
Yep, that's true.

00:00:05.230 --> 00:00:08.430
So which of the following
statements are also true?

00:00:08.430 --> 00:00:10.870
And it says to choose two answers.

00:00:10.870 --> 00:00:14.550
So pause this video and see
if you can work through that.

00:00:14.550 --> 00:00:16.760
All right, now let's go
through them one by one.

00:00:16.760 --> 00:00:21.510
So this first one says,
"Three is a multiple of 15."

00:00:21.510 --> 00:00:24.300
Now, in order for three to be
able to be a multiple of 15,

00:00:24.300 --> 00:00:26.280
that means that we can multiply 15

00:00:26.280 --> 00:00:28.600
by some whole number to get to three.

00:00:28.600 --> 00:00:32.040
But a multiple of 15,
we're thinking 15, 30, 45,

00:00:32.040 --> 00:00:33.370
it's not gonna be three.

00:00:33.370 --> 00:00:35.600
What whole number can I
multiple 15 by to get three?

00:00:35.600 --> 00:00:38.470
If I multiply 15 by
one, I'm already at 15.

00:00:38.470 --> 00:00:40.330
So this is not going to be our choice.

00:00:40.330 --> 00:00:42.710
And since it say pick two answers, well,

00:00:42.710 --> 00:00:44.440
we might be able to figure out it's these.

00:00:44.440 --> 00:00:45.750
But let's just read them

00:00:45.750 --> 00:00:47.710
to make sure that they make sense.

00:00:47.710 --> 00:00:49.790
15 is a multiple of three.

00:00:49.790 --> 00:00:52.690
So that means I can multiply
three times some whole number

00:00:52.690 --> 00:00:55.170
to get to 15, and we know
what that whole number is.

00:00:55.170 --> 00:00:56.003
It's five.

00:00:56.003 --> 00:00:56.836
They tell us right over there.

00:00:56.836 --> 00:00:58.730
Five times three is 15.

00:00:58.730 --> 00:01:02.250
So 15 is a multiple of three.

00:01:02.250 --> 00:01:04.310
15 would also be a multiple
of five 'cause I can

00:01:04.310 --> 00:01:06.710
multiply five by the whole
number three to get to 15.

00:01:06.710 --> 00:01:07.963
So I like this choice.

00:01:08.930 --> 00:01:11.860
Five is a factor of 15.

00:01:11.860 --> 00:01:14.010
Well, a factor, factors of a number

00:01:14.010 --> 00:01:15.580
are numbers that you can multiply together

00:01:15.580 --> 00:01:16.950
to get that number.

00:01:16.950 --> 00:01:20.600
So five is a factor of 15,
and three is a factor of 15,

00:01:20.600 --> 00:01:22.800
because five times three is 15.

00:01:22.800 --> 00:01:25.380
So this one is also true.

00:01:25.380 --> 00:01:26.220
It would've also been true

00:01:26.220 --> 00:01:27.890
if they'd said three is a factor of 15,

00:01:27.890 --> 00:01:30.130
or if they'd said 15
is a multiple of five.

00:01:30.130 --> 00:01:32.210
Any of those would have
been true statements

00:01:32.210 --> 00:01:36.693
based on what we know, that
five times three is equal to 15.


Comparing decimals example

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:02.900
- [Instructor] So we have
four numbers listed here,

00:00:02.900 --> 00:00:04.260
and what I would like you to do

00:00:04.260 --> 00:00:07.180
is get out some pencil and
paper and pause this video

00:00:07.180 --> 00:00:09.160
and see if you can order these numbers

00:00:09.160 --> 00:00:10.400
from least to greatest.

00:00:10.400 --> 00:00:12.220
So the least would be at the left,

00:00:12.220 --> 00:00:13.820
and then keep going greater, and greater,

00:00:13.820 --> 00:00:15.310
and greater until you get
to the greatest number.

00:00:15.310 --> 00:00:18.260
So pause the video and have a go at that.

00:00:18.260 --> 00:00:19.960
All right, now let's tackle this together.

00:00:19.960 --> 00:00:21.100
And the way that I like to do it

00:00:21.100 --> 00:00:23.220
is I start at the, I guess you could say,

00:00:23.220 --> 00:00:25.010
the most significant place value

00:00:25.010 --> 00:00:27.520
or the largest place
value, compare the numbers.

00:00:27.520 --> 00:00:29.180
And then keep moving to the right

00:00:29.180 --> 00:00:31.360
to smaller and smaller place values.

00:00:31.360 --> 00:00:33.200
So we can start in the ones place.

00:00:33.200 --> 00:00:36.670
This number has zero ones,
this number has zero ones,

00:00:36.670 --> 00:00:38.370
this number has zero ones,

00:00:38.370 --> 00:00:40.360
and that number has zero ones.

00:00:40.360 --> 00:00:43.040
So the ones place really
doesn't help us much.

00:00:43.040 --> 00:00:46.030
But then we can move to the tenths place.

00:00:46.030 --> 00:00:48.430
This number has 7/10.

00:00:48.430 --> 00:00:50.630
This number has 0/10,

00:00:50.630 --> 00:00:52.890
so just from that we know
that the second number

00:00:52.890 --> 00:00:54.140
is less than the first number.

00:00:54.140 --> 00:00:56.740
This has 7/10, this has 0/10.

00:00:56.740 --> 00:00:57.880
It doesn't matter what's happening

00:00:57.880 --> 00:01:00.290
in the places after that,
to the right of that.

00:01:00.290 --> 00:01:02.540
This number over here also has 7/10,

00:01:02.540 --> 00:01:03.830
just like the first number.

00:01:03.830 --> 00:01:06.530
And this last number also has 7/10.

00:01:06.530 --> 00:01:10.260
So we know from comparing the
ones and then the tenths place

00:01:10.260 --> 00:01:12.060
is that this number right over here

00:01:12.060 --> 00:01:14.380
is the smallest of the four numbers.

00:01:14.380 --> 00:01:18.410
They all have zero ones,
but this one also has 0/10.

00:01:18.410 --> 00:01:23.290
So I'll list that here, 0.074.

00:01:23.290 --> 00:01:26.400
Now let's move to the hundredths place.

00:01:26.400 --> 00:01:29.030
So this number has 0/100.

00:01:29.030 --> 00:01:30.870
We've already used this number.

00:01:30.870 --> 00:01:34.050
This number has 7/100.

00:01:34.050 --> 00:01:36.185
And then this number,
it might not be obvious,

00:01:36.185 --> 00:01:39.290
but the hundredths place
you can view as being zero,

00:01:39.290 --> 00:01:42.010
the hundredths place, you
can just put a zero there

00:01:42.010 --> 00:01:42.843
and not change the value.

00:01:42.843 --> 00:01:45.120
So this also has 0/100.

00:01:45.120 --> 00:01:49.410
So these three numbers,
same ones, same tenths,

00:01:49.410 --> 00:01:54.410
but this number, 0.77, has 7/100,

00:01:54.730 --> 00:01:57.380
while the other two had 0/100.

00:01:57.380 --> 00:02:00.420
So this is going to be the
largest of our four numbers.

00:02:00.420 --> 00:02:02.590
This is larger than these other two

00:02:02.590 --> 00:02:05.150
because of what we see
in the hundredths place.

00:02:05.150 --> 00:02:06.430
It doesn't matter what's happening

00:02:06.430 --> 00:02:09.200
in the thousandths place
or anything beyond that.

00:02:09.200 --> 00:02:13.500
So we put the 0.77 right over there.

00:02:13.500 --> 00:02:16.200
And now we are tasked, and
we've used this number,

00:02:16.200 --> 00:02:18.530
and now we have to
compare these two numbers,

00:02:18.530 --> 00:02:22.210
which were equal in the ones,
tenths, and hundredths place,

00:02:22.210 --> 00:02:24.410
so we have to go to the thousandths place.

00:02:24.410 --> 00:02:26.920
This number has 7/1000.

00:02:26.920 --> 00:02:30.150
This number has 0/1000.

00:02:30.150 --> 00:02:35.150
So this number is smaller
than this first number here.

00:02:35.380 --> 00:02:37.630
So I'll write this next, 0.7,

00:02:38.970 --> 00:02:40.730
and then the third smallest,

00:02:40.730 --> 00:02:43.270
or the next to largest
number is this one over here,

00:02:43.270 --> 00:02:47.390
0.707, and we're done.

00:02:47.390 --> 00:02:49.360
So the main idea is you wanna compare

00:02:49.360 --> 00:02:50.860
the most significant place values,

00:02:50.860 --> 00:02:52.550
the largest places values first,

00:02:52.550 --> 00:02:55.390
and then based on that,
keep moving to the right

00:02:55.390 --> 00:02:59.653
to compare less and less
significant place values.


Estimating with multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:01.770
- [Instructor] In this video,
we're gonna get a little bit

00:00:01.770 --> 00:00:04.920
of practice estimating
with multiplication.

00:00:04.920 --> 00:00:07.570
So over here it says question
mark is, and then you have

00:00:07.570 --> 00:00:09.450
this squiggly equal sign.

00:00:09.450 --> 00:00:12.760
And so you can view that
squiggly equal sign as being what

00:00:12.760 --> 00:00:14.440
does this roughly equal to?

00:00:14.440 --> 00:00:16.460
It doesn't have to be exactly right.

00:00:16.460 --> 00:00:21.460
So what is roughly equal to 58 times six?

00:00:21.640 --> 00:00:23.360
And so which of these
choices would you pick?

00:00:23.360 --> 00:00:26.410
Pause the video and see
if you can answer that.

00:00:26.410 --> 00:00:29.210
So once again, we don't have
to get this exactly we just

00:00:29.210 --> 00:00:32.270
need to estimate what 58 times six is.

00:00:32.270 --> 00:00:35.480
And the way that I would
tackle it is hey, can I rewrite

00:00:35.480 --> 00:00:40.300
58 as a number that is easy
to multiply by six or easy

00:00:40.300 --> 00:00:41.710
to multiply in general?

00:00:41.710 --> 00:00:43.730
And the easiest thing I can think of is 58

00:00:43.730 --> 00:00:46.520
is awfully close to 60.

00:00:46.520 --> 00:00:49.210
So if you say that that is
close to 60 if we're just

00:00:49.210 --> 00:00:53.880
estimating it, well what
would 60 times six be?

00:00:53.880 --> 00:00:57.630
Well 60 times six is the same
thing as, actually let me

00:00:57.630 --> 00:00:58.463
just write it this way.

00:00:58.463 --> 00:01:03.370
You can view 60 times six
as six times 10 times six,

00:01:04.760 --> 00:01:08.500
which you could also view
as six times six times 10.

00:01:08.500 --> 00:01:10.280
And what's six times six?

00:01:10.280 --> 00:01:11.660
That's 36.

00:01:11.660 --> 00:01:15.500
And then 36 times 10 is 360.

00:01:15.500 --> 00:01:17.410
So I would go with that
choice right over there.

00:01:17.410 --> 00:01:20.090
Notice, these other choices
aren't anywhere close to that

00:01:20.090 --> 00:01:22.620
so we feel pretty good
about our estimation.

00:01:22.620 --> 00:01:24.560
Let's do another example.

00:01:24.560 --> 00:01:28.470
So here we're asked
what's three times 2,890?

00:01:28.470 --> 00:01:29.650
What's that roughly equal to?

00:01:29.650 --> 00:01:31.260
How would you estimate this?

00:01:31.260 --> 00:01:33.783
Pause the video and see if
you can figure that out.

00:01:34.960 --> 00:01:39.320
Alright, so 2,890 is not an
easy thing to maybe multiply

00:01:39.320 --> 00:01:43.310
in our heads, but I can say
hey, that's kinda close to 3,000

00:01:43.310 --> 00:01:47.990
so I'm gonna say that this
is going to be approximately

00:01:47.990 --> 00:01:52.907
equal to, or roughly equal
to, three times 3,000.

00:01:54.150 --> 00:01:56.100
Now what's three times three?

00:01:56.100 --> 00:01:57.090
That's nine.

00:01:57.090 --> 00:01:58.770
So what's three times 3,000?

00:01:58.770 --> 00:02:01.220
We'll it's gonna be nine thousands.

00:02:01.220 --> 00:02:03.500
So it would be this
choice right over here.

00:02:03.500 --> 00:02:05.880
If what I just did was a little
bit confusing, you can view

00:02:05.880 --> 00:02:10.880
three times 3,000 as three
times three times 1,000.

00:02:11.090 --> 00:02:13.430
This part right over there is 3,000.

00:02:13.430 --> 00:02:16.480
Three times three is the nine,

00:02:16.480 --> 00:02:19.700
and you have nine thousands, 9,000.

00:02:19.700 --> 00:02:21.260
Let's do one more example.

00:02:21.260 --> 00:02:24.150
This is going to involve
fairly large numbers.

00:02:24.150 --> 00:02:28.610
So what is, if you had to
estimate 5,111 times nine.

00:02:30.530 --> 00:02:32.220
Which of these would you pick?

00:02:32.220 --> 00:02:33.820
Pause the video and answer that.

00:02:34.730 --> 00:02:37.730
Alright, so I would view
this, this is roughly,

00:02:37.730 --> 00:02:41.070
I would estimate it, as 5,000 times 9.

00:02:41.070 --> 00:02:46.070
And so 5,000 times nine, that's
the same thing as five times

00:02:49.033 --> 00:02:54.033
1,000 times nine, which is
the same thing as five times

00:02:55.160 --> 00:02:58.070
nine times 1,000.

00:02:58.070 --> 00:03:02.100
Five times nine is 45, so
it's gonna be 45 thousands,

00:03:02.100 --> 00:03:07.100
or 45,000 which is this
choice right over there.


Estimating 2 digit multiplication example

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:01.320
- [Instructor] So, we are asked,

00:00:01.320 --> 00:00:04.060
question mark is roughly equal

00:00:04.060 --> 00:00:06.500
to this squiggly equal
sign right over here,

00:00:06.500 --> 00:00:08.140
this means roughly equal to,

00:00:08.140 --> 00:00:10.310
so not exactly equal to

00:00:10.310 --> 00:00:12.920
44 times 78.

00:00:12.920 --> 00:00:16.700
So one way to think about
it is, 44 times 78 is

00:00:16.700 --> 00:00:19.170
roughly equal to what?

00:00:19.170 --> 00:00:21.150
So they're really asking us to estimate.

00:00:21.150 --> 00:00:22.600
So I encourage you to pause this video

00:00:22.600 --> 00:00:25.500
and try to think through
how you would estimate

00:00:25.500 --> 00:00:27.300
what 44 times 78 is.

00:00:27.300 --> 00:00:28.990
Once again, you don't
have to get it exactly,

00:00:28.990 --> 00:00:30.680
you just want to get it roughly.

00:00:30.680 --> 00:00:32.274
And think about which of these choices

00:00:32.274 --> 00:00:35.823
are closest to that estimation of yours.

00:00:36.940 --> 00:00:38.470
All right now, let's
work through it together.

00:00:38.470 --> 00:00:41.627
So for me, the best way to
estimate is to think about,

00:00:41.627 --> 00:00:43.547
"Hey, can I think of these numbers,

00:00:43.547 --> 00:00:44.757
"can I estimate these numbers,

00:00:44.757 --> 00:00:46.950
"or think about what are
they roughly equal to?"

00:00:46.950 --> 00:00:48.700
And when I think about what
they're roughly equal to I want

00:00:48.700 --> 00:00:50.140
to think about numbers that are easy

00:00:50.140 --> 00:00:52.390
to multiply potentially in my head.

00:00:52.390 --> 00:00:56.580
So for example 44 is
reasonably close to 40,

00:00:56.580 --> 00:00:57.507
so I could say that,

00:00:57.507 --> 00:01:02.507
"Hey this is approximately
equal to or roughly equal to 40,

00:01:03.837 --> 00:01:07.850
"so that's my estimate of
44, or 44 is roughly 40."

00:01:07.850 --> 00:01:09.097
And then 78 I could say,

00:01:09.097 --> 00:01:12.317
"Hey, that's pretty close
to 80" and so I could say,

00:01:12.317 --> 00:01:16.860
"This is roughly equal to 40 times 80,"

00:01:16.860 --> 00:01:18.530
and this is pretty straight forward.

00:01:18.530 --> 00:01:20.230
Some of you might already recognize

00:01:20.230 --> 00:01:23.390
that four times eight is 32,

00:01:23.390 --> 00:01:26.900
and then we have four tens
and then we have eight tens,

00:01:26.900 --> 00:01:30.070
so it would be four times
eight times 10 twice.

00:01:30.070 --> 00:01:32.440
But if what I just said
is confusing to you,

00:01:32.440 --> 00:01:34.430
we could think about it this way.

00:01:34.430 --> 00:01:38.890
40 times 80 is the same
thing as four times 10,

00:01:38.890 --> 00:01:42.750
that's 40, times eight
times 10, that's 80,

00:01:42.750 --> 00:01:45.470
and this is equal to, I'm just
gonna switch the order here

00:01:45.470 --> 00:01:47.450
of this multiplication which we can do,

00:01:47.450 --> 00:01:52.040
this is equal to four times
eight times 10 times 10,

00:01:52.040 --> 00:01:55.040
four times eight is 32,

00:01:55.040 --> 00:01:57.490
10 times 10 is 100.

00:01:57.490 --> 00:02:00.710
So we can say that this
is equal to 32 hundreds,

00:02:00.710 --> 00:02:04.470
or we could view that as 3,200,

00:02:04.470 --> 00:02:06.780
which we could also view as 32 hundreds.

00:02:06.780 --> 00:02:09.980
And so that is this
choice right over here.

00:02:09.980 --> 00:02:13.260
Now this isn't exactly
what 44 times 78 is,

00:02:13.260 --> 00:02:14.950
but it's roughly, and this is useful.

00:02:14.950 --> 00:02:17.640
Sometimes in life, you just
have to get a rough sense

00:02:17.640 --> 00:02:19.510
of what something is going to be.

00:02:19.510 --> 00:02:22.200
Sometimes when you even are
trying to find this exact

00:02:22.200 --> 00:02:25.580
product, your brain wants to
check does that make sense?

00:02:25.580 --> 00:02:28.450
Let say you went through some
process to figure out this,

00:02:28.450 --> 00:02:31.040
and you got some number

00:02:31.040 --> 00:02:35.110
that is close to 20, and you're
like, "Wait, wait a minute"

00:02:35.110 --> 00:02:36.890
But roughly when I
think about 40 times 80,

00:02:36.890 --> 00:02:38.410
this should be close to 3,200,

00:02:38.410 --> 00:02:39.960
I must have done something wrong.

00:02:39.960 --> 00:02:42.623
So it also helps you
keep a check on yourself.


Compare with multiplication examples

WEBVTT
Kind: captions
Language: en

00:00:00.230 --> 00:00:01.430
- [Instructor] This here is a screenshot

00:00:01.430 --> 00:00:03.360
from this exercise on Khan Academy.

00:00:03.360 --> 00:00:08.360
It says, the number 48 is
six times as many as eight.

00:00:09.410 --> 00:00:13.860
Write this comparison as
a multiplication equation.

00:00:13.860 --> 00:00:17.170
So, pause this video and see
if you can have a go at that.

00:00:17.170 --> 00:00:18.740
All right, so, it sounds very fancy.

00:00:18.740 --> 00:00:21.720
This comparison as a
multiplication equation.

00:00:21.720 --> 00:00:23.000
But really they're just asking us

00:00:23.000 --> 00:00:24.640
to write this as an equation.

00:00:24.640 --> 00:00:27.030
And if we were on Khan Academy
we would type it in here

00:00:27.030 --> 00:00:27.870
but I'm just going to do it,

00:00:27.870 --> 00:00:28.790
this is a screenshot,

00:00:28.790 --> 00:00:30.890
so I'm just going to do it by digital pen.

00:00:31.797 --> 00:00:34.440
So, they're saying the number 48,

00:00:34.440 --> 00:00:36.150
the number 48,

00:00:36.150 --> 00:00:38.087
and when they say is,

00:00:38.087 --> 00:00:40.170
another way of interpreting is to say,

00:00:40.170 --> 00:00:41.930
hey, that's equal to something.

00:00:41.930 --> 00:00:43.410
Now, what does that equal to?

00:00:43.410 --> 00:00:46.270
That's equal to six
times as many as eight.

00:00:46.270 --> 00:00:47.450
Well, what's another way to write

00:00:47.450 --> 00:00:49.080
six times as many as eight?

00:00:49.080 --> 00:00:52.130
Well, we can write that
as six times eight.

00:00:52.130 --> 00:00:53.170
So this right over here,

00:00:53.170 --> 00:00:54.950
this is that comparison

00:00:54.950 --> 00:00:57.070
written as a multiplication equation.

00:00:57.070 --> 00:00:58.340
That's all they're asking for.

00:00:58.340 --> 00:01:00.880
And so you would type
that in right over there.

00:01:00.880 --> 00:01:02.130
Let's do another example.

00:01:03.310 --> 00:01:06.550
Here, we are told, or we are asked,

00:01:06.550 --> 00:01:09.330
what statement represents the equation

00:01:09.330 --> 00:01:13.400
seven times four is equal to 28?

00:01:13.400 --> 00:01:14.380
So, pause this video

00:01:14.380 --> 00:01:16.940
and think about which
choice would you pick.

00:01:16.940 --> 00:01:17.773
So, let's see.

00:01:17.773 --> 00:01:21.940
Choice A says, 28 is seven more than four.

00:01:21.940 --> 00:01:23.020
First of all, that's not true.

00:01:23.020 --> 00:01:25.050
28 is not seven more than four.

00:01:25.050 --> 00:01:28.380
But if you were to write this
comparison as an equation,

00:01:28.380 --> 00:01:31.510
this would be that 28 is,

00:01:31.510 --> 00:01:34.510
so is equal to seven more than four

00:01:34.510 --> 00:01:36.090
is seven plus four.

00:01:36.090 --> 00:01:37.280
Not only is that not true,

00:01:37.280 --> 00:01:38.880
but that's not the original equation.

00:01:38.880 --> 00:01:40.430
So we rule that out.

00:01:40.430 --> 00:01:43.570
28 is seven times as many as four.

00:01:43.570 --> 00:01:44.410
Well, this is interesting.

00:01:44.410 --> 00:01:45.640
28,

00:01:45.640 --> 00:01:47.220
28 is,

00:01:47.220 --> 00:01:48.460
is equal to

00:01:48.460 --> 00:01:50.720
seven times as many as four,

00:01:50.720 --> 00:01:53.350
seven times four.

00:01:53.350 --> 00:01:56.410
Well, the only difference
between this and this

00:01:56.410 --> 00:01:59.130
is just we've swapped the
sides on the equality,

00:01:59.130 --> 00:02:01.140
which is completely okay.

00:02:01.140 --> 00:02:02.940
So, 28 is equal to seven times four,

00:02:02.940 --> 00:02:05.350
that's the same equation, really,

00:02:05.350 --> 00:02:07.930
as seven times four is equal to 28.

00:02:07.930 --> 00:02:10.290
So I like this choice.

00:02:10.290 --> 00:02:11.680
Now let's just look at the other choices.

00:02:11.680 --> 00:02:15.560
Seven is four times as many as 28.

00:02:15.560 --> 00:02:16.930
That plays with your head a little bit,

00:02:16.930 --> 00:02:18.780
but let's just write it as an equation.

00:02:18.780 --> 00:02:19.660
Seven

00:02:20.610 --> 00:02:21.443
is

00:02:21.443 --> 00:02:23.070
so that would be equal to

00:02:23.070 --> 00:02:24.820
four times as many as 28

00:02:24.820 --> 00:02:27.240
that would be four times 28.

00:02:27.240 --> 00:02:28.820
One, this is not true.

00:02:28.820 --> 00:02:32.280
Four times 28 is over a
100, so it's not seven.

00:02:32.280 --> 00:02:34.520
But this is also very different
than what we see up here,

00:02:34.520 --> 00:02:35.930
so we'd rule that out.

00:02:35.930 --> 00:02:38.030
Seven is four less than 28.

00:02:38.030 --> 00:02:39.100
Well, this equation would be

00:02:39.100 --> 00:02:40.580
seven is,

00:02:40.580 --> 00:02:41.580
so is equal to,

00:02:41.580 --> 00:02:43.760
four less than 28,

00:02:43.760 --> 00:02:47.340
so four less than 28
would be 28 minus four.

00:02:47.340 --> 00:02:49.050
Once again, not only is this not true,

00:02:49.050 --> 00:02:52.210
28 minus four is 24, not seven,

00:02:52.210 --> 00:02:54.030
but this is not that original equation.

00:02:54.030 --> 00:02:55.430
So we'd rule that one out as well.

00:02:55.430 --> 00:02:57.173
So we like our choice B.