Adding multi digit numbers with regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.971 --> 00:00:03.701
- [Instructor] What we're
going to do in this video

00:00:03.701 --> 00:00:05.784
is add 48,029 to 233,930.

00:00:07.962 --> 00:00:10.005
And like always, pause this
video, and I really encourage

00:00:10.005 --> 00:00:11.625
you to try to figure it out on your own,

00:00:11.625 --> 00:00:13.349
and let's see if we get the same answer.

00:00:13.349 --> 00:00:15.067
And if we don't, why?

00:00:15.067 --> 00:00:17.494
Alright, so the way I'm
going to tackle this,

00:00:17.494 --> 00:00:19.129
I assume that you've had a go at it,

00:00:19.129 --> 00:00:20.826
I'm going to take the larger number.

00:00:20.826 --> 00:00:21.914
I'm going to write it on top.

00:00:21.914 --> 00:00:23.247
I'm really doing the standard method.

00:00:23.247 --> 00:00:27.129
There's multiple ways where you
can add multi-digit numbers,

00:00:27.129 --> 00:00:30.056
but what I'm going to do is
really the most typical method.

00:00:30.056 --> 00:00:32.087
And then I'm going to write
the smaller number below it,

00:00:32.087 --> 00:00:34.328
but I'm going to match up the place value.

00:00:34.328 --> 00:00:36.589
So I'm going to write the ones place

00:00:36.589 --> 00:00:38.798
in the same column as the
ones place on the top number.

00:00:38.798 --> 00:00:42.131
So this is the 10,000s place, so 48,029.

00:00:44.847 --> 00:00:47.645
So nine ones is zero ones,
two tens, three tens,

00:00:47.645 --> 00:00:49.433
so on and so forth.

00:00:49.433 --> 00:00:51.600
And now I am ready to add.

00:00:52.463 --> 00:00:55.815
So let's start in the ones place.

00:00:55.815 --> 00:00:57.394
And if I'm adding numbers,

00:00:57.394 --> 00:00:59.661
it's always a good idea to
start in the ones place.

00:00:59.661 --> 00:01:02.828
Zero ones plus nine ones is nine ones.

00:01:03.759 --> 00:01:06.813
Then I can go into the tens place.

00:01:06.813 --> 00:01:09.980
Three tens plus two tens is five tens.

00:01:10.958 --> 00:01:13.314
This is going well, alright.

00:01:13.314 --> 00:01:16.054
Then I go to the hundreds place.

00:01:16.054 --> 00:01:18.086
Nine hundreds plus zero hundreds,

00:01:18.086 --> 00:01:19.758
well, that's just going
to be nine hundreds.

00:01:19.758 --> 00:01:21.265
So far, so good.

00:01:21.265 --> 00:01:23.810
Alright, now something
interesting is going to happen

00:01:23.810 --> 00:01:25.570
in the thousands place.

00:01:25.570 --> 00:01:28.322
Three thousands plus eight thousands,

00:01:28.322 --> 00:01:30.099
well, that would be 11 thousands.

00:01:30.099 --> 00:01:34.266
But we can rewrite 11
thousands as 1000 and one 10000

00:01:36.755 --> 00:01:39.437
sometimes you might see this
described as carrying the one.

00:01:39.437 --> 00:01:44.018
Three plus eight is 11
where you carry the one.

00:01:44.018 --> 00:01:46.467
But all you're really doing is regrouping.

00:01:46.467 --> 00:01:50.896
3000 plus 8000 is 11000, we
write the 1000 here and we

00:01:50.896 --> 00:01:54.548
write the 10000 right over
there, the 10000 place.

00:01:54.548 --> 00:01:58.715
And now so we have one 10000
plus three 10000s plus four

00:02:00.021 --> 00:02:03.342
10000s so one plus three
plus four that's eight 10000s

00:02:03.342 --> 00:02:04.175
or 80000.

00:02:06.066 --> 00:02:08.642
And then last but not least
we have these two hundred

00:02:08.642 --> 00:02:10.447
thousands right over there and we're done.

00:02:10.447 --> 00:02:13.447
281,959 did you get the same answer?


Adding multi digit numbers with place value

WEBVTT
Kind: captions
Language: en

00:00:01.923 --> 00:00:03.471
- [Instructor] What we're
going to do in this video is

00:00:03.471 --> 00:00:05.997
get some practice adding
multiple digit numbers,

00:00:05.997 --> 00:00:09.065
but the point of it, isn't
just to get the answer,

00:00:09.065 --> 00:00:12.871
but to understand why the
method we use actually works.

00:00:12.871 --> 00:00:16.121
So we're going to add 40,762 to 30,473.

00:00:20.980 --> 00:00:23.032
And you can pause the video and
try to solve it on your own,

00:00:23.032 --> 00:00:24.714
but I encourage you to watch this one

00:00:24.714 --> 00:00:28.817
because it's really about
understanding how things happen.

00:00:28.817 --> 00:00:30.463
So what I'm going to do is first,

00:00:30.463 --> 00:00:32.156
think of this in terms of place value.

00:00:32.156 --> 00:00:33.983
So let me write out my
place values, let's see,

00:00:33.983 --> 00:00:36.351
this goes all the way up
to the ten thousands place.

00:00:36.351 --> 00:00:40.518
So let's see ten thousands
and then we have a thousands,

00:00:43.689 --> 00:00:45.715
then we have a hundreds--

00:00:45.715 --> 00:00:47.258
I could write these words
out but this is a little bit

00:00:47.258 --> 00:00:48.612
faster to write it this way.

00:00:48.612 --> 00:00:51.401
And then we could have a tens place

00:00:51.401 --> 00:00:54.560
and then we can have a ones place.

00:00:54.560 --> 00:00:56.328
I wanna do that in a different color.

00:00:56.328 --> 00:00:58.749
So then we have a ones place.

00:00:58.749 --> 00:01:00.453
Let me make a table here

00:01:00.453 --> 00:01:02.622
and I'm gonna express
both of these numbers

00:01:02.622 --> 00:01:05.132
in terms of ten thousands, thousands,

00:01:05.132 --> 00:01:07.246
hundreds, tens, and ones

00:01:07.246 --> 00:01:09.150
and then I'm gonna, at the same time,

00:01:09.150 --> 00:01:11.457
use what's sometimes known
as the standard method

00:01:11.457 --> 00:01:13.698
or the standard algorithm.

00:01:13.698 --> 00:01:16.784
Algorithm is a fancy word for a system,

00:01:16.784 --> 00:01:19.009
a way of doing something.

00:01:19.009 --> 00:01:21.104
But let's first represent these numbers.

00:01:21.104 --> 00:01:24.311
Here I have four ten thousands.

00:01:24.311 --> 00:01:25.940
One, two, three, four.

00:01:25.940 --> 00:01:28.028
Here I have three ten thousands.

00:01:28.028 --> 00:01:29.464
One, two, three,

00:01:29.464 --> 00:01:31.746
I'm gonna add these two
together eventually.

00:01:31.746 --> 00:01:33.296
In both of these numbers,

00:01:33.296 --> 00:01:35.129
I have zero thousands,

00:01:36.139 --> 00:01:38.534
so I have nothing in
this column right now.

00:01:38.534 --> 00:01:40.777
Here, I have seven hundreds.

00:01:40.777 --> 00:01:43.071
One, two, three, four, five, six, seven.

00:01:43.071 --> 00:01:45.237
Here I have four hundreds.

00:01:45.237 --> 00:01:47.696
One, two, three, four.

00:01:47.696 --> 00:01:49.702
Then I go to the tens place,

00:01:49.702 --> 00:01:50.932
here I have six tens.

00:01:50.932 --> 00:01:54.073
One, two, three, four, five, six.

00:01:54.073 --> 00:01:56.687
Here I have seven tens.

00:01:56.687 --> 00:01:59.789
One, two, three, four, five, six, seven.

00:01:59.789 --> 00:02:03.651
And then last, but not
least, here I have two ones.

00:02:03.651 --> 00:02:04.484
One, two.

00:02:04.484 --> 00:02:06.587
And here I have three ones.

00:02:06.587 --> 00:02:07.920
One, two, three.

00:02:08.783 --> 00:02:11.091
Now, let's just rewrite
this number up here,

00:02:11.091 --> 00:02:14.105
this is four ten thousands,

00:02:14.105 --> 00:02:16.741
so this is forty thousand,

00:02:16.741 --> 00:02:19.040
I have zero thousands.

00:02:19.040 --> 00:02:19.873
Right over there.

00:02:19.873 --> 00:02:23.138
And then I have seven hundreths.

00:02:23.138 --> 00:02:25.015
Seven hundreds.

00:02:25.015 --> 00:02:26.348
I have six tens.

00:02:27.833 --> 00:02:29.335
Six tens.

00:02:29.335 --> 00:02:31.002
And I have two ones.

00:02:32.268 --> 00:02:33.861
I'm just rewriting the number,

00:02:33.861 --> 00:02:36.371
but let me write the
tens in that blue color.

00:02:36.371 --> 00:02:38.871
So six tens and then two ones.

00:02:41.660 --> 00:02:43.369
Having trouble switching colors.

00:02:43.369 --> 00:02:44.202
Two ones.

00:02:44.202 --> 00:02:45.358
So there you have it.

00:02:45.358 --> 00:02:47.244
This and this are just different ways of

00:02:47.244 --> 00:02:48.894
representing the same number

00:02:48.894 --> 00:02:52.959
and then this down here, I
have three ten thousands,

00:02:52.959 --> 00:02:55.542
and then I have zero thousands,

00:02:57.199 --> 00:02:59.282
I have four hundreds here

00:03:01.378 --> 00:03:03.628
and then I have seven tens,

00:03:05.463 --> 00:03:06.380
seven tens,

00:03:07.659 --> 00:03:09.909
and then I have three ones.

00:03:11.826 --> 00:03:14.651
And so now, let's add up everything.

00:03:14.651 --> 00:03:16.721
So I can add it here and then I can also

00:03:16.721 --> 00:03:18.809
add up things right over here.

00:03:18.809 --> 00:03:20.186
So in the standard method,

00:03:20.186 --> 00:03:21.945
we would start at the lowest place,

00:03:21.945 --> 00:03:23.848
and we'd say okay, two
ones plus three ones

00:03:23.848 --> 00:03:25.341
is equal to five ones,

00:03:25.341 --> 00:03:28.194
and similarly, two ones
plus three ones would be

00:03:28.194 --> 00:03:31.007
one, two, three, four, five ones.

00:03:31.007 --> 00:03:33.358
Fair enough, nothing fancy there.

00:03:33.358 --> 00:03:34.974
Now let's go to the tens place.

00:03:34.974 --> 00:03:38.631
Well, in the tens place we have six tens

00:03:38.631 --> 00:03:41.360
plus seven tens, and
in the standard method

00:03:41.360 --> 00:03:44.039
what you say is, "that's 13 tens,"

00:03:44.039 --> 00:03:47.622
but 13 tens is the same
thing as three tens

00:03:48.828 --> 00:03:50.202
and one hundred.

00:03:50.202 --> 00:03:52.750
So what you do is, you would regroup.

00:03:52.750 --> 00:03:55.171
You would say, "hey
look, this is three tens

00:03:55.171 --> 00:03:57.376
"and one hundreds."

00:03:57.376 --> 00:03:59.677
Sometimes people say, "Oh
you're carrying the one.

00:03:59.677 --> 00:04:02.209
"Six plus seven is 13, carry the one."

00:04:02.209 --> 00:04:03.621
And it seems somewhat magical,

00:04:03.621 --> 00:04:06.847
but all you're doing is
you're taking ten of the tens

00:04:06.847 --> 00:04:08.875
and you're regrouping it as a hundred.

00:04:08.875 --> 00:04:10.199
It'll be a little clearer here.

00:04:10.199 --> 00:04:14.199
So we have six tens and
then we have seven tens,

00:04:15.768 --> 00:04:16.862
you add 'em all together,

00:04:16.862 --> 00:04:19.996
you get one, two, three,
four, five, six, seven,

00:04:19.996 --> 00:04:23.980
eight, nine, ten, eleven,
twelve, thirteen tens.

00:04:23.980 --> 00:04:26.940
All we're doing right
over is we're saying,

00:04:26.940 --> 00:04:29.773
"Look this is equal to a hundred."

00:04:30.842 --> 00:04:35.009
So let's just convert that
into a hundred right over here.

00:04:39.186 --> 00:04:41.875
Let's just convert that
and so what we do is,

00:04:41.875 --> 00:04:44.131
we just write the three in the tens place

00:04:44.131 --> 00:04:47.819
and then we add an extra
one in the hundreds place.

00:04:47.819 --> 00:04:50.096
And so what are we gonna have
in the hundreds place now?

00:04:50.096 --> 00:04:51.595
And actually, let me do it here

00:04:51.595 --> 00:04:53.104
because it's a little more interesting.

00:04:53.104 --> 00:04:57.236
So I have one, two, three,
four, five, six, seven, eight,

00:04:57.236 --> 00:04:59.982
nine, ten, eleven, twelve hundreds.

00:04:59.982 --> 00:05:01.278
So I can write 'em down here.

00:05:01.278 --> 00:05:04.719
One, two, three, four, five,
six, seven, eight, nine, ten,

00:05:04.719 --> 00:05:06.219
eleven and twelve,

00:05:07.323 --> 00:05:09.870
but with the same thing,
we don't have a digit for

00:05:09.870 --> 00:05:12.849
the number twelve in our
traditional number system,

00:05:12.849 --> 00:05:16.025
so what I could do is I
could take ten of these

00:05:16.025 --> 00:05:19.570
and I can convert is to a thousand.

00:05:19.570 --> 00:05:23.079
So I'm gonna take ten of those
and give myself a thousand.

00:05:23.079 --> 00:05:25.086
No we're gonna do the
exact same thing over here.

00:05:25.086 --> 00:05:28.035
One plus seven plus four is twelve.

00:05:28.035 --> 00:05:29.124
So you'd write--

00:05:29.124 --> 00:05:33.435
That's two hundreds because
this is twelve hundreds,

00:05:33.435 --> 00:05:36.837
that's two hundreds plus a thousand,

00:05:36.837 --> 00:05:38.520
so we just regrouped again.

00:05:38.520 --> 00:05:39.722
Now in the thousands place,

00:05:39.722 --> 00:05:42.226
one plus zero plus zero is one thousand

00:05:42.226 --> 00:05:44.483
and you see that right over
here, you have one thousand.

00:05:44.483 --> 00:05:47.010
And then finally, in
the ten thousands place,

00:05:47.010 --> 00:05:49.110
four ten thousands plus
three ten thousands

00:05:49.110 --> 00:05:51.116
is seven ten thousands.

00:05:51.116 --> 00:05:53.389
Four ten thousands plus
three ten thousands

00:05:53.389 --> 00:05:56.999
is one, two, three, four,
five, six, seven ten thousands.

00:05:56.999 --> 00:05:59.416
And so this number is 71,235.

00:06:05.015 --> 00:06:05.848
71,235.

00:06:08.710 --> 00:06:09.957
So hopefully it all makes sense

00:06:09.957 --> 00:06:13.074
how these two things will fit together.

00:06:13.074 --> 00:06:14.746
That what's going on over here--

00:06:14.746 --> 00:06:16.232
You're not just magically carrying numbers

00:06:16.232 --> 00:06:18.632
or magically regrouping,
you're just representing

00:06:18.632 --> 00:06:21.407
the same number in different ways.


Understanding place value when subtracting

WEBVTT
Kind: captions
Language: en

00:00:01.111 --> 00:00:02.628
- [Instructor] What we're
going to do in this video

00:00:02.628 --> 00:00:05.523
is get some practice
subtracting multi-digit numbers

00:00:05.523 --> 00:00:10.151
I'm going to use 1000
minus 528 as our example.

00:00:10.151 --> 00:00:12.385
But really to understand
different methods and how they

00:00:12.385 --> 00:00:15.317
all fit together, why
it actually makes sense.

00:00:15.317 --> 00:00:18.257
So if you wanted to visualize
what this difference

00:00:18.257 --> 00:00:22.506
means, imagine something
that has a length of 1000

00:00:22.506 --> 00:00:26.673
some type of units, so it's
length right over here is 1000

00:00:28.519 --> 00:00:32.085
and we were to take away 528 from that.

00:00:32.085 --> 00:00:36.795
So 528 from that is what
we take away and so this

00:00:36.795 --> 00:00:40.467
difference would be well
what do we have left over.

00:00:40.467 --> 00:00:42.603
So this is equal to question mark.

00:00:42.603 --> 00:00:44.740
And I'm going to do it two different ways.

00:00:44.740 --> 00:00:47.047
I'm going to do it using
a table with place value

00:00:47.047 --> 00:00:49.314
and I'm also going to do
it using what's sometimes

00:00:49.314 --> 00:00:51.845
called the standard method
it's the way that people often

00:00:51.845 --> 00:00:54.990
learn to subtract numbers
like this especially if we're

00:00:54.990 --> 00:00:57.020
going to have to do some regrouping.

00:00:57.020 --> 00:00:59.066
So I'm actually going to
do them simultaneously

00:00:59.066 --> 00:01:01.156
for your benefit, alright.

00:01:01.156 --> 00:01:04.117
So let me just write out our
table with our place values.

00:01:04.117 --> 00:01:08.967
So first of all you have your
thousands place, thousands.

00:01:08.967 --> 00:01:11.849
And let me square off
the numbers here that are

00:01:11.849 --> 00:01:15.447
in the thousands place so
that 1000 right over there,

00:01:15.447 --> 00:01:17.561
that's one in the thousands place.

00:01:17.561 --> 00:01:22.088
Then you have your hundreds
place, hundreds place,

00:01:22.088 --> 00:01:25.700
and this number I have zero
hundreds right now here I have

00:01:25.700 --> 00:01:26.867
five hundreds.

00:01:27.847 --> 00:01:32.014
And then you have your
tens place, tens, zero tens

00:01:33.543 --> 00:01:37.710
right there in the tens place,
two tens right over there.

00:01:38.920 --> 00:01:42.822
And then of course you
have your ones place.

00:01:42.822 --> 00:01:46.470
Here I have zero ones,
here I have eight ones.

00:01:46.470 --> 00:01:48.455
Now let me also rewrite these
numbers and I'm going to do it

00:01:48.455 --> 00:01:50.324
using the standard method.

00:01:50.324 --> 00:01:53.991
So I have 1000 and then
I have zero hundreds

00:01:56.024 --> 00:01:59.852
I have zero tens and I have zero ones.

00:01:59.852 --> 00:02:04.019
And from that I am going to
subtract five hundreds, five

00:02:06.149 --> 00:02:10.316
hundreds, two tens, two tens,
and eight ones, eight ones.

00:02:16.897 --> 00:02:19.474
So let's do both of
these at the same time,

00:02:19.474 --> 00:02:23.343
make a little bit of a
table right over here.

00:02:23.343 --> 00:02:25.010
So that is my table.

00:02:26.976 --> 00:02:29.249
So let's start with what
we originally have, we have

00:02:29.249 --> 00:02:31.389
1000 that's what we're subtracting from.

00:02:31.389 --> 00:02:33.488
Well on this table I would
just represent as that

00:02:33.488 --> 00:02:37.456
as 1000, now we want to take
five hundreds and two tens

00:02:37.456 --> 00:02:41.489
and eight ones from it, how do we do that?

00:02:41.489 --> 00:02:44.113
'Cause right now we have no
hundreds, we have no tens,

00:02:44.113 --> 00:02:46.389
and we have no ones.

00:02:46.389 --> 00:02:48.465
And with the standard method
we have the same problem

00:02:48.465 --> 00:02:51.553
'cause we start in the ones
place we say hey we want

00:02:51.553 --> 00:02:53.599
to take eight ones from zero
ones, similar problem here.

00:02:53.599 --> 00:02:55.584
How do we take eight ones here?

00:02:55.584 --> 00:02:57.918
Similarly we want to take
two tens from zero tens,

00:02:57.918 --> 00:03:01.702
how do we do that here and
the answer is regrouping.

00:03:01.702 --> 00:03:04.484
What we want to do is break
up this thousand and so we can

00:03:04.484 --> 00:03:06.764
start to fill in these
other categories it's like

00:03:06.764 --> 00:03:09.953
exchanging monies that's
sometimes an example used.

00:03:09.953 --> 00:03:13.089
So 1000 is how many hundreds?

00:03:13.089 --> 00:03:15.587
Well if we get rid of this
thousands I can break it up

00:03:15.587 --> 00:03:20.116
into ten hundreds, so one,
two, three, four, five, six,

00:03:20.116 --> 00:03:22.116
seven, eight, nine, ten.

00:03:22.993 --> 00:03:25.445
And so that's equivalent
of I could get rid of this

00:03:25.445 --> 00:03:29.123
1000 or this one in the
thousands place and give myself

00:03:29.123 --> 00:03:30.206
ten hundreds.

00:03:31.981 --> 00:03:33.603
Well that starts to solve my
problem 'cause I could now

00:03:33.603 --> 00:03:36.672
take five hundreds from this,
I could take five from ten.

00:03:36.672 --> 00:03:40.190
Five hundreds from ten
hundreds, but I still have

00:03:40.190 --> 00:03:43.905
the problem in the tens
and the ones place.

00:03:43.905 --> 00:03:46.784
And so what I could do is I
could break up one of these

00:03:46.784 --> 00:03:50.545
hundreds and into ten
tens so let me do that.

00:03:50.545 --> 00:03:53.793
So I'm going to take, (mumbles)
so I'm going to take that

00:03:53.793 --> 00:03:58.065
one away and then that 100
is ten tens, one, two, three,

00:03:58.065 --> 00:04:01.482
four, five, six, seven, eight, nine, ten.

00:04:02.399 --> 00:04:04.605
And if I did that here,
well if I take away one of

00:04:04.605 --> 00:04:08.772
the hundreds I'm now going
to have nine hundreds left.

00:04:09.749 --> 00:04:13.916
Nine hundreds left, but now I
have ten, now I have ten tens.

00:04:17.686 --> 00:04:20.170
So I'm in good shape now
I can take some tens here

00:04:20.170 --> 00:04:22.711
but I still don't have any
ones, remember I want to take

00:04:22.711 --> 00:04:24.694
eight ones from here, so you
can imagine what's going on.

00:04:24.694 --> 00:04:28.086
I could take one of my tens
and that's going to give me

00:04:28.086 --> 00:04:31.746
one, two, three, four, five,
six, seven, eight, nine,

00:04:31.746 --> 00:04:36.334
ten ones and so over here I
could take away one of my tens

00:04:36.334 --> 00:04:40.501
so I'm now going to have nine
tens and I'm going to break

00:04:43.122 --> 00:04:45.539
that into ten ones, ten ones.

00:04:47.096 --> 00:04:49.894
And so now things are
pretty straightforward.

00:04:49.894 --> 00:04:51.183
What do I do?

00:04:51.183 --> 00:04:54.362
Well I can now take my eight
ones from the ten ones.

00:04:54.362 --> 00:04:57.763
So ten minus eight,
that is going to be two.

00:04:57.763 --> 00:04:59.873
How would I represent that over here?

00:04:59.873 --> 00:05:02.881
I'm going to do the subtraction
in this yellow color.

00:05:02.881 --> 00:05:06.497
I want to take away eight of
these ones, so I take away

00:05:06.497 --> 00:05:10.664
one, two, three, four, five,
six, seven, eight, and I am

00:05:11.971 --> 00:05:16.562
left with that two right over
there, that two is that two.

00:05:16.562 --> 00:05:19.325
Now I can move on to the tens place.

00:05:19.325 --> 00:05:21.694
If I have nine tens and I
take away two tens, I'm going

00:05:21.694 --> 00:05:25.861
to be left with seven, I'm going
to be left with seven tens.

00:05:28.193 --> 00:05:30.978
How would we see it over here?

00:05:30.978 --> 00:05:35.521
Well I have nine tens left
over, I'm going to take away

00:05:35.521 --> 00:05:37.843
two of them, so take a one
two and I am left with seven.

00:05:37.843 --> 00:05:39.280
Is that seven?

00:05:39.280 --> 00:05:41.291
One, two, three, four, five,
six, seven, yep that is seven

00:05:41.291 --> 00:05:42.865
tens right over there.

00:05:42.865 --> 00:05:46.272
We have two ones, seven tens
and so this seven is exactly

00:05:46.272 --> 00:05:49.425
this seven right over there same colors.

00:05:49.425 --> 00:05:54.129
I think you see where this
is going and the whole idea

00:05:54.129 --> 00:05:55.728
is not just to get the answer
but to understand how we got

00:05:55.728 --> 00:05:57.151
this answer.

00:05:57.151 --> 00:05:59.635
So in the hundreds place if I
have nine hundreds and I take

00:05:59.635 --> 00:06:02.155
away five hundreds then I'm
going to be left with four

00:06:02.155 --> 00:06:05.382
hundreds, same idea over here.

00:06:05.382 --> 00:06:09.569
I have nine hundreds, I take
away one, two, three, four

00:06:09.569 --> 00:06:13.486
five, I am going to be
left with four hundreds,

00:06:14.796 --> 00:06:17.894
this four and this four is the same.

00:06:17.894 --> 00:06:20.289
And so there you get the
general idea, with the standard

00:06:20.289 --> 00:06:22.864
method it sometimes seems
like magic of how we're

00:06:22.864 --> 00:06:25.268
regrouping things but all
we're doing is we're taking

00:06:25.268 --> 00:06:27.728
that thousand and saying
hey that's ten hundreds

00:06:27.728 --> 00:06:30.785
and then we take one of
those hundreds and we say

00:06:30.785 --> 00:06:34.192
hey that's ten tens and we take
one of those tens and we say

00:06:34.192 --> 00:06:37.545
that's ten ones and then
we are able to subtract.


Estimating adding decimals

WEBVTT
Kind: captions
Language: en

00:00:00.526 --> 00:00:01.359
- [Instructor] What we're going to do

00:00:01.359 --> 00:00:02.862
in this video is get some practice

00:00:02.862 --> 00:00:05.737
estimating adding decimals.

00:00:05.737 --> 00:00:09.504
Here it says 12.93 plus 6.1

00:00:09.504 --> 00:00:12.234
is approximately equal to,

00:00:12.234 --> 00:00:14.881
this little squiggly equal
sign means approximately

00:00:14.881 --> 00:00:18.561
equal to, so try to estimate
this and pick the right choice.

00:00:18.561 --> 00:00:21.012
Pause the video and try to do that.

00:00:21.012 --> 00:00:23.150
Alright, now let's work
through this together.

00:00:23.150 --> 00:00:26.103
We could work this out by
hand, but the whole goal here

00:00:26.103 --> 00:00:29.761
is estimation, so if I were
doing this in my head I'd say,

00:00:29.761 --> 00:00:33.928
well, look, 12.93 is a
little bit less than 13,

00:00:34.913 --> 00:00:38.680
and 6.1 is a little bit more than six.

00:00:38.680 --> 00:00:42.132
And it's actually nice because
when I rounded I went down

00:00:42.132 --> 00:00:45.793
on this number and I
went up on this number.

00:00:45.793 --> 00:00:48.706
I went up 700ths to get to 13

00:00:48.706 --> 00:00:51.540
and I went down 1/10th to
get to six, so I feel good

00:00:51.540 --> 00:00:53.630
when I add them up that we're
going to get pretty close

00:00:53.630 --> 00:00:55.224
to the actual answer.

00:00:55.224 --> 00:00:58.773
13 plus six is equal
to 19, and lucky for us

00:00:58.773 --> 00:01:00.905
that that is one of the choices.

00:01:00.905 --> 00:01:05.104
There's multiple ways that you
could try to estimate things.

00:01:05.104 --> 00:01:08.360
For example, you could do a
little finer of an estimation.

00:01:08.360 --> 00:01:12.869
You'd say, this is
approximately 12.9 plus 6.1,

00:01:12.869 --> 00:01:16.965
and then you'd say, 12 plus
six is 18, and then, 9/10ths

00:01:16.965 --> 00:01:20.315
plus 1/10th is another one,
so you'd say this is 18

00:01:20.315 --> 00:01:23.284
plus one, and you'd say,
okay, that, once again gets me

00:01:23.284 --> 00:01:26.608
pretty close to 19,
or, this gets me to 19.

00:01:26.608 --> 00:01:28.258
Let's do another example.

00:01:28.258 --> 00:01:32.810
Here we're told five plus one plus 4.91

00:01:32.810 --> 00:01:36.385
is approximately equal to
what, so, pause this video

00:01:36.385 --> 00:01:37.883
and see if you can figure that out.

00:01:37.883 --> 00:01:41.403
Try to estimate it, try to
do it without any paper.

00:01:41.403 --> 00:01:44.334
The way I would do it is
actually very similar to the way

00:01:44.334 --> 00:01:45.844
we just did the last one.

00:01:45.844 --> 00:01:49.671
5.1 is approximately equal to five,

00:01:49.671 --> 00:01:51.207
and,

00:01:51.207 --> 00:01:54.151
is approximately equal
to five, and 4.91 is also

00:01:54.151 --> 00:01:56.043
approximately equal to five.

00:01:56.043 --> 00:01:58.643
I rounded to the nearest
whole number in either case,

00:01:58.643 --> 00:02:02.874
and so, that would be equal to 10.

00:02:02.874 --> 00:02:06.069
Once again, it makes sense
because we rounded down here

00:02:06.069 --> 00:02:09.838
and we rounded up here, we
rounded down to go from 5.1

00:02:09.838 --> 00:02:12.083
to five, and we rounded up to go

00:02:12.083 --> 00:02:14.724
from 4.91 to five.

00:02:14.724 --> 00:02:18.082
If you're rounding down in
both cases, or rounding up

00:02:18.082 --> 00:02:21.624
in both cases your estimate
might get a little bit further

00:02:21.624 --> 00:02:23.833
from your actual answer,
so you might want to be

00:02:23.833 --> 00:02:25.755
a little bit careful.

00:02:25.755 --> 00:02:28.291
Alright, so in this last
example, once again,

00:02:28.291 --> 00:02:30.101
pause the video and see if you can

00:02:30.101 --> 00:02:31.709
work through it on your own.

00:02:31.709 --> 00:02:33.353
We're just going to do
the same thing here.

00:02:33.353 --> 00:02:35.368
This is going to be approximately equal

00:02:35.368 --> 00:02:38.892
to three plus, I rounded down, 3.14,

00:02:38.892 --> 00:02:40.751
rounded to the nearest
whole number, three.

00:02:40.751 --> 00:02:44.085
5.92 rounded to the nearest
whole number is six.

00:02:44.085 --> 00:02:46.431
And once again, just like I
mentioned in the previous ones,

00:02:46.431 --> 00:02:49.747
this is good because I am rounding down

00:02:49.747 --> 00:02:53.160
from 3.14 to three, I took away 1400ths,

00:02:53.160 --> 00:02:55.966
and I add 800ths to 5.92,

00:02:55.966 --> 00:02:57.632
so it's going to get me reasonably close

00:02:57.632 --> 00:03:00.927
to my actual answer, and
so, three plus six is equal

00:03:00.927 --> 00:03:04.442
to nine, which is, indeed,
one of the answers here.

00:03:04.442 --> 00:03:06.085
And this is actually
interesting, you're not always

00:03:06.085 --> 00:03:07.646
going to see a multiple
choice, so if you're trying

00:03:07.646 --> 00:03:10.500
to estimate things, but in
this case, you know that just

00:03:10.500 --> 00:03:14.055
three plus five is going
to be eight, and so this

00:03:14.055 --> 00:03:15.610
can't be the choice.

00:03:15.610 --> 00:03:19.343
Then you're going to have
eight plus whatever 1400ths

00:03:19.343 --> 00:03:22.714
plus 9200ths is, and you
know that's more than one,

00:03:22.714 --> 00:03:26.147
so you know our answer's going
to be a little bit over nine.

00:03:26.147 --> 00:03:29.510
Once again, nine is a good approximation.


Estimating subtracting decimals

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:02.409
- [Instructor] Alright,
now let's get some practice

00:00:02.409 --> 00:00:05.101
estimating, subtracting decimals.

00:00:05.101 --> 00:00:08.184
So, over here it says 12.93 minus 6.1

00:00:09.816 --> 00:00:12.274
is approximately equal to what?

00:00:12.274 --> 00:00:14.820
This squiggly-looking equal sign

00:00:14.820 --> 00:00:18.730
you can view as roughly equal
to or approximately equal to.

00:00:18.730 --> 00:00:21.340
So, pause this video and see
if you can figure it out.

00:00:21.340 --> 00:00:23.152
And, you shouldn't
really need paper for it.

00:00:23.152 --> 00:00:24.335
The whole point of estimation

00:00:24.335 --> 00:00:26.716
is to be able to do it quickly
and get close to the answer.

00:00:26.716 --> 00:00:29.310
You don't have to get the exact answer.

00:00:29.310 --> 00:00:30.143
Well, let's think about it.

00:00:30.143 --> 00:00:32.565
The way my brain would do it is 12.93--

00:00:32.565 --> 00:00:33.758
Let's just round each of them

00:00:33.758 --> 00:00:34.984
to the nearest whole number.

00:00:34.984 --> 00:00:37.824
So, this is going to
be approximately 12.93

00:00:37.824 --> 00:00:39.794
if I round to the nearest one

00:00:39.794 --> 00:00:40.870
or to the nearest whole number,

00:00:40.870 --> 00:00:42.973
I'm gonna round up to 13.

00:00:42.973 --> 00:00:45.395
And 6.1, if I round to
the nearest whole number,

00:00:45.395 --> 00:00:47.450
I'm gonna round down to six.

00:00:47.450 --> 00:00:50.522
So, I could say this
is roughly 13 minus six

00:00:50.522 --> 00:00:52.229
which is equal to seven,

00:00:52.229 --> 00:00:55.896
and that is indeed one
of the choices there.

00:00:57.070 --> 00:00:58.396
And, when you're doing estimation,

00:00:58.396 --> 00:01:01.058
you might have gotten a
slightly different result,

00:01:01.058 --> 00:01:02.661
but I'm guessing that the way that

00:01:02.661 --> 00:01:04.626
this question was set up,

00:01:04.626 --> 00:01:07.206
that if you do any type
of reasonable estimate,

00:01:07.206 --> 00:01:09.329
that you're going to have a number

00:01:09.329 --> 00:01:12.661
that is closer to seven than
any of the other choices.

00:01:12.661 --> 00:01:15.449
And to be clear, sometimes you
could do this in your head.

00:01:15.449 --> 00:01:18.153
For example, 12.93 minus 6.1,

00:01:18.153 --> 00:01:22.659
you could say 12 minus
six is going to be six.

00:01:22.659 --> 00:01:26.132
And then you have 93 hundredths minus,

00:01:26.132 --> 00:01:27.895
you could do this as 10 hundredths.

00:01:27.895 --> 00:01:30.235
So, it's gonna be 93
hundredths minus 10 hundredths

00:01:30.235 --> 00:01:31.860
is 83 hundredths,

00:01:31.860 --> 00:01:32.693
plus .83.

00:01:33.813 --> 00:01:34.646
So, just like that,

00:01:34.646 --> 00:01:36.110
you could do something like this and say,

00:01:36.110 --> 00:01:39.408
alright, the answer here
is gonna be 6.83 exactly.

00:01:39.408 --> 00:01:42.291
And that, once again, is
approximately equal to seven.

00:01:42.291 --> 00:01:43.843
But, the whole point of this exercise

00:01:43.843 --> 00:01:46.099
is to get comfortable estimating things

00:01:46.099 --> 00:01:46.969
because a lot of times in life

00:01:46.969 --> 00:01:48.916
you don't need the exact answer.

00:01:48.916 --> 00:01:50.232
You just need a rough answer,

00:01:50.232 --> 00:01:53.811
just to know the ball park
of what you're talking about.

00:01:53.811 --> 00:01:56.323
Let's do another example.

00:01:56.323 --> 00:01:58.073
So here, we are asked

00:01:58.974 --> 00:02:00.974
what is 56.75 minus 46.9

00:02:02.502 --> 00:02:03.998
approximately equal to?

00:02:03.998 --> 00:02:07.588
Once again, pause this video
and try to work it out.

00:02:07.588 --> 00:02:09.483
Well, here I'm gonna do the same thing.

00:02:09.483 --> 00:02:11.436
I'm just gonna round each
to the nearest whole number

00:02:11.436 --> 00:02:12.567
and then subtract.

00:02:12.567 --> 00:02:13.451
And, I'm gonna write it out,

00:02:13.451 --> 00:02:15.556
but you could do this in your head.

00:02:15.556 --> 00:02:19.094
So 56.75 rounded to the
nearest whole number is 57

00:02:19.094 --> 00:02:20.103
round up,

00:02:20.103 --> 00:02:24.048
minus 46.9, I would also round up to 47.

00:02:24.048 --> 00:02:25.854
And, in our head this is
pretty straightforward.

00:02:25.854 --> 00:02:28.021
This is going to be equal to ten.

00:02:28.021 --> 00:02:31.565
And, that is indeed is one of the choices.

00:02:31.565 --> 00:02:33.044
And, normally in life
when you're estimating,

00:02:33.044 --> 00:02:34.455
you don't have choices like this.

00:02:34.455 --> 00:02:36.325
I guess this is just the easiest way for--

00:02:36.325 --> 00:02:38.461
This if off of the Khan Academy exercises.

00:02:38.461 --> 00:02:40.213
So, for someone to be able to grade it

00:02:40.213 --> 00:02:41.678
because estimations can be different,

00:02:41.678 --> 00:02:43.427
so you have to pick
the one that's at least

00:02:43.427 --> 00:02:46.359
closest to what you came
up as your estimate.

00:02:46.359 --> 00:02:49.540
Now, let's do one more example.

00:02:49.540 --> 00:02:51.139
And, I will write in purple.

00:02:51.139 --> 00:02:52.314
Alright.

00:02:52.314 --> 00:02:53.766
So, pause the video again.

00:02:53.766 --> 00:02:55.638
See if you can figure this out.

00:02:55.638 --> 00:02:59.486
Alright, so 49.7, I'm
gonna round up to 50,

00:02:59.486 --> 00:03:02.132
4.16 I'm gonna round down to four.

00:03:02.132 --> 00:03:06.221
So, that's going to be
approximately equal to 30--

00:03:06.221 --> 00:03:10.454
Oh sorry, that's gonna be
approximately equal to 46.

00:03:10.454 --> 00:03:13.717
And, we do indeed see
46 as one of the choices

00:03:13.717 --> 00:03:14.550
right over there.

00:03:14.550 --> 00:03:15.771
So, hopefully you feel comfortable.

00:03:15.771 --> 00:03:17.375
There's nothing fancy going on here.

00:03:17.375 --> 00:03:19.403
The whole point of this
is to just get comfortable

00:03:19.403 --> 00:03:21.817
doing things like this in
your head and estimating

00:03:21.817 --> 00:03:24.405
and rounding numbers so you
can get a ball park sense

00:03:24.405 --> 00:03:27.257
of what this difference is going to be.


Estimating decimal multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.611 --> 00:00:02.255
- [Instructor] Let's
now get some practice.

00:00:02.255 --> 00:00:04.628
Estimating multiplying with decimals.

00:00:04.628 --> 00:00:08.193
So first here, we have 7.8 times 307,

00:00:08.193 --> 00:00:10.510
is approximately equal to what?

00:00:10.510 --> 00:00:12.334
When you see the squiggly equal sign,

00:00:12.334 --> 00:00:15.088
that means approximately equal to what.

00:00:15.088 --> 00:00:17.324
So pause this video, and
see if you can figure it out

00:00:17.324 --> 00:00:18.324
on your own.

00:00:19.332 --> 00:00:21.752
Alright, so the way that I
would think about doing it,

00:00:21.752 --> 00:00:23.761
even if I was trying to do this in my head

00:00:23.761 --> 00:00:25.407
at the supermarket or something,

00:00:25.407 --> 00:00:27.375
I'd say, well, okay,
I'd probably need paper

00:00:27.375 --> 00:00:28.928
to do this properly,

00:00:28.928 --> 00:00:32.817
but gee, 7.8 is awfully close to eight,

00:00:32.817 --> 00:00:36.140
and 307 is pretty close to 300,

00:00:36.140 --> 00:00:38.392
so maybe I can estimate
this by multiplying

00:00:38.392 --> 00:00:39.942
eight times 300.

00:00:39.942 --> 00:00:41.281
Now, this isn't going to be exact,

00:00:41.281 --> 00:00:42.766
it's definitely going to be off,

00:00:42.766 --> 00:00:44.058
but it's gonna give me a good sense

00:00:44.058 --> 00:00:47.451
of roughly what 7.8 times 307 is.

00:00:47.451 --> 00:00:49.532
So what is eight times 300?

00:00:49.532 --> 00:00:52.388
Well, eight times three is 24,

00:00:52.388 --> 00:00:55.482
and so eight times 300 would be 2,400,

00:00:55.482 --> 00:00:57.882
we got these two more
zeros, two more zeros.

00:00:57.882 --> 00:00:59.870
And so there you have it, and luckily,

00:00:59.870 --> 00:01:01.236
the people who wrote this question

00:01:01.236 --> 00:01:03.648
estimated in a very similar way.

00:01:03.648 --> 00:01:05.262
Two people when they
estimate might not get

00:01:05.262 --> 00:01:08.724
the exact same answer, but
in this case, we happen to.

00:01:08.724 --> 00:01:11.016
Let's do another example.

00:01:11.016 --> 00:01:15.099
So here, we're trying to
estimate 99.87 times 19.

00:01:16.074 --> 00:01:19.159
So pause the video again
and see if you can come up

00:01:19.159 --> 00:01:21.581
with a good estimation.

00:01:21.581 --> 00:01:24.851
Alright, so once again, not
easy to do this in your head,

00:01:24.851 --> 00:01:28.925
but this, 99 and 87/100 is pretty close,

00:01:28.925 --> 00:01:30.327
let me do this in a new color,

00:01:30.327 --> 00:01:33.755
this is pretty close to 100 times,

00:01:33.755 --> 00:01:36.466
and I could multiply 100 times 19,

00:01:36.466 --> 00:01:37.839
that's actually not so difficult,

00:01:37.839 --> 00:01:40.526
so for example, I could say 100 times 19

00:01:40.526 --> 00:01:43.585
is equal to 1,900, but notice,

00:01:43.585 --> 00:01:44.903
we don't see that choice here,

00:01:44.903 --> 00:01:46.682
we could say, look, 1,900 is definitely

00:01:46.682 --> 00:01:49.443
much closer to 2,000
than any of these other,

00:01:49.443 --> 00:01:52.063
so that might be a good approximation.

00:01:52.063 --> 00:01:53.798
Now, how did they get 2,000?

00:01:53.798 --> 00:01:55.849
Well, they rounded both of these numbers,

00:01:55.849 --> 00:02:00.094
they said, this is
approximately 100 times 20.

00:02:00.094 --> 00:02:03.902
So, it's not that it's
the right thing to do,

00:02:03.902 --> 00:02:05.670
to round this 19 up to 20,

00:02:05.670 --> 00:02:07.081
if you could do 100 times 19,

00:02:07.081 --> 00:02:08.039
this is actually gonna give you

00:02:08.039 --> 00:02:09.665
a slightly more accurate result

00:02:09.665 --> 00:02:11.587
than doing 100 times 20,

00:02:11.587 --> 00:02:15.373
but 100 times 20 is even easier
to estimate in your head.

00:02:15.373 --> 00:02:17.546
But either way, the closest choice here,

00:02:17.546 --> 00:02:18.754
and that's why, I guess, these had to be

00:02:18.754 --> 00:02:22.297
multiple choice questions, is 2,000.

00:02:22.297 --> 00:02:24.380
Let's do another example.

00:02:25.972 --> 00:02:30.055
So here, we are asked to
multiply 2.21 times 5.1,

00:02:31.706 --> 00:02:33.739
and we wanna know what
it approximately equals,

00:02:33.739 --> 00:02:35.141
so once again, we are estimating,

00:02:35.141 --> 00:02:38.250
so pause this video and
try to figure it out.

00:02:38.250 --> 00:02:39.644
Well, we're just gonna do the same thing

00:02:39.644 --> 00:02:41.280
we did in the last two examples.

00:02:41.280 --> 00:02:42.722
2.21, and, well, first of all,

00:02:42.722 --> 00:02:44.341
this is hard to do in my head,

00:02:44.341 --> 00:02:47.273
and so 2.21, well, that's approximately,

00:02:47.273 --> 00:02:49.465
if I round to the nearest two,

00:02:49.465 --> 00:02:51.479
or to the nearest one, I should say,

00:02:51.479 --> 00:02:54.814
this is gonna be two times five,

00:02:54.814 --> 00:02:56.894
which is equal to 10, so this would be

00:02:56.894 --> 00:03:00.021
approximately equal to
10, and that is a choice.

00:03:00.021 --> 00:03:01.068
Now, some of you might say,

00:03:01.068 --> 00:03:03.547
"Whoa, there's ways to
get better estimations

00:03:03.547 --> 00:03:05.585
"that you could still do with your head,"

00:03:05.585 --> 00:03:09.201
so for example, you could say
that this is pretty close to,

00:03:09.201 --> 00:03:10.992
this is approximate,
let me do it over here,

00:03:10.992 --> 00:03:14.742
this is approximately
equal to two times 5.1,

00:03:15.910 --> 00:03:18.164
this is still pretty
straightforward to do in your head,

00:03:18.164 --> 00:03:19.747
this would be 10.2.

00:03:21.354 --> 00:03:23.263
And so, but you'd still
say that this is by far

00:03:23.263 --> 00:03:24.361
the closest one.

00:03:24.361 --> 00:03:26.110
Or you could even say something like,

00:03:26.110 --> 00:03:29.943
this is approximately
equal to 2.2 times five.

00:03:31.114 --> 00:03:32.394
What is this going to be equal to?

00:03:32.394 --> 00:03:34.704
Well, two times five is 10,

00:03:34.704 --> 00:03:37.929
and 2/10 times five is another whole,

00:03:37.929 --> 00:03:40.549
and so this is going to be equal to 11,

00:03:40.549 --> 00:03:42.827
so all of these might be
things that you could estimate

00:03:42.827 --> 00:03:44.526
that you might be able to do in your head,

00:03:44.526 --> 00:03:47.353
but the important thing to
realize is however you do it,

00:03:47.353 --> 00:03:50.549
by far, 10 is going to
be the closest response

00:03:50.549 --> 00:03:51.832
to what you're getting at,

00:03:51.832 --> 00:03:53.931
and 10 would be a very natural estimation

00:03:53.931 --> 00:03:56.549
if you try to simplify
both of these numbers

00:03:56.549 --> 00:03:58.697
when making that estimate.


Approximating dividing by decimals

WEBVTT
Kind: captions
Language: en

00:00:00.876 --> 00:00:02.015
- [Instructor] What we're
going to do in this video

00:00:02.015 --> 00:00:03.694
is get a little bit of practice

00:00:03.694 --> 00:00:06.537
estimating dividing with decimals.

00:00:06.537 --> 00:00:09.064
So for example, we wanna
figure out approximately,

00:00:09.064 --> 00:00:12.576
that's what these kind of
squiggly equal sign means.

00:00:12.576 --> 00:00:14.862
This means approximately equal.

00:00:14.862 --> 00:00:19.632
So what does 80 divided by
1.94 approximately equal?

00:00:19.632 --> 00:00:21.353
So we wanna estimate what this is.

00:00:21.353 --> 00:00:24.581
So pause this video and see
if you can figure it out.

00:00:24.581 --> 00:00:26.252
So before we even look at these choices,

00:00:26.252 --> 00:00:28.840
how would we try to do this in our head?

00:00:28.840 --> 00:00:32.450
Well, we could say 1.94, it's
hard to do that in our head

00:00:32.450 --> 00:00:36.159
to divide it into 80 but
it's awfully close to two.

00:00:36.159 --> 00:00:37.590
So we could say this is close to,

00:00:37.590 --> 00:00:42.276
this is approximately
equal to 80 divided by two

00:00:42.276 --> 00:00:45.176
'cause once again, 1.94
is awfully close to two

00:00:45.176 --> 00:00:46.930
and then this is easy to figure out.

00:00:46.930 --> 00:00:48.602
That is going to be equal to 40.

00:00:48.602 --> 00:00:50.184
So our first expression, you could say

00:00:50.184 --> 00:00:52.427
is approximately equal to 40.

00:00:52.427 --> 00:00:53.487
Is it exactly equal to 40?

00:00:53.487 --> 00:00:55.404
No but it's pretty close.

00:00:55.404 --> 00:00:57.528
So I would select that choice.

00:00:57.528 --> 00:00:59.767
Let's do another one.

00:00:59.767 --> 00:01:00.828
So pause this video

00:01:00.828 --> 00:01:03.598
and see if you can figure
out the approximation.

00:01:03.598 --> 00:01:07.431
What is 209 divided by
three roughly equal to?

00:01:08.901 --> 00:01:12.079
So once again, here we don't
have any clear decimals.

00:01:12.079 --> 00:01:13.019
If we were to divide it out,

00:01:13.019 --> 00:01:16.031
we would get a decimal
answer for our quotient

00:01:16.031 --> 00:01:18.643
but what we could do is
well, is either one of these,

00:01:18.643 --> 00:01:20.726
is for example 209,

00:01:20.726 --> 00:01:24.685
is it close to a multiple of
three that we might recognize?

00:01:24.685 --> 00:01:26.724
Well, you might immediately recognize,

00:01:26.724 --> 00:01:28.804
well, if you think about
multiples of three,

00:01:28.804 --> 00:01:30.435
you think in terms of,

00:01:30.435 --> 00:01:35.032
well, three times six is
18, three times seven is 21.

00:01:35.032 --> 00:01:37.282
Well, this is close to 210.

00:01:38.135 --> 00:01:42.743
So this is approximately
equal to 210 divided by three.

00:01:42.743 --> 00:01:44.700
Now, why is this interesting?

00:01:44.700 --> 00:01:47.956
Well, 210 is just 21 times 10.

00:01:47.956 --> 00:01:51.423
So if 21 divided by three is seven,

00:01:51.423 --> 00:01:55.909
210 divided by three is
going to be equal to 70.

00:01:55.909 --> 00:01:59.053
So once again, 209 is pretty close to 210.

00:01:59.053 --> 00:02:00.645
Remember, we're just estimating

00:02:00.645 --> 00:02:02.396
and so 209 divided by three

00:02:02.396 --> 00:02:05.418
would approximately be equal to 70

00:02:05.418 --> 00:02:08.151
which is that choice right over there.

00:02:08.151 --> 00:02:10.560
Let's do one more example.

00:02:10.560 --> 00:02:12.030
So once again, pause the video

00:02:12.030 --> 00:02:15.533
and figure out what 6.86 divided by 1.12

00:02:15.533 --> 00:02:17.700
is approximately equal to.

00:02:19.736 --> 00:02:22.386
Alright, so here I would just try to round

00:02:22.386 --> 00:02:23.653
to the nearest whole number

00:02:23.653 --> 00:02:26.228
and see if that helps
my division in my head.

00:02:26.228 --> 00:02:28.728
So 6.86 is approximately seven

00:02:29.908 --> 00:02:32.472
if we round up to the nearest whole number

00:02:32.472 --> 00:02:34.162
and if I say seven divided by,

00:02:34.162 --> 00:02:36.746
and if we round down, 1.12,

00:02:36.746 --> 00:02:37.930
seven divided by one,

00:02:37.930 --> 00:02:40.248
this is much easier for
us to do in our head.

00:02:40.248 --> 00:02:42.205
So that is going to be seven.

00:02:42.205 --> 00:02:43.880
Now, once again, it's
very important to realize

00:02:43.880 --> 00:02:45.144
these are estimations.

00:02:45.144 --> 00:02:48.616
It's not going to be
exactly equal to seven

00:02:48.616 --> 00:02:50.777
but it's definitely gonna
be much closer to seven

00:02:50.777 --> 00:02:52.527
than 70, 700 or 7000.


Estimate multiplying multi digit numbers

WEBVTT
Kind: captions
Language: en

00:00:00.225 --> 00:00:01.699
- [Instructor] What I would like to do

00:00:01.699 --> 00:00:03.527
in this video is get some practice

00:00:03.527 --> 00:00:06.711
estimating the product
of multi-digit numbers

00:00:06.711 --> 00:00:08.734
and there's no better way of getting

00:00:08.734 --> 00:00:11.061
practice than just trying it ourselves.

00:00:11.061 --> 00:00:12.385
And so right over here, it says

00:00:12.385 --> 00:00:14.385
estimate 29 times 3,198.

00:00:16.328 --> 00:00:19.384
Why don't you pause this video
and try to estimate this?

00:00:19.384 --> 00:00:20.647
And of course you can do this by

00:00:20.647 --> 00:00:23.014
multiplying this out on
paper or using a calculator,

00:00:23.014 --> 00:00:25.696
but this is a useful skill,
try to do it in your head.

00:00:25.696 --> 00:00:28.010
See if you can estimate
what this is going to be.

00:00:28.010 --> 00:00:30.132
Well, before even
looking at these answers,

00:00:30.132 --> 00:00:33.913
I would say this is going to
be approximately equal to ...

00:00:33.913 --> 00:00:37.618
Let's see, 29 is awfully close to 30,

00:00:37.618 --> 00:00:39.636
and then I could multiply that times,

00:00:39.636 --> 00:00:41.842
I could either multiply
that times 32 hundred,

00:00:41.842 --> 00:00:44.367
which is awfully close to 3,198,

00:00:44.367 --> 00:00:47.361
or if I wanted an even
more rough approximation,

00:00:47.361 --> 00:00:48.831
I could say that's roughly equal

00:00:48.831 --> 00:00:51.524
to 30 times three thousand.

00:00:51.524 --> 00:00:53.567
So if I did 30 times three thousand,

00:00:53.567 --> 00:00:56.925
three times three is equal to nine,

00:00:56.925 --> 00:00:59.945
and then I have one,
two, three, four zeroes.

00:00:59.945 --> 00:01:02.260
One, two, three, four zeroes.

00:01:02.260 --> 00:01:04.043
And actually my
approximation, it turns out,

00:01:04.043 --> 00:01:06.509
is here, right over here, 90 thousand.

00:01:06.509 --> 00:01:08.622
Now if I wanted a slightly
better approximation,

00:01:08.622 --> 00:01:09.976
I could have said this is approximately

00:01:09.976 --> 00:01:11.976
equal to 30 times 3,200,

00:01:15.509 --> 00:01:17.498
and this you could also do in your head.

00:01:17.498 --> 00:01:20.255
You could say, well,
what is three times 32?

00:01:20.255 --> 00:01:22.705
Well, that is gonna be 96, and then

00:01:22.705 --> 00:01:24.892
you have one, two, three zeroes.

00:01:24.892 --> 00:01:26.672
One, two, three zeroes, so this

00:01:26.672 --> 00:01:28.415
would be a slightly better approximation,

00:01:28.415 --> 00:01:30.364
and if this is what you
got, the closest answer

00:01:30.364 --> 00:01:33.742
here is still going to
be equal to 90 thousand.

00:01:33.742 --> 00:01:35.668
Let's do another example.

00:01:35.668 --> 00:01:39.585
So here, we are asked to
estimate 137 times 18.

00:01:41.958 --> 00:01:43.281
So pause this video again and see

00:01:43.281 --> 00:01:44.464
if you can come up with an estimate.

00:01:44.464 --> 00:01:46.240
Try to do it in your head.

00:01:46.240 --> 00:01:48.999
Well, there's once again many
ways of trying to tackle it.

00:01:48.999 --> 00:01:50.395
The way I would tackle it, I would say,

00:01:50.395 --> 00:01:54.476
well, that's pretty close to 140 times 20,

00:01:54.476 --> 00:01:58.643
and then this would be
equal to 14 times two is 28,

00:01:59.608 --> 00:02:01.421
and then we have two zeroes here,

00:02:01.421 --> 00:02:03.811
so that would be roughly 28 hundred,

00:02:03.811 --> 00:02:07.578
but when I look over here,
there is no 28 hundred,

00:02:07.578 --> 00:02:11.075
and so maybe the closest
one right over here

00:02:11.075 --> 00:02:13.428
is two thousand, so that would be my,

00:02:13.428 --> 00:02:15.688
that could be an approximation.

00:02:15.688 --> 00:02:17.191
Another way, it looks actually

00:02:17.191 --> 00:02:18.569
the way that they did it, they even

00:02:18.569 --> 00:02:20.089
did a coarser approximation,
they rounded this

00:02:20.089 --> 00:02:22.309
to the nearest hundred, and so they said

00:02:22.309 --> 00:02:25.510
this is approximately
equal to one hundred times

00:02:25.510 --> 00:02:27.588
and they rounded this to the nearest ten,

00:02:27.588 --> 00:02:29.949
one hundred times 20
which is even easier to do

00:02:29.949 --> 00:02:32.610
in your head, which is
equal to two thousand,

00:02:32.610 --> 00:02:36.777
which is this choice that
they got right over here.


Approximating multi digit division

WEBVTT
Kind: captions
Language: en

00:00:01.019 --> 00:00:02.302
- [Instructor] What we
wanna do in this video

00:00:02.302 --> 00:00:04.160
is get some practice estimating

00:00:04.160 --> 00:00:06.743
multi-digit division problems.

00:00:06.743 --> 00:00:10.743
Here, we're asked to
estimate 794 divided by 18.

00:00:12.098 --> 00:00:13.938
Now, if you wanted to get the exact answer

00:00:13.938 --> 00:00:15.796
you'd probably have to do,
in fact you would have to do,

00:00:15.796 --> 00:00:18.252
some long division or
maybe use a calculator.

00:00:18.252 --> 00:00:19.372
But, the whole point here

00:00:19.372 --> 00:00:21.055
is to get some practice estimating.

00:00:21.055 --> 00:00:23.621
Pause this video and try to
see if you can figure this out

00:00:23.621 --> 00:00:26.953
without even using any pencil or paper.

00:00:26.953 --> 00:00:29.252
All right, so now let's do this together.

00:00:29.252 --> 00:00:31.202
I'll tell you what my brain does.

00:00:31.202 --> 00:00:35.369
My brain says that look,
794 is awfully close to 800

00:00:37.186 --> 00:00:39.436
and then 18 is close to 20.

00:00:40.296 --> 00:00:43.213
I would say this is going
to be roughly the same thing

00:00:43.213 --> 00:00:48.078
as 800, is gonna be roughly
equal to 800 divided by 20.

00:00:48.078 --> 00:00:49.270
Is it going to be exactly equal?

00:00:49.270 --> 00:00:52.248
No, but these are the numbers
closest to the numbers there

00:00:52.248 --> 00:00:54.972
that it's easy for me
to divide in my head.

00:00:54.972 --> 00:00:57.623
For example, I would then say,

00:00:57.623 --> 00:00:59.330
if I was just doing
this purely in my head,

00:00:59.330 --> 00:01:02.857
I would say, well, 100
divided by 20 is five,

00:01:02.857 --> 00:01:05.857
so 800 divided by 20 is equal to 40.

00:01:07.921 --> 00:01:11.715
Another way to think
about it, 800 over 20.

00:01:11.715 --> 00:01:14.298
If we divide the numerator
and the denominator by 10,

00:01:14.298 --> 00:01:19.266
it's the same thing as 80
over 2, which is equal to 40.

00:01:19.266 --> 00:01:23.180
So 794 divided by 18
definitely isn't exactly 40,

00:01:23.180 --> 00:01:25.212
but it's going to be close to 40,

00:01:25.212 --> 00:01:27.594
and especially of these
choices right over here.

00:01:27.594 --> 00:01:29.353
It's definitely gonna be much closer to 40

00:01:29.353 --> 00:01:31.853
than 80, 4, or definitely 400.

00:01:32.753 --> 00:01:34.281
So, we're gonna go with that.

00:01:34.281 --> 00:01:35.948
Let's get another example here,

00:01:35.948 --> 00:01:38.182
make sure we get enough practice.

00:01:38.182 --> 00:01:42.349
Once again, we are asked to
estimate 722 divided by 68.

00:01:44.474 --> 00:01:45.804
Why don't you have a go at it.

00:01:45.804 --> 00:01:47.551
Pause the video and try to
see if you can figure out

00:01:47.551 --> 00:01:51.429
what it is and try to do it in your head.

00:01:51.429 --> 00:01:52.782
Well, what I would do in my head,

00:01:52.782 --> 00:01:57.286
I would say, well look, 722,
this is awfully close to 700,

00:01:57.286 --> 00:01:59.786
and 68 is awfully close to 70.

00:02:01.547 --> 00:02:03.791
And then, this is quite
straightforward to do in your head,

00:02:03.791 --> 00:02:07.541
700 divided by 70 is
going to be equal to 10.

00:02:08.789 --> 00:02:10.598
And, good for us, there's a choice here

00:02:10.598 --> 00:02:13.534
that is awfully close
to what we approximated.

00:02:13.534 --> 00:02:15.485
Now, if you don't find
your exact answer here,

00:02:15.485 --> 00:02:17.711
that's okay 'cause remember,
it's an approximation.

00:02:17.711 --> 00:02:19.406
You should find the answer that is closest

00:02:19.406 --> 00:02:21.023
to what you estimated.

00:02:21.023 --> 00:02:22.140
But, whoever wrote this question

00:02:22.140 --> 00:02:23.886
had a very similar thought process.

00:02:23.886 --> 00:02:28.096
They said, okay, 722, pretty close to 700.

00:02:28.096 --> 00:02:30.415
And 68, pretty close to 70.

00:02:30.415 --> 00:02:32.009
And so it's gonna be approximately equal

00:02:32.009 --> 00:02:33.759
to 700 divided by 70.


Rounding decimals on the number line

WEBVTT
Kind: captions
Language: en

00:00:00.555 --> 00:00:02.470
- [Instructor] So we are
asked to drag the point

00:00:02.470 --> 00:00:05.691
to 12.5 on the number line, so let's see.

00:00:05.691 --> 00:00:10.305
Let's see, this is 12, and 12.5
is halfway between 12 and 13

00:00:10.305 --> 00:00:13.690
then say say what is 12.5
rounded to the nearest 10?

00:00:13.690 --> 00:00:15.436
Well, what's cool about
this is you can see

00:00:15.436 --> 00:00:19.704
on the number line that our
tens are in this blue hash.

00:00:19.704 --> 00:00:22.690
So we have 10 and then 20
and we're between 10 and 20

00:00:22.690 --> 00:00:24.038
and which one are we closest to?

00:00:24.038 --> 00:00:26.237
Which is literally the nearest 10?

00:00:26.237 --> 00:00:29.045
Well, you can see we are
closer to 10 than we are to 20

00:00:29.045 --> 00:00:32.941
so you would say 10, and this
helps us build an intuition

00:00:32.941 --> 00:00:36.050
for what rounding to the
nearest 10 even means

00:00:36.050 --> 00:00:38.018
because you might know a rule like,

00:00:38.018 --> 00:00:40.539
hey look, you go one place
less than the tens place,

00:00:40.539 --> 00:00:41.807
which would be the one's place

00:00:41.807 --> 00:00:46.151
and if it's less than five
there you round down to 10.

00:00:46.151 --> 00:00:48.504
If it's five or greater
you round up to 20.

00:00:48.504 --> 00:00:51.629
But you see why over here,
we are just closer to 10.

00:00:51.629 --> 00:00:54.827
Let's do another example here.

00:00:54.827 --> 00:00:58.160
So here it says drag the point to 0.136.

00:00:59.428 --> 00:01:03.595
So this is 0.13, this is
0.14, so this is 13 hundredths

00:01:04.497 --> 00:01:07.542
and this is 14 hundredths, now let's see.

00:01:07.542 --> 00:01:08.873
There's one, two, there, four,

00:01:08.873 --> 00:01:11.973
five, six, six, seven, eight, nine, ten.

00:01:11.973 --> 00:01:15.882
So these are, you could think
of it as the ten thousandths

00:01:15.882 --> 00:01:18.163
between these two hundredths.

00:01:18.163 --> 00:01:21.165
So we want to go 13 hundredths

00:01:21.165 --> 00:01:23.616
and then another six thousandths.

00:01:23.616 --> 00:01:27.783
So let's see, we go one,
two, three, four, five, six.

00:01:29.080 --> 00:01:32.146
Just like that, and then
they say what is this number?

00:01:32.146 --> 00:01:36.313
136 thousandths or 0.136 rounded
to the nearest hundredth.

00:01:39.251 --> 00:01:42.065
Well, we have our hundredths in blue here,

00:01:42.065 --> 00:01:43.220
and which one are we closer to?

00:01:43.220 --> 00:01:47.156
Well, we're closer to 14 hundredths, 0.14.

00:01:47.156 --> 00:01:49.571
And that's consistent with
what we've seen in other cases

00:01:49.571 --> 00:01:51.587
where you look at the thousandths place

00:01:51.587 --> 00:01:53.876
and if it's five or greater you round up.

00:01:53.876 --> 00:01:57.161
In this case, you would
round up to 14 hundredths.

00:01:57.161 --> 00:02:00.654
Let's do a few more examples here.

00:02:00.654 --> 00:02:04.725
So let's say that, so they're asking us

00:02:04.725 --> 00:02:08.392
which point is at 44.197
on the number line?

00:02:09.743 --> 00:02:13.258
So let's see, that's going
to be between 44 and 45,

00:02:13.258 --> 00:02:17.425
so it's right over here, 44.197,
so that would be point C.

00:02:18.461 --> 00:02:21.878
And then they say, what is 44.197 rounded

00:02:22.728 --> 00:02:24.774
to the nearest whole number?

00:02:24.774 --> 00:02:26.901
Well, there's a couple of
ways to think about it.

00:02:26.901 --> 00:02:29.486
You could just look at the tenths place

00:02:29.486 --> 00:02:33.301
and say look, that's less than
five so we round down to 44

00:02:33.301 --> 00:02:35.532
or a more intuitive way
of thinking about it

00:02:35.532 --> 00:02:38.395
is like look, point C is
the number we care about.

00:02:38.395 --> 00:02:39.645
That is 44.197.

00:02:41.294 --> 00:02:43.562
What's the closest whole number to it?

00:02:43.562 --> 00:02:46.046
Is it closer to 44 or is it closer to 45?

00:02:46.046 --> 00:02:48.774
Well, it's clearly closer to 44,

00:02:48.774 --> 00:02:52.372
so that's another reason
why 44 makes a lot of sense.

00:02:52.372 --> 00:02:54.455
Let's do another example.

00:02:55.716 --> 00:02:58.966
Let's say that we want to, so they say,

00:03:00.152 --> 00:03:02.828
what is A rounded to
the nearest thousandth?

00:03:02.828 --> 00:03:06.142
What is A rounded to
the nearest hundredth?

00:03:06.142 --> 00:03:09.056
So A is right over here,
let's get our bearings.

00:03:09.056 --> 00:03:11.851
So 0.7, that's seven hundredths.

00:03:11.851 --> 00:03:15.859
This is eight hundredths right over here.

00:03:15.859 --> 00:03:18.044
And then between them, let's see,

00:03:18.044 --> 00:03:21.013
this looks like it's
zero, so you can view this

00:03:21.013 --> 00:03:24.343
as 70 thousandths, 71
thousandths, 72 thousandths,

00:03:24.343 --> 00:03:27.921
73 thousandths, 74 thousandths, and so on.

00:03:27.921 --> 00:03:31.930
So this is between 78
thousands and 79 thousandths.

00:03:31.930 --> 00:03:33.640
So if we round to the nearest thousandth

00:03:33.640 --> 00:03:35.776
it looks closer, in fact,
it's definitely closer

00:03:35.776 --> 00:03:39.010
to 78 thousandths than
it is to 79 thousandths.

00:03:39.010 --> 00:03:43.010
So I would say this is
0.078, or 78 thousandths.

00:03:43.915 --> 00:03:45.907
That's closer, that's
the nearest thousandth.

00:03:45.907 --> 00:03:49.617
What is A rounded to
the nearest hundredth?

00:03:49.617 --> 00:03:52.277
Well, what are the hundredths
that it's in-between?

00:03:52.277 --> 00:03:56.053
It's in-between seven
hundredths and eight hundredths.

00:03:56.053 --> 00:03:57.514
And which one is it closer to?

00:03:57.514 --> 00:04:01.181
Well, it's closer to
eight hundredths, 0.08.

00:04:02.803 --> 00:04:04.267
So once again, the whole
point of this video

00:04:04.267 --> 00:04:06.017
is to appreciate when people say round

00:04:06.017 --> 00:04:08.943
to the nearest whole
number, to the nearest 10,

00:04:08.943 --> 00:04:10.969
to the nearest thousandth,
to the nearest hundredth,

00:04:10.969 --> 00:04:13.604
you can think of it on a
number line and just say,

00:04:13.604 --> 00:04:16.234
well, what is the closest hundredth to it?

00:04:16.234 --> 00:04:18.412
Or what hundredth is it closest to?

00:04:18.412 --> 00:04:21.638
Which thousandth is it
closest to on the number line?


Multiplying using area models and the standard algorithm

WEBVTT
Kind: captions
Language: en

00:00:00.566 --> 00:00:01.820
- [Instructor] What we're
going to do in this video

00:00:01.820 --> 00:00:04.070
is multiply the numbers 352

00:00:04.908 --> 00:00:05.741
and 481.

00:00:08.856 --> 00:00:10.528
And we're gonna do it
in two different ways,

00:00:10.528 --> 00:00:13.593
but realize that the
underlying ideas are the same.

00:00:13.593 --> 00:00:15.961
So first let's just appreciate that 352

00:00:15.961 --> 00:00:17.633
can be rewritten as

00:00:17.633 --> 00:00:19.300
300 plus 50 plus two

00:00:22.114 --> 00:00:24.204
or we could think of it as

00:00:24.204 --> 00:00:25.121
two plus 50

00:00:27.083 --> 00:00:27.916
plus 300,

00:00:30.102 --> 00:00:30.935
plus 300.

00:00:31.867 --> 00:00:33.980
You add these three numbers together,

00:00:33.980 --> 00:00:36.534
you're going to have 352.

00:00:36.534 --> 00:00:38.206
And same idea,

00:00:38.206 --> 00:00:40.289
481 that's four hundreds,

00:00:41.178 --> 00:00:43.761
four hundreds, plus eight tens,

00:00:45.444 --> 00:00:46.444
which is 80,

00:00:47.673 --> 00:00:49.299
so plus 80,

00:00:49.299 --> 00:00:50.878
and then we have one one,

00:00:50.878 --> 00:00:52.433
so plus one.

00:00:52.433 --> 00:00:55.220
And you might have be
familiar with multiplying

00:00:55.220 --> 00:00:57.449
like this in the past,
setting up this grid.

00:00:57.449 --> 00:00:59.330
And it's essentially we're applying

00:00:59.330 --> 00:01:00.351
the distributive property.

00:01:00.351 --> 00:01:01.930
We're gonna take the two
and multiply it times 400

00:01:01.930 --> 00:01:03.254
plus 80 plus one.

00:01:03.254 --> 00:01:05.530
So we're gonna multiply two
times each of those numbers,

00:01:05.530 --> 00:01:08.804
and actually I'm gonna just
draw some quick lines here.

00:01:08.804 --> 00:01:10.336
So we have that,

00:01:10.336 --> 00:01:12.008
and then we'll have

00:01:12.008 --> 00:01:13.099
let me do it like this,

00:01:13.099 --> 00:01:14.599
then we have this,

00:01:16.048 --> 00:01:18.631
and then let me set up my grid.

00:01:21.273 --> 00:01:23.130
I'm having trouble drawing straight lines,

00:01:23.130 --> 00:01:24.709
okay there we go.

00:01:24.709 --> 00:01:27.705
Then one more in this direction.

00:01:27.705 --> 00:01:28.819
There you go.

00:01:28.819 --> 00:01:30.236
And then in this direction,

00:01:30.236 --> 00:01:32.163
let me draw some horizontal lines

00:01:32.163 --> 00:01:35.181
to have a neat clean grid here.

00:01:35.181 --> 00:01:36.551
There we go.

00:01:36.551 --> 00:01:38.989
Now first we'll multiply two times 400

00:01:38.989 --> 00:01:41.056
plus 80 plus one.

00:01:41.056 --> 00:01:43.056
So two times 400 is 800.

00:01:45.096 --> 00:01:47.906
Two times, let me do it
in that same blue color,

00:01:47.906 --> 00:01:49.903
so this is 800.

00:01:49.903 --> 00:01:52.295
Two times 80 is 160,

00:01:52.295 --> 00:01:54.795
and then two times one is two.

00:01:55.963 --> 00:01:58.726
And then we can multiply 50 times these.

00:01:58.726 --> 00:02:01.211
So what's 50 times 400?

00:02:01.211 --> 00:02:03.463
Well five times four is 20,

00:02:03.463 --> 00:02:06.064
and then we have another
one, two, three zeros.

00:02:06.064 --> 00:02:08.874
So one, two, three, so that's 20,000.

00:02:08.874 --> 00:02:12.519
50 times 80, five times eight is 40,

00:02:12.519 --> 00:02:14.795
and then we have two zeros,

00:02:14.795 --> 00:02:16.420
just like that.

00:02:16.420 --> 00:02:18.696
And then we have 50 times
one which is of course

00:02:18.696 --> 00:02:20.693
going to be equal to 50.

00:02:20.693 --> 00:02:22.179
And then we go to the 300,

00:02:22.179 --> 00:02:23.641
which we will distribute and multiply

00:02:23.641 --> 00:02:24.849
times each of these,

00:02:24.849 --> 00:02:26.567
each of these numbers.

00:02:26.567 --> 00:02:29.191
300 times 400, three times four is 12,

00:02:29.191 --> 00:02:30.584
and then we have four zeros.

00:02:30.584 --> 00:02:32.929
One, two, three, four.

00:02:32.929 --> 00:02:34.508
We get 120,000.

00:02:34.508 --> 00:02:37.364
300 times 80, three times eight is 24,

00:02:37.364 --> 00:02:39.942
and then we have one, two, three zeros.

00:02:39.942 --> 00:02:42.612
One, two, three, so we get 24,000.

00:02:42.612 --> 00:02:46.467
And then 300 times one is
of course equal to 300.

00:02:46.467 --> 00:02:48.742
And then what we wanna do is add up

00:02:48.742 --> 00:02:49.973
all of these numbers.

00:02:49.973 --> 00:02:52.388
So let's actually add up the rows first.

00:02:52.388 --> 00:02:54.756
So if we add up the rows,

00:02:54.756 --> 00:02:58.866
let me draw another line
going straight down like that.

00:02:58.866 --> 00:03:00.677
And so if we sum this up,

00:03:00.677 --> 00:03:02.860
this is going to be 962.

00:03:02.860 --> 00:03:05.739
800 plus 160 is 960 plus two,

00:03:05.739 --> 00:03:06.989
so this is 962.

00:03:08.595 --> 00:03:11.846
This right over here is 24,050.

00:03:11.846 --> 00:03:12.679
24,050.

00:03:15.376 --> 00:03:18.812
And then this right over
here is what 100 and,

00:03:18.812 --> 00:03:19.645
144,300?

00:03:22.040 --> 00:03:22.873
144,300.

00:03:25.360 --> 00:03:27.860
120,000 plus 24,000 is 144,000

00:03:28.866 --> 00:03:31.049
plus 300, there you have it.

00:03:31.049 --> 00:03:34.648
And then you would add up
these numbers just like that

00:03:34.648 --> 00:03:36.970
to get your final answer.

00:03:36.970 --> 00:03:38.804
Now I'm gonna hold off
doing that for a second

00:03:38.804 --> 00:03:41.428
as we see the other way of
multiplying these numbers.

00:03:41.428 --> 00:03:43.054
So the other way of doing it,

00:03:43.054 --> 00:03:44.656
we could've said,

00:03:44.656 --> 00:03:47.406
481, and this is sometimes called

00:03:48.348 --> 00:03:50.182
the standard algorithm,

00:03:50.182 --> 00:03:51.015
481 times

00:03:53.688 --> 00:03:56.855
300 and 50, let me do the same colors,

00:03:59.006 --> 00:03:59.839
352.

00:04:04.091 --> 00:04:05.809
And in the standard algorithm,

00:04:05.809 --> 00:04:09.803
the way that we do it is
we start with this two

00:04:09.803 --> 00:04:12.868
in the ones place and then
we multiply it times 481.

00:04:12.868 --> 00:04:14.772
So two times one is two,

00:04:14.772 --> 00:04:16.605
two times eight is 16.

00:04:18.418 --> 00:04:20.670
So we put the six here
and then we sometimes

00:04:20.670 --> 00:04:22.690
we will say we carry the one,

00:04:22.690 --> 00:04:25.244
but really regrouping that as hundreds.

00:04:25.244 --> 00:04:27.032
That's 10 tens which is hundreds.

00:04:27.032 --> 00:04:29.633
And then two times four is eight,

00:04:29.633 --> 00:04:31.816
which is really 800, plus one,

00:04:31.816 --> 00:04:34.323
so that's nine or really 900.

00:04:34.323 --> 00:04:35.809
You see a pattern here.

00:04:35.809 --> 00:04:39.873
This 962 is the exact
same thing as that 962

00:04:39.873 --> 00:04:40.871
right over there.

00:04:40.871 --> 00:04:41.893
Why?

00:04:41.893 --> 00:04:44.656
Well because we multiplied
the 200 times the one,

00:04:44.656 --> 00:04:47.628
times the 80, times the 400.

00:04:47.628 --> 00:04:48.697
We saw that over here,

00:04:48.697 --> 00:04:50.368
and then we just added 'em
all together to be 962.

00:04:50.368 --> 00:04:53.921
That's all the standard
algorithm did just now.

00:04:53.921 --> 00:04:55.941
And then we move over to the five.

00:04:55.941 --> 00:04:58.542
But this is really five tens or 50,

00:04:58.542 --> 00:05:00.190
and that's why in the standard algorithm

00:05:00.190 --> 00:05:01.700
we put a zero here

00:05:01.700 --> 00:05:04.788
before saying all right
what's five times one?

00:05:04.788 --> 00:05:06.251
It's five.

00:05:06.251 --> 00:05:08.039
What's five times eight?

00:05:08.039 --> 00:05:08.872
It is 40.

00:05:10.059 --> 00:05:11.870
We regroup the four.

00:05:11.870 --> 00:05:13.449
Let's delete this from before.

00:05:13.449 --> 00:05:14.981
What's five times four?

00:05:14.981 --> 00:05:17.564
Well that's 20 plus four is 24.

00:05:18.790 --> 00:05:23.434
Notice 24,050, that's exactly
what we had over here,

00:05:23.434 --> 00:05:25.941
and it makes sense
because we're taking a 50

00:05:25.941 --> 00:05:28.588
and we're multiplying it times 481

00:05:28.588 --> 00:05:31.421
which is exactly what
we did right over there.

00:05:31.421 --> 00:05:33.047
And so you might guess what's gonna happen

00:05:33.047 --> 00:05:34.742
when we take this three

00:05:34.742 --> 00:05:37.064
and we multiply it times 481.

00:05:37.064 --> 00:05:40.036
That's really a 300 times 481.

00:05:40.036 --> 00:05:41.917
Let me delete that so
I don't get confused.

00:05:41.917 --> 00:05:44.123
So because it's a 300,

00:05:44.123 --> 00:05:45.655
in the standard algorithm,

00:05:45.655 --> 00:05:48.047
we put two zeros here first.

00:05:48.047 --> 00:05:49.393
And when I say algorithm it just means

00:05:49.393 --> 00:05:51.576
a method of doing something.

00:05:51.576 --> 00:05:55.802
And so we'll say three times one is three.

00:05:55.802 --> 00:05:57.802
Three times eight is 24.

00:05:58.937 --> 00:06:02.854
And then three times four
is 12 plus two is 14.

00:06:03.906 --> 00:06:04.997
And so notice,

00:06:04.997 --> 00:06:06.247
I have 144,300.

00:06:07.830 --> 00:06:10.291
And the standard way of
doing it at this point

00:06:10.291 --> 00:06:11.429
we just add 'em all up.

00:06:11.429 --> 00:06:12.706
So whether we're doing it here or here,

00:06:12.706 --> 00:06:14.239
we just add everything up.

00:06:14.239 --> 00:06:16.259
So two plus zero plus zero is two.

00:06:16.259 --> 00:06:18.674
Six plus five is 11.

00:06:18.674 --> 00:06:20.415
Regroup that one.

00:06:20.415 --> 00:06:23.016
So one plus nine plus three is 13.

00:06:23.016 --> 00:06:25.663
And then one plus four plus four is nine.

00:06:25.663 --> 00:06:27.288
Two plus four is six.

00:06:27.288 --> 00:06:29.123
Then we have a one right over there,

00:06:29.123 --> 00:06:30.623
so we get 169,312.

00:06:33.070 --> 00:06:34.858
And so when you just learn this method,

00:06:34.858 --> 00:06:37.528
the standard algorithm we
some people might call it,

00:06:37.528 --> 00:06:39.293
it might seem like hey
this just seems like

00:06:39.293 --> 00:06:40.268
a little bit of magic.

00:06:40.268 --> 00:06:42.521
But all you're doing is you're
going to each of these places

00:06:42.521 --> 00:06:44.146
and you're distributing it,

00:06:44.146 --> 00:06:47.025
you're multiplying it
times 400 plus 80 plus one

00:06:47.025 --> 00:06:48.047
to get this.

00:06:48.047 --> 00:06:51.507
Then you're multiplying 50
times 400 plus 80 plus one,

00:06:51.507 --> 00:06:53.690
and then 300 times 400 plus 80 plus one,

00:06:53.690 --> 00:06:56.690
exactly what we did right over here.


Evaluating exponent expressions with variables

WEBVTT
Kind: captions
Language: en

00:00:00.977 --> 00:00:03.530
- [Instructor] We are asked
to evaluate the expression

00:00:03.530 --> 00:00:07.212
five to the x power minus
three to the x power

00:00:07.212 --> 00:00:09.613
for x equals two.

00:00:09.613 --> 00:00:12.723
So pause this video, and
see if you can figure out,

00:00:12.723 --> 00:00:16.563
what does this expression
equal when x equals two?

00:00:16.563 --> 00:00:18.050
All right, now let's work
through this together.

00:00:18.050 --> 00:00:20.495
So what we want to do is
everywhere we see an x,

00:00:20.495 --> 00:00:22.080
we want to replace it with a two.

00:00:22.080 --> 00:00:25.098
So this expression, for
x equals two, would be

00:00:25.098 --> 00:00:27.810
five to the second power

00:00:27.810 --> 00:00:30.918
minus three to the second power.

00:00:30.918 --> 00:00:33.308
Well, what's that going to be equal to?

00:00:33.308 --> 00:00:35.221
Well, five to the second power,

00:00:35.221 --> 00:00:39.105
that's the same thing as five times five.

00:00:39.105 --> 00:00:42.384
And then from that, we
are going to subtract

00:00:42.384 --> 00:00:43.884
three times three,

00:00:44.826 --> 00:00:46.830
three times three.

00:00:46.830 --> 00:00:48.512
And now order of operations would tell us

00:00:48.512 --> 00:00:50.894
to do the multiplication
or do the exponents first,

00:00:50.894 --> 00:00:52.522
which is this multiplication.

00:00:52.522 --> 00:00:55.221
But just to make it clear,
I'll put some parentheses here.

00:00:55.221 --> 00:00:58.020
And this is going to be equal to,

00:00:58.020 --> 00:00:59.853
five times five is 25,

00:01:00.856 --> 00:01:01.689
minus

00:01:02.530 --> 00:01:03.363
nine,

00:01:04.298 --> 00:01:05.798
which is equal to,

00:01:06.783 --> 00:01:07.870
what's 25 minus nine?

00:01:07.870 --> 00:01:09.167
It is equal to 16.

00:01:09.167 --> 00:01:12.916
So that's what that expression
equals for x equals two.

00:01:12.916 --> 00:01:15.395
Let's do another example.

00:01:15.395 --> 00:01:16.978
So now we are asked

00:01:18.104 --> 00:01:21.615
what is the value of y
squared minus x to the fourth

00:01:21.615 --> 00:01:24.400
when y is equal to nine and x equals two?

00:01:24.400 --> 00:01:25.655
So once again, pause this video,

00:01:25.655 --> 00:01:28.008
and see if you can evaluate that.

00:01:28.008 --> 00:01:29.508
All right, so here

00:01:30.420 --> 00:01:32.878
we have variables as the bases,

00:01:32.878 --> 00:01:34.536
as opposed to being the exponents,

00:01:34.536 --> 00:01:36.084
and we have two different variables.

00:01:36.084 --> 00:01:38.064
But all we have to do
is wherever we see a y,

00:01:38.064 --> 00:01:39.673
we substitute it with a nine.

00:01:39.673 --> 00:01:43.905
And wherever we see an x,
we substitute it with a two.

00:01:43.905 --> 00:01:47.162
So y squared is going
to be the same thing as

00:01:47.162 --> 00:01:48.162
nine squared

00:01:49.851 --> 00:01:50.684
minus,

00:01:52.057 --> 00:01:53.274
minus

00:01:53.274 --> 00:01:55.177
x, which is two.

00:01:55.177 --> 00:01:57.201
That minus looks a little
bit funny, let me see.

00:01:57.201 --> 00:02:00.930
So this is gonna be nine squared minus x,

00:02:00.930 --> 00:02:02.831
which is two,

00:02:02.831 --> 00:02:03.664
two to the

00:02:04.552 --> 00:02:06.117
fourth power.

00:02:06.117 --> 00:02:09.007
Now what is this going to be equal to?

00:02:09.007 --> 00:02:12.632
Well, nine squared is nine times nine.

00:02:12.632 --> 00:02:16.549
So this whole thing is
going to be equal to 81.

00:02:18.130 --> 00:02:20.568
This whole thing right over
here is nine times nine.

00:02:20.568 --> 00:02:23.196
Nine times nine is that right over there.

00:02:23.196 --> 00:02:27.077
And then from that,
we're going to subtract

00:02:27.077 --> 00:02:28.991
two to the fourth power.

00:02:28.991 --> 00:02:30.259
Well, what's two to the fourth power?

00:02:30.259 --> 00:02:34.355
That is two times two,
times two, times two.

00:02:34.355 --> 00:02:37.154
So this is going to be,
two times two is four,

00:02:37.154 --> 00:02:40.092
four times two is eight,
eight times two is 16.

00:02:40.092 --> 00:02:41.759
So it's 81 minus 16.

00:02:43.047 --> 00:02:45.291
Now what is that going to be equal to?

00:02:45.291 --> 00:02:47.791
Let's see, 81 minus six is 75,

00:02:49.194 --> 00:02:52.737
and then minus another
10 is going to be 65.

00:02:52.737 --> 00:02:54.022
So there you have it,

00:02:54.022 --> 00:02:56.126
y squared minus x to the fourth,

00:02:56.126 --> 00:02:59.746
when y is equal to nine and
x equals two, is equal to 65.

00:02:59.746 --> 00:03:00.833
And we're done.


Comparing exponent expressions

WEBVTT
Kind: captions
Language: en

00:00:00.622 --> 00:00:02.392
- [Instructor] So we are asked to order

00:00:02.392 --> 00:00:04.839
the expressions from least to greatest

00:00:04.839 --> 00:00:07.367
and this is from the
exercises on Khan Academy

00:00:07.367 --> 00:00:09.152
and if we're doing it on Kahn Academy,

00:00:09.152 --> 00:00:11.219
we would drag these little tiles around

00:00:11.219 --> 00:00:13.999
from least to greatest, least on the left,

00:00:13.999 --> 00:00:15.066
greatest on the right.

00:00:15.066 --> 00:00:16.432
I can't drag it around 'cause this

00:00:16.432 --> 00:00:18.142
is just a picture, so
I'm gonna evaluate each

00:00:18.142 --> 00:00:20.091
of these, and then I'm gonna rewrite them

00:00:20.091 --> 00:00:21.912
from least to greatest.

00:00:21.912 --> 00:00:23.793
So let's start with two to the third minus

00:00:23.793 --> 00:00:25.104
two to the first.

00:00:25.104 --> 00:00:26.082
What is that going to be?

00:00:26.082 --> 00:00:27.456
Two to the third minus two to the first.

00:00:27.456 --> 00:00:28.990
And if you feel really confident,

00:00:28.990 --> 00:00:30.073
just pause this video and try

00:00:30.073 --> 00:00:31.304
to figure out the whole thing.

00:00:31.304 --> 00:00:33.467
Order them from least to greatest.

00:00:33.467 --> 00:00:37.817
Well two to the third, that
is two times two times two,

00:00:37.817 --> 00:00:40.373
and then two to the first,
well that's just two.

00:00:40.373 --> 00:00:43.649
So two times two is
four, times two is eight,

00:00:43.649 --> 00:00:48.059
minus two, this is going
to be equal to six.

00:00:48.059 --> 00:00:49.858
So this expression right over here could

00:00:49.858 --> 00:00:52.970
be evaluated as being equal to six.

00:00:52.970 --> 00:00:55.104
Now, what about this right over here?

00:00:55.104 --> 00:00:57.508
What is this equal to?

00:00:57.508 --> 00:01:00.425
Well let's see, we have two squared

00:01:01.285 --> 00:01:03.238
plus three to the zero.

00:01:03.238 --> 00:01:07.029
Two squared is two times two and anything

00:01:07.029 --> 00:01:10.862
to the zero power is
going to be equal to one.

00:01:12.076 --> 00:01:13.114
It's an interesting thing to think

00:01:13.114 --> 00:01:15.039
about what zero to the zeroth power should

00:01:15.039 --> 00:01:17.278
be but that'll be a
topic for another video.

00:01:17.278 --> 00:01:19.068
But here we have three
to the zeroth power,

00:01:19.068 --> 00:01:21.972
which is clearly equal to one.

00:01:21.972 --> 00:01:23.767
And so we have two times two plus one.

00:01:23.767 --> 00:01:27.496
This is four plus one,
which is equal to five.

00:01:27.496 --> 00:01:30.189
So the second tile is equal to five.

00:01:30.189 --> 00:01:33.464
And then three squared,
well three squared,

00:01:33.464 --> 00:01:35.146
that's just three times three.

00:01:35.146 --> 00:01:38.063
Three times three is equal to nine.

00:01:38.994 --> 00:01:40.653
So if I were to order them from least

00:01:40.653 --> 00:01:42.767
to greatest, the smallest of these

00:01:42.767 --> 00:01:46.600
is two squared plus three
to the zeroth power.

00:01:47.465 --> 00:01:50.226
That one is equal to five,
so I'd put that on the left.

00:01:50.226 --> 00:01:52.276
Then we have this thing that's equal

00:01:52.276 --> 00:01:56.870
to six, two to the third power
minus two to the first power.

00:01:56.870 --> 00:02:00.629
And then the largest value
here is three squared.

00:02:00.629 --> 00:02:04.455
So we would put that tile, three squared.

00:02:04.455 --> 00:02:07.892
We will put that tile on
the right, and we're done.


Exponents of decimals

WEBVTT
Kind: captions
Language: en

00:00:00.747 --> 00:00:02.176
- [Instructor] What we're
going to do in this video

00:00:02.176 --> 00:00:06.793
is get some practice evaluating
exponents of decimals.

00:00:06.793 --> 00:00:10.793
So let's say that I have
0.2 to the third power.

00:00:13.316 --> 00:00:14.698
Pause this video, see
if you can figure out

00:00:14.698 --> 00:00:17.206
what that is going to be.

00:00:17.206 --> 00:00:19.376
Well, this would just mean
that if I take something

00:00:19.376 --> 00:00:22.118
to the third power, that means
I take three of that number

00:00:22.118 --> 00:00:23.771
and I multiply them together.

00:00:23.771 --> 00:00:26.438
So it's 0.2 times 0.2 times 0.2.

00:00:31.213 --> 00:00:33.376
Well, what is this going to be equal to?

00:00:33.376 --> 00:00:37.543
Well, if I take 0.2 times
0.2, that is going to be 0.04.

00:00:41.068 --> 00:00:44.243
One way to think about
it, two times two is four

00:00:44.243 --> 00:00:46.723
and then I have one
number behind the decimal

00:00:46.723 --> 00:00:47.781
to the right of the decimal here.

00:00:47.781 --> 00:00:49.782
I have another digit to
the right of the decimal

00:00:49.782 --> 00:00:53.316
right over here, so my product
is going to have two digits

00:00:53.316 --> 00:00:56.518
to the right of the
decimal, so it'd be 0.04.

00:00:56.518 --> 00:01:00.653
And then if I were to
multiply that times 0.2,

00:01:00.653 --> 00:01:02.751
so if I were to multiply that together

00:01:02.751 --> 00:01:04.172
what is that going to be equal to?

00:01:04.172 --> 00:01:07.339
Well, four times two is equal to eight

00:01:08.256 --> 00:01:11.423
and now I have one, two, three numbers

00:01:12.625 --> 00:01:14.243
to the right of the decimal point,

00:01:14.243 --> 00:01:18.853
so my product is going to
have one, two, three numbers

00:01:18.853 --> 00:01:21.448
to the right of the decimal point.

00:01:21.448 --> 00:01:23.831
So now that we've had a little
bit of practice with that,

00:01:23.831 --> 00:01:26.077
let's do another example.

00:01:26.077 --> 00:01:30.244
So let's say that I were to
ask you, what is 0.9 squared?

00:01:36.464 --> 00:01:40.279
Pause this video and see
if you can figure that out.

00:01:40.279 --> 00:01:44.446
All right, well this is just
going to be 0.9 times 0.9.

00:01:46.688 --> 00:01:48.533
And what's that going to be equal to?

00:01:48.533 --> 00:01:51.176
Well, you could just say nine times nine

00:01:51.176 --> 00:01:55.412
is going to be equal to
81, and so, let's see,

00:01:55.412 --> 00:01:57.401
in the two numbers that I'm multiplying

00:01:57.401 --> 00:02:01.277
I have a total of one, two
numbers, or two digits,

00:02:01.277 --> 00:02:03.059
to the right of the decimal point

00:02:03.059 --> 00:02:06.279
so my answer's going
to have one, two digits

00:02:06.279 --> 00:02:07.678
to the right of the decimal point.

00:02:07.678 --> 00:02:09.518
So put the decimal right over there

00:02:09.518 --> 00:02:11.904
and I'll put the zero, so 0.81.

00:02:11.904 --> 00:02:15.139
Another way to think about it
is nine-tenths of nine-tenths

00:02:15.139 --> 00:02:18.812
is 81 hundredths, but there you go.

00:02:18.812 --> 00:02:22.612
Using exponents, or taking
exponents of decimals

00:02:22.612 --> 00:02:25.141
is the same as when we're
taking it of integers.

00:02:25.141 --> 00:02:27.188
It's just in this case you just have to

00:02:27.188 --> 00:02:30.345
do a little bit of decimal multiplication.


Identifying transformation described with other algebra and geometry concepts

WEBVTT
Kind: captions
Language: en

00:00:00.636 --> 00:00:02.517
- [Instructor] We're told
that a certain mapping

00:00:02.517 --> 00:00:06.766
in the xy-plane has the
following two properties.

00:00:06.766 --> 00:00:09.552
Each point on the line y is equal to

00:00:09.552 --> 00:00:12.552
three x minus two maps to itself.

00:00:13.569 --> 00:00:15.938
Any point P not on the line

00:00:15.938 --> 00:00:17.771
maps to a new point P'

00:00:18.771 --> 00:00:22.509
in such a way that the
perpendicular bisector of

00:00:22.509 --> 00:00:25.504
the segment PP' is the line

00:00:25.504 --> 00:00:29.220
y is equal to three x minus two.

00:00:29.220 --> 00:00:31.031
Which of the following statements is true?

00:00:31.031 --> 00:00:33.283
So is this describing a
reflection, a rotation,

00:00:33.283 --> 00:00:35.025
or a translation?

00:00:35.025 --> 00:00:37.114
So pause this video and see
if you can work through it

00:00:37.114 --> 00:00:38.114
on your own.

00:00:39.181 --> 00:00:42.246
All right so let me just
try to visualize this.

00:00:42.246 --> 00:00:45.149
So, and I'll just do a very quick,

00:00:45.149 --> 00:00:47.447
so if that's my y-axis,

00:00:47.447 --> 00:00:50.907
and that this right
over here is my x-axis.

00:00:50.907 --> 00:00:54.088
Three x minus two might
look something like this.

00:00:54.088 --> 00:00:55.969
The line three x minus two would look

00:00:55.969 --> 00:00:57.339
something like that.

00:00:57.339 --> 00:00:58.686
And so what we're saying is,

00:00:58.686 --> 00:00:59.986
or what they're telling us,

00:00:59.986 --> 00:01:01.728
is any point on this
after the transformation

00:01:01.728 --> 00:01:03.539
maps to itself.

00:01:03.539 --> 00:01:06.047
Now that by itself is a pretty good clue

00:01:06.047 --> 00:01:08.090
that we're likely dealing
with a reflection.

00:01:08.090 --> 00:01:09.297
Because remember with a reflection

00:01:09.297 --> 00:01:11.434
you reflect over a line,

00:01:11.434 --> 00:01:13.245
but if a point sits on the line,

00:01:13.245 --> 00:01:16.310
well it's just gonna
continue to sit on the line.

00:01:16.310 --> 00:01:18.167
But let's just make sure
that the second point

00:01:18.167 --> 00:01:20.722
is consistent with it being a reflection.

00:01:20.722 --> 00:01:23.508
So any point P not on the line,

00:01:23.508 --> 00:01:26.712
so let's see, point P, right over here,

00:01:26.712 --> 00:01:29.081
it maps to a new point P' in such a way

00:01:29.081 --> 00:01:32.308
that the perpendicular bisector of PP'

00:01:32.308 --> 00:01:35.489
is the line y equals three x minus two.

00:01:35.489 --> 00:01:37.115
So I need to connect,

00:01:37.115 --> 00:01:38.032
so the line

00:01:39.530 --> 00:01:40.900
three x minus two,

00:01:40.900 --> 00:01:42.525
y is equal to three x minus two,

00:01:42.525 --> 00:01:44.383
would be the perpendicular bisector

00:01:44.383 --> 00:01:46.890
of the segment between P and what?

00:01:46.890 --> 00:01:50.002
Well let's see I'd have to
draw a perpendicular line.

00:01:50.002 --> 00:01:52.928
It would have to have the
same length on both sides

00:01:52.928 --> 00:01:55.598
of the line y equals three x minus two.

00:01:55.598 --> 00:01:58.524
So P' would have to be right over there.

00:01:58.524 --> 00:02:02.378
So once again this is consistent
with being a reflection.

00:02:02.378 --> 00:02:06.976
P' is equidistant on the
other side of the line as P.

00:02:06.976 --> 00:02:09.414
So I definitely feel good that this

00:02:09.414 --> 00:02:12.943
is going to be a
reflection right over here.

00:02:12.943 --> 00:02:15.358
Let's do another example.

00:02:15.358 --> 00:02:17.518
So here we are told,

00:02:17.518 --> 00:02:19.259
and I'll switch my colors up,

00:02:19.259 --> 00:02:20.931
a certain mapping in the
plane has the following

00:02:20.931 --> 00:02:21.929
two properties.

00:02:21.929 --> 00:02:24.275
Point O maps to itself.

00:02:24.275 --> 00:02:27.131
Every point V on a circle C

00:02:27.131 --> 00:02:28.298
centered at O,

00:02:29.545 --> 00:02:32.796
all right, maps to a new point W

00:02:32.796 --> 00:02:33.713
on circle C

00:02:35.792 --> 00:02:39.716
so that the counterclockwise
angle from segment OV

00:02:39.716 --> 00:02:42.433
to OW measures 137 degrees.

00:02:42.433 --> 00:02:44.778
So is this a reflection,
rotation, or translation?

00:02:44.778 --> 00:02:48.330
Pause this video and try to
figure it out on your own.

00:02:48.330 --> 00:02:50.560
All right, so let's see.

00:02:50.560 --> 00:02:53.160
We're talking about circle

00:02:53.160 --> 00:02:54.577
centered at O.

00:02:54.577 --> 00:02:55.807
So let's see, let me just say,

00:02:55.807 --> 00:02:57.595
so I have this point O.

00:02:57.595 --> 00:03:00.103
It maps to itself on its transformation.

00:03:00.103 --> 00:03:03.586
Now every point V on
circle C centered at O.

00:03:03.586 --> 00:03:05.420
So let's see, let's say this is circle C

00:03:05.420 --> 00:03:07.789
centered at point O,

00:03:07.789 --> 00:03:10.807
so I'm gonna try to draw a
decent looking circle here.

00:03:10.807 --> 00:03:12.665
You get the idea.

00:03:12.665 --> 00:03:16.125
This is not the best hand-drawn
circle ever, all right.

00:03:16.125 --> 00:03:18.981
So every point, let's
just pick a point V here.

00:03:18.981 --> 00:03:21.651
So let's say that that is the point V,

00:03:21.651 --> 00:03:24.786
on a circle centered at
O maps to a new point W

00:03:24.786 --> 00:03:26.434
on the circle C.

00:03:26.434 --> 00:03:28.803
So maybe it maps to a new point W on,

00:03:28.803 --> 00:03:30.242
actually let me keep reading,

00:03:30.242 --> 00:03:33.958
W on circle C so that the
counterclockwise angle

00:03:33.958 --> 00:03:35.041
from OV to OW

00:03:36.442 --> 00:03:38.671
measures 137 degrees.

00:03:38.671 --> 00:03:41.921
Okay so we wanna know the angle from OV

00:03:42.781 --> 00:03:45.114
to OW going counterclockwise

00:03:45.962 --> 00:03:47.212
is 137 degrees.

00:03:49.260 --> 00:03:52.510
So this right over here is 137 degrees.

00:03:53.370 --> 00:03:57.201
And so this would be the segment OW.

00:03:57.201 --> 00:03:59.871
W would go right over there.

00:03:59.871 --> 00:04:02.541
And so what this looks like is well

00:04:02.541 --> 00:04:05.003
if we're talking about angles

00:04:05.003 --> 00:04:07.580
and we are rotating something,

00:04:07.580 --> 00:04:09.531
this point corresponds to this point,

00:04:09.531 --> 00:04:12.062
it's essentially the
point has been rotated

00:04:12.062 --> 00:04:13.229
by 137 degrees

00:04:14.337 --> 00:04:15.587
around point O.

00:04:16.729 --> 00:04:20.562
So this right over here
is clearly a rotation.

00:04:21.489 --> 00:04:22.696
This is a rotation.

00:04:22.696 --> 00:04:24.391
Sometimes reading this language at first

00:04:24.391 --> 00:04:25.854
is a little bit daunting.

00:04:25.854 --> 00:04:27.317
It was a little bit
daunting to me when I first

00:04:27.317 --> 00:04:28.734
(laughing) read it.

00:04:28.734 --> 00:04:30.730
But when you actually just break it down

00:04:30.730 --> 00:04:33.795
and you actually try to
visualize what's going on,

00:04:33.795 --> 00:04:35.165
you'll say well okay
look they're just taking

00:04:35.165 --> 00:04:38.370
point V and they're
rotating it by 137 degrees

00:04:38.370 --> 00:04:40.204
around point O.

00:04:40.204 --> 00:04:43.594
And so this would be a rotation.

00:04:43.594 --> 00:04:45.761
Let's do one more example.

00:04:47.100 --> 00:04:49.237
So here we are told,

00:04:49.237 --> 00:04:50.305
so they're talking about,

00:04:50.305 --> 00:04:53.881
again a certain mapping in the xy-plane.

00:04:53.881 --> 00:04:56.226
Each circle O with radius r

00:04:56.226 --> 00:04:58.966
and centered at x y is
mapped to a circle O'

00:04:58.966 --> 00:05:01.961
with radius r and centered at x plus 11

00:05:01.961 --> 00:05:03.878
and then y minus seven.

00:05:05.050 --> 00:05:06.930
So once again pause this
video, what is this?

00:05:06.930 --> 00:05:09.183
Reflection, rotation, or translation?

00:05:09.183 --> 00:05:11.667
All right so you might be tempted,

00:05:11.667 --> 00:05:13.037
if they're talking about circles

00:05:13.037 --> 00:05:14.291
like we did in the last example,

00:05:14.291 --> 00:05:15.499
maybe they're talking about a rotation.

00:05:15.499 --> 00:05:17.449
But look, what they're really saying is,

00:05:17.449 --> 00:05:20.073
is that if I have a circle,

00:05:20.073 --> 00:05:22.557
let's say I have a circle right over here,

00:05:22.557 --> 00:05:24.392
centered right over here.

00:05:24.392 --> 00:05:26.969
After it, so this is x comma y,

00:05:26.969 --> 00:05:28.455
centered at x comma y.

00:05:28.455 --> 00:05:32.287
It's mapped to a new circle
O' with the same radius.

00:05:32.287 --> 00:05:34.191
So if this is the radius,

00:05:34.191 --> 00:05:36.629
it's mapped to a new circle
with the same radius,

00:05:36.629 --> 00:05:38.796
but now it is centered at,

00:05:40.228 --> 00:05:43.432
now it is centered at x plus 11,

00:05:43.432 --> 00:05:46.149
so our new x-coordinate
is gonna be 11 larger,

00:05:46.149 --> 00:05:50.166
x plus 11, and our y-coordinate
is gonna be seven less.

00:05:50.166 --> 00:05:52.627
But we have the exact same radius.

00:05:52.627 --> 00:05:54.926
We have the exact same radius.

00:05:54.926 --> 00:05:57.062
So our circle would still,

00:05:57.062 --> 00:05:59.361
so we have the exact same
radius right over here.

00:05:59.361 --> 00:06:02.356
So what just happened to the circle?

00:06:02.356 --> 00:06:04.144
Well we kept the radius the same,

00:06:04.144 --> 00:06:05.955
and we just shifted,

00:06:05.955 --> 00:06:08.649
we just shifted our center

00:06:08.649 --> 00:06:10.832
to the right by 11,

00:06:10.832 --> 00:06:14.249
plus 11, and we shifted it down by seven.

00:06:15.592 --> 00:06:17.565
We shifted it down by seven.

00:06:17.565 --> 00:06:20.259
So this is clearly a translation.

00:06:20.259 --> 00:06:22.697
So we would select that right over there.

00:06:22.697 --> 00:06:23.947
And we're done.