Introduction to factoring higher degree polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.450
- [Narrator] When we first
learned algebra together,

00:00:02.450 --> 00:00:06.160
we started factoring polynomials,
especially quadratics.

00:00:06.160 --> 00:00:09.810
We recognized that an
expression like x squared

00:00:09.810 --> 00:00:12.920
could be written as x times x.

00:00:12.920 --> 00:00:17.920
We also recognized that a
polynomial like three x squared

00:00:18.600 --> 00:00:22.930
plus four x, that in
this situation both terms

00:00:22.930 --> 00:00:24.440
have the common factor of x

00:00:24.440 --> 00:00:25.860
and you could factor that out

00:00:25.860 --> 00:00:27.540
and so you could rewrite this as

00:00:27.540 --> 00:00:31.750
x times three x plus four.

00:00:31.750 --> 00:00:34.300
And we also learned to do fancier things.

00:00:34.300 --> 00:00:36.260
We learned to factor things like

00:00:36.260 --> 00:00:40.610
x squared plus seven x plus 12.

00:00:40.610 --> 00:00:43.050
We were able to say, "Hey!
What two numbers would add up

00:00:43.050 --> 00:00:44.820
to seven, and if I were to multiply them

00:00:44.820 --> 00:00:47.040
I'd get 12, and in those early videos,

00:00:47.040 --> 00:00:48.400
we show why that worked, and we'd say

00:00:48.400 --> 00:00:51.330
well, three and four, so
maybe this can be factored as

00:00:51.330 --> 00:00:56.330
x plus three times x plus four.

00:00:56.420 --> 00:00:58.590
If this is unfamiliar
to you, I encourage you

00:00:58.590 --> 00:00:59.910
to go review that in some of the

00:00:59.910 --> 00:01:03.300
introductory factoring
quadratics on Khan Academy.

00:01:03.300 --> 00:01:04.705
It should be review at this point

00:01:04.705 --> 00:01:06.910
in your journey.

00:01:06.910 --> 00:01:09.680
We also looked at things
like differences of squares.

00:01:09.680 --> 00:01:11.200
X squared minus nine.

00:01:11.200 --> 00:01:13.139
We'd say "Hey, that's x
squared minus three squared,

00:01:13.139 --> 00:01:16.330
so we could factor that as x plus three

00:01:16.330 --> 00:01:18.820
times x minus three. And
we looked at other types

00:01:18.820 --> 00:01:20.580
of quadratics.

00:01:20.580 --> 00:01:24.100
Now, as we go deeper into
our algebra journeys,

00:01:24.100 --> 00:01:25.830
we're going to build on this to

00:01:25.830 --> 00:01:29.010
factor higher degree polynomials.

00:01:29.010 --> 00:01:30.810
Third degree, fourth
degree, fifth degree, which

00:01:30.810 --> 00:01:33.980
will be very useful in
your mathematical careers.

00:01:33.980 --> 00:01:36.610
But we're going to start
doing it by really looking

00:01:36.610 --> 00:01:38.570
at some of the structures,
some of the patterns

00:01:38.570 --> 00:01:41.370
that we seen in introductory algebra.

00:01:41.370 --> 00:01:44.340
For example, let's say someone
walks up to you on the street

00:01:44.340 --> 00:01:46.845
and says, "Can you factor x to the third

00:01:46.845 --> 00:01:51.845
plus seven x squared plus 12 x?"

00:01:52.350 --> 00:01:54.461
Well, at first your wanna say
"Oh, this is a third-degree

00:01:54.461 --> 00:01:56.680
polynomial that seems
kind of intimidating",

00:01:56.680 --> 00:01:58.760
until you realize, "Hey,
all of these terms have

00:01:58.760 --> 00:02:01.830
the common factor x, so
if I factor that out,

00:02:01.830 --> 00:02:06.830
then it becomes x times x
squared plus seven x plus 12."

00:02:07.380 --> 00:02:10.160
And then, this is exactly
what we saw over here,

00:02:10.160 --> 00:02:12.170
so we could rewrite all of this as

00:02:12.170 --> 00:02:17.170
x times x plus three times x plus four.

00:02:17.600 --> 00:02:18.870
So we're going to see
that we might be able

00:02:18.870 --> 00:02:20.710
to do some simple factoring like this,

00:02:20.710 --> 00:02:23.210
and even factoring multiple times.

00:02:23.210 --> 00:02:26.220
We might also start to
appreciate structure

00:02:26.220 --> 00:02:27.290
that brings us back to

00:02:27.290 --> 00:02:30.100
some of what we saw in
our introductory algebra.

00:02:30.100 --> 00:02:35.060
So, for example, you might
see something like this,

00:02:35.060 --> 00:02:37.670
where, once again, someone
walks up to you on the street

00:02:37.670 --> 00:02:41.130
and says "Hey, you factor
this, a to the fourth power

00:02:41.130 --> 00:02:45.047
plus seven a squared plus
12," and at first you're like

00:02:45.047 --> 00:02:49.350
"Wow! There's a fourth
power here, what do I do?"

00:02:49.350 --> 00:02:52.970
Until you say "Well, what
if I were to rewrite this

00:02:52.970 --> 00:02:57.970
as a squared squared plus
seven a squared plus 12."

00:03:02.040 --> 00:03:06.160
And now, this a squared is
looking an awful lot like

00:03:06.160 --> 00:03:07.740
this x over here.

00:03:07.740 --> 00:03:11.060
If this were an x, than
this would be x squared.

00:03:11.060 --> 00:03:13.720
If this were an x, than
this would just be an x.

00:03:13.720 --> 00:03:16.160
And then these expressions
would be the same.

00:03:16.160 --> 00:03:18.600
So when I factor it,
everywhere I see an x,

00:03:18.600 --> 00:03:20.880
I could replace with an a squared.

00:03:20.880 --> 00:03:22.784
So I could factor this out
really looking at the same

00:03:22.784 --> 00:03:27.350
structure we have here
as a squared plus three

00:03:27.350 --> 00:03:29.820
times a squared plus four.

00:03:29.820 --> 00:03:31.920
Now, I'm going really
faster, this is really

00:03:31.920 --> 00:03:34.760
the introductory video,
the overview video.

00:03:34.760 --> 00:03:37.390
Don't worry if this is a
little bit much too fast.

00:03:37.390 --> 00:03:39.030
This is really just to
give you a sense of things.

00:03:39.030 --> 00:03:41.730
Later in this unit,
we're going to dig deeper

00:03:41.730 --> 00:03:44.130
into each of these cases.

00:03:44.130 --> 00:03:45.960
But just to give you a
sense of where we're going,

00:03:45.960 --> 00:03:48.510
I'll give you another example
that builds off of what you

00:03:48.510 --> 00:03:52.500
likely saw in your
introductory algebra learning.

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So, building off of the structure here,

00:03:54.610 --> 00:03:56.930
if someone were to walk up to you again,

00:03:56.930 --> 00:03:58.540
a lot of people are walking up to you,

00:03:58.540 --> 00:04:02.958
and say "Factor four x to the sixth

00:04:02.958 --> 00:04:06.600
minus nine y to the fourth."

00:04:06.600 --> 00:04:09.260
Well, at first, this
looks quite intimidating,

00:04:09.260 --> 00:04:11.750
until you realize that "Hey, I could write

00:04:11.750 --> 00:04:14.870
both of these as squares.
I could write this

00:04:14.870 --> 00:04:19.320
first one as two x to the third squared

00:04:19.320 --> 00:04:21.570
minus, and I could write this second term

00:04:21.570 --> 00:04:24.530
as three y squared squared."

00:04:24.530 --> 00:04:26.960
And now, this is just a
difference of squares.

00:04:26.960 --> 00:04:31.960
So it'd be, two x to the
third, plus three y squared

00:04:32.910 --> 00:04:37.910
times two x to the third
minus three y squared.

00:04:38.230 --> 00:04:40.010
We'll also see things like this where

00:04:40.010 --> 00:04:42.500
we're going to be
factoring multiple times.

00:04:42.500 --> 00:04:45.430
So, once again, someone
walks up to you in the street

00:04:45.430 --> 00:04:48.020
and they, you're a very popular person.

00:04:48.020 --> 00:04:50.227
Someone walks up to you
on the street and says

00:04:50.227 --> 00:04:53.820
"Factor x to the fourth
minus y to the fourth."

00:04:53.820 --> 00:04:55.280
Well, based on what we just saw,

00:04:55.280 --> 00:04:57.460
you could realize that
this is the same thing as

00:04:57.460 --> 00:05:02.200
x squared squared minus y squared squared,

00:05:02.200 --> 00:05:04.160
and you say "Okay, this is
a difference of squares,

00:05:04.160 --> 00:05:06.360
just like this was a
difference of squares."

00:05:06.360 --> 00:05:09.770
So it's going to be the sum of x squared

00:05:09.770 --> 00:05:12.870
and y squared, x squared plus y squared,

00:05:12.870 --> 00:05:14.370
times the difference of them.

00:05:14.370 --> 00:05:16.460
X squared minus y squared.

00:05:16.460 --> 00:05:19.660
Now this is fun, because this
too a difference of squares.

00:05:19.660 --> 00:05:21.600
So we can rewrite this whole thing as,

00:05:21.600 --> 00:05:25.410
I'll rewrite this first part, x squared,

00:05:25.410 --> 00:05:28.096
x squared plus y squared,
and then we can factor

00:05:28.096 --> 00:05:29.590
this as a difference of squares,

00:05:29.590 --> 00:05:31.360
just as we factored this up here.

00:05:31.360 --> 00:05:35.200
And we get x plus y times x minus y.

00:05:35.200 --> 00:05:36.360
So I'll leave you there.

00:05:36.360 --> 00:05:38.940
I've just bombarded you
with a bunch of information,

00:05:38.940 --> 00:05:41.170
but this is really just
to get you warmed up.

00:05:41.170 --> 00:05:43.281
Don't stress about it
because we're gonna go deep

00:05:43.281 --> 00:05:45.810
into each of these and there's
gonna be plenty of chances

00:05:45.810 --> 00:05:48.830
to practice it on Khan Academy
to make sure you understand

00:05:48.830 --> 00:05:51.430
where all of this is coming from.

00:05:51.430 --> 00:05:52.403
Enjoy.


Geometric series word problems: hike

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:01.650
- [Instructor] We're told Sloan went

00:00:01.650 --> 00:00:03.770
on a four-day hiking trip.

00:00:03.770 --> 00:00:07.080
Each day, she walked 20%
more than the distance

00:00:07.080 --> 00:00:09.540
that she walked the day before.

00:00:09.540 --> 00:00:13.000
She walked a total of 27 kilometers.

00:00:13.000 --> 00:00:15.080
What is the distance Sloan walked

00:00:15.080 --> 00:00:16.840
in the first day of the trip?

00:00:16.840 --> 00:00:18.410
And it says to round our final answer

00:00:18.410 --> 00:00:19.640
to the nearest kilometer.

00:00:19.640 --> 00:00:21.900
So, like always, have a go with this

00:00:21.900 --> 00:00:22.840
and see if you can figure out

00:00:22.840 --> 00:00:25.163
how much she walked on the first day.

00:00:26.050 --> 00:00:27.670
All right, well let's just call the amount

00:00:27.670 --> 00:00:30.150
that she walked on the first day a.

00:00:30.150 --> 00:00:33.190
And then using a, let's
see if we can set up

00:00:33.190 --> 00:00:36.320
an expression for how
much she walked in total.

00:00:36.320 --> 00:00:38.720
And then, that should be equal to 27.

00:00:38.720 --> 00:00:41.760
And then hopefully, we're going
to be able to solve for a.

00:00:41.760 --> 00:00:44.970
So the first day, she walks a kilometers.

00:00:44.970 --> 00:00:46.610
Now, how about the second day?

00:00:46.610 --> 00:00:50.560
Well, they tell us that each
day, she walked 20% more

00:00:50.560 --> 00:00:53.270
than the distance she
walked the day before.

00:00:53.270 --> 00:00:56.020
So, on the next day, she's
going to walk 20% more

00:00:56.020 --> 00:01:01.020
than a kilometers, so that's 1.2 times a.

00:01:01.330 --> 00:01:04.800
And what about the day
after that, her third day?

00:01:04.800 --> 00:01:08.290
Well, that's just gonna be 1.2
times this, the second day,

00:01:08.290 --> 00:01:10.290
and so that's going to be 1.2 times 1.2,

00:01:12.468 --> 00:01:14.840
or we can say 1.2 squared times a.

00:01:15.770 --> 00:01:17.450
And then how much on the fourth day?

00:01:17.450 --> 00:01:19.180
And that's, she went on
a four-day hiking trip,

00:01:19.180 --> 00:01:20.920
so that's the last day.

00:01:20.920 --> 00:01:23.730
Well, that's gonna be
1.2 times the third day.

00:01:23.730 --> 00:01:28.190
So, that's going to be 1.2
to the third power times a.

00:01:28.190 --> 00:01:30.360
So, this is an expression in a

00:01:30.360 --> 00:01:32.520
on how much she walked over the four days.

00:01:32.520 --> 00:01:36.080
And we know that she walked
a total of 27 kilometers,

00:01:36.080 --> 00:01:39.350
so this is going to be
equal to 27 kilometers.

00:01:39.350 --> 00:01:42.580
Now, you could solve for a over here.

00:01:42.580 --> 00:01:44.570
You could factor out the a

00:01:44.570 --> 00:01:48.780
and you could say a times one plus 1.2

00:01:48.780 --> 00:01:53.610
plus 1.2 squared plus
1.2 to the third power

00:01:53.610 --> 00:01:55.130
is equal to 27.

00:01:55.130 --> 00:01:58.900
And then you could say
that a is equal to 27

00:01:58.900 --> 00:02:03.290
over one plus 1.2 plus 1.2 squared

00:02:03.290 --> 00:02:05.460
plus 1.2 to the third power,

00:02:05.460 --> 00:02:08.770
and we would need a
calculator to evaluate this.

00:02:08.770 --> 00:02:10.350
But I'm gonna do a different technique,

00:02:10.350 --> 00:02:13.690
a technique that would work
even if you had 20 terms here.

00:02:13.690 --> 00:02:16.140
You, in theory, could
also do this with 20 terms

00:02:16.140 --> 00:02:19.713
but it gets a lot more complicated,
or if you had 200 terms.

00:02:21.110 --> 00:02:22.740
So the other way to approach this

00:02:22.740 --> 00:02:26.860
is use the formula for a
finite geometric series.

00:02:26.860 --> 00:02:28.950
What does it evaluate to?

00:02:28.950 --> 00:02:31.440
And just as a reminder, the sum

00:02:32.420 --> 00:02:37.140
of first n terms,

00:02:37.140 --> 00:02:41.253
it's going to be the first
term, which we could call a,

00:02:42.200 --> 00:02:47.200
minus the first term
times our common ratio.

00:02:47.380 --> 00:02:49.660
In this case, our common ratio is 1.2

00:02:49.660 --> 00:02:53.110
because every successive
term is 1.2 times the first.

00:02:53.110 --> 00:02:57.040
So our first term times our
common ratio to the nth power,

00:02:57.040 --> 00:03:01.270
all of that over one
minus the common ratio.

00:03:01.270 --> 00:03:03.750
In other videos, we explain
where this comes from,

00:03:03.750 --> 00:03:06.200
we prove this, but here,
we can just apply it.

00:03:06.200 --> 00:03:08.930
We already know what our a is,
I used that as our variable.

00:03:08.930 --> 00:03:11.140
Our common ratio in this situation

00:03:11.140 --> 00:03:13.233
is going to be equal to 1.2.

00:03:14.520 --> 00:03:18.760
And our n is going to be equal to four.

00:03:18.760 --> 00:03:20.100
Another way I like to think about it

00:03:20.100 --> 00:03:23.670
is it's our first term, which
we see right over there,

00:03:23.670 --> 00:03:27.310
minus the term that we did not get to.

00:03:27.310 --> 00:03:29.120
If we were to have a fifth term,

00:03:29.120 --> 00:03:31.830
it would have been that fifth
term that we're subtracting

00:03:31.830 --> 00:03:34.090
because we aren't getting
to a fourth power here,

00:03:34.090 --> 00:03:35.970
the fifth term would have
been the fourth power,

00:03:35.970 --> 00:03:38.880
all of that over one
minus the common ratio.

00:03:38.880 --> 00:03:42.680
And so, this left-hand
side of our equation,

00:03:42.680 --> 00:03:46.360
we could rewrite as our first term

00:03:46.360 --> 00:03:50.760
minus our first term times
our common ratio, 1.2,

00:03:51.620 --> 00:03:53.750
to the fourth power.

00:03:53.750 --> 00:03:58.070
All of that over one
minus our common ratio.

00:03:58.070 --> 00:04:00.960
And then that could be equal to 27.

00:04:00.960 --> 00:04:02.300
Let me scroll down a little bit

00:04:02.300 --> 00:04:05.980
so we have some more
space to then solve this.

00:04:05.980 --> 00:04:09.143
And so, let's see, I can
simplify this a little bit.

00:04:10.210 --> 00:04:13.573
This is going to be equal to negative 0.2.

00:04:14.440 --> 00:04:17.210
Our numerator, we can factor out a.

00:04:17.210 --> 00:04:20.030
And so, this is going to be equal to a

00:04:20.030 --> 00:04:24.640
times one minus 1.2 to the fourth power.

00:04:24.640 --> 00:04:26.010
And let's see, we can multiply

00:04:26.010 --> 00:04:28.790
both the numerator and the
denominator by negative one.

00:04:28.790 --> 00:04:33.070
And so, this would get us to a times,

00:04:33.070 --> 00:04:36.580
a times, and I'll put the
a out of the fraction,

00:04:36.580 --> 00:04:38.500
a time, so I'll just swap the order here

00:04:38.500 --> 00:04:39.980
and get rid of this negative.

00:04:39.980 --> 00:04:43.110
1.2 to the fourth power minus one

00:04:43.110 --> 00:04:47.890
over 0.2 is equal to 27.

00:04:47.890 --> 00:04:51.130
Again, all I did is I took
the a out of the fraction

00:04:51.130 --> 00:04:53.560
so it's out here, and I multiplied

00:04:53.560 --> 00:04:55.120
the numerator and the
denominator by negative.

00:04:55.120 --> 00:04:57.810
The numerator multiplied by
negative would swap these two.

00:04:57.810 --> 00:05:00.530
And then multiplying
negative 0.2 times negative

00:05:00.530 --> 00:05:02.260
is just positive 0.2.

00:05:02.260 --> 00:05:05.190
And so now, I can just multiply both sides

00:05:05.190 --> 00:05:08.050
times the reciprocal of this.

00:05:08.050 --> 00:05:09.540
So, I'll do up here.

00:05:09.540 --> 00:05:14.190
So, 0.2 over 1.2 to the fourth minus one.

00:05:14.190 --> 00:05:18.140
And then here, 0.2 over

00:05:18.140 --> 00:05:21.410
1.2 to the fourth minus one.

00:05:21.410 --> 00:05:23.300
That cancels with that,
that cancels with that,

00:05:23.300 --> 00:05:24.760
that's exactly why I did that.

00:05:24.760 --> 00:05:28.380
And we're left with a is equal to,

00:05:28.380 --> 00:05:31.120
it is equal to, I'll
just write it in yellow,

00:05:31.120 --> 00:05:34.760
27 times 0.2,

00:05:34.760 --> 00:05:38.910
all of that over 1.2 to
the fourth minus one.

00:05:38.910 --> 00:05:40.520
And this expression should give us

00:05:40.520 --> 00:05:43.600
the exact same value as
that expression we just saw,

00:05:43.600 --> 00:05:45.730
but this is useful even
if we had a hundred terms,

00:05:45.730 --> 00:05:46.930
we could use this.

00:05:46.930 --> 00:05:49.380
And so, I'll get the calculator out.

00:05:49.380 --> 00:05:50.890
This will give us...

00:05:52.150 --> 00:05:55.370
So, actually, I'll evaluate
this denominator first,

00:05:55.370 --> 00:05:59.370
so I'll have 1.2 to the fourth power,

00:05:59.370 --> 00:06:02.650
which is equal to minus one, is equal to,

00:06:02.650 --> 00:06:03.920
and that's in the denominator,

00:06:03.920 --> 00:06:06.260
so I could just take the reciprocal of it,

00:06:06.260 --> 00:06:08.970
and then multiply that, times 27

00:06:09.840 --> 00:06:12.610
times 0.2

00:06:12.610 --> 00:06:15.070
is equal to 5.029.

00:06:15.070 --> 00:06:16.700
Now, they want us to round our answer

00:06:16.700 --> 00:06:17.950
to the nearest kilometer,

00:06:17.950 --> 00:06:21.810
so this is going to be
approximately equal to,

00:06:21.810 --> 00:06:23.950
approximately equal to five kilometers.

00:06:23.950 --> 00:06:27.440
That's how much approximately
that she would have traveled

00:06:27.440 --> 00:06:29.753
on the first day of her hiking trip.


Sign of average rate of change of polynomial

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.710
- [Instructor] So we are
given this function h of x,

00:00:04.710 --> 00:00:07.530
and we're asked, over which interval

00:00:07.530 --> 00:00:12.140
does h have a positive
average rate of change?

00:00:12.140 --> 00:00:16.280
So like always, pause this
video and have a go at it

00:00:16.280 --> 00:00:19.180
before we do this together.

00:00:19.180 --> 00:00:21.560
All right, now let's work
through this together.

00:00:21.560 --> 00:00:23.680
And to start, let's just remind ourselves

00:00:23.680 --> 00:00:26.830
what an average rate of change even is.

00:00:26.830 --> 00:00:30.980
You can view it as, the change
in your value of the function

00:00:30.980 --> 00:00:34.970
for a given change in
the underlying variable

00:00:34.970 --> 00:00:36.400
for a given change in x.

00:00:36.400 --> 00:00:38.760
We could also view this as,

00:00:38.760 --> 00:00:40.290
if we wanna figure out the interval,

00:00:40.290 --> 00:00:43.700
we could say our x final
minus our x initial,

00:00:43.700 --> 00:00:47.230
and in the numerator, it would
be the value of our function

00:00:47.230 --> 00:00:51.000
at the x final, minus
the value of the function

00:00:51.000 --> 00:00:52.700
at our x initial.

00:00:52.700 --> 00:00:55.210
Now they aren't asking
us to calculate this

00:00:55.210 --> 00:00:57.320
for all of these different intervals.

00:00:57.320 --> 00:01:00.230
They're just asking us
whether it is positive.

00:01:00.230 --> 00:01:01.810
And if you look over here,

00:01:01.810 --> 00:01:05.530
as long as our x final is
greater than x initial,

00:01:05.530 --> 00:01:08.360
in order to have a positive
average rate of change,

00:01:08.360 --> 00:01:12.670
we just need to figure
out whether h at x final

00:01:12.670 --> 00:01:17.450
is greater than h at x initial.

00:01:17.450 --> 00:01:20.410
If the value of the function
at the higher endpoint

00:01:20.410 --> 00:01:22.870
is larger than the value of the function

00:01:22.870 --> 00:01:23.720
at the lower endpoint,

00:01:23.720 --> 00:01:26.930
then we have a positive
average rate of change.

00:01:26.930 --> 00:01:31.050
So let's see if that's happening
for any of these choices.

00:01:31.050 --> 00:01:34.260
So let's see, h of zero,

00:01:34.260 --> 00:01:38.450
this endpoint, is going
to be equal to zero.

00:01:40.600 --> 00:01:43.100
If I just say 1/8 times zero minus zero,

00:01:43.100 --> 00:01:46.411
and h of two is equal to 1/8

00:01:46.411 --> 00:01:49.936
times two to the third power is eight.

00:01:49.936 --> 00:01:54.010
So 1/8 times eight is one minus four.

00:01:54.010 --> 00:01:56.870
So that's going to be,
this is negative three.

00:01:56.870 --> 00:01:59.520
And so we don't have a situation

00:01:59.520 --> 00:02:04.040
where h at our higher
endpoint is actually larger.

00:02:04.040 --> 00:02:06.210
This is a negative average rate of change.

00:02:06.210 --> 00:02:08.010
So I'll rule this one out.

00:02:08.010 --> 00:02:10.270
And actually just to
help us visualize this,

00:02:10.270 --> 00:02:12.860
I did go to Desmos and
graph this function.

00:02:12.860 --> 00:02:14.490
And we can visually see that we have

00:02:14.490 --> 00:02:16.300
a negative average rate of change

00:02:16.300 --> 00:02:18.940
from x equals zero to x equals two.

00:02:18.940 --> 00:02:21.290
At x equals zero, this
is where our function is,

00:02:21.290 --> 00:02:24.150
at x equals two, this is
where our function is,

00:02:24.150 --> 00:02:26.740
and so you can see, at x equals two,

00:02:26.740 --> 00:02:28.280
our function has a lower value.

00:02:28.280 --> 00:02:30.520
You could also think of
the average rate of change

00:02:30.520 --> 00:02:32.630
as the slope of the line

00:02:32.630 --> 00:02:35.460
that connects the two
endpoints on the function.

00:02:35.460 --> 00:02:37.500
And so you can see it
has a negative slope,

00:02:37.500 --> 00:02:39.330
so we have a negative
average rate of change

00:02:39.330 --> 00:02:41.060
between those two points.

00:02:41.060 --> 00:02:42.800
Now what about between these two?

00:02:42.800 --> 00:02:45.740
So h of zero, we already
calculated as zero,

00:02:45.740 --> 00:02:48.480
and what is h of eight?

00:02:48.480 --> 00:02:50.060
Well, let's see that's 1/8

00:02:50.060 --> 00:02:52.610
times eight to the third power.

00:02:52.610 --> 00:02:55.410
Well, if I do eight to third power,

00:02:55.410 --> 00:02:57.160
but then divide by eight,
that's the same thing

00:02:57.160 --> 00:02:58.380
as eight to the second power.

00:02:58.380 --> 00:03:00.130
So that's going to be 64

00:03:00.130 --> 00:03:02.840
minus eight to the second power minus 64,

00:03:02.840 --> 00:03:04.400
so that's equal to zero.

00:03:04.400 --> 00:03:07.280
So here we have a zero
average rate of change

00:03:07.280 --> 00:03:08.880
'cause this numerator's going to be zero,

00:03:08.880 --> 00:03:09.840
so we can rule that out.

00:03:09.840 --> 00:03:11.520
And you see it right over here.

00:03:11.520 --> 00:03:14.210
When x is equal to zero,
our function is there,

00:03:14.210 --> 00:03:16.220
when x is equal to eight,
our function is there,

00:03:16.220 --> 00:03:19.060
and you can see that the slope of the line

00:03:19.060 --> 00:03:21.640
that connects those two points is zero.

00:03:21.640 --> 00:03:23.850
So you have zero average rate of change

00:03:23.850 --> 00:03:25.690
between those two points.

00:03:25.690 --> 00:03:27.410
Now what about choice C?

00:03:27.410 --> 00:03:30.150
So let's see, h of six

00:03:30.150 --> 00:03:32.740
is going to be equal to

00:03:32.740 --> 00:03:36.450
1/8 times six to the third power.

00:03:36.450 --> 00:03:39.890
So let's see, 36 times six is 180

00:03:39.890 --> 00:03:43.203
plus 36, so that is going to be 216.

00:03:44.990 --> 00:03:48.740
216 minus 36,

00:03:48.740 --> 00:03:52.480
216 is six times six times six,

00:03:52.480 --> 00:03:55.130
and then if we divide that by eight,

00:03:55.130 --> 00:03:56.880
that is going to be the same thing as,

00:03:56.880 --> 00:04:01.830
this is 36, and then we have 6/8 of 36.

00:04:01.830 --> 00:04:03.850
So this is going to simplify

00:04:03.850 --> 00:04:08.490
to 3/4 times 36 minus 36,

00:04:08.490 --> 00:04:10.920
which is going to be
equal to negative nine.

00:04:10.920 --> 00:04:12.260
You could have done it with a calculator

00:04:12.260 --> 00:04:13.640
or done some long division,

00:04:13.640 --> 00:04:16.240
but hopefully what I just
did makes some sense.

00:04:16.240 --> 00:04:17.530
It's a little bit of arithmetic.

00:04:17.530 --> 00:04:22.020
And so, at h of six, we have
our function is negative nine,

00:04:22.020 --> 00:04:24.410
and then h of eight, I'll draw a line here

00:04:24.410 --> 00:04:25.890
so we don't make it too messy,

00:04:25.890 --> 00:04:28.840
h of eight we already
know is equal to zero.

00:04:28.840 --> 00:04:31.410
So our function at this endpoint is higher

00:04:31.410 --> 00:04:33.690
than the value of our
function at this endpoint.

00:04:33.690 --> 00:04:36.290
So we do have a positive
average rate of change.

00:04:36.290 --> 00:04:38.180
So I would pick that
choice right over there.

00:04:38.180 --> 00:04:39.820
And you could see it visually.

00:04:39.820 --> 00:04:43.000
h of six, when x is equal to six,

00:04:43.000 --> 00:04:45.430
our value of our function
is negative nine.

00:04:45.430 --> 00:04:47.160
And when x is equal to eight,

00:04:47.160 --> 00:04:48.920
our value of our function is zero.

00:04:48.920 --> 00:04:52.770
And so, the line that
connects those two points

00:04:52.770 --> 00:04:54.810
definitely has a positive slope.

00:04:54.810 --> 00:04:57.060
So we have a positive
average rate of change

00:04:57.060 --> 00:04:58.610
over that interval.

00:04:58.610 --> 00:05:00.300
Now, if we were just
doing this on our own,

00:05:00.300 --> 00:05:01.720
we'd be done, but we could just

00:05:01.720 --> 00:05:03.440
check this one right over here.

00:05:03.440 --> 00:05:08.060
If we compare h of zero,
we already know is zero,

00:05:08.060 --> 00:05:11.670
and h of six, we already know
is equal to negative nine.

00:05:11.670 --> 00:05:13.830
So this is a negative
average rate of change

00:05:13.830 --> 00:05:16.740
because at the higher
endpoint right over here,

00:05:16.740 --> 00:05:18.860
we have a lower value of our function,

00:05:18.860 --> 00:05:19.760
so we'd rule this out.

00:05:19.760 --> 00:05:20.660
And you see it right over here,

00:05:20.660 --> 00:05:23.630
if you go from x equals
zero where the function is

00:05:23.630 --> 00:05:25.970
to x equals six where the function is,

00:05:25.970 --> 00:05:27.640
it looks something,

00:05:27.640 --> 00:05:30.910
it looks something like that.

00:05:30.910 --> 00:05:32.810
Clearly that line has a negative slope.

00:05:32.810 --> 00:05:35.713
So we have a negative
average rate of change.


Multiplying and dividing by 10, 100, 1000

WEBVTT
Kind: captions
Language: en

00:00:00.170 --> 00:00:01.860
- [Instructor] In this video,
we're gonna think about

00:00:01.860 --> 00:00:03.410
what happens when we multiply

00:00:03.410 --> 00:00:07.443
or divide by 10, 100, or 1,000.

00:00:08.890 --> 00:00:10.410
Let's just start with an example.

00:00:10.410 --> 00:00:15.410
Let's say we wanna figure
out what 237 times 10 is.

00:00:16.470 --> 00:00:19.840
Pause this video and see
if you can have a go at it.

00:00:19.840 --> 00:00:21.470
All right, well, one way to think about it

00:00:21.470 --> 00:00:26.450
is this is 237 10s, so if
I put my place values here,

00:00:26.450 --> 00:00:31.450
so this is the 1,000s place,
this is the 100s place,

00:00:31.900 --> 00:00:34.410
I'm just not writing it out
in words just to save space,

00:00:34.410 --> 00:00:36.680
sometimes you'll see it written
out as the word thousands,

00:00:36.680 --> 00:00:40.930
and hundreds, this is the 10s place,

00:00:40.930 --> 00:00:44.050
and then this is the 1s place.

00:00:44.050 --> 00:00:45.620
Well, if I were to just say seven 10s,

00:00:45.620 --> 00:00:47.660
I would put the seven over there.

00:00:47.660 --> 00:00:51.950
If I were to say 37 10s, I
would write it like this,

00:00:51.950 --> 00:00:55.520
because 30 10s is three 100s,

00:00:55.520 --> 00:00:58.360
and if I were to say 237 10s,

00:00:58.360 --> 00:00:59.880
I would write it just like that.

00:00:59.880 --> 00:01:04.880
Because 200 10s is two 1,000s.

00:01:04.920 --> 00:01:07.080
And so what just happened?

00:01:07.080 --> 00:01:11.410
Well, all of my digits
shifted one place to the left.

00:01:11.410 --> 00:01:14.830
So what used to be in the 1s
place is now in the 10s place.

00:01:14.830 --> 00:01:18.010
What used to be in the 10s
place is now in the 100s place,

00:01:18.010 --> 00:01:20.323
and what used to be in the 100s place

00:01:20.323 --> 00:01:23.530
is now in the 1,000s place,
but what's in the 1s place?

00:01:23.530 --> 00:01:25.690
We have to put something in the 1s place.

00:01:25.690 --> 00:01:28.670
Well, we no longer have
any 1s in this product,

00:01:28.670 --> 00:01:31.010
so we would just put a zero there.

00:01:31.010 --> 00:01:34.953
So 237 times 10 is 2,370.

00:01:37.780 --> 00:01:38.960
Now, one way to think about it is,

00:01:38.960 --> 00:01:41.580
when I multiplied by 10,
I really just shifted,

00:01:41.580 --> 00:01:43.950
I put a zero right over here
at the end of this number,

00:01:43.950 --> 00:01:46.220
but it's important to realize
that what really happened

00:01:46.220 --> 00:01:49.690
is I shifted all my digits
one place to the left.

00:01:49.690 --> 00:01:51.480
But what if we were to go
in the other direction?

00:01:51.480 --> 00:01:53.620
What if we were to take, I don't know,

00:01:53.620 --> 00:01:57.850
450 and we were to divide it by 10?

00:01:57.850 --> 00:02:00.320
Well, you can imagine if multiplying by 10

00:02:00.320 --> 00:02:02.870
shifts the digits one place to the left,

00:02:02.870 --> 00:02:05.290
then dividing by 10 is
going to shift the digits

00:02:05.290 --> 00:02:09.020
one place to the right, and so we have

00:02:09.020 --> 00:02:12.810
a four in the 100s place, we
have a five in the 10s place,

00:02:12.810 --> 00:02:14.800
and we have nothing in the 1s place.

00:02:14.800 --> 00:02:17.160
And so what used to be in the 100s place

00:02:17.160 --> 00:02:18.960
will now be in the 10s place,

00:02:18.960 --> 00:02:21.770
so I will put that four right over there,

00:02:21.770 --> 00:02:24.250
and then what used to be in the 10s place

00:02:24.250 --> 00:02:26.370
is now going to be in the 1s place,

00:02:26.370 --> 00:02:28.900
so I will put that five over there,

00:02:28.900 --> 00:02:30.690
and we have nothing in the 1s place--

00:02:30.690 --> 00:02:32.390
If we did, and this is a little bit

00:02:32.390 --> 00:02:36.200
out of this video's purpose,
but you could start to think

00:02:36.200 --> 00:02:39.050
what might happen if this was not a zero.

00:02:39.050 --> 00:02:44.050
But we have 450 divided by 10
is 45, and it is indeed 45.

00:02:44.150 --> 00:02:45.420
Now, another way to think about it,

00:02:45.420 --> 00:02:48.230
other than the fact that
all of the digits shifted

00:02:48.230 --> 00:02:51.567
one place to the right is that we removed

00:02:51.567 --> 00:02:54.763
this right zero here
when we divided by 10.

00:02:55.610 --> 00:02:59.590
Now, let's extend that, let's
say that we had 359 times 100.

00:03:03.560 --> 00:03:05.310
Pause this video and see
if you can figure out

00:03:05.310 --> 00:03:07.060
what that is going to be.

00:03:07.060 --> 00:03:09.510
Well, there's two ways that
you could think about it.

00:03:09.510 --> 00:03:14.000
You could just think of it in
terms of this is 359 hundreds,

00:03:14.000 --> 00:03:15.830
so let's put some places here.

00:03:15.830 --> 00:03:18.754
So let's say this is the 10,000s place,

00:03:18.754 --> 00:03:21.204
this is the 1,000s place,
this is the 100s place,

00:03:24.223 --> 00:03:26.003
this is the 10s place,
and this is the 1s place.

00:03:27.890 --> 00:03:29.980
So if you were to say nine 100s,

00:03:29.980 --> 00:03:31.410
you would go right over there,

00:03:31.410 --> 00:03:33.760
if you were to say 59
hundreds, it would be like that

00:03:33.760 --> 00:03:36.590
because 50 hundreds are five 1,000s,

00:03:36.590 --> 00:03:40.237
and if you were to say 359 hundreds,

00:03:40.237 --> 00:03:43.237
well, 300 hundreds is three 10,000s.

00:03:44.677 --> 00:03:46.693
And of course, in this scenario now,

00:03:46.693 --> 00:03:49.230
you would have no 10s and no 1s.

00:03:49.230 --> 00:03:51.650
And so what we see happened is

00:03:51.650 --> 00:03:55.430
all of the digits shifted
two places to the left.

00:03:55.430 --> 00:03:58.970
What was in the 1s place
is now in the 100s place.

00:03:58.970 --> 00:04:01.880
What was in the 10s place
is now in the 1,000s place,

00:04:01.880 --> 00:04:05.130
and what was in the 100s place
is now in the 10,000s place

00:04:05.130 --> 00:04:10.130
and so this product is 35,900.

00:04:10.380 --> 00:04:12.130
Now, another way you could think about it

00:04:12.130 --> 00:04:17.130
is this is the same thing
as 359 times 10 times 10.

00:04:19.320 --> 00:04:22.460
And we already know that
every time you multiply by 10,

00:04:22.460 --> 00:04:25.420
you shift your digits
one place to the left.

00:04:25.420 --> 00:04:27.990
So if you're going to
multiply by 10 twice,

00:04:27.990 --> 00:04:29.880
you're going to shift all your digits

00:04:29.880 --> 00:04:34.260
two places to the left, which
is exactly what happened here.

00:04:34.260 --> 00:04:38.330
So based on that, what do you think--

00:04:38.330 --> 00:04:40.160
or based on everything we've looked at,

00:04:40.160 --> 00:04:44.590
what do you think, if
I were to take 75,000

00:04:46.150 --> 00:04:49.980
and I wanted to divide it by 100,

00:04:49.980 --> 00:04:51.510
what do you think this is going to be?

00:04:51.510 --> 00:04:53.420
Pause the video and see if you

00:04:53.420 --> 00:04:55.840
can work through it on your own.

00:04:55.840 --> 00:04:58.980
All right, well, we've already
seen if you divide by 10,

00:04:58.980 --> 00:05:02.240
you shift the digits
one place to the right,

00:05:02.240 --> 00:05:04.270
and dividing by 100 is
really the same thing

00:05:04.270 --> 00:05:08.270
as dividing by 10 twice, so
we really just need to shift

00:05:08.270 --> 00:05:11.060
all our digits two places to the right.

00:05:11.060 --> 00:05:14.940
So what used to be in the 10,000s place

00:05:14.940 --> 00:05:17.900
is now going to go to the 100s place.

00:05:17.900 --> 00:05:20.300
What used to be in the 1,000s place

00:05:20.300 --> 00:05:24.250
is now going to be in the 10s place,

00:05:24.250 --> 00:05:27.220
and what used to be in the 100s place

00:05:27.220 --> 00:05:29.870
is now going to go into the 1s place.

00:05:29.870 --> 00:05:33.390
And so what we are going to get is,

00:05:33.390 --> 00:05:36.273
instead of seven 10,000s,
we're going to get seven 100s,

00:05:37.520 --> 00:05:42.520
instead of five 1,000s,
we're going to get five 10s,

00:05:42.940 --> 00:05:47.940
and then instead of zero 100s,
we will now have zero 1s.

00:05:48.220 --> 00:05:50.230
So this is equal to 750.

00:05:50.230 --> 00:05:51.510
Another way to think about it,

00:05:51.510 --> 00:05:54.260
and we've talked about it
before, if you're dividing by 100

00:05:54.260 --> 00:05:56.510
and if you have some of
these trailing zeros here,

00:05:56.510 --> 00:05:58.550
well, then, you would remove two of these

00:05:58.550 --> 00:06:01.170
right-trailing zeros,
but it's really important

00:06:01.170 --> 00:06:03.430
to know why that happened.

00:06:03.430 --> 00:06:06.080
Now, you've hopefully seen a pattern,

00:06:06.080 --> 00:06:08.690
so I'll ask you two more questions.

00:06:08.690 --> 00:06:11.440
Let's say that someone were to
walk up to you on the street

00:06:11.440 --> 00:06:15.790
and say, "Hey, you, what
is 164 times 1,000?"

00:06:21.210 --> 00:06:22.577
and they were also to ask you,

00:06:22.577 --> 00:06:27.577
"What is 98,000, I'll
just say 98,000 divi--

00:06:28.957 --> 00:06:33.450
"I'll say 198,000 divided by 1,000?"

00:06:33.450 --> 00:06:35.733
Pause this video and
see if you can do this.

00:06:36.590 --> 00:06:38.330
All right, you've probably
seen the pattern now.

00:06:38.330 --> 00:06:42.870
Just as multiplying by 100
was multiplying by 10 twice,

00:06:42.870 --> 00:06:46.000
multiplying by 1,000 is
multiplying by 10 three times,

00:06:46.000 --> 00:06:50.860
so this is equal to 164,
and I'll color-code this,

00:06:50.860 --> 00:06:55.700
times 10 times 10 times 10,
so what we're going to do

00:06:55.700 --> 00:06:59.990
is shift all the digits
three places to the left.

00:06:59.990 --> 00:07:04.880
So instead of 164, this four
is no longer in the 1s place,

00:07:04.880 --> 00:07:07.530
it is now in the 1,000s place.

00:07:07.530 --> 00:07:11.270
It is now in the 1,000s
place, and that makes sense.

00:07:11.270 --> 00:07:13.560
If you take four 1s and
you multiply it by 1,000,

00:07:13.560 --> 00:07:15.470
you're going to get four 1,000s.

00:07:15.470 --> 00:07:18.390
What was in the 10s place
is now in the 10,000s place,

00:07:18.390 --> 00:07:20.896
and what was in the 100s place is now

00:07:20.896 --> 00:07:23.200
going to be in the 100,000s place,

00:07:23.200 --> 00:07:25.070
and so, in order to represent this number,

00:07:25.070 --> 00:07:27.170
I'll have to put something
in the 100s place,

00:07:27.170 --> 00:07:28.690
we'll have no 100s now,

00:07:28.690 --> 00:07:31.410
I have to put something in the
10s place, I have no 10s now,

00:07:31.410 --> 00:07:33.210
and I have to put
something in the 1s place.

00:07:33.210 --> 00:07:36.900
So it's going to be
164,000, and once again,

00:07:36.900 --> 00:07:38.650
that's consistent with the pattern we saw.

00:07:38.650 --> 00:07:43.080
We've added three zeros to
the right of this number.

00:07:43.080 --> 00:07:45.940
So similarly, what's
going to happen over here?

00:07:45.940 --> 00:07:48.100
Well, we're going to shift all the digits

00:07:48.100 --> 00:07:52.640
three places to the right, so the one,

00:07:52.640 --> 00:07:56.210
it's going to go one, two,
three places to the right,

00:07:56.210 --> 00:07:58.350
so what was in the 100,000s place

00:07:58.350 --> 00:08:00.310
is going to go into the 100s place,

00:08:00.310 --> 00:08:03.030
what was in the 10,000s place

00:08:03.030 --> 00:08:06.170
is going to go one, two,
three places to the right,

00:08:06.170 --> 00:08:08.310
it's going to go into the 10s place,

00:08:08.310 --> 00:08:12.420
and what was in the 1,000s place

00:08:12.420 --> 00:08:15.540
is going to shift one, two,
three digits to the right,

00:08:15.540 --> 00:08:17.180
it's going to be in the 1s place,

00:08:17.180 --> 00:08:18.260
or another way to think about it

00:08:18.260 --> 00:08:20.730
is when we divide by 1,000,
especially when we have

00:08:20.730 --> 00:08:23.800
these three trailing zeros,
those are going to go away.

00:08:23.800 --> 00:08:28.800
So instead of 100,000 here,
this is just going to be 100.

00:08:29.120 --> 00:08:33.243
Instead of 90,000 here, or nine 10,000s,

00:08:33.243 --> 00:08:35.280
this is just going to be nine 10s.

00:08:35.280 --> 00:08:39.790
And instead of eight 1,000s,
this is just going to be eight,

00:08:39.790 --> 00:08:41.903
so it's equal to 198.


Multiplication as repeated addition

WEBVTT
Kind: captions
Language: en

00:00:00.210 --> 00:00:01.640
- [Instructor] So as
some of you already know,

00:00:01.640 --> 00:00:05.120
I really enjoy eating a good avocado,

00:00:05.120 --> 00:00:06.480
which, despite its appearance

00:00:06.480 --> 00:00:08.010
that it looks like a vegetable,

00:00:08.010 --> 00:00:09.930
but it's actually a fruit.

00:00:09.930 --> 00:00:14.000
And let's say that I eat
two avocados per day,

00:00:14.000 --> 00:00:18.800
and I eat two avocados
per day for six days.

00:00:18.800 --> 00:00:20.840
Now, there's a couple of
ways that I could think

00:00:20.840 --> 00:00:23.060
about how many avocados did I eat?

00:00:23.060 --> 00:00:25.450
I could say, hey, I eat two a day,

00:00:25.450 --> 00:00:27.550
and I'm going to do that for six days,

00:00:27.550 --> 00:00:29.940
so I'm gonna add six twos together.

00:00:29.940 --> 00:00:34.940
So it'll be two, plus two,
plus two, plus two, plus two,

00:00:35.780 --> 00:00:38.590
plus two, I have six
twos right over there.

00:00:38.590 --> 00:00:40.930
And then I can add them together.

00:00:40.930 --> 00:00:42.800
And we could say two plus two is four.

00:00:42.800 --> 00:00:44.290
You add another two, you get to six.

00:00:44.290 --> 00:00:45.520
You add another two, you get to eight.

00:00:45.520 --> 00:00:46.980
Yet another two to get to 10.

00:00:46.980 --> 00:00:49.690
Yet another two, you get to 12.

00:00:49.690 --> 00:00:53.970
And that all is fine,
but there's an easier way

00:00:53.970 --> 00:00:56.490
to express this repeated addition.

00:00:56.490 --> 00:01:00.570
One way is to view it as multiplication.

00:01:00.570 --> 00:01:03.360
Instead of just writing out six twos

00:01:03.360 --> 00:01:06.560
and adding them together,
mathematicians have come up

00:01:06.560 --> 00:01:08.780
with a neater way of writing that.

00:01:08.780 --> 00:01:11.490
Let's say, okay, we're going
to add up a bunch of twos.

00:01:11.490 --> 00:01:13.490
How many twos are we going to add up?

00:01:13.490 --> 00:01:15.540
We're going to have six of those twos,

00:01:15.540 --> 00:01:17.950
and we need to come up with
some type of a symbol for it.

00:01:17.950 --> 00:01:20.940
So we will use this X-looking thing.

00:01:20.940 --> 00:01:24.100
And so, six times two can be viewed

00:01:24.100 --> 00:01:27.770
as repeated addition in
exactly this same way.

00:01:27.770 --> 00:01:32.150
So six times two would be equal to 12.

00:01:32.150 --> 00:01:33.960
And we could go the other way around.

00:01:33.960 --> 00:01:38.960
If someone were to ask you,
what is four times three?

00:01:39.870 --> 00:01:42.600
Pause this video and see
if you can write it out

00:01:42.600 --> 00:01:46.790
as repeated addition, like we saw up here.

00:01:46.790 --> 00:01:48.890
Well, one way to interpret this is to say,

00:01:48.890 --> 00:01:51.660
this is four threes, so
we could say this is equal

00:01:51.660 --> 00:01:56.530
to three, plus three,
plus three, plus three.

00:01:56.530 --> 00:01:59.520
And three plus three is
six, six plus three is nine,

00:01:59.520 --> 00:02:02.280
nine plus three is equal to 12.

00:02:02.280 --> 00:02:03.770
You might be familiar with skip counting,

00:02:03.770 --> 00:02:07.240
and you would say three, six, nine, 12.

00:02:07.240 --> 00:02:08.710
Just out of curiosity,

00:02:08.710 --> 00:02:13.380
what do you think three
times four is going to be?

00:02:13.380 --> 00:02:15.800
Pause this video and try to represent it

00:02:15.800 --> 00:02:18.753
as repeated addition, and then
see what you come up with.

00:02:19.730 --> 00:02:23.470
Well, we can interpret
this as three fours.

00:02:23.470 --> 00:02:25.160
And so, we could say this is going

00:02:25.160 --> 00:02:28.520
to be four, plus four, plus four.

00:02:28.520 --> 00:02:32.669
And if we skip count fours,
we'd have four, eight, 12,

00:02:32.669 --> 00:02:35.570
(chuckles) I was about to
go to 16, four, eight, 12.

00:02:35.570 --> 00:02:37.440
So this is going to be 12.

00:02:37.440 --> 00:02:41.140
So this is interesting,
at least for this example,

00:02:41.140 --> 00:02:43.690
for these two examples,
I got to the same thing.

00:02:43.690 --> 00:02:46.620
Four times three got me the same result

00:02:46.620 --> 00:02:49.390
as three times four, interesting.

00:02:49.390 --> 00:02:51.470
I wonder if that's always true.

00:02:51.470 --> 00:02:54.780
But anyway, big picture from this video is

00:02:54.780 --> 00:02:59.780
that you can view multiplication
as repeated addition.


Multiplying and dividing by powers of 10

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:01.413
- [Instructor] In another video,

00:00:01.413 --> 00:00:03.460
we introduce ourselves to the idea

00:00:03.460 --> 00:00:05.450
of powers of 10.

00:00:05.450 --> 00:00:09.100
We saw that if I were to
say 10 to the first power,

00:00:09.100 --> 00:00:12.370
that means that we are just
really going to take one 10.

00:00:12.370 --> 00:00:15.340
If we have 10 to the second power,

00:00:15.340 --> 00:00:17.850
that means that we're gonna take two 10s,

00:00:17.850 --> 00:00:19.600
so a 10 and a 10,

00:00:19.600 --> 00:00:21.200
and we're gonna multiply them

00:00:21.200 --> 00:00:23.880
and so that's going to be 100.

00:00:23.880 --> 00:00:26.640
If we take 10 to the third power,

00:00:26.640 --> 00:00:31.210
that's going to be three
10s multiplied together,

00:00:31.210 --> 00:00:33.410
which is equal to 1000,

00:00:33.410 --> 00:00:35.760
which is also one
followed by three zeroes.

00:00:35.760 --> 00:00:37.580
So you're already
starting to see a pattern.

00:00:37.580 --> 00:00:38.890
What we're going to do in this video

00:00:38.890 --> 00:00:42.480
is think about patterns when
we multiply arbitrary things

00:00:42.480 --> 00:00:44.180
or divide arbitrary things

00:00:44.180 --> 00:00:46.660
by powers of 10.

00:00:46.660 --> 00:00:49.850
So let's start with a number.

00:00:49.850 --> 00:00:53.960
Let's say I will start with 2.3.

00:00:55.720 --> 00:01:00.720
And let's first just multiply
by 10 to the first power.

00:01:00.740 --> 00:01:03.790
Well, that's the same thing
as just multiplying it by 10,

00:01:03.790 --> 00:01:07.100
and we've seen already
when you multiply by 10,

00:01:07.100 --> 00:01:10.110
you shift all the digits
one place to the left.

00:01:10.110 --> 00:01:11.660
So the two, which is the ones place,

00:01:11.660 --> 00:01:13.250
one's up in the 10s place,

00:01:13.250 --> 00:01:16.020
and then three, which
is in the tenths place

00:01:16.020 --> 00:01:17.990
will end up in the ones place.

00:01:17.990 --> 00:01:21.910
So this is just going to be equal to 23.

00:01:21.910 --> 00:01:24.360
And it's always good to do a
little bit of a reality check.

00:01:24.360 --> 00:01:26.790
If I just had two and if
were to multiply it by 10,

00:01:26.790 --> 00:01:28.270
you'd say okay, that's about 20,

00:01:28.270 --> 00:01:32.330
so it makes sense that 2.3 times 10 is 23.

00:01:32.330 --> 00:01:33.560
But let's keep going.

00:01:33.560 --> 00:01:37.880
Now let's multiply 2.3 not
by 10 to the first power,

00:01:37.880 --> 00:01:39.000
which is just 10,

00:01:39.000 --> 00:01:42.730
but let's multiply it times
10 to the second power.

00:01:42.730 --> 00:01:44.150
What is that going to be?

00:01:44.150 --> 00:01:46.933
Pause this video and see
if you can figure that out.

00:01:48.130 --> 00:01:50.450
All right, well 10 to the second power,

00:01:50.450 --> 00:01:53.230
we already know that's equal to 100,

00:01:53.230 --> 00:01:57.210
and so when you multiply by 100
or you multiply by 10 twice,

00:01:57.210 --> 00:02:00.910
you're just going to shift
every digit two places

00:02:00.910 --> 00:02:01.990
to the left.

00:02:01.990 --> 00:02:04.470
So let me draw some places here.

00:02:04.470 --> 00:02:06.840
So the thing that is in
the ones place will go

00:02:06.840 --> 00:02:08.280
to the hundreds place,

00:02:08.280 --> 00:02:10.590
and the thing that is in the tenths place

00:02:10.590 --> 00:02:13.550
will go to the 10s place.

00:02:13.550 --> 00:02:18.513
And so this two will now be two hundreds.

00:02:19.580 --> 00:02:23.450
This 3/10 will now be three 10s

00:02:23.450 --> 00:02:26.820
and we now have zero ones.

00:02:26.820 --> 00:02:30.490
And then if we were to
multiply by 10 again,

00:02:30.490 --> 00:02:33.890
so if we were to say 2.3

00:02:33.890 --> 00:02:37.300
times 10 to the third power,

00:02:37.300 --> 00:02:40.160
well then we're going to
shift everything three places

00:02:40.160 --> 00:02:41.220
to the left.

00:02:41.220 --> 00:02:43.030
10 to the third power.

00:02:43.030 --> 00:02:45.963
This is the same thing
as multiplying by 1000.

00:02:46.910 --> 00:02:49.320
So 2.3 times 10 to the third,

00:02:49.320 --> 00:02:52.410
the two is going to be shifted
three places to the left,

00:02:52.410 --> 00:02:55.900
so the two is going to become
2000, which makes sense.

00:02:55.900 --> 00:02:57.620
So it's going to be 2000.

00:02:57.620 --> 00:03:02.030
The 3/10 is going to shift
two places to the left,

00:03:02.030 --> 00:03:03.970
so it's going to be 300,

00:03:03.970 --> 00:03:08.970
and then we now have
zero 10s and zero ones.

00:03:09.660 --> 00:03:11.440
So the pattern that you've probably seen

00:03:11.440 --> 00:03:16.430
is if you multiply a number
times 10 to some power,

00:03:16.430 --> 00:03:21.050
you are just shifting the digits
to the left by that power.

00:03:21.050 --> 00:03:24.120
And if we divide by a power of 10,

00:03:24.120 --> 00:03:25.290
the same thing would happen,

00:03:25.290 --> 00:03:27.610
but we would now be
shifting our digits' places

00:03:27.610 --> 00:03:28.600
to the right.

00:03:28.600 --> 00:03:33.600
So for example, what is 2.3 divided by,

00:03:33.830 --> 00:03:36.560
divided by 10 to the first power?

00:03:36.560 --> 00:03:39.130
Pause this video and
try to figure that out.

00:03:39.130 --> 00:03:41.610
Well 10 to the first power
is the same thing as 10,

00:03:41.610 --> 00:03:43.450
so when we divide by 10,

00:03:43.450 --> 00:03:46.360
all of our digits are just
going to shift one place

00:03:46.360 --> 00:03:47.193
to the right,

00:03:47.193 --> 00:03:49.890
so this two is going to
end up in the tenths place

00:03:49.890 --> 00:03:53.100
and the three is going to end
up in the hundredths place.

00:03:53.100 --> 00:03:55.523
So this is going to give us 0.23.

00:03:58.230 --> 00:03:59.870
Two is now in the tenths,

00:03:59.870 --> 00:04:01.990
three is now in the hundredths,

00:04:01.990 --> 00:04:02.930
but we could keep going.

00:04:02.930 --> 00:04:07.930
What if we were to say 2.3 divided by 10

00:04:08.220 --> 00:04:09.440
to the second power?

00:04:09.440 --> 00:04:12.300
Pause this video, try to
figure out what that is.

00:04:12.300 --> 00:04:13.900
Well in this situation,

00:04:13.900 --> 00:04:17.570
we are going to shift all the
digits two places to the right

00:04:17.570 --> 00:04:19.020
and so let me put my places here,

00:04:19.020 --> 00:04:20.320
so that's the ones,

00:04:20.320 --> 00:04:23.910
tenths, hundredths, thousandths.

00:04:23.910 --> 00:04:26.630
And so the thing that's
in the ones place is two.

00:04:26.630 --> 00:04:27.890
It won't just go to the tenths.

00:04:27.890 --> 00:04:29.870
It'll go to the hundredths place,

00:04:29.870 --> 00:04:31.280
which you don't quite see here.

00:04:31.280 --> 00:04:32.520
It's right over there.

00:04:32.520 --> 00:04:36.030
So the two is going to show up here.

00:04:36.030 --> 00:04:38.890
And then the three is going to shift

00:04:38.890 --> 00:04:39.907
two places to the right,

00:04:39.907 --> 00:04:41.970
and so it's going to end up there,

00:04:41.970 --> 00:04:44.160
and we have zero ones

00:04:44.160 --> 00:04:46.260
and we have zero 10s.

00:04:46.260 --> 00:04:48.970
So you're probably already
seeing the pattern here again.

00:04:48.970 --> 00:04:50.320
Whatever the exponent is,

00:04:50.320 --> 00:04:52.357
if you're dividing by 10 to that power,

00:04:52.357 --> 00:04:55.120
you're going to shift that power,

00:04:55.120 --> 00:04:56.870
the exponent that many times

00:04:56.870 --> 00:04:59.790
you're going to shift the
digit that many places

00:04:59.790 --> 00:05:00.743
to the right.


Introduction to powers of 10

WEBVTT
Kind: captions
Language: en

00:00:00.220 --> 00:00:01.053
- [Instructor] In this video,

00:00:01.053 --> 00:00:02.960
I'm going to introduce you to a new type

00:00:02.960 --> 00:00:05.170
of mathematical notation

00:00:05.170 --> 00:00:07.380
that will seem fancy at first,

00:00:07.380 --> 00:00:09.580
but hopefully you'll
appreciate is pretty useful

00:00:09.580 --> 00:00:12.000
and also pretty straightforward.

00:00:12.000 --> 00:00:15.090
So let's just start with some
things that we already know.

00:00:15.090 --> 00:00:17.130
So I could have just a 10.

00:00:17.130 --> 00:00:20.320
I could take two 10s and
multiply them together,

00:00:20.320 --> 00:00:24.380
so 10 times 10, which
you know is equal to 100.

00:00:24.380 --> 00:00:27.050
I could take three 10s and
multiply them together.

00:00:27.050 --> 00:00:31.740
10 times 10 times 10 is equal to 1000.

00:00:32.880 --> 00:00:36.580
And I could do this
with any number of 10s.

00:00:36.580 --> 00:00:39.340
But at some point, if I'm
doing this with enough 10s,

00:00:39.340 --> 00:00:40.990
it gets pretty hard to write.

00:00:40.990 --> 00:00:45.660
So for example, let's say I
were to do this with 10 10s.

00:00:45.660 --> 00:00:48.510
So if I were to go 10 times 10

00:00:48.510 --> 00:00:53.510
times 10 times 10,
that's four, that's five,

00:00:54.030 --> 00:00:57.590
that's six, that's seven, that's eight,

00:00:57.590 --> 00:00:59.910
that's nine, that is 10 10s.

00:00:59.910 --> 00:01:01.880
Let's see, one, two, three, four,

00:01:01.880 --> 00:01:04.630
five, six, seven, eight, nine, 10.

00:01:04.630 --> 00:01:06.030
This is going to be equal to,

00:01:06.030 --> 00:01:07.930
even the number that it's equal to

00:01:07.930 --> 00:01:10.170
is going to be quite hard to write.

00:01:10.170 --> 00:01:12.700
It's going to be one
followed by 10 zeroes.

00:01:12.700 --> 00:01:15.020
One, two, three, four, five,

00:01:15.020 --> 00:01:17.710
six, seven, eight, nine, 10.

00:01:17.710 --> 00:01:18.720
We put the commas there

00:01:18.720 --> 00:01:20.880
so it's just a little bit easier to read.

00:01:20.880 --> 00:01:23.470
This right over here is 10 billion,

00:01:23.470 --> 00:01:25.630
and it's already getting
kinda hard to write,

00:01:25.630 --> 00:01:27.970
and imagine if we have 30 10s

00:01:27.970 --> 00:01:30.020
that we were multiplying together.

00:01:30.020 --> 00:01:33.190
So mathematicians have
come up with a notation

00:01:33.190 --> 00:01:36.930
and some ideas to be able
to write things like this

00:01:36.930 --> 00:01:39.770
a little bit more elegantly.

00:01:39.770 --> 00:01:41.390
So the way they do this

00:01:41.390 --> 00:01:45.780
is through something known as exponents.

00:01:45.780 --> 00:01:47.253
Exponents.

00:01:48.230 --> 00:01:50.970
And so 10 times 10,

00:01:50.970 --> 00:01:54.070
we can rewrite as being equal to,

00:01:54.070 --> 00:01:56.830
if I have two 10s and I'm
multiplying them together,

00:01:56.830 --> 00:01:59.000
I could write this as 10

00:01:59.000 --> 00:02:01.950
to the second power.

00:02:01.950 --> 00:02:03.080
That's how someone would say it.

00:02:03.080 --> 00:02:05.450
They would say 10 to the second power.

00:02:05.450 --> 00:02:06.940
That looks fancy,

00:02:06.940 --> 00:02:09.800
but all that means is let's take two 10s

00:02:09.800 --> 00:02:11.800
and multiply them together

00:02:11.800 --> 00:02:14.500
and we're going to get 100.

00:02:14.500 --> 00:02:17.730
Just so you're familiar with
some of the parts of this,

00:02:17.730 --> 00:02:19.920
the two would be called the exponent

00:02:19.920 --> 00:02:22.090
and the 10 would be the base.

00:02:22.090 --> 00:02:25.640
So 10 to the second power is 10 times 10

00:02:25.640 --> 00:02:27.380
is equal to 100.

00:02:27.380 --> 00:02:31.840
So how would you write 10
times 10 times 10 or 1000?

00:02:31.840 --> 00:02:33.700
How would you write that using exponents?

00:02:33.700 --> 00:02:36.770
Pause this video and see
if you can figure that out.

00:02:36.770 --> 00:02:39.140
Well, as exactly as you
might have imagined,

00:02:39.140 --> 00:02:41.400
we're taking a certain number of 10s

00:02:41.400 --> 00:02:42.970
and we see we're taking three 10s

00:02:42.970 --> 00:02:45.110
and we're multiplying them together.

00:02:45.110 --> 00:02:48.020
So this would be 10 to the third power.

00:02:48.020 --> 00:02:50.440
10 is the base, three is the exponent.

00:02:50.440 --> 00:02:53.420
We would read this as
10 to the third power.

00:02:53.420 --> 00:02:54.900
If you ever saw 10 to the third power,

00:02:54.900 --> 00:02:58.110
that means hey, let me
multiply 10 times 10 times 10.

00:02:58.110 --> 00:02:59.540
That's the same thing as 1000.

00:02:59.540 --> 00:03:02.840
So this is really another
way of writing 1000.

00:03:02.840 --> 00:03:05.230
And what about this
number here, 10 billion?

00:03:05.230 --> 00:03:07.700
What's a way that we could
write it using exponents?

00:03:07.700 --> 00:03:09.500
Pause the video and figure that out.

00:03:10.630 --> 00:03:11.953
Well as you might have imagined,

00:03:11.953 --> 00:03:14.800
we were taking 10 10s and
multiplying them together.

00:03:14.800 --> 00:03:19.430
This is 10 to the 10th power.

00:03:19.430 --> 00:03:20.960
And we can go the other way around.

00:03:20.960 --> 00:03:23.170
If someone were to walk
up to you on the street

00:03:23.170 --> 00:03:28.170
and say what is 10 to the fifth power?

00:03:28.610 --> 00:03:29.443
What is that?

00:03:29.443 --> 00:03:31.250
What number that you're probably familiar

00:03:31.250 --> 00:03:32.523
with would this be?

00:03:33.410 --> 00:03:36.020
Well this would mean that
we're going to take five 10s

00:03:36.020 --> 00:03:37.170
and multiply them together.

00:03:37.170 --> 00:03:42.170
So 10 times 10 times 10 times 10 times 10.

00:03:43.410 --> 00:03:45.249
And so 10 times 10 is 100,

00:03:45.249 --> 00:03:46.960
100 times 10 is 1000,

00:03:46.960 --> 00:03:49.550
1000 times 10 is 10000,

00:03:49.550 --> 00:03:53.307
10000 times 10 is equal to 100000.

00:03:55.800 --> 00:03:56.840
So there you have it.

00:03:56.840 --> 00:03:58.920
You have the basics of exponents

00:03:58.920 --> 00:04:00.980
when we're dealing with 10,

00:04:00.980 --> 00:04:02.410
and I know what you were thinking.

00:04:02.410 --> 00:04:05.630
Can I put another number
here instead of 10?

00:04:05.630 --> 00:04:07.840
And the simple answer is, you can,

00:04:07.840 --> 00:04:10.633
but we're not gonna cover
that just yet in this video.


Multiplication on the number line

WEBVTT
Kind: captions
Language: en

00:00:00.210 --> 00:00:01.530
- [Instructor] What we're
going to do in this video

00:00:01.530 --> 00:00:05.290
is think about different ways
to represent multiplication.

00:00:05.290 --> 00:00:07.700
And especially connect it to the notions

00:00:07.700 --> 00:00:10.460
of skip counting and the number line.

00:00:10.460 --> 00:00:15.460
So if we were to think about
what four times two means,

00:00:15.680 --> 00:00:17.820
we've already seen in other videos,

00:00:17.820 --> 00:00:20.200
you could view this as four groups of two.

00:00:20.200 --> 00:00:22.000
So we could have four groups,

00:00:22.000 --> 00:00:24.670
so one group, two groups,

00:00:24.670 --> 00:00:26.810
three groups, and four groups,

00:00:26.810 --> 00:00:29.160
and each of them have two of something.

00:00:29.160 --> 00:00:30.870
I'll just put two little circles here.

00:00:30.870 --> 00:00:32.220
So you have two there,

00:00:32.220 --> 00:00:33.540
you have two there,

00:00:33.540 --> 00:00:35.320
you have two there,

00:00:35.320 --> 00:00:36.770
and you have two there,

00:00:36.770 --> 00:00:38.440
and you could also view that

00:00:38.440 --> 00:00:42.220
as four twos, or four twos added together.

00:00:42.220 --> 00:00:45.260
So we could view it as two plus two

00:00:45.260 --> 00:00:47.530
plus two plus two.

00:00:47.530 --> 00:00:50.510
And this, of course, is going
to be two plus two is four,

00:00:50.510 --> 00:00:53.040
four plus two is six,
six plus two is eight.

00:00:53.040 --> 00:00:54.560
We see that over here,

00:00:54.560 --> 00:00:55.770
we could even skip count.

00:00:55.770 --> 00:00:58.380
Two, four, six, eight.

00:00:58.380 --> 00:01:00.500
Four times two is equal to eight.

00:01:00.500 --> 00:01:03.380
We can also think about
that on a number line.

00:01:03.380 --> 00:01:05.940
So I'm gonna make a little
bit of a number line here.

00:01:05.940 --> 00:01:10.940
And so we could imagine four
times two being, all right.

00:01:10.950 --> 00:01:13.940
So this is one times two,

00:01:13.940 --> 00:01:15.810
two times two,

00:01:15.810 --> 00:01:17.560
three times two,

00:01:17.560 --> 00:01:19.530
and four times two.

00:01:19.530 --> 00:01:21.880
So we started at zero,

00:01:21.880 --> 00:01:26.250
and we took four hops of
two along the number line

00:01:26.250 --> 00:01:27.910
to end up at eight.

00:01:27.910 --> 00:01:32.650
We went from zero to
two, four, six, eight.

00:01:32.650 --> 00:01:36.000
We just skip counted our way to eight.

00:01:36.000 --> 00:01:37.610
So if I were to ask a
similar question, actually,

00:01:37.610 --> 00:01:40.530
let me draw a little series of hops,

00:01:40.530 --> 00:01:41.900
and I want you to think
about it the other way.

00:01:41.900 --> 00:01:46.030
What multiplication does that represent?

00:01:46.030 --> 00:01:48.350
So let's say I start here

00:01:48.350 --> 00:01:50.530
and then I'm going to hop like this.

00:01:50.530 --> 00:01:52.240
So I'm gonna go there

00:01:52.240 --> 00:01:55.020
and then I'm gonna go there.

00:01:55.020 --> 00:01:57.030
I'm taking equal jumps every time.

00:01:57.030 --> 00:01:59.300
Then I'm gonna go there.

00:01:59.300 --> 00:02:02.539
Then I'm gonna go there.

00:02:02.539 --> 00:02:06.250
Then I'm going to go over there.

00:02:06.250 --> 00:02:08.130
So what would that represent,

00:02:08.130 --> 00:02:09.630
if we used the same type of ideas

00:02:09.630 --> 00:02:11.210
that we just thought about?

00:02:11.210 --> 00:02:16.210
Well I went from zero to
four, eight, 12, 16, 20.

00:02:16.380 --> 00:02:18.360
I'm skip counting by four.

00:02:18.360 --> 00:02:22.120
So you could imagine this is
probably something times four.

00:02:22.120 --> 00:02:23.450
Now how many hops did I take?

00:02:23.450 --> 00:02:28.350
I took one, two, three,
four, five hops of four.

00:02:28.350 --> 00:02:30.820
So this is five times four.

00:02:30.820 --> 00:02:34.340
And we can see that we ended up at 20.

00:02:34.340 --> 00:02:37.360
We could also view this
as being the same thing

00:02:37.360 --> 00:02:42.360
as five fours or four plus four plus four

00:02:42.520 --> 00:02:44.920
plus four plus four, and
you see that over here.

00:02:44.920 --> 00:02:48.110
You have, we're starting
at zero, we're adding four,

00:02:48.110 --> 00:02:50.060
then another four, then another four,

00:02:50.060 --> 00:02:51.470
then another four, and another four.

00:02:51.470 --> 00:02:53.690
We have five fours here.

00:02:53.690 --> 00:02:55.490
Let's do one more.

00:02:55.490 --> 00:02:57.010
So I'm gonna have a number line here

00:02:57.010 --> 00:02:59.440
and think about what it would mean

00:02:59.440 --> 00:03:04.440
to say, do something like
seven times three, times three.

00:03:08.650 --> 00:03:12.270
Well, we could view that
as seven hops of three

00:03:12.270 --> 00:03:14.520
starting at zero, seven equal hops.

00:03:14.520 --> 00:03:16.900
So one,

00:03:16.900 --> 00:03:18.290
two,

00:03:18.290 --> 00:03:19.430
three,

00:03:19.430 --> 00:03:21.290
four,

00:03:21.290 --> 00:03:22.990
five,

00:03:22.990 --> 00:03:25.890
six, and seven.

00:03:25.890 --> 00:03:29.500
We end up at 21, so this is equal to 21.

00:03:29.500 --> 00:03:31.910
You could also view this
as we took seven threes

00:03:31.910 --> 00:03:34.210
and added them together.

00:03:34.210 --> 00:03:35.890
And you could also view
it as skip counting.

00:03:35.890 --> 00:03:38.919
You went from zero to
three, six, nine, 12,

00:03:38.919 --> 00:03:41.030
15, 18, and 21.

00:03:41.030 --> 00:03:42.270
Now just out of interest,

00:03:42.270 --> 00:03:43.920
what if we went the other way around?

00:03:43.920 --> 00:03:47.410
What if we were to take
three hops of seven?

00:03:47.410 --> 00:03:48.763
What would that be?

00:03:49.750 --> 00:03:51.210
Well, we would start here

00:03:51.210 --> 00:03:53.080
and so we would take our first hop

00:03:53.080 --> 00:03:55.790
of seven right over there.

00:03:55.790 --> 00:03:58.640
Get to seven, then if we
take another hop of seven,

00:03:58.640 --> 00:03:59.840
we get to 14,

00:03:59.840 --> 00:04:01.940
and then if we take another hop of seven,

00:04:01.940 --> 00:04:05.670
we get to 21, interesting.

00:04:05.670 --> 00:04:07.280
At least for this situation,

00:04:07.280 --> 00:04:09.320
whether we took seven hops of three

00:04:09.320 --> 00:04:12.910
or three hops of seven, we
got to the exact same value.

00:04:12.910 --> 00:04:15.530
I encourage you to think about
whether that's always going

00:04:15.530 --> 00:04:17.000
to be the case.

00:04:17.000 --> 00:04:18.843
I'll see you in a future video.


Introduction to multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:02.010
- [Instructor] Our
squirrel friend here likes

00:00:02.010 --> 00:00:03.960
to collect acorns, because, really,

00:00:03.960 --> 00:00:06.430
that's how he is able to live.

00:00:06.430 --> 00:00:11.430
And let's say everyday, he
collects exactly three acorns.

00:00:11.620 --> 00:00:14.660
And so what I'm curious
about is how many acorns

00:00:14.660 --> 00:00:19.170
will he have after doing
this for five days?

00:00:19.170 --> 00:00:22.400
So one way to think about
it is every day he is able

00:00:22.400 --> 00:00:24.690
to collect a group of three acorns.

00:00:24.690 --> 00:00:27.210
So you could view this
as maybe what he's able

00:00:27.210 --> 00:00:29.250
to collect in day one.

00:00:29.250 --> 00:00:32.310
And then in day two, he's
able to collect a second group

00:00:32.310 --> 00:00:33.610
of three acorns.

00:00:33.610 --> 00:00:36.260
In day three, he's able
to collect another group

00:00:36.260 --> 00:00:37.420
of three acorns.

00:00:37.420 --> 00:00:39.790
And every day it's a
equal number of acorns

00:00:39.790 --> 00:00:40.623
that he's collecting.

00:00:40.623 --> 00:00:42.450
On the fourth day, another three.

00:00:42.450 --> 00:00:44.920
On the fifth day, another three.

00:00:44.920 --> 00:00:47.800
And so if you were curious
how many total acorns

00:00:47.800 --> 00:00:51.630
he's collected, well, you
could just count them up,

00:00:51.630 --> 00:00:53.410
or you could think about,

00:00:53.410 --> 00:00:57.160
well, he's got five
groups of three acorns.

00:00:57.160 --> 00:00:59.510
Five equal groups of three acorns.

00:00:59.510 --> 00:01:02.200
So you could say five groups

00:01:03.540 --> 00:01:08.510
of three acorns, three acorns.

00:01:08.510 --> 00:01:11.800
And so the total amount would be five,

00:01:11.800 --> 00:01:14.283
we could view this as five threes.

00:01:16.070 --> 00:01:18.550
Now, five threes, you could view this

00:01:18.550 --> 00:01:21.500
as five threes added together.

00:01:21.500 --> 00:01:26.020
Three plus three, plus three,
plus three, plus three.

00:01:26.020 --> 00:01:27.370
And if you wanted to calculate this,

00:01:27.370 --> 00:01:29.610
you could skip count by three.

00:01:29.610 --> 00:01:34.270
So this would be three, six, nine, 12, 15,

00:01:35.640 --> 00:01:37.770
because we add three, we get to six,

00:01:37.770 --> 00:01:39.000
we add another three, we get to nine,

00:01:39.000 --> 00:01:40.300
we add another three, we get to 12,

00:01:40.300 --> 00:01:43.140
we add another three, we get to 15.

00:01:43.140 --> 00:01:45.860
And so this would be a way of recognizing

00:01:45.860 --> 00:01:47.740
that you have 15 acorns,

00:01:47.740 --> 00:01:50.570
but we're starting to touch
on one of the most fundamental

00:01:50.570 --> 00:01:52.960
ideas in all of mathematics.

00:01:52.960 --> 00:01:54.390
In fact, we actually are applying it,

00:01:54.390 --> 00:01:55.950
we just haven't used the world,

00:01:55.950 --> 00:01:58.440
and that's, we are multiplying.

00:01:58.440 --> 00:02:01.373
We are using multiplication.

00:02:02.570 --> 00:02:04.280
All multiplication is

00:02:04.280 --> 00:02:09.280
is this notion of multiple
equal groups of something.

00:02:10.150 --> 00:02:13.680
So, here, one way to
express what we just did,

00:02:13.680 --> 00:02:16.230
is we just, when we said five threes,

00:02:16.230 --> 00:02:18.940
that's the same thing as five,

00:02:18.940 --> 00:02:22.420
and I'm going to introduce
a new symbol to you,

00:02:22.420 --> 00:02:25.203
five times three.

00:02:26.140 --> 00:02:28.590
So all of these things are equivalent.

00:02:28.590 --> 00:02:30.860
You're already used to seeing equal groups

00:02:30.860 --> 00:02:32.400
and multiple equal groups,

00:02:32.400 --> 00:02:35.590
and you're used to adding
something multiple times,

00:02:35.590 --> 00:02:37.140
and you're used to skip counting,

00:02:37.140 --> 00:02:40.240
and all of that is, in
some way shape or form,

00:02:40.240 --> 00:02:42.920
you have been doing multiplication.

00:02:42.920 --> 00:02:45.520
So when someone says five times three,

00:02:45.520 --> 00:02:49.130
you could view that is
five groups of three,

00:02:49.130 --> 00:02:50.790
or you could view that as five threes,

00:02:50.790 --> 00:02:51.990
or you could view that
as three plus three,

00:02:51.990 --> 00:02:54.070
plus three, plus three,

00:02:54.070 --> 00:02:57.910
or you could view that as 15.

00:02:57.910 --> 00:02:59.280
I'll leave ya there.

00:02:59.280 --> 00:03:01.020
There's a lot of practice on Khan Academy

00:03:01.020 --> 00:03:01.853
to work through this

00:03:01.853 --> 00:03:04.830
and make sure you get the underlying idea.

00:03:04.830 --> 00:03:07.880
But as you'll see, this is
perhaps one of the most useful

00:03:07.880 --> 00:03:11.563
concepts that you might
learn in your entire lives.


Multiplying and dividing decimals by 10

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:01.790
- [Instructor] We've already learned

00:00:01.790 --> 00:00:03.600
that when we multiply by 10,

00:00:03.600 --> 00:00:05.560
let's say we took the number 53

00:00:05.560 --> 00:00:08.290
and we were to multiply it by 10,

00:00:08.290 --> 00:00:11.040
it has the effect of
shifting all the digits

00:00:11.040 --> 00:00:12.800
once place to the left.

00:00:12.800 --> 00:00:14.760
So, this should be a review for you.

00:00:14.760 --> 00:00:17.493
But this was going to be 530.

00:00:18.890 --> 00:00:22.680
And we could see that what
used to be in the tens place

00:00:22.680 --> 00:00:26.370
has been shifted to the
left, to the hundreds place.

00:00:26.370 --> 00:00:27.960
And what used to be in the ones place

00:00:27.960 --> 00:00:31.520
has been shifted to the
left to the tens place now.

00:00:31.520 --> 00:00:33.580
And we saw, if you divide by 10,

00:00:33.580 --> 00:00:35.220
you have the opposite effect.

00:00:35.220 --> 00:00:38.930
And so, let's say if I had 120,

00:00:38.930 --> 00:00:41.140
I could say, let's divide by 10,

00:00:41.140 --> 00:00:46.140
I could also say this is the
same thing as 120 times 1/10.

00:00:46.220 --> 00:00:48.120
What is going to happen there?

00:00:48.120 --> 00:00:49.700
Well, in this situation,

00:00:49.700 --> 00:00:53.870
all the digits are going to
shift one place to the right.

00:00:53.870 --> 00:00:55.700
So, what used to be in the tens place

00:00:55.700 --> 00:00:57.900
will now be in the ones place;

00:00:57.900 --> 00:00:59.400
what used to be in the hundreds place

00:00:59.400 --> 00:01:01.030
will now be in the tens place.

00:01:01.030 --> 00:01:03.560
So, this is just going to be equal to 12.

00:01:03.560 --> 00:01:04.850
So, that was all review.

00:01:04.850 --> 00:01:06.530
But now were going to
extend this a little bit

00:01:06.530 --> 00:01:10.140
by thinking about things
that have place values

00:01:10.140 --> 00:01:12.530
representing less than a
one, I guess you could say,

00:01:12.530 --> 00:01:14.660
or we're gonna deal with decimals.

00:01:14.660 --> 00:01:17.700
So, just to get ourselves warmed up,

00:01:17.700 --> 00:01:20.363
let's see if we could
figure out what 3.015

00:01:22.870 --> 00:01:25.690
times 10 is.

00:01:25.690 --> 00:01:28.433
Pause this video, and see
if you can figure that out.

00:01:29.860 --> 00:01:32.390
Well, the exact same
thing is going to happen.

00:01:32.390 --> 00:01:35.910
All of our digits are going to
shift one place to the left.

00:01:35.910 --> 00:01:39.450
So, right now, we have a
three in the ones place,

00:01:39.450 --> 00:01:42.930
we have a zero in the tenths place,

00:01:42.930 --> 00:01:45.860
we have a one in the hundredths place,

00:01:45.860 --> 00:01:49.470
and we have a five in the thousandths,

00:01:49.470 --> 00:01:51.060
thousandths place.

00:01:51.060 --> 00:01:54.230
But now, they're all going to
shift one place to the left.

00:01:54.230 --> 00:01:57.190
So, the three is now going
to into the tens place,

00:01:57.190 --> 00:01:59.630
it's going to shift one place to the left.

00:01:59.630 --> 00:02:02.940
So, we're going to
have, in the tens place,

00:02:02.940 --> 00:02:05.510
we are now going to have our three.

00:02:05.510 --> 00:02:09.420
And now, this zero, which
was in the tenths place,

00:02:09.420 --> 00:02:13.540
is now going to shift into the ones place.

00:02:13.540 --> 00:02:17.030
So, the zero is going to go right over,

00:02:17.030 --> 00:02:18.980
the zero's going to go right over there.

00:02:18.980 --> 00:02:22.630
That is now in the ones place.

00:02:22.630 --> 00:02:25.090
And them we'll put our decimal.

00:02:25.090 --> 00:02:30.090
And now, what's going to
go into the tenths place

00:02:30.160 --> 00:02:33.050
and the hundredths place?

00:02:33.050 --> 00:02:37.010
And actually, I'll rewrite
the thousandths place as well.

00:02:37.010 --> 00:02:39.660
This one is going to shift
one place to the left,

00:02:39.660 --> 00:02:42.170
into the tenths place,

00:02:42.170 --> 00:02:44.050
so into the tenths place.

00:02:44.050 --> 00:02:46.900
And then the five that was
in the thousandths place

00:02:46.900 --> 00:02:49.180
is now going to shift
one place to the left,

00:02:49.180 --> 00:02:51.630
into the hundredths place,

00:02:51.630 --> 00:02:54.120
so into the hundredths
place, just like that.

00:02:54.120 --> 00:02:56.710
Now, we could put a
trailing zero over here,

00:02:56.710 --> 00:02:59.140
but that's not going to change
the value of this number,

00:02:59.140 --> 00:03:00.530
so I'll just leave it like that.

00:03:00.530 --> 00:03:01.860
And so, there you have it.

00:03:01.860 --> 00:03:05.000
We see that every digit has
shifted one place to the left.

00:03:05.000 --> 00:03:08.493
So, this is equal to 30.15.

00:03:09.440 --> 00:03:11.340
I'll just put that zero there for kicks.

00:03:11.340 --> 00:03:13.820
And so, we could think
about the other way around.

00:03:13.820 --> 00:03:18.330
What if I were to take 67.5,

00:03:19.680 --> 00:03:21.500
and if I were to divide it by 10,

00:03:21.500 --> 00:03:23.500
or another way of thinking about it is

00:03:23.500 --> 00:03:26.260
if I were to multiply this by 1/10.

00:03:26.260 --> 00:03:27.680
Pause the video, and see if you can

00:03:27.680 --> 00:03:29.430
figure out what that's going to be.

00:03:30.810 --> 00:03:33.200
Well, now, every digit is going to shift

00:03:33.200 --> 00:03:34.980
one place to the right.

00:03:34.980 --> 00:03:37.280
So, the six is going to
be in the ones place,

00:03:37.280 --> 00:03:41.280
the seven is going to go
into the tenths place,

00:03:41.280 --> 00:03:45.230
and then the five is going to
go into the hundredths place.

00:03:45.230 --> 00:03:46.110
So, let's write that out.

00:03:46.110 --> 00:03:49.240
So, I have six is going
to go into the ones place,

00:03:49.240 --> 00:03:50.730
and then we're gonna
have our decimal point,

00:03:50.730 --> 00:03:53.450
our seven is going to go
into the tenths place,

00:03:53.450 --> 00:03:57.750
and then our five is going to
go into the hundredths place.

00:03:57.750 --> 00:03:59.973
So, there you have it, we get 6.75.


Multiplying and dividing decimals by 10, 100, 1000

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:01.980
- [Instructor] In this video,

00:00:01.980 --> 00:00:04.260
we're going to get a little
bit of practice multiplying

00:00:04.260 --> 00:00:05.690
and dividing decimals

00:00:05.690 --> 00:00:08.930
by 10, 100, and 1000.

00:00:08.930 --> 00:00:10.720
So let's just start with
a little bit of a warmup.

00:00:10.720 --> 00:00:15.720
If I were to say, what is 2.05 times 10?

00:00:17.280 --> 00:00:20.110
Pause this video and see
if you can figure that out.

00:00:20.110 --> 00:00:21.650
Well, in previous videos,

00:00:21.650 --> 00:00:24.170
we've already said that
when you multiply by 10,

00:00:24.170 --> 00:00:27.470
you shift each of the digits
one place to the left.

00:00:27.470 --> 00:00:29.610
And so this is going to be equal to,

00:00:29.610 --> 00:00:30.690
instead of two ones,

00:00:30.690 --> 00:00:32.870
we're now going to have two 10s,

00:00:32.870 --> 00:00:34.580
so two in the tens place.

00:00:34.580 --> 00:00:36.580
And then, instead of zero tenths,

00:00:36.580 --> 00:00:38.563
we're now going to have zero ones.

00:00:39.790 --> 00:00:42.170
And instead of five hundredths,

00:00:42.170 --> 00:00:44.490
we're now going to have five tenths,

00:00:44.490 --> 00:00:47.620
so this is equal to 20.5.

00:00:47.620 --> 00:00:49.650
Now what if we were to
go the other way around?

00:00:49.650 --> 00:00:51.910
What if we were to say 2.05

00:00:53.820 --> 00:00:55.390
divided by 10?

00:00:55.390 --> 00:00:58.073
Pause the video and see if
you can figure that out.

00:00:59.320 --> 00:01:00.430
Well here,

00:01:00.430 --> 00:01:03.280
all of our digits are going to
shift one place to the right

00:01:03.280 --> 00:01:04.710
because we're dividing by 10.

00:01:04.710 --> 00:01:07.720
You could also view that
as multiplying by 1/10.

00:01:07.720 --> 00:01:10.970
And so our two ones are
going to become two tenths,

00:01:10.970 --> 00:01:13.540
so this is going to be zero point,

00:01:13.540 --> 00:01:15.610
so we're gonna have two tenths now.

00:01:15.610 --> 00:01:19.220
Our zero tenths are
going be zero hundredths,

00:01:19.220 --> 00:01:20.910
and then our five hundredths

00:01:20.910 --> 00:01:23.200
are going to be five thousandths,

00:01:23.200 --> 00:01:25.220
we've covered that in other videos,

00:01:25.220 --> 00:01:29.470
but now let's do this
with, say, 100 or 1000.

00:01:29.470 --> 00:01:31.870
So if I were to ask you

00:01:31.870 --> 00:01:36.340
what is 57

00:01:36.340 --> 00:01:38.240
divided by 1000?

00:01:40.480 --> 00:01:43.680
Pause this video and see
if you can work that out.

00:01:43.680 --> 00:01:45.580
All right, now let's do this together.

00:01:46.540 --> 00:01:49.550
So when you divide by 1000,

00:01:49.550 --> 00:01:53.660
that's the same thing as
dividing by 10 three times.

00:01:53.660 --> 00:01:55.340
Or it's the same thing as multiplying

00:01:55.340 --> 00:01:57.730
by 1/10 three times,

00:01:57.730 --> 00:01:59.020
or you could just say hey,

00:01:59.020 --> 00:02:00.720
that means I'm going to shift each

00:02:00.720 --> 00:02:04.390
of these digits three places to the right,

00:02:04.390 --> 00:02:06.520
and so let me create some places here.

00:02:06.520 --> 00:02:08.810
So that's 10s, ones,

00:02:08.810 --> 00:02:12.150
tenths, hundredths, thousandths,

00:02:12.150 --> 00:02:17.150
and so our five was in the
10s place, it's five 10s,

00:02:17.880 --> 00:02:19.020
so it was here,

00:02:19.020 --> 00:02:22.488
but we're going to shift
three places to the right.

00:02:22.488 --> 00:02:24.613
One, two, three.

00:02:24.613 --> 00:02:26.388
So our five will go there.

00:02:26.388 --> 00:02:30.655
So what was five 10s
is now five hundredths.

00:02:30.655 --> 00:02:34.752
And then our seven is similarly going to,

00:02:34.752 --> 00:02:36.732
it was in the ones place,

00:02:36.732 --> 00:02:39.261
but we're gonna shift
three places to the right.

00:02:39.261 --> 00:02:41.140
One, two, three.

00:02:41.140 --> 00:02:42.420
And there you have it.

00:02:42.420 --> 00:02:46.840
What was just 57 is now 57 thousandths.

00:02:46.840 --> 00:02:48.040
And to make that very clear,

00:02:48.040 --> 00:02:49.720
I'll put a zero in the tenths place

00:02:49.720 --> 00:02:51.970
and a zero in the ones place,

00:02:51.970 --> 00:02:53.350
and that makes sense.

00:02:53.350 --> 00:02:54.620
So as another example,

00:02:54.620 --> 00:02:56.850
let's say someone walks
up to you on the street

00:02:56.850 --> 00:03:00.040
and says I started with 1.032

00:03:02.800 --> 00:03:06.300
and I multiplied it by something

00:03:06.300 --> 00:03:09.813
and I was able to get 103.2.

00:03:13.320 --> 00:03:17.150
What did they multiply by to get 103.2?

00:03:17.150 --> 00:03:19.150
Pause this video and try to work it out.

00:03:20.230 --> 00:03:21.530
Well to understand this,

00:03:21.530 --> 00:03:22.363
we just have to think well,

00:03:22.363 --> 00:03:24.630
how much did each digit

00:03:24.630 --> 00:03:25.980
get shifted by?

00:03:25.980 --> 00:03:29.310
So what was in the ones
place got shifted not just

00:03:29.310 --> 00:03:33.810
to the 10s place, but it got
shifted to the hundreds place.

00:03:33.810 --> 00:03:37.700
So it got shifted two places to the left.

00:03:37.700 --> 00:03:41.530
The zero, similarly, got
shifted two places to the left.

00:03:41.530 --> 00:03:43.760
From the tenths to the 10s,

00:03:43.760 --> 00:03:44.630
and if you look at it,

00:03:44.630 --> 00:03:47.430
every digit got shifted
two places to the left.

00:03:47.430 --> 00:03:50.490
And we must have multiplied by 10 twice,

00:03:50.490 --> 00:03:52.970
so you could say times 10 times 10,

00:03:52.970 --> 00:03:57.410
or you could just rewrite
this as times 100.

00:03:57.410 --> 00:04:00.970
Let's just do one more example for kicks.

00:04:00.970 --> 00:04:04.530
Let's say that someone were to say

00:04:04.530 --> 00:04:09.530
what is 0.015 times 100?

00:04:12.060 --> 00:04:15.350
Pause this video and see
if you can figure that out.

00:04:15.350 --> 00:04:17.550
All right, well, like we've done before,

00:04:17.550 --> 00:04:19.620
we're just going to shift every digit

00:04:19.620 --> 00:04:22.330
two places to the left.

00:04:22.330 --> 00:04:26.130
So the one, which is in
the hundredths place,

00:04:26.130 --> 00:04:28.770
is going to end up in the ones place,

00:04:28.770 --> 00:04:32.750
and then the five, which is
in the thousandths place,

00:04:32.750 --> 00:04:35.530
is going to end up in the tenths place.

00:04:35.530 --> 00:04:38.640
So this is going to be equal to 1.5,

00:04:42.700 --> 00:04:43.633
and we're done.


Skip counting equal groups

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:01.360
- [Instructor] What we have here

00:00:01.360 --> 00:00:03.410
are pictures of running pigs,

00:00:03.410 --> 00:00:06.220
and we can try to figure out
how many running pigs there are

00:00:06.220 --> 00:00:08.280
by just counting the pigs,

00:00:08.280 --> 00:00:10.820
but we're gonna start
building some new muscles

00:00:10.820 --> 00:00:12.830
and this muscle's going to involve,

00:00:12.830 --> 00:00:16.750
hey, if we group the pigs
into equal groupings,

00:00:16.750 --> 00:00:19.670
can that help us figure out
how many pigs there are?

00:00:19.670 --> 00:00:22.820
And you can see that I
have groupings of seven.

00:00:22.820 --> 00:00:26.900
Each of these groups
are seven running pigs,

00:00:26.900 --> 00:00:29.130
so how many total groups do I have?

00:00:29.130 --> 00:00:32.930
Well, I have one, two, three groups

00:00:32.930 --> 00:00:35.300
of seven running pigs each.

00:00:35.300 --> 00:00:37.330
Three equal groups.

00:00:37.330 --> 00:00:39.340
If I said, what's the total
number of running pigs

00:00:39.340 --> 00:00:40.820
that I'm dealing with?

00:00:40.820 --> 00:00:45.440
Well, I have three sevens,
three groups of seven, and so,

00:00:45.440 --> 00:00:49.200
that's the same thing as
seven plus seven plus seven,

00:00:49.200 --> 00:00:50.800
and if you wanted to
figure out what that is,

00:00:50.800 --> 00:00:51.910
you could skip count.

00:00:51.910 --> 00:00:56.360
If we skip count by seven,
we go seven then 14 then 21,

00:00:58.200 --> 00:01:01.430
and so, three sevens, which is
seven plus seven plus seven,

00:01:01.430 --> 00:01:04.480
is 21, so there's 21 pigs here,

00:01:04.480 --> 00:01:05.960
and you can verify on your own

00:01:05.960 --> 00:01:09.910
that if you count this, you
will indeed see 21 pigs.

00:01:09.910 --> 00:01:11.183
Let's do another example.

00:01:12.400 --> 00:01:15.200
Here, I'm looking at a picture
of a bunch of blue seals,

00:01:15.200 --> 00:01:17.800
not something that you are
likely to see in the wild.

00:01:17.800 --> 00:01:20.640
And these blue seals have,
once again, been grouped.

00:01:20.640 --> 00:01:21.640
Let's see how many groups there are.

00:01:21.640 --> 00:01:26.640
There are one, two, three,
four, five, six, seven groups,

00:01:27.840 --> 00:01:29.700
and how many are in each group?

00:01:29.700 --> 00:01:32.040
Well, we could see that
they're all groups of

00:01:32.040 --> 00:01:33.730
one, two, three, four, five.

00:01:33.730 --> 00:01:35.710
They're equal groups.

00:01:35.710 --> 00:01:36.840
What are dealing with?

00:01:36.840 --> 00:01:39.730
Well, we're dealing with
seven groups of five,

00:01:39.730 --> 00:01:44.140
so I could write seven groups of five.

00:01:44.140 --> 00:01:47.200
I could write seven fives.

00:01:47.200 --> 00:01:52.200
I could write five plus
five plus five plus five

00:01:52.950 --> 00:01:57.950
plus five plus five plus five,
so that is seven fives there.

00:01:58.390 --> 00:01:59.940
And if I wanted to
figure out where that is,

00:01:59.940 --> 00:02:00.810
I could skip count.

00:02:00.810 --> 00:02:03.000
I could skip count seven fives,

00:02:03.000 --> 00:02:08.000
so that'd be five, 10, 15, 20, 25, 30, 35.

00:02:11.220 --> 00:02:12.650
If I were to add all these fives up,

00:02:12.650 --> 00:02:15.250
five plus five is 10, add
another five, you get to 15.

00:02:15.250 --> 00:02:16.490
Add another five, 20.

00:02:16.490 --> 00:02:17.630
Add another five, 25.

00:02:17.630 --> 00:02:18.550
Add another five, 30.

00:02:18.550 --> 00:02:22.203
Add another five, 35, which
is exactly what you see there.

00:02:23.290 --> 00:02:26.320
We're starting to see that
sometimes grouping things

00:02:26.320 --> 00:02:29.950
can help us appreciate how
many things there are there.

00:02:29.950 --> 00:02:31.030
And I know what some of
you all are thinking,

00:02:31.030 --> 00:02:33.410
well, hey, couldn't I just count things?

00:02:33.410 --> 00:02:34.740
Well, you could,

00:02:34.740 --> 00:02:37.650
but what if you had 100 groups of five?

00:02:37.650 --> 00:02:38.540
Well then, it's gonna take you

00:02:38.540 --> 00:02:40.150
a little bit more time to count.

00:02:40.150 --> 00:02:42.130
But the techniques that we're building

00:02:42.130 --> 00:02:43.680
will help you one day be able to

00:02:43.680 --> 00:02:46.403
think about those things quite easily.