Squaring binomials of the form (x+a)Ã  Â²

WEBVTT
Kind: captions
Language: en

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- [Voiceover] Let's see
if we can figure out what

00:00:01.541 --> 00:00:05.788
x plus seven, let me write
that a little bit neater,

00:00:05.788 --> 00:00:09.485
x plus seven squared is.

00:00:09.485 --> 00:00:10.643
And I encourage you to pause the video

00:00:10.643 --> 00:00:13.223
and work through it on your own.

00:00:13.223 --> 00:00:15.295
Alright, now let's work
through this together.

00:00:15.295 --> 00:00:16.494
So we just have to remember,

00:00:16.494 --> 00:00:19.664
we're squaring the entire binomial.

00:00:19.664 --> 00:00:22.813
So this thing is going
to be the same thing as:

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x plus seven times x plus seven.

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I'm gonna write the second

00:00:27.040 --> 00:00:28.440
x plus seven in a different color,

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which is going to be helpful when

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we actually multiply things out.

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When we see it like this,

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then we can multiply these out the way

00:00:34.312 --> 00:00:36.559
we would multiply any binomials.

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And I'll first do it
the, I guess you can say,

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the slower way, but
the more intuitive way,

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applying the distributive property twice.

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And then we'll think
about maybe some shortcuts

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or some patterns we might
be able to recognize,

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especially when we are squaring binomials.

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So let's start with just applying

00:00:50.479 --> 00:00:52.103
the distributive property twice.

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So let's distribute
this yellow x plus seven

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over this magenta x plus seven.

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So we can multiply it by
the x, this magenta x,

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so it's going to be x, let
me do it in that same color.

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So it's going to be magenta x

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times x plus seven

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plus magenta seven

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times yellow x plus seven.

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X plus seven, and now we can apply the

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distributive property again.

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We can take this magenta x

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and distribute it over the x plus seven.

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So x times x is x-squared.

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X times seven is seven x.

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And then we can do it again over here.

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This seven, let me do
it in a different color,

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so this seven times that x is going to be

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plus another seven x

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and then the seven times the seven

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is going to be 49.

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And we're in the home stretch.
We can then simplify it.

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This is going to be x-squared

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and then these two middle
terms we can add together.

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Seven x, let me do this in orange,

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seven x plus seven x is going to be 14 x

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plus 14 x plus 49.

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Plus 49.

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And we're done.

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Now the key question is do
we see some patterns here?

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Do we see some patterns
that we can generalize

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and that might help us square binomials

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a little bit faster in the future?

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Well, when we first looked at
just multiplying binomials,

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we saw a pattern like

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x plus a times x plus b

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is going to be equal to x-squared,

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let me write it this way,

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is going to be equal to x-squared plus

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a plus b

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x plus b-squared.

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And so, if both a and
b are the same thing,

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we can say that x plus a

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times x plus a

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is going to be equal to x-squared,

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and this is the case when we have a

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coefficient of one on both
of these x's, x-squared's.

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Now in this case, a and b are both a.

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So it's going to be a plus a times x,

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or we can just say plus two a x.

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Let me be clear what I just did.

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Instead of writing a plus b,

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I can just view this as a plus a times x,

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and then plus a-squared,

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or that's the same thing as

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x-squared plus two a x

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plus a-squared.

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This is a general way of expressing

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a squared binomial like this.

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A squared binomial where the coefficients

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on both x's are one.

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We can see that's exactly
what we saw over here.

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In this, in the example
we did, seven is our a.

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So we got x-squared right
over there let me circle it.

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So we have this blue x-squared

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that corresponds to that over there.

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And then seven is our a,

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so two a x , two times seven is 14 x.

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Notice we have the 14 x right over there.

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So this 14 x corresponds to two a x,

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and then finally if a is seven,

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a-squared is 49.

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A-squared is 49.

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So in general if you are
squaring a binomial, you could,

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a fast way of doing it is
to do this pattern here,

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and we can do another example real fast,

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just to make sure that
we've understood things.

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If I were to tell you what is x minus,

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I'll throw a negative in here,

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x minus three squared,

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I encourage you to pause the video

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and think about it.

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Think about expressing
this using this pattern.

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Well this is going to
be, in this case our a,

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we have to be careful,

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our a is going to be negative three,

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so that is our a right over there.

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So this is going to be equal to x-squared.

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Now two a x, let me do it
in the same colors actually,

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just for fun.

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So it's going to be x-squared.

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Now what is two times a times x?

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A is negative three, so two
times a is negative six.

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So it's going to be negative six x.

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So, minus six x, that's two
times a is the coefficient.

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And then we have our x there.

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And then plus a-squared.

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Well if a is negative three,

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what is negative three
times negative three?

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It's going to be positive nine.

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And just like that, when
we looked at this pattern,

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we were able to very quickly figure out

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what this binomial squared actually is.

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And I encourage you, you can do it again,

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with applying the
distributive property twice

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to verify that this is
indeed the same thing as

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x minus three squared.


Multiplying binomials intro

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:01.320
- [Voiceover] Let's see
if we can figure out

00:00:01.320 --> 00:00:05.020
the product of x minus
four and x plus seven.

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And we want to write that product
in standard quadratic form

00:00:08.420 --> 00:00:10.550
which is just a fancy way of saying a form

00:00:10.550 --> 00:00:13.950
where you have some coefficient
on the second degree term,

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a x squared plus some coefficient b

00:00:16.610 --> 00:00:19.520
on the first degree term
plus the constant term.

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So this right over here would
be standard quadratic form.

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So that's the form that we
want to express this product in

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and encourage you to pause the video

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and try to work through it on your own.

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Alright, now let's work through this.

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And the key when we're multiplying
two binomials like this,

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or actually when you're
multiplying any polynomials,

00:00:34.820 --> 00:00:37.420
is just to remember the
distributive property

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that we all by this point know quite well.

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So what we could view this is as

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is we could distribute this x minus four,

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this entire expression
over the x and the seven.

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So we could say that
this is the same thing

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as x minus four times x

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plus x minus four times seven.

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So let's write that.

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So x minus four times x,
or we could write this

00:00:58.820 --> 00:01:03.820
as x times x minus four.

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That's distributing, or multiplying
the x minus four times x

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that's right there.

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Plus seven times x minus four.

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Times x minus four.

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Notice all we did is
distribute the x minus four.

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We took this whole thing
and we multiplied it

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by each term over here.

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We multiplied x by x minus four

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and we multiplied seven by x minus four.

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Now, we see that we have these,

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I guess you can call
them two seperate terms.

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And to simplify each of them,
or to multiply them out,

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we just have to distribute.

00:01:35.490 --> 00:01:37.790
In this first we're going to
have to distribute this blue x.

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And over here we have to
distribute this blue seven.

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So let's do that.

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So here we can say x times
x is going to be x squared.

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X times, we have a negative here,

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so we can say negative four is
going to be negative four x.

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And just like, that we get
x squared minus four x.

00:01:55.350 --> 00:02:00.150
And then over here we have seven times x

00:02:00.150 --> 00:02:03.290
so that's going to be plus seven x.

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And then we have seven
times the negative four

00:02:06.820 --> 00:02:09.720
which is negative 28.

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And we are almost done.

00:02:11.750 --> 00:02:13.220
We can simplify it a little bit more.

00:02:13.220 --> 00:02:15.020
We have two first degree terms here.

00:02:15.020 --> 00:02:18.990
If I have negative four xs
and to that I add seven xs,

00:02:18.990 --> 00:02:20.390
what is that going to be?

00:02:20.390 --> 00:02:21.720
Well those two terms together,

00:02:21.720 --> 00:02:24.590
these two terms together are going to be

00:02:24.590 --> 00:02:27.920
negative four plus seven xs.

00:02:27.920 --> 00:02:32.920
Negative four plus, plus seven.

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Negative four plus seven xs.

00:02:37.120 --> 00:02:38.510
So all I'm doing here,
I'm making it very clear

00:02:38.510 --> 00:02:40.220
that I'm adding these two coefficients,

00:02:40.220 --> 00:02:42.050
and then we have all of the other terms.

00:02:42.050 --> 00:02:43.220
We have the x squared.

00:02:43.220 --> 00:02:46.190
X squared plus this and then we have,

00:02:46.190 --> 00:02:48.420
and then we have the minus,

00:02:48.420 --> 00:02:50.480
and then we have the minus 28.

00:02:50.480 --> 00:02:51.990
And we're at the home stretch!

00:02:51.990 --> 00:02:55.020
This would simply to x squared.

00:02:55.020 --> 00:02:57.650
Now negative four plus seven is three,

00:02:57.650 --> 00:03:00.790
so this is going to be plus three x.

00:03:00.790 --> 00:03:04.590
That's what these two middle
terms simplify to, to three x.

00:03:04.590 --> 00:03:06.620
And then we have minus 28.

00:03:06.620 --> 00:03:09.050
Minus 28.

00:03:09.050 --> 00:03:11.920
And just like that, we are done!

00:03:11.920 --> 00:03:15.250
And a fun thing to think about,
since it's in the same form.

00:03:15.250 --> 00:03:18.400
If we were to compare a is one,

00:03:18.400 --> 00:03:21.650
b is three, and c is -28,

00:03:21.650 --> 00:03:23.950
but it's interesting here
to look at the pattern

00:03:23.950 --> 00:03:26.590
when we multiplied these two binomials.

00:03:26.590 --> 00:03:28.790
Especially these two binomials
where the coefficient

00:03:28.790 --> 00:03:31.650
on the x term was a one.

00:03:31.650 --> 00:03:34.020
Notice we have x times x,

00:03:34.020 --> 00:03:36.710
that what actually forms the
x squared term over here.

00:03:36.710 --> 00:03:39.650
We have negative four, let
me do this in a new color.

00:03:39.650 --> 00:03:43.350
We have negative four times,
that's not a new color.

00:03:43.350 --> 00:03:45.740
We have,

00:03:45.740 --> 00:03:50.420
we have negative four times seven,

00:03:50.420 --> 00:03:53.590
which is going to be negative 28.

00:03:53.590 --> 00:03:55.480
And then how did we get this middle term?

00:03:55.480 --> 00:03:57.020
How did we get this three x?

00:03:57.020 --> 00:04:01.220
Well, you had the negative
four x plus the seven x.

00:04:01.220 --> 00:04:04.380
Or the negative four
plus the seven times x.

00:04:04.380 --> 00:04:07.610
You had the negative four plus the seven,

00:04:07.610 --> 00:04:10.690
plus the seven times x.

00:04:10.690 --> 00:04:12.390
So I hope you see a little
bit of a pattern here.

00:04:12.390 --> 00:04:13.820
If you're multiplying two binomials

00:04:13.820 --> 00:04:16.250
where the coefficients on
the x term are both one.

00:04:16.250 --> 00:04:17.950
It's going to be x squared.

00:04:17.950 --> 00:04:20.190
And then the last term, the constant term,

00:04:20.190 --> 00:04:21.920
is going to be the product
of these two constants.

00:04:21.920 --> 00:04:23.720
Negative four and seven.

00:04:23.720 --> 00:04:26.520
And then the first degree
term right over here,

00:04:26.520 --> 00:04:28.520
it's coefficient is going to be the sum

00:04:28.520 --> 00:04:31.790
of these two constants,
negative four and seven.

00:04:31.790 --> 00:04:33.290
Now this might, you could do

00:04:33.290 --> 00:04:34.950
this pattern if you practice it.

00:04:34.950 --> 00:04:36.250
It's just something that will help you

00:04:36.250 --> 00:04:37.985
multiply binomials a little bit faster.

00:04:37.985 --> 00:04:39.250
But it's super important that

00:04:39.250 --> 00:04:40.790
you realize where this came from.

00:04:40.790 --> 00:04:42.280
This came from nothing more than

00:04:42.280 --> 00:04:44.870
applying the distributive property twice.


Breakthrough Junior Challenge - Sal and Priscilla

WEBVTT
Kind: captions
Language: en

00:00:00.505 --> 00:00:02.002
- HI, I'm Priscilla Chan.

00:00:02.002 --> 00:00:04.348
- And I'm Sal Khan, and
we're here to announce

00:00:04.348 --> 00:00:06.588
the Breakthrough Junior Challenge.

00:00:06.588 --> 00:00:07.761
For those of you who don't know,

00:00:07.761 --> 00:00:10.721
the Breakthrough Prize,
is a prize to the world's

00:00:10.721 --> 00:00:12.930
top mathematicians and scientists

00:00:12.930 --> 00:00:14.437
for discoveries that are going to

00:00:14.437 --> 00:00:16.480
change the world around us.

00:00:16.480 --> 00:00:18.686
And the Breakthrough Junior Challenge

00:00:18.686 --> 00:00:21.484
is to reach out to high school students,

00:00:21.484 --> 00:00:23.318
with the recognition that
the biggest breakthroughs

00:00:23.318 --> 00:00:25.861
start with an insight, with a
new way of looking at things.

00:00:25.861 --> 00:00:28.020
It could be existing
discoveries or theories,

00:00:28.020 --> 00:00:29.205
it could be a recent one,

00:00:29.205 --> 00:00:32.441
or it could be something that
you discover on your own.

00:00:32.441 --> 00:00:33.663
- We want to hear from you.

00:00:33.663 --> 00:00:36.914
All you scientists, tinkers,
innovators in the world,

00:00:36.914 --> 00:00:38.539
send us your video.

00:00:38.539 --> 00:00:41.465
We want to hear from you by October 7th,

00:00:41.465 --> 00:00:43.636
in a short video where you explain

00:00:43.636 --> 00:00:46.178
what you're excited about these days.

00:00:46.178 --> 00:00:47.641
- And what do they win?

00:00:47.641 --> 00:00:50.602
- This is pretty exciting in itself.

00:00:50.602 --> 00:00:52.355
You'll win a college scholarship,

00:00:52.355 --> 00:00:54.630
your teacher will get a prize for being

00:00:54.630 --> 00:00:56.801
such an important mentor in your life,

00:00:56.801 --> 00:01:00.470
and your school gets a
brand new lab for everyone

00:01:00.470 --> 00:01:02.897
to be able to work hands-on
and learn about science.

00:01:02.897 --> 00:01:04.534
- [Sal] And they get to fly out here.

00:01:04.534 --> 00:01:06.229
- Oh yea, this is the most glamorous

00:01:06.229 --> 00:01:08.154
science event in the world.

00:01:08.154 --> 00:01:09.480
Best science fair ever.

00:01:09.480 --> 00:01:11.117
- And we can't wait to
see what you produce.

00:01:11.117 --> 00:01:13.369
We suspect that you're
going to blow our minds,

00:01:13.369 --> 00:01:15.377
and open many, many more people's minds

00:01:15.377 --> 00:01:18.001
to the magic of science
that will eventually lead

00:01:18.001 --> 00:01:19.464
to future breakthroughs.

00:01:19.464 --> 00:01:20.300
- [Priscilla] We're so excited.

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We can't wait to hear from you.

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(soft rock music)


Practice finding patterns in numbers

WEBVTT
Kind: captions
Language: en

00:00:00.628 --> 00:00:01.835
- [Voiceover] What I want to in this video

00:00:01.835 --> 00:00:05.202
is get some practice figuring
out patterns and numbers.

00:00:05.202 --> 00:00:07.524
In particular, patterns that take us from

00:00:07.524 --> 00:00:10.903
one number to a next number in a sequence.

00:00:10.903 --> 00:00:13.596
So over here, in this magenta color,

00:00:13.596 --> 00:00:18.263
I go from 4 to 25 to 46 to 67.

00:00:18.263 --> 00:00:20.393
So what's the pattern here?

00:00:20.393 --> 00:00:22.831
How did I get from 4 to 25

00:00:22.831 --> 00:00:25.502
and can I get the same way from 25 to 46

00:00:25.502 --> 00:00:30.308
and 46 to 67, and I could just
keep going on and on and on?

00:00:30.308 --> 00:00:31.887
Well there's a couple of
ways to think about it.

00:00:31.887 --> 00:00:33.986
When I see 4 and 25, let's see,

00:00:33.986 --> 00:00:38.026
25 isn't an obvious multiple of 4.

00:00:38.026 --> 00:00:43.026
Another way to go from
4 to 25, I could add 21.

00:00:43.041 --> 00:00:48.041
Let's see, if I add 21, 4 plus 21 is 25.

00:00:48.196 --> 00:00:51.052
If I were to go from 25 to 46,

00:00:51.052 --> 00:00:53.025
well I could just add 21 again.

00:00:53.025 --> 00:00:55.835
It looks like to go from
one number to the next

00:00:55.835 --> 00:00:57.483
I'm just adding.

00:00:57.483 --> 00:00:59.573
I wrote 12 by accident, 21.

00:00:59.573 --> 00:01:02.684
I'm just adding 21 over and over again.

00:01:02.684 --> 00:01:07.166
That's going to be 46 plus 21 is 67.

00:01:07.166 --> 00:01:11.159
And if I were to keep going, if I add 21

00:01:11.159 --> 00:01:13.110
I'm going to get to 89.

00:01:13.110 --> 00:01:18.110
If I add 21 to that I'm going to get 110,

00:01:20.865 --> 00:01:22.955
and I could keep going
and going and going.

00:01:22.955 --> 00:01:27.157
I could just keep adding
21 over and over again.

00:01:27.157 --> 00:01:31.522
The pattern here is I'm adding 21.

00:01:31.522 --> 00:01:34.471
Now what about over here, in green?

00:01:34.471 --> 00:01:36.607
When I look at it at first,
it's tempting to say,

00:01:36.607 --> 00:01:40.276
3 plus 3 is 6.

00:01:40.276 --> 00:01:43.480
But then I'm not adding 3 anymore to get

00:01:43.480 --> 00:01:46.290
from 6 to 12, I'm adding 6.

00:01:46.290 --> 00:01:48.147
And then to get from 12 to 24,

00:01:48.147 --> 00:01:51.096
I'm not adding 6 anymore, I added 12.

00:01:51.096 --> 00:01:54.463
So every time I'm adding twice as much.

00:01:54.463 --> 00:01:58.389
But maybe an easier pattern might be,

00:01:58.389 --> 00:01:59.884
another way to go from 3 to 6,

00:01:59.884 --> 00:02:03.738
isn't to add 3, but to multiply it by 2.

00:02:03.738 --> 00:02:06.501
So I multiply by 2 to go from 3 to 6,

00:02:06.501 --> 00:02:10.030
and if I multiply by 2
again, I go from 6 to 12.

00:02:10.030 --> 00:02:11.818
6 times 2 is 12.

00:02:11.818 --> 00:02:14.929
If I multiply by 2 again, I'll go to 24.

00:02:14.929 --> 00:02:19.457
2 times 12 is 24 and I could
keep going on and on and on.

00:02:19.457 --> 00:02:23.520
2 times 24 is 48, 96, I
could go on and on and on.

00:02:23.520 --> 00:02:26.353
The pattern here, it's
not adding a fixed amount,

00:02:26.353 --> 00:02:30.927
it's multiplying each
number by a certain amount,

00:02:30.927 --> 00:02:33.760
by 2 in this case, to get the next number.

00:02:33.760 --> 00:02:38.102
So 3 times 2 is 6, 6 times
2 is 12, 12 times 2 is 24.

00:02:38.102 --> 00:02:40.076
Alright, now let's look at this last one.

00:02:40.076 --> 00:02:42.908
The first two terms here
are the same, 3 and 6.

00:02:42.908 --> 00:02:46.117
The first two numbers here.

00:02:46.117 --> 00:02:47.833
I could say, maybe this is times 2,

00:02:47.833 --> 00:02:51.223
but then to go from 6 to 9,
I'm not multiplying by 2.

00:02:51.223 --> 00:02:53.429
But maybe I am just adding 3 here.

00:02:53.429 --> 00:02:55.937
So 3 to 6, I just added 3.

00:02:55.937 --> 00:02:59.327
Then 6 to 9, I add 3 again,

00:02:59.327 --> 00:03:02.276
and then 9 to 12, I add 3 again.

00:03:02.276 --> 00:03:03.506
So this one actually does look like

00:03:03.506 --> 00:03:06.432
I'm just adding 3 every time.

00:03:06.432 --> 00:03:09.633
The whole point here is to see,
is there something I can do,

00:03:09.633 --> 00:03:12.142
can I do the same something
over and over again

00:03:12.142 --> 00:03:13.814
to get from one number to

00:03:13.814 --> 00:03:16.786
the next number in a sequence like this?

00:03:16.786 --> 00:03:18.637
What you want to make sure
is even if you think you know

00:03:18.637 --> 00:03:20.480
how to go from the first
number to the second number,

00:03:20.480 --> 00:03:22.175
you've got to make sure
that that same way works

00:03:22.175 --> 00:03:24.961
to go from the second
number to the third number,

00:03:24.961 --> 00:03:27.074
and the third number to the fourth number.

00:03:27.074 --> 00:03:28.258
But here we figured it out.

00:03:28.258 --> 00:03:31.323
In this first set of numbers,
we just add 21 every time.

00:03:31.323 --> 00:03:33.274
This one we multiply by 2 every time.

00:03:33.274 --> 00:03:37.083
This one we add 3 every time.


Introduction to even and odd numbers

WEBVTT
Kind: captions
Language: en

00:00:00.556 --> 00:00:02.668
- What I'm going to introduce
you to in this video

00:00:02.668 --> 00:00:07.171
is a way to classify numbers
as either being even,

00:00:07.171 --> 00:00:09.831
or being odd.

00:00:09.831 --> 00:00:12.466
So what does it mean to be even?

00:00:12.466 --> 00:00:14.494
Well an even number is one

00:00:14.494 --> 00:00:17.420
where if you had that many doughnuts,

00:00:17.420 --> 00:00:21.159
you could split it evenly
between two people.

00:00:21.159 --> 00:00:25.208
So even numbers are numbers
that are multiples of two.

00:00:25.208 --> 00:00:28.583
So if you have two doughnuts,
you could split them.

00:00:28.583 --> 00:00:29.516
If you had two people,

00:00:29.516 --> 00:00:31.860
you could give one
doughnut to each person.

00:00:31.860 --> 00:00:32.997
Four doughnuts,

00:00:32.997 --> 00:00:35.366
you could split that
evenly with two people.

00:00:35.366 --> 00:00:37.061
They could each get two doughnuts.

00:00:37.061 --> 00:00:38.220
And then we could keep going.

00:00:38.220 --> 00:00:39.916
All of the multiples of two,

00:00:39.916 --> 00:00:41.751
these are even numbers.

00:00:41.751 --> 00:00:45.140
8, 10, 12, and of course
we could keep going

00:00:45.140 --> 00:00:47.033
on, and on, and on, and on.

00:00:47.033 --> 00:00:49.738
An easy way to spot an even number

00:00:49.738 --> 00:00:52.930
is that it's ones place
is going to be even.

00:00:52.930 --> 00:00:57.930
So, for example, the
number 32 is an even number

00:00:58.050 --> 00:01:00.287
because in the ones place you have a two.

00:01:00.287 --> 00:01:01.573
The number

00:01:01.573 --> 00:01:05.503
5,977,354,

00:01:05.503 --> 00:01:09.341
5,977,354,

00:01:09.341 --> 00:01:13.506
well that's an even number,
because the ones place is even.

00:01:13.506 --> 00:01:15.488
Now there's one number
that's an interesting one,

00:01:15.488 --> 00:01:16.595
why some people sometimes say,

00:01:16.595 --> 00:01:18.091
"Well is this one really even?"

00:01:18.091 --> 00:01:20.747
And that's the number zero.

00:01:20.747 --> 00:01:23.486
And the number zero is even

00:01:23.486 --> 00:01:25.750
because it is a multiple of two.

00:01:25.750 --> 00:01:27.131
How is it a multiple of two?

00:01:27.131 --> 00:01:31.044
Well, zero times two is equal to zero.

00:01:31.044 --> 00:01:32.971
So zero is a multiple of two,

00:01:32.971 --> 00:01:36.744
and so zero, for sure, is an even number.

00:01:36.744 --> 00:01:37.742
And then that actually makes

00:01:37.742 --> 00:01:41.283
our looking at the ones
place idea hold up.

00:01:41.283 --> 00:01:43.349
Because a number like...

00:01:43.349 --> 00:01:47.981
a number like 150 is an even number,

00:01:47.981 --> 00:01:49.925
and we can look at the
ones place of it and see,

00:01:49.925 --> 00:01:51.069
we have a zero there.

00:01:51.069 --> 00:01:52.950
And a zero is an even number.

00:01:52.950 --> 00:01:55.225
So if, in the ones place,
you have an even number,

00:01:55.225 --> 00:01:58.172
you are looking, the
whole thing is going to be

00:01:58.172 --> 00:02:00.008
an even number.

00:02:00.008 --> 00:02:02.110
So what are odd numbers?

00:02:02.110 --> 00:02:03.143
Well one way to think about them

00:02:03.143 --> 00:02:05.399
is they're the numbers that aren't even.

00:02:05.399 --> 00:02:07.565
So not multiples of two.

00:02:07.565 --> 00:02:10.669
Not multiples...

00:02:10.669 --> 00:02:12.945
not multiples of two.

00:02:12.945 --> 00:02:15.637
So what are some examples of odd numbers?

00:02:15.637 --> 00:02:19.455
Well one, three, five,

00:02:19.455 --> 00:02:21.461
seven, nine,

00:02:21.461 --> 00:02:23.730
and of course you can
go on, and on, and on.

00:02:23.730 --> 00:02:25.819
And just like we could
look at the ones place

00:02:25.819 --> 00:02:27.909
to spot an even number, you
can also look at the ones place

00:02:27.909 --> 00:02:29.569
to spot an odd number.

00:02:29.569 --> 00:02:31.531
The number 59,

00:02:31.531 --> 00:02:33.830
well I have a nine over
here in the ones place.

00:02:33.830 --> 00:02:38.172
Nine is odd, so this is
going to be an odd number.

00:02:38.172 --> 00:02:42.259
The number 1,441

00:02:42.259 --> 00:02:44.209
has a one in the ones place.

00:02:44.209 --> 00:02:45.776
That is an odd number.

00:02:45.776 --> 00:02:47.912
So this whole thing is going to be odd.

00:02:47.912 --> 00:02:49.073
And so what's another way of...

00:02:49.073 --> 00:02:50.820
why is it called odd?

00:02:50.820 --> 00:02:55.806
Well, it'd be hard to
split 1,441 doughnuts.

00:02:55.806 --> 00:02:58.449
You can't split it
evenly between two people

00:02:58.449 --> 00:02:59.788
without breaking up a doughnut.

00:02:59.788 --> 00:03:01.610
If you wanted to leave
the doughnuts whole,

00:03:01.610 --> 00:03:03.064
one person would have to get

00:03:03.064 --> 00:03:05.766
one extra doughnut than the other person.

00:03:05.766 --> 00:03:07.148
You can't split it evenly.

00:03:07.148 --> 00:03:08.634
The easiest way I think about odd,

00:03:08.634 --> 00:03:11.606
it's not a multiple of two, it's not even.

00:03:11.606 --> 00:03:13.742
So another way, it is not...

00:03:13.742 --> 00:03:16.992
it is not even.

00:03:16.992 --> 00:03:20.626
So now that we know what an
odd or an even number is,

00:03:20.626 --> 00:03:23.145
let's think about what
happens when we operate

00:03:23.145 --> 00:03:25.165
on odd or even numbers.

00:03:25.165 --> 00:03:27.712
So let's think, in particular,

00:03:27.712 --> 00:03:31.074
actually let me move over
to the side right over here.

00:03:31.074 --> 00:03:34.255
Let me move over....

00:03:34.255 --> 00:03:35.567
right over here.

00:03:35.567 --> 00:03:37.030
Let's think, whoops.

00:03:37.030 --> 00:03:38.713
Fell off the screen.

00:03:38.713 --> 00:03:43.066
Let's think about what happens
if I take an even number,

00:03:43.066 --> 00:03:47.269
an even number, plus another even number.

00:03:47.269 --> 00:03:51.595
Is this number, the sum, is
it going to be even or odd?

00:03:51.595 --> 00:03:53.050
Well you could take some examples.

00:03:53.050 --> 00:03:56.457
If I say two plus six,

00:03:56.457 --> 00:03:58.633
that is going to be equal to eight.

00:03:58.633 --> 00:04:00.349
Well eight is an even number.

00:04:00.349 --> 00:04:05.349
If I say 14 plus four,

00:04:05.425 --> 00:04:07.533
well that equals 18.

00:04:07.533 --> 00:04:09.628
Once again, eight in the ones place.

00:04:09.628 --> 00:04:11.345
This is an even number.

00:04:11.345 --> 00:04:14.627
If I say 150...

00:04:14.627 --> 00:04:17.692
156 plus...

00:04:17.692 --> 00:04:20.827
plus 100...

00:04:20.827 --> 00:04:22.395
100 and...

00:04:22.395 --> 00:04:23.826
why don't we do a simpler number,

00:04:23.826 --> 00:04:27.092
plus four,

00:04:27.092 --> 00:04:30.450
that is equal to 160.

00:04:30.450 --> 00:04:33.624
Zero in the ones place,
this also is an even number.

00:04:33.624 --> 00:04:35.748
So it looks like all the
examples that I've done so far,

00:04:35.748 --> 00:04:39.242
when I add an even to an
even, I get an even number.

00:04:39.242 --> 00:04:40.903
And I encourage you to
keep trying this out.

00:04:40.903 --> 00:04:42.526
I encourage you to keep trying,

00:04:42.526 --> 00:04:45.384
pick some even number,
then another even number,

00:04:45.384 --> 00:04:48.346
add them together and you'll
see that you keep getting

00:04:48.346 --> 00:04:50.098
even numbers.

00:04:50.098 --> 00:04:51.091
And it makes sense.

00:04:51.091 --> 00:04:53.497
Because if you had one number
that's a multiple of two,

00:04:53.497 --> 00:04:56.447
and you add it to another
number that's a multiple of two,

00:04:56.447 --> 00:04:57.944
it makes sense that the sum

00:04:57.944 --> 00:05:01.311
is going to be a multiple of two.

00:05:01.311 --> 00:05:05.224
Now what happens if you
add an odd and an odd?

00:05:05.224 --> 00:05:06.925
What happens if you add an odd number

00:05:06.925 --> 00:05:09.043
plus another odd number?

00:05:09.043 --> 00:05:10.064
Well let's try.

00:05:10.064 --> 00:05:12.432
What is one plus three?

00:05:12.432 --> 00:05:13.988
Well that's equal to four.

00:05:13.988 --> 00:05:17.111
You actually get an even number.

00:05:17.111 --> 00:05:19.185
You add two odd numbers,
you get an even number.

00:05:19.185 --> 00:05:21.046
Well maybe that was
just that special case.

00:05:21.046 --> 00:05:24.065
What if I add 15 plus seven?

00:05:24.065 --> 00:05:25.632
These are both odd numbers,

00:05:25.632 --> 00:05:27.921
when I add them together I get 22.

00:05:27.921 --> 00:05:30.068
I get another even number.

00:05:30.068 --> 00:05:31.309
This is interesting.

00:05:31.309 --> 00:05:36.309
If I take, let me say 19 plus 3.

00:05:36.568 --> 00:05:39.707
Actually that gets us 22 again.

00:05:39.707 --> 00:05:41.966
We have an even number.

00:05:41.966 --> 00:05:43.741
This is interesting.

00:05:43.741 --> 00:05:46.424
What about, what about, let's see...

00:05:46.424 --> 00:05:51.424
23 plus 5, well that gets us to 28.

00:05:51.509 --> 00:05:53.564
Once again, get an even number.

00:05:53.564 --> 00:05:56.443
So the pattern that
seems to be forming here

00:05:56.443 --> 00:05:59.682
is that if I have an even
plus an even I get an even,

00:05:59.682 --> 00:06:01.644
but also if I have an odd plus an odd,

00:06:01.644 --> 00:06:04.163
then I also get an even number.

00:06:04.163 --> 00:06:05.881
And once again, I encourage you to try out

00:06:05.881 --> 00:06:09.004
as many numbers as you can
to see if this pattern holds.

00:06:09.004 --> 00:06:10.629
And you will see that it is true.

00:06:10.629 --> 00:06:14.260
An odd plus an odd number is an even.

00:06:14.260 --> 00:06:17.385
Now let's think about
one last combination.

00:06:17.385 --> 00:06:19.939
What about an even,

00:06:19.939 --> 00:06:23.225
what about an even plus an odd?

00:06:23.225 --> 00:06:24.978
An even plus an odd.

00:06:24.978 --> 00:06:28.762
So let's say I have two plus one.

00:06:28.762 --> 00:06:31.920
Two plus one, well what
is that going to be?

00:06:31.920 --> 00:06:33.859
Well that's going to be equal to three.

00:06:33.859 --> 00:06:35.519
That's going to be an odd number.

00:06:35.519 --> 00:06:39.745
Well what about, what
about four plus three?

00:06:39.745 --> 00:06:43.076
Four plus three, well
that's equal to seven.

00:06:43.076 --> 00:06:45.757
That's equal to an odd number.

00:06:45.757 --> 00:06:48.266
And so it looks like if I
have an even plus an odd,

00:06:48.266 --> 00:06:49.995
and I've only tried out two cases here,

00:06:49.995 --> 00:06:52.097
but I encourage you to try
out many, many, many more

00:06:52.097 --> 00:06:54.139
to make sure that you
feel good about this,

00:06:54.139 --> 00:06:58.702
but an even and an odd is
always going to give you an odd.

00:06:58.702 --> 00:07:02.034
So this is always going
to give you an odd.

00:07:02.034 --> 00:07:03.299
So let's just remind ourselves,

00:07:03.299 --> 00:07:07.061
even numbers, they're multiples
of two, including zero.

00:07:07.061 --> 00:07:09.324
Odd numbers are just numbers
that aren't multiples of two,

00:07:09.324 --> 00:07:10.473
that aren't even.

00:07:10.473 --> 00:07:14.653
And an even plus an even is
going to be equal to an even.

00:07:14.653 --> 00:07:19.123
An odd plus an odd is also
going to be equal to an even.

00:07:19.123 --> 00:07:23.036
We saw that, we saw that multiple times.

00:07:23.036 --> 00:07:24.881
But an even plus an odd?

00:07:24.881 --> 00:07:27.815
That is odd.


Properties and patterns for multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.812 --> 00:00:03.020
We want to figure out how
many balloons we have here.

00:00:03.020 --> 00:00:04.720
And obviously, we
could just count these.

00:00:04.720 --> 00:00:06.190
But now, we have other
ways of thinking about it,

00:00:06.190 --> 00:00:07.648
especially because
they're arranged

00:00:07.648 --> 00:00:12.030
in this nice array, this
nice grid pattern here.

00:00:12.030 --> 00:00:14.830
And the reason why it's
useful to not just always have

00:00:14.830 --> 00:00:17.180
to count it, but to be able
to use little multiplication

00:00:17.180 --> 00:00:19.260
with the number of rows
and the number of columns

00:00:19.260 --> 00:00:21.094
is that you might
run into things

00:00:21.094 --> 00:00:22.510
and you will run
into things where

00:00:22.510 --> 00:00:25.340
it's very hard to count each
of the objects individually,

00:00:25.340 --> 00:00:28.250
but it might be a little
bit easier to count the rows

00:00:28.250 --> 00:00:29.940
and to count the columns.

00:00:29.940 --> 00:00:32.229
So for example,
right over here, we

00:00:32.229 --> 00:00:37.900
see that we have
1, 2, 3, 4 rows.

00:00:37.900 --> 00:00:47.230
And we have 1, 2, 3,
4, 5, 6, 7 columns.

00:00:47.230 --> 00:00:53.950
So you could view this
4 as an array of objects

00:00:53.950 --> 00:00:55.300
where we have 4 rows.

00:00:55.300 --> 00:00:56.570
Let me write that down.

00:00:56.570 --> 00:01:00.975
We have 4 rows and
we have 7 columns.

00:01:04.700 --> 00:01:07.420
And you might already
remember that we

00:01:07.420 --> 00:01:09.130
can calculate the
total number of objects

00:01:09.130 --> 00:01:12.420
by multiplying the rows
times the columns-- 4 rows

00:01:12.420 --> 00:01:14.320
times 7 columns.

00:01:14.320 --> 00:01:17.340
Now, why does this work?

00:01:17.340 --> 00:01:21.850
Why will this give us the
actual number of objects?

00:01:21.850 --> 00:01:24.760
Well, we could view
this-- we have 4 rows.

00:01:24.760 --> 00:01:26.730
So we have 4 groups of things.

00:01:26.730 --> 00:01:29.764
And how many things are
in each of those rows?

00:01:29.764 --> 00:01:30.930
Well, the number of columns.

00:01:30.930 --> 00:01:36.210
We have 7 things in each of
those 4 rows-- so 4 groups

00:01:36.210 --> 00:01:37.250
of 7.

00:01:37.250 --> 00:01:39.000
Or you could view it
the other way around.

00:01:39.000 --> 00:01:41.630
You could view that
each column is a group.

00:01:41.630 --> 00:01:43.700
So then you have 7 groups.

00:01:43.700 --> 00:01:45.651
And how many objects
do you have in each?

00:01:45.651 --> 00:01:47.150
Well, that's what
the rows tell you.

00:01:47.150 --> 00:01:51.250
You have 4 things in
each of those columns.

00:01:51.250 --> 00:01:53.759
And we already know that
both of these quantities

00:01:53.759 --> 00:01:56.300
are going to find the exact same
number, the number of things

00:01:56.300 --> 00:01:57.890
that we have right over here.

00:01:57.890 --> 00:01:59.780
So these two things
are equivalent.

00:01:59.780 --> 00:02:04.592
4 times 7 is equal to 7 times 4.

00:02:04.592 --> 00:02:06.050
And there's a bunch
of ways that we

00:02:06.050 --> 00:02:07.510
can calculate
either one of these.

00:02:07.510 --> 00:02:09.680
We can skip count by 4.

00:02:09.680 --> 00:02:19.027
We say 4 times 4, 8,
12, 16, 20, 24, 28.

00:02:19.027 --> 00:02:19.610
And let's see.

00:02:19.610 --> 00:02:20.901
Let me make sure that that's 7.

00:02:20.901 --> 00:02:26.490
So 4 times 1, 2, 3, 4,
5, 6, 7-- so we get 28.

00:02:26.490 --> 00:02:29.640
We could just calculate that
there are 28 objects here.

00:02:29.640 --> 00:02:32.580
And likewise, we
could skip count by 7.

00:02:32.580 --> 00:02:38.300
So we could go 7-- 7 times 2
is 14, times 3 is 21, times 4

00:02:38.300 --> 00:02:40.870
is 28, just adding
seven every time.

00:02:40.870 --> 00:02:44.416
And so we could get
28 the other way.

00:02:44.416 --> 00:02:45.790
Let's do that in
that same color.

00:02:45.790 --> 00:02:49.327
We could get 28 the other way.

00:02:49.327 --> 00:02:51.660
But if you had a situation
where you didn't know-- where

00:02:51.660 --> 00:02:53.250
you either didn't want
to do these techniques

00:02:53.250 --> 00:02:54.890
or it had been hard
to do these techniques

00:02:54.890 --> 00:02:57.550
or you didn't know what 4 times
7 was off the top of your head,

00:02:57.550 --> 00:03:00.110
which you should know at
some point in the very, very

00:03:00.110 --> 00:03:01.030
near future.

00:03:01.030 --> 00:03:03.030
Is there any way to break
this down to something

00:03:03.030 --> 00:03:05.520
that maybe you do know or maybe
that's a little bit easier

00:03:05.520 --> 00:03:06.820
to compute?

00:03:06.820 --> 00:03:09.420
Well, you could
realize that 7 columns

00:03:09.420 --> 00:03:12.690
is the same thing as 5
columns and then 2 columns.

00:03:12.690 --> 00:03:19.370
So you could view 7
columns as 5 columns--

00:03:19.370 --> 00:03:23.470
so this is 5 columns right
over here-- plus 2 columns.

00:03:31.320 --> 00:03:40.740
So that's just like
saying that 4 times 7

00:03:40.740 --> 00:03:47.390
is the same thing
as 4 times 5 plus 2.

00:03:52.900 --> 00:03:57.450
And all I do is I replace
the seven with a 5 plus 2.

00:03:57.450 --> 00:04:01.540
7 has been replaced
with a 5 plus 2.

00:04:01.540 --> 00:04:03.370
Now, why is this interesting?

00:04:03.370 --> 00:04:07.200
Well, now, I can break this
up into two separate arrays.

00:04:07.200 --> 00:04:09.760
So I could say,
well, look, there's

00:04:09.760 --> 00:04:13.590
the array that has 4 rows and
2 columns right over here.

00:04:13.590 --> 00:04:18.649
And then there's the array that
has 4 rows and 5 columns right

00:04:18.649 --> 00:04:20.380
over here.

00:04:20.380 --> 00:04:22.079
So how many objects
are in this one,

00:04:22.079 --> 00:04:24.150
in the yellow one
right over here?

00:04:24.150 --> 00:04:29.680
Well, there's 4 times 5 objects.

00:04:29.680 --> 00:04:35.955
So there's 4 times 5
objects in the yellow grid

00:04:35.955 --> 00:04:37.360
or yellow array.

00:04:37.360 --> 00:04:41.930
And how many in this
orange-ish looking thing?

00:04:41.930 --> 00:04:44.050
Well, there's going
to be 4 times 2.

00:04:49.880 --> 00:04:52.310
And if we take the sum
of the 4 times the 5

00:04:52.310 --> 00:04:54.580
and the 4 times the 2,
what are we going to get?

00:04:54.580 --> 00:04:56.800
Well, we're going to
get the 4 times the 7.

00:04:56.800 --> 00:05:00.680
We're going to get the
4 times the 5 plus 2.

00:05:00.680 --> 00:05:03.649
So if we take sum
of these things--

00:05:03.649 --> 00:05:05.440
and we want to do the
multiplication first,

00:05:05.440 --> 00:05:07.189
so I'll just put a
parentheses around that

00:05:07.189 --> 00:05:09.440
to emphasize that--
this is going

00:05:09.440 --> 00:05:14.490
to be the same thing as
these things up here.

00:05:14.490 --> 00:05:16.870
And so you might say, oh,
well, I know what 4 times 5 is.

00:05:16.870 --> 00:05:18.330
4 times 5 is 20.

00:05:20.890 --> 00:05:23.680
4 times 2 is 8.

00:05:23.680 --> 00:05:26.360
20 plus 8 is 28.

00:05:26.360 --> 00:05:30.460
And you might say,
OK, Sal, I get it.

00:05:30.460 --> 00:05:36.830
4 times 7 is 28, which is the
same thing as 4 times 5 plus 2.

00:05:36.830 --> 00:05:41.170
And I see that that's the same
thing as 4 times 5 plus 4 times

00:05:41.170 --> 00:05:41.670
2.

00:05:41.670 --> 00:05:44.003
And actually, this is called
the distributive property--

00:05:44.003 --> 00:05:47.200
that 4 times 5 plus 2 is
the same thing as 4 times 5

00:05:47.200 --> 00:05:49.190
plus 4 times 2.

00:05:49.190 --> 00:05:52.472
But I could just do one
of these first techniques

00:05:52.472 --> 00:05:53.180
you talked about.

00:05:53.180 --> 00:05:55.560
Why is this
distributive property

00:05:55.560 --> 00:05:59.310
that you just showed me
useful for computing or doing

00:05:59.310 --> 00:06:01.040
multiplication problems?

00:06:01.040 --> 00:06:07.210
Well, let me give you a
slightly more difficult one.

00:06:07.210 --> 00:06:20.850
Let's imagine you wanted
to multiply 6 times 36.

00:06:20.850 --> 00:06:24.580
Actually, I don't need to
write that parentheses.

00:06:24.580 --> 00:06:26.020
So how could you do this?

00:06:26.020 --> 00:06:29.850
Well, you could decompose
36 into two products

00:06:29.850 --> 00:06:31.410
or into two numbers
where it's easier

00:06:31.410 --> 00:06:33.800
to find the product
of that and 6.

00:06:33.800 --> 00:06:37.620
So for example 36, is the
same thing as 30 plus 6.

00:06:37.620 --> 00:06:44.118
So this is going to be
equal to 6 times 30 plus 6.

00:06:49.734 --> 00:06:50.900
And what's this going to be?

00:06:50.900 --> 00:06:51.650
Well, we just saw.

00:06:51.650 --> 00:06:54.990
6 times these two things
added together first,

00:06:54.990 --> 00:07:07.300
this is going to be equal to
6 times 30 plus 6 times 6.

00:07:13.510 --> 00:07:20.750
Notice, we distributed the
6-- 6 times 30 plus 6 times 6.

00:07:20.750 --> 00:07:22.520
Now, why is this useful?

00:07:22.520 --> 00:07:24.410
Why was this useful at all?

00:07:24.410 --> 00:07:26.160
I'm going to put
parentheses to emphasize.

00:07:26.160 --> 00:07:28.490
We're going to do the
multiplication first.

00:07:28.490 --> 00:07:30.700
In general, when you see
multiplication and addition

00:07:30.700 --> 00:07:32.359
in a row like this,
or division, you

00:07:32.359 --> 00:07:34.650
want to do your multiplication
and division first, then

00:07:34.650 --> 00:07:36.710
do your addition
and subtraction.

00:07:36.710 --> 00:07:37.690
So what's 6 times 30?

00:07:37.690 --> 00:07:39.200
Well, this is
easier to calculate.

00:07:39.200 --> 00:07:41.410
6 times 3, we know to be 18.

00:07:41.410 --> 00:07:45.220
So 6 times 30 is
going to be 180.

00:07:45.220 --> 00:07:49.140
And 6 times 6-- well, we
know that's going to be 36.

00:07:49.140 --> 00:07:53.200
So this is going
to be 180 plus 36.

00:07:53.200 --> 00:07:54.390
And what's that going to be?

00:07:54.390 --> 00:07:59.790
180 plus 36-- well,
0 plus 6 is 6.

00:07:59.790 --> 00:08:01.950
8 plus 3 is 11.

00:08:01.950 --> 00:08:03.880
1 plus 1 is 2.

00:08:03.880 --> 00:08:10.390
So you just figured out that
6 times 36 is equal to 216.

00:08:10.390 --> 00:08:13.115
And what we just did with
the distributive property,

00:08:13.115 --> 00:08:14.740
this is actually
going to be how you're

00:08:14.740 --> 00:08:18.040
going to multiply all sorts
of larger numbers, way larger

00:08:18.040 --> 00:08:19.340
than what we just saw.

00:08:19.340 --> 00:08:21.620
So the distributive property,
which hopefully you're

00:08:21.620 --> 00:08:24.850
pretty convinced by, based
on how we broke things up,

00:08:24.850 --> 00:08:26.750
is a super useful
thing as you want

00:08:26.750 --> 00:08:28.429
to compute larger
and larger numbers.

00:08:28.429 --> 00:08:30.470
And you're going to find
it even more useful when

00:08:30.470 --> 00:08:33.870
you go even further in
your mathematical career.


Order doesn't matter when purely multiplying

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.720
So if you look at each
of these 4 by 6 grids,

00:00:03.720 --> 00:00:06.800
it's pretty clear that there's
24 of these green circle

00:00:06.800 --> 00:00:08.150
things in each of them.

00:00:08.150 --> 00:00:10.940
But what I want to show
you is that you can get 24

00:00:10.940 --> 00:00:13.850
as the product of three numbers
in multiple different ways.

00:00:13.850 --> 00:00:17.140
And it actually doesn't matter
which products you take first

00:00:17.140 --> 00:00:19.760
or what order you
actually do them in.

00:00:19.760 --> 00:00:21.190
So let's think about this first.

00:00:21.190 --> 00:00:22.970
So the way that
I've colored it in,

00:00:22.970 --> 00:00:25.270
I have these three groups of 4.

00:00:25.270 --> 00:00:27.030
If you look at the
blue highlighting,

00:00:27.030 --> 00:00:30.273
this is one group of 4, two
groups of 4, three groups of 4.

00:00:30.273 --> 00:00:32.189
Actually, let me make
it a little bit clearer.

00:00:32.189 --> 00:00:37.890
One group of 4, two groups
of 4, and three groups of 4.

00:00:37.890 --> 00:00:43.380
So these three columns you
could view as 3 times 4.

00:00:43.380 --> 00:00:47.020
Now, we have another 3
times 4 right over here.

00:00:47.020 --> 00:00:48.930
This is also 3 times 4.

00:00:48.930 --> 00:00:52.000
We have one group of
4, two groups of 4,

00:00:52.000 --> 00:00:54.220
and three groups of 4.

00:00:54.220 --> 00:00:58.630
So you could view these
combined as 2 times 3 times 4.

00:00:58.630 --> 00:00:59.840
We have one 3 times 4.

00:00:59.840 --> 00:01:01.890
And then we have
another 3 times 4.

00:01:01.890 --> 00:01:07.480
So the whole thing we could
view as-- let me give myself

00:01:07.480 --> 00:01:13.180
some more space-- as 2 times--
let me do that in blue-- 2

00:01:13.180 --> 00:01:17.050
times 3 times 4.

00:01:17.050 --> 00:01:18.890
That's the total
number of balls here.

00:01:18.890 --> 00:01:21.620
And you could see it based
on how it was colored.

00:01:21.620 --> 00:01:24.950
And of course, if you did 3
times 4 first, you get 12.

00:01:24.950 --> 00:01:26.540
And then you multiply
that times 2,

00:01:26.540 --> 00:01:29.810
you get 24, which is the total
number of these green circle

00:01:29.810 --> 00:01:31.000
things.

00:01:31.000 --> 00:01:34.140
And I encourage you now to
look at these other two.

00:01:34.140 --> 00:01:36.310
Pause the video and
think about what

00:01:36.310 --> 00:01:37.810
these would be the
product of, first

00:01:37.810 --> 00:01:39.226
looking at the
blue grouping, then

00:01:39.226 --> 00:01:41.237
looking at the purple
grouping in the same way

00:01:41.237 --> 00:01:43.820
that we did right over here, and
verify that the product still

00:01:43.820 --> 00:01:45.700
equals 24.

00:01:45.700 --> 00:01:47.560
Well, I assume that
you've paused the video.

00:01:47.560 --> 00:01:52.100
So you see here in this first, I
guess you could call it a zone,

00:01:52.100 --> 00:01:55.160
we have two groups of 4.

00:01:55.160 --> 00:01:58.410
So this is 2 times
4 right over here.

00:01:58.410 --> 00:02:01.030
We have one group of
4, another group of 4.

00:02:01.030 --> 00:02:02.540
That's 2 times 4.

00:02:02.540 --> 00:02:04.760
We have one group of
4, another group of 4.

00:02:04.760 --> 00:02:07.970
So this is also 2 times 4 if
we look in this purple zone.

00:02:07.970 --> 00:02:10.470
One group of 4,
another group of 4.

00:02:10.470 --> 00:02:12.510
So this is also 2 times 4.

00:02:12.510 --> 00:02:15.160
So we have three 2 times 4's.

00:02:15.160 --> 00:02:18.060
So if we look at each of
these, or all together,

00:02:18.060 --> 00:02:24.490
this is 3 times 2 times
4, so 3 times 2 times 4.

00:02:27.730 --> 00:02:29.920
Notice I did a different order.

00:02:29.920 --> 00:02:32.540
And here I did 3 times 4 first.

00:02:32.540 --> 00:02:35.570
Here I'm doing 2 times 4 first.

00:02:35.570 --> 00:02:37.895
But just like before,
2 times 4 is 8.

00:02:37.895 --> 00:02:41.220
8 times 3 is still equal
to 24, as it needs to,

00:02:41.220 --> 00:02:44.960
because we have exactly 24
of these green circle things.

00:02:44.960 --> 00:02:47.650
Once again, pause the video
and try to do the same here.

00:02:47.650 --> 00:02:49.520
Look at the groupings
in blue, then look

00:02:49.520 --> 00:02:51.140
at the groupings
in purple, and try

00:02:51.140 --> 00:02:57.770
to express these 24 as some
kind of product of 2, 3, and 4.

00:02:57.770 --> 00:03:00.480
Well, you see first we
have these groupings of 3.

00:03:00.480 --> 00:03:02.810
So we have one grouping
of 3 in this purple zone,

00:03:02.810 --> 00:03:05.300
two groupings of 3
in this purple zone.

00:03:05.300 --> 00:03:08.030
So you could do
that as 2 times 3.

00:03:08.030 --> 00:03:10.860
And we have one 3 and another 3.

00:03:10.860 --> 00:03:13.970
So in this purple zone,
this is another 2 times 3.

00:03:13.970 --> 00:03:17.030
We have another 2 times 3.

00:03:17.030 --> 00:03:17.530
Whoops.

00:03:17.530 --> 00:03:19.090
I wrote 2 times 2.

00:03:19.090 --> 00:03:20.050
2 times 3.

00:03:20.050 --> 00:03:22.160
We have another 2 times 3.

00:03:24.930 --> 00:03:30.050
And then finally, we
have a fourth 2 times 3.

00:03:30.050 --> 00:03:34.530
So how many 2 times
3's do we have here?

00:03:34.530 --> 00:03:39.000
Well, we have one, two,
three, four 2 times 3's.

00:03:39.000 --> 00:03:48.430
So this whole thing could be
written as 4 times 2 times 3.

00:03:48.430 --> 00:03:50.240
Now, what's this
going to be equal to?

00:03:50.240 --> 00:03:52.130
Well, it needs to
be equal to 24.

00:03:52.130 --> 00:03:57.480
And we can verify 2 times 3
is 6 times 4 is, indeed, 24.

00:03:57.480 --> 00:04:00.860
So the whole idea of what
I'm trying to show here

00:04:00.860 --> 00:04:05.100
is that the order in which
you multiply does not matter.

00:04:05.100 --> 00:04:07.630
Let me make this very clear.

00:04:07.630 --> 00:04:10.990
Let me pick a different example,
a completely new example.

00:04:10.990 --> 00:04:20.360
So let's say that I
have 4 times 5 times 6.

00:04:20.360 --> 00:04:22.570
You can do this multiplication
in multiple ways.

00:04:22.570 --> 00:04:24.960
You could do 4 times 5 first.

00:04:24.960 --> 00:04:30.740
Or you could do 4
times 5 times 6 first.

00:04:30.740 --> 00:04:31.740
And you can verify that.

00:04:31.740 --> 00:04:33.156
I encourage you
to pause the video

00:04:33.156 --> 00:04:35.262
and verify that these two
things are equivalent.

00:04:35.262 --> 00:04:37.470
And this is actually called
the associative property.

00:04:37.470 --> 00:04:40.450
It doesn't matter how you
associate these things, which

00:04:40.450 --> 00:04:42.830
of these that you do first.

00:04:42.830 --> 00:04:44.410
Also, order does not matter.

00:04:44.410 --> 00:04:46.950
And we've seen that
multiple times.

00:04:46.950 --> 00:04:52.050
Whether you do this or
you do 5 times 4 times

00:04:52.050 --> 00:04:55.150
6-- notice I swapped the 5
and 4-- this doesn't matter.

00:04:55.150 --> 00:05:00.220
Or whether you do this
or 6 times 5 times 4,

00:05:00.220 --> 00:05:00.970
it doesn't matter.

00:05:00.970 --> 00:05:03.320
Here I swapped the
6 and the 5 times 4.

00:05:03.320 --> 00:05:06.150
All of these are going to
get the exact same value.

00:05:06.150 --> 00:05:08.190
And I encourage you
to pause the video.

00:05:08.190 --> 00:05:09.840
So when we're talking
about which one

00:05:09.840 --> 00:05:14.456
we do first, whether we do the 4
times 5 first or the 5 times 6,

00:05:14.456 --> 00:05:16.080
that's called the
associative property.

00:05:16.080 --> 00:05:19.150
It's kind of fancy word for
a reasonably simple thing.

00:05:19.150 --> 00:05:21.400
And when we're saying
that order doesn't matter,

00:05:21.400 --> 00:05:25.390
when it doesn't matter whether
we do 4 times 5 or 5 times 4,

00:05:25.390 --> 00:05:27.310
that's called the
commutative property.

00:05:27.310 --> 00:05:29.670
And once again, fancy word
for a very simple thing.

00:05:29.670 --> 00:05:33.530
It's just saying it doesn't
matter what order I do it in.


Introduction to types of quadrilaterals

WEBVTT
Kind: captions
Language: en

00:00:00.443 --> 00:00:02.206
- [Voiceover] What I wanna
talk about in this video

00:00:02.206 --> 00:00:05.260
is shapes with four sides,

00:00:05.260 --> 00:00:08.964
and the mathy word for
shapes with four sides

00:00:08.964 --> 00:00:10.798
is quadrilateral.

00:00:10.798 --> 00:00:12.062
Quadrilateral.

00:00:12.062 --> 00:00:14.559
And any time you see
quad as part of a word,

00:00:14.559 --> 00:00:17.426
it's a good, it's a good sign

00:00:17.426 --> 00:00:19.806
that you're dealing with
the number four somehow.

00:00:19.806 --> 00:00:22.859
So a quadrilateral is
something that has four sides.

00:00:22.859 --> 00:00:26.726
So that is a quadrilateral,

00:00:26.726 --> 00:00:29.755
this is a quadrilateral,

00:00:29.755 --> 00:00:32.159
this is a quadrilateral.

00:00:32.159 --> 00:00:33.378
They all have four sides.

00:00:33.378 --> 00:00:36.396
This is a quadrilateral.

00:00:36.396 --> 00:00:39.705
even this is a,

00:00:39.705 --> 00:00:42.456
let me make it look even weirder,

00:00:42.456 --> 00:00:45.579
even this is a quadrilateral.

00:00:45.579 --> 00:00:47.564
So what's not a quadrilateral?

00:00:47.564 --> 00:00:49.619
Well, a triangle would
not be a quadrilateral.

00:00:49.619 --> 00:00:50.524
It has three sides,

00:00:50.524 --> 00:00:51.638
one, two, three.

00:00:51.638 --> 00:00:53.635
So let's cross that out.

00:00:53.635 --> 00:00:55.714
A pentagon that has five sides,

00:00:55.714 --> 00:00:57.014
that would not be a quadrilateral.

00:00:57.014 --> 00:00:59.782
It has one, two, three, four, five sides.

00:00:59.782 --> 00:01:03.264
A circle that has, I guess
you could say, no sides,

00:01:03.264 --> 00:01:06.735
it's just one big curve, it's a circle.

00:01:06.735 --> 00:01:09.359
That's not going to be a quadrilateral.

00:01:09.359 --> 00:01:11.414
If you had six sides, seven
sides, a hundred sides,

00:01:11.414 --> 00:01:14.618
none of that is going
to be a quadrilateral.

00:01:14.618 --> 00:01:15.353
So now let's think about

00:01:15.353 --> 00:01:17.164
the different types of quadrilaterals,

00:01:17.164 --> 00:01:19.950
or the different categories
of quadrilaterals.

00:01:19.950 --> 00:01:23.085
So one is the parallelogram.

00:01:23.085 --> 00:01:26.766
So a parallelogram is a quadrilateral,

00:01:26.766 --> 00:01:28.066
and as we learn more math,

00:01:28.066 --> 00:01:31.211
we'll learn other ways
of thinking about this.

00:01:31.211 --> 00:01:35.367
It's a quadrilateral where
opposite sides are parallel.

00:01:35.367 --> 00:01:36.841
And parallel is just another way of saying

00:01:36.841 --> 00:01:39.268
that they're going in the same direction.

00:01:39.268 --> 00:01:41.061
So what do I mean by that?

00:01:41.061 --> 00:01:43.209
So something like this,

00:01:43.209 --> 00:01:45.229
something like this,

00:01:45.229 --> 00:01:47.423
would be a parallelogram.

00:01:47.423 --> 00:01:48.538
Why?

00:01:48.538 --> 00:01:52.315
Because this side is
opposite to this side,

00:01:52.315 --> 00:01:54.213
and they're pointed in the same direction.

00:01:54.213 --> 00:01:55.374
They're going,

00:01:55.374 --> 00:01:58.323
they're going, if I were to draw an arrow,

00:01:58.323 --> 00:01:59.623
if I were to draw an arrow here,

00:01:59.623 --> 00:02:04.139
those arrows are pointed in the same way.

00:02:04.139 --> 00:02:06.322
So those two sides are pointed,

00:02:06.322 --> 00:02:09.154
are parallel, is the word we say.

00:02:09.154 --> 00:02:11.941
And these two sides,

00:02:11.941 --> 00:02:15.296
these two sides right
over here are parallel.

00:02:15.296 --> 00:02:17.397
So this is a parallelogram.

00:02:17.397 --> 00:02:20.020
So what are other
examples of parallelogram?

00:02:20.020 --> 00:02:21.994
Well even your classic square,

00:02:21.994 --> 00:02:23.759
even your classic square,

00:02:23.759 --> 00:02:25.012
is a parallelogram.

00:02:25.012 --> 00:02:26.893
And we'll talk more about
what makes a square special.

00:02:26.893 --> 00:02:28.832
It's a special type of parallelogram

00:02:28.832 --> 00:02:31.595
because this side

00:02:31.595 --> 00:02:34.392
is going in the same
direction as that side,

00:02:34.392 --> 00:02:38.595
and this side, and this side,

00:02:38.595 --> 00:02:40.894
whoops, let me do that in yellow,

00:02:40.894 --> 00:02:44.794
and this side is parallel to that side.

00:02:44.794 --> 00:02:47.511
So what's not a parallelogram?

00:02:47.511 --> 00:02:49.833
Well, something like,

00:02:49.833 --> 00:02:53.065
something like this

00:02:53.065 --> 00:02:55.026
would not be a parallelogram.

00:02:55.026 --> 00:02:55.582
You might say, "Wait,

00:02:55.582 --> 00:02:57.882
"I see two opposite sides are parallel."

00:02:57.882 --> 00:03:02.224
You might say, "Look,
this is parallel to this."

00:03:02.224 --> 00:03:06.454
But then you would see that
this is not parallel to this.

00:03:06.454 --> 00:03:08.441
One way to think about some
things that are not parallel

00:03:08.441 --> 00:03:10.298
is if the lines kept going,

00:03:10.298 --> 00:03:12.677
they would cross each other at some point,

00:03:12.677 --> 00:03:14.279
while these lines,

00:03:14.279 --> 00:03:15.998
these lines right over here,

00:03:15.998 --> 00:03:18.180
they're never going to cross each other.

00:03:18.180 --> 00:03:19.933
So this one right over here

00:03:19.933 --> 00:03:21.720
is not a parallelogram.

00:03:21.720 --> 00:03:25.099
It has one set of opposite
sides being parallel,

00:03:25.099 --> 00:03:26.619
but not the other.

00:03:26.619 --> 00:03:29.616
Another example of something
that is not a parallelogram

00:03:29.616 --> 00:03:31.972
would be this one right over here,

00:03:31.972 --> 00:03:34.734
because none of the
opposite sides are parallel.

00:03:34.734 --> 00:03:38.299
So parallelogram, opposite
sides are parallel.

00:03:38.299 --> 00:03:39.948
So now let's talk about,

00:03:39.948 --> 00:03:44.050
let's talk about more
types of four sided shapes,

00:03:44.050 --> 00:03:46.256
or quadrilaterals.

00:03:46.256 --> 00:03:50.529
So the next one we'll
talk about is the rhombus.

00:03:50.529 --> 00:03:54.208
So the rhombus is a type of parallelogram.

00:03:54.208 --> 00:03:57.262
The opposite sides need to be parallel,

00:03:57.262 --> 00:04:00.999
but that's not, by itself,
that doesn't make it a rhombus.

00:04:00.999 --> 00:04:03.020
The opposite sides need to be parallel,

00:04:03.020 --> 00:04:06.816
and all the sides have to be equal.

00:04:06.816 --> 00:04:09.500
So for example,

00:04:09.500 --> 00:04:12.810
this that I'm drawing,

00:04:12.810 --> 00:04:16.321
that is a parallelogram,
but it is not a rhombus.

00:04:16.321 --> 00:04:18.897
It's a parallelogram because that side,

00:04:18.897 --> 00:04:21.277
these opposite sides are parallel.

00:04:21.277 --> 00:04:24.598
If they kept going, they
would not cross each other.

00:04:24.598 --> 00:04:28.556
And these two opposite sides are parallel.

00:04:28.556 --> 00:04:30.355
So it's a parallelogram,
but it's not a rhombus

00:04:30.355 --> 00:04:34.621
because the blue sides are
longer than the yellow sides.

00:04:34.621 --> 00:04:36.746
So that is not a rhombus.

00:04:36.746 --> 00:04:39.985
A rhombus would have to look like this.

00:04:39.985 --> 00:04:42.608
A rhombus would have to look like that.

00:04:42.608 --> 00:04:44.384
So opposite sides are parallel

00:04:44.384 --> 00:04:47.182
and all the sides are the same length.

00:04:47.182 --> 00:04:50.584
And now you might say, "Well
maybe a square is a rhombus,"

00:04:50.584 --> 00:04:51.721
and I want you to think about that.

00:04:51.721 --> 00:04:53.762
Is a square a rhombus?

00:04:53.762 --> 00:04:56.156
Are all the sides the same length,

00:04:56.156 --> 00:04:58.942
and are the opposite sides parallel?

00:04:58.942 --> 00:05:00.820
Well we already said
that the opposite sides

00:05:00.820 --> 00:05:01.933
of a square are parallel.

00:05:01.933 --> 00:05:03.825
A square is a parallelogram.

00:05:03.825 --> 00:05:07.644
And all the sides of a
square are the same length.

00:05:07.644 --> 00:05:11.034
So a square is a rhombus.

00:05:11.034 --> 00:05:14.691
So one way to think about
rhombuses, or rhombi,

00:05:14.691 --> 00:05:16.479
is they're squares,

00:05:16.479 --> 00:05:18.610
and you could kind of view them as

00:05:18.610 --> 00:05:21.273
kind of a pushed over version of squares,

00:05:21.273 --> 00:05:22.399
if a square was moving

00:05:22.399 --> 00:05:24.698
really, really, really fast in a cartoon,

00:05:24.698 --> 00:05:28.774
that's what my brain thinks
of when I think of a rhombus.

00:05:28.774 --> 00:05:30.500
So now let's think of rectangle.

00:05:30.500 --> 00:05:32.775
And you might have heard the
word rectangle in the past,

00:05:32.775 --> 00:05:33.959
but let's think a little bit more

00:05:33.959 --> 00:05:36.293
about what makes a rectangle.

00:05:36.293 --> 00:05:39.810
So a rectangle is going
to be a parallelogram,

00:05:39.810 --> 00:05:42.736
but that by itself does
not make it a rectangle.

00:05:42.736 --> 00:05:46.523
So for example, this right
over here is a rectangle.

00:05:46.523 --> 00:05:47.373
Why is it?

00:05:47.373 --> 00:05:49.532
Well it's definitely a parallelogram.

00:05:49.532 --> 00:05:52.527
This side and that side are parallel.

00:05:52.527 --> 00:05:54.942
They'll never intersect.

00:05:54.942 --> 00:05:59.528
And this side and this side are parallel.

00:05:59.528 --> 00:06:01.826
They're never, they're
never going to intersect

00:06:01.826 --> 00:06:03.606
if you kept going on and on and on,

00:06:03.606 --> 00:06:07.286
or they're never going
to cross each other.

00:06:07.286 --> 00:06:09.620
But what makes it a rectangle?

00:06:09.620 --> 00:06:10.560
It's definitely a parallelogram,

00:06:10.560 --> 00:06:13.880
but what makes it, why do
we use the word rectangle?

00:06:13.880 --> 00:06:16.063
Well one way to think about it is

00:06:16.063 --> 00:06:19.058
they way that they come
together at the corners.

00:06:19.058 --> 00:06:22.343
So in a rectangle, the things come in to,

00:06:22.343 --> 00:06:24.642
I guess you could call
them square corners.

00:06:24.642 --> 00:06:26.963
And that's called a right angle.

00:06:26.963 --> 00:06:29.134
That's called a right
angle right over there.

00:06:29.134 --> 00:06:30.585
So this is what makes a rectangle.

00:06:30.585 --> 00:06:34.057
It's a parallelogram where all
the corners are right angles.

00:06:34.057 --> 00:06:35.902
You could put a little square there,

00:06:35.902 --> 00:06:37.702
if you wanna think about it that way.

00:06:37.702 --> 00:06:41.416
So for example, this right over here

00:06:41.416 --> 00:06:43.052
would not be a rectangle.

00:06:43.052 --> 00:06:43.444
Why?

00:06:43.444 --> 00:06:45.596
Cause you can't, if you
put a square here, notice.

00:06:45.596 --> 00:06:48.702
That does not, a square
doesn't fit in the corners

00:06:48.702 --> 00:06:50.957
the way that it fits right over here.

00:06:50.957 --> 00:06:55.067
That you have, the square
does not fit over here.

00:06:55.067 --> 00:06:57.806
This is a parallelogram,
but not a rectangle.

00:06:57.806 --> 00:07:02.149
A rectangle is a parallelogram
that has square corners.

00:07:02.149 --> 00:07:04.273
But what about our square?

00:07:04.273 --> 00:07:06.316
Is a square a rectangle?

00:07:06.316 --> 00:07:08.754
Well let's draw it out.

00:07:08.754 --> 00:07:09.397
Well let's think about it.

00:07:09.397 --> 00:07:11.661
A square, opposite sides are parallel.

00:07:11.661 --> 00:07:13.576
We've already said it's a parallelogram.

00:07:13.576 --> 00:07:16.919
And a square has,

00:07:16.919 --> 00:07:19.578
the corners are square.

00:07:19.578 --> 00:07:21.401
That's where when people
say make a square corner,

00:07:21.401 --> 00:07:22.711
that's where it comes from.

00:07:22.711 --> 00:07:24.059
The corners are square.

00:07:24.059 --> 00:07:25.580
They are at right angles.

00:07:25.580 --> 00:07:27.668
So the square is a rectangle.

00:07:27.668 --> 00:07:31.291
So the square is a really
interesting quadrilateral,

00:07:31.291 --> 00:07:35.355
because a square falls in
to all of these categories.

00:07:35.355 --> 00:07:38.337
The square, a square is a square,

00:07:38.337 --> 00:07:39.657
it's a rhombus,

00:07:39.657 --> 00:07:41.212
it's the type of rhombus

00:07:41.212 --> 00:07:44.805
where the corners are right angles,

00:07:44.805 --> 00:07:46.825
or you could say where
the corners are square.

00:07:46.825 --> 00:07:48.601
This one right over here is not a square,

00:07:48.601 --> 00:07:49.472
this one is square.

00:07:49.472 --> 00:07:52.107
They're both rhombuses,
or they're both rhombi.

00:07:52.107 --> 00:07:54.162
A square is also a rectangle.

00:07:54.162 --> 00:07:58.038
It's a parallelogram where
the corners are right angles,

00:07:58.038 --> 00:07:59.641
where they are square.

00:07:59.641 --> 00:08:01.637
A square is definitely a parallelogram,

00:08:01.637 --> 00:08:05.126
and everything we've talked
about is a quadrilateral.


Arithmetic word problems with volume

WEBVTT
Kind: captions
Language: en

00:00:00.557 --> 00:00:04.429
- [Voiceover] Li Min squeezed
three lemons to make lemonade.

00:00:04.429 --> 00:00:08.804
She squeezed 42 millimeters of lemon juice

00:00:08.804 --> 00:00:10.972
from the first lemon.

00:00:10.972 --> 00:00:15.418
36 milliliters of juice
from the second lemon,

00:00:16.203 --> 00:00:20.046
and 45 milliliters of juice

00:00:20.046 --> 00:00:21.829
from the third lemon.

00:00:21.829 --> 00:00:23.630
How much lemon juice did Li MIn

00:00:23.630 --> 00:00:25.830
squeeze from her three lemons?

00:00:25.830 --> 00:00:27.968
Well we just have to add
how many milliliters she got

00:00:27.968 --> 00:00:30.139
from each of them, add them all together.

00:00:30.139 --> 00:00:33.254
So from the first one,
she got 42 milliliters

00:00:33.254 --> 00:00:36.862
42, and short for milliliters I can write

00:00:36.862 --> 00:00:41.227
lowercase m for mili, and
uppercase L for liters.

00:00:41.227 --> 00:00:43.633
42 milliliters from the first lemon.

00:00:43.633 --> 00:00:47.467
36 milliliters from the second lemon,

00:00:49.373 --> 00:00:52.546
and then 45 milliliters
from the third lemon.

00:00:55.020 --> 00:00:58.188
So we can just add these
three quantities together.

00:00:58.188 --> 00:01:00.500
So let's do that, let's add them together.

00:01:00.500 --> 00:01:02.797
So we're gonna get something
in terms of milliliters.

00:01:02.797 --> 00:01:04.909
So let's go first to the ones place.

00:01:04.909 --> 00:01:08.404
Two plus six is eight,
eight plus five is 13.

00:01:08.404 --> 00:01:11.207
13 is three ones and one 10,

00:01:11.207 --> 00:01:13.500
so I could write that
one 10 in the 10s place.

00:01:13.500 --> 00:01:16.755
So one plus four is five,
plus three is eight,

00:01:16.755 --> 00:01:20.315
plus four is 12, so it's gonna be 12 10s.

00:01:20.315 --> 00:01:22.997
Which is the same thing as 120.

00:01:22.997 --> 00:01:26.570
So in total, Li Min
squeezed 123 milliliters

00:01:26.570 --> 00:01:29.311
from her three lemons.

00:01:29.311 --> 00:01:31.752
Let's do another problem like this.

00:01:33.321 --> 00:01:37.055
Sarah poured five cans of motor oil

00:01:37.055 --> 00:01:38.736
into the container below.

00:01:38.736 --> 00:01:42.016
So this container below has 55,

00:01:42.016 --> 00:01:44.352
let's see this is halfway, it's filled up

00:01:44.352 --> 00:01:46.664
after she poured the five cans,

00:01:46.664 --> 00:01:50.089
and it's filled up
halfway between 40 and 50,

00:01:50.089 --> 00:01:54.344
so this mark right here, this
looks like that would be 45.

00:01:54.344 --> 00:01:57.831
So it's filled up to 45 milliliters.

00:01:57.831 --> 00:02:01.151
So she poured five cans,
and those five cans combined

00:02:01.151 --> 00:02:03.792
ended up being 45 milliliters.

00:02:03.792 --> 00:02:06.311
All of the cans held
the same amount of oil,

00:02:06.311 --> 00:02:07.448
and I guess we can assume that

00:02:07.448 --> 00:02:11.214
this container below was empty

00:02:11.214 --> 00:02:13.703
before she filled any of the cans in.

00:02:13.703 --> 00:02:16.031
So all of the cans held
the same amount of oil.

00:02:16.031 --> 00:02:19.040
How many milliliters of
motor oil were in each can?

00:02:19.040 --> 00:02:23.276
So she had five cans, so one,

00:02:23.276 --> 00:02:27.679
one can, two cans, three cans,

00:02:27.679 --> 00:02:31.456
four cans, and five cans.

00:02:31.456 --> 00:02:34.428
They each had a certain
amount of oil in them.

00:02:34.428 --> 00:02:36.790
They each had a certain
amount of oil in them.

00:02:36.790 --> 00:02:39.459
And when she put them
all in, so let's say they

00:02:39.459 --> 00:02:42.180
each have a question mark of oil in them.

00:02:42.180 --> 00:02:46.434
So she had five cans, and
they each had a question mark

00:02:46.434 --> 00:02:49.080
amount, say question mark
milliliters of oil in it.

00:02:49.080 --> 00:02:51.920
And when you multiplied by five,

00:02:51.920 --> 00:02:54.864
when you put all five together
you got 45 milliliters.

00:02:54.864 --> 00:02:57.959
You got 45 milliliters.

00:02:57.959 --> 00:03:01.753
So five times what amount of oil per can

00:03:01.753 --> 00:03:03.881
would get us to 45 in total?

00:03:03.881 --> 00:03:06.632
Well five times nine is 45.

00:03:06.632 --> 00:03:08.701
So this thing right over here,

00:03:08.701 --> 00:03:12.321
five times nine is 45.

00:03:12.321 --> 00:03:16.800
So Sarah had nine milliliters in each can,

00:03:16.800 --> 00:03:18.763
they each had the same
amount, they say that.

00:03:18.763 --> 00:03:21.994
All the cans held the same amount of oil.

00:03:21.994 --> 00:03:25.719
So if she had nine milliliters
in each one, notice,

00:03:25.719 --> 00:03:28.987
if in each of the five cans
you had nine milliliters,

00:03:28.987 --> 00:03:32.337
five times nine is 45.

00:03:32.337 --> 00:03:35.949
So she had nine milliliters
of motor oil in each can.


Mass problems

WEBVTT
Kind: captions
Language: en

00:00:00.288 --> 00:00:01.504
- [Voiceover] Let's see
if we can get a sense

00:00:01.504 --> 00:00:03.455
of different masses,
and we're gonna do that

00:00:03.455 --> 00:00:05.759
by doing the arithmetic
word problems with mass

00:00:05.759 --> 00:00:07.464
on Kahn Academy.

00:00:07.464 --> 00:00:10.497
So it says which is the
closest to the mass of a cow?

00:00:10.497 --> 00:00:14.940
So it's either seven
kilograms or 700 kilograms.

00:00:15.816 --> 00:00:19.130
So let's think about what
seven kilograms would be.

00:00:19.130 --> 00:00:22.805
So one kilogram if you're
used to dealing with pounds,

00:00:22.805 --> 00:00:25.500
one kilogram is 2.2 pounds.

00:00:25.500 --> 00:00:29.531
So seven kilograms is
like 15 or 16 pounds.

00:00:29.531 --> 00:00:34.070
That would be the weight
of maybe a large cat,

00:00:34.070 --> 00:00:38.283
or a several-month-old baby.

00:00:38.283 --> 00:00:40.717
That would be seven kilograms.

00:00:40.717 --> 00:00:44.939
So most cows are not the
size of just a large cat,

00:00:44.939 --> 00:00:46.957
they are much bigger than that.

00:00:46.957 --> 00:00:50.719
700 kilograms, that actually
seems a little bit more right.

00:00:50.719 --> 00:00:55.231
That's going to be closer to 1500 pounds.

00:00:55.231 --> 00:00:58.864
Cows are really, really big
animals, and they weigh a lot.

00:00:58.864 --> 00:01:01.303
So I would go with 700 kilograms,

00:01:01.303 --> 00:01:03.543
because seven kilograms
is just ridiculous.

00:01:03.543 --> 00:01:05.862
That's the size of a,

00:01:05.862 --> 00:01:08.919
as I said, a large cat.

00:01:08.919 --> 00:01:10.961
Let's keep going.

00:01:12.303 --> 00:01:15.597
All of Glinda's magic
wands have the same mass.

00:01:15.597 --> 00:01:18.653
Glinda knows the mass of one magic wand.

00:01:18.653 --> 00:01:21.597
What is the total mass
of seven magic wands?

00:01:21.597 --> 00:01:25.177
So it tells us the mass of
one magic wand is six grams.

00:01:25.177 --> 00:01:28.312
And just so you know,
six grams is not a lot.

00:01:28.312 --> 00:01:31.216
So you might, in the last exercise we saw,

00:01:31.216 --> 00:01:35.777
we were talking about seven
kilograms, or 700 kilograms.

00:01:35.777 --> 00:01:39.711
When you put a kilo in front
of gram, that means 1000 grams.

00:01:39.711 --> 00:01:43.709
So seven kilograms is 7,000 grams.

00:01:43.709 --> 00:01:46.630
Here we're just taking
about straight-up grams.

00:01:46.630 --> 00:01:50.109
So grams are not much mass at all.

00:01:50.109 --> 00:01:52.517
If you were to think
about a teaspoon of sugar,

00:01:52.517 --> 00:01:55.482
that's about four or five grams of sugar.

00:01:55.482 --> 00:01:58.005
So this wand is very very light.

00:01:58.005 --> 00:01:59.496
But anyway, what they're asking us is

00:01:59.496 --> 00:02:02.054
what would be the mass
of seven magic wands?

00:02:02.054 --> 00:02:04.030
Well if each of them are six grams,

00:02:04.030 --> 00:02:07.605
and we have seven of them,
seven times six is 42.

00:02:07.605 --> 00:02:11.405
So we're gonna have 42 grams.

00:02:11.405 --> 00:02:13.997
Let's keep going, this is fun.

00:02:13.997 --> 00:02:18.269
All crayons at Caryn's Crayon
Factory have the same mass.

00:02:18.269 --> 00:02:21.161
Caryn weighs seven crayons and finds that

00:02:21.161 --> 00:02:23.370
they have the total mass of 56 grams.

00:02:23.370 --> 00:02:25.681
What is the mass of one of the crayons?

00:02:25.681 --> 00:02:28.769
So seven crayons have a mass of 56 grams.

00:02:28.769 --> 00:02:31.376
So the mass of each of
them, let's just take 56

00:02:31.376 --> 00:02:34.561
and divide it by seven,
by the seven crayons.

00:02:34.561 --> 00:02:37.089
Well 56 divided by seven is eight.

00:02:37.089 --> 00:02:39.249
Seven times eight is 56.

00:02:39.249 --> 00:02:41.844
So the mass of each crayon is eight grams.

00:02:41.844 --> 00:02:43.096
And you could even check that.

00:02:43.096 --> 00:02:45.987
Seven crayons times
eight grams per crayon,

00:02:45.987 --> 00:02:49.253
is going to be 56 grams.

00:02:49.253 --> 00:02:51.767
This is surprisingly fun.

00:02:51.767 --> 00:02:54.703
Tina has five pencils.

00:02:54.703 --> 00:02:57.247
All of her pencils have the same mass.

00:02:57.247 --> 00:03:00.462
Tina finds the mass of
one pencil shown below.

00:03:00.462 --> 00:03:02.695
And it's eight grams.

00:03:02.695 --> 00:03:05.609
What is the total mass of five pencils?

00:03:05.609 --> 00:03:08.183
Well if one pencil is eight grams,

00:03:08.183 --> 00:03:10.871
five pencils are gonna be
five times eight grams.

00:03:10.871 --> 00:03:12.810
Well five times eight is 40.

00:03:12.810 --> 00:03:15.614
It's going to be 40 grams.

00:03:17.844 --> 00:03:19.619
All right, this one seems similar.

00:03:21.449 --> 00:03:24.562
All of Sophia's Squirrel's
acorns have the same mass.

00:03:24.562 --> 00:03:26.597
Sophia's squirrel finds the total mass

00:03:26.597 --> 00:03:28.701
of her four acorns shown below.

00:03:28.701 --> 00:03:30.962
What is the mass of one of the acorns?

00:03:30.962 --> 00:03:32.402
So here, actually this
is a little different

00:03:32.402 --> 00:03:34.202
than the last one, they find the mass

00:03:34.202 --> 00:03:37.394
of all four acorns are 20 grams.

00:03:37.394 --> 00:03:39.524
So what's the mass of each of them?

00:03:39.524 --> 00:03:42.028
Well, four times what is 20?

00:03:42.028 --> 00:03:44.585
Or 20 divided by four is what?

00:03:44.585 --> 00:03:47.682
20 divided by four is five.

00:03:47.682 --> 00:03:51.153
So five, 10, 15, 20.

00:03:51.153 --> 00:03:53.057
So if each of them are five grams,

00:03:53.057 --> 00:03:56.977
five grams times four is
gonna have a mass of 20 grams.


Intuition for grams

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:01.951
- So let's talk about the unit

00:00:01.951 --> 00:00:05.208
of measurement called the "gram."

00:00:05.208 --> 00:00:10.208
And the gram is a unit of
measurement for measuring mass.

00:00:10.743 --> 00:00:13.430
Now, what is mass?

00:00:13.430 --> 00:00:15.615
Well, you might have seen
in everyday language,

00:00:15.615 --> 00:00:17.895
someone might look at a feather,

00:00:17.895 --> 00:00:20.875
let me see if I can draw a good feather.

00:00:20.875 --> 00:00:22.824
So a feather like this.

00:00:22.824 --> 00:00:26.561
And say, "Hey, this is not very massive."

00:00:26.561 --> 00:00:28.931
This is not massive.

00:00:28.931 --> 00:00:31.531
And notice, you have that
word "mass" in there.

00:00:31.531 --> 00:00:35.405
And then they might point
to some big boulder,

00:00:35.405 --> 00:00:40.249
some big heavy boulder, let
me draw a big heavy boulder.

00:00:40.249 --> 00:00:43.140
And say, "This is massive."

00:00:43.140 --> 00:00:47.128
This is massive.

00:00:47.128 --> 00:00:48.097
And so, just with those examples,

00:00:48.097 --> 00:00:50.651
you might get an intuition
for what mass is.

00:00:50.651 --> 00:00:55.651
It's how much stuff there is,
and it's related to weight.

00:00:56.380 --> 00:00:57.605
And I said "related to weight,"

00:00:57.605 --> 00:00:59.217
not the same thing as weight,

00:00:59.217 --> 00:01:00.890
because later on when you take Physics,

00:01:00.890 --> 00:01:03.793
you will learn that mass
and weight are related,

00:01:03.793 --> 00:01:06.869
but that they're not
exactly the same thing.

00:01:06.869 --> 00:01:09.197
Now, if you're on a given planet,

00:01:09.197 --> 00:01:13.347
and the same distance from
the center of that planet,

00:01:13.347 --> 00:01:15.759
roughly speaking, mass and weight

00:01:15.759 --> 00:01:17.921
will be strongly related to each other.

00:01:17.921 --> 00:01:20.219
Something with more mass is
going to have more weight.

00:01:20.219 --> 00:01:21.798
Two things with the same mass,

00:01:21.798 --> 00:01:23.628
once again if I'm on the same planet,

00:01:23.628 --> 00:01:25.292
at the same distance from
the center of the planet,

00:01:25.292 --> 00:01:27.301
they're going to have the same weight.

00:01:27.301 --> 00:01:30.109
So you'll often hear people say,

00:01:30.109 --> 00:01:32.871
"How many grams does that thing weigh?"

00:01:32.871 --> 00:01:33.825
They really should be saying,

00:01:33.825 --> 00:01:35.961
"How many grams does that thing mass,"

00:01:35.961 --> 00:01:38.144
but it's okay, because as long
as we're staying on earth,

00:01:38.144 --> 00:01:41.429
roughly you know, at sea level,
plus or minus a little bit,

00:01:41.429 --> 00:01:43.553
or even if we're a little bit
further away from sea level,

00:01:43.553 --> 00:01:46.405
mass and weight, it's not crazy

00:01:46.405 --> 00:01:48.105
to hold them a little bit interchangeable.

00:01:48.105 --> 00:01:51.064
And one way to gauge something's mass

00:01:51.064 --> 00:01:53.409
is to hold it if you can, and say,

00:01:53.409 --> 00:01:54.431
"Okay, how much does this weigh,"

00:01:54.431 --> 00:01:56.344
because the more something weighs,

00:01:56.344 --> 00:01:57.890
the more its mass is going to be.

00:01:57.890 --> 00:02:01.327
The more something's mass is going to be,

00:02:01.327 --> 00:02:03.660
the more it's going to weigh, once again,

00:02:03.660 --> 00:02:05.239
assuming we're on the planet, etc., etc.

00:02:05.239 --> 00:02:06.888
When you take Physics
class, you'll learn that

00:02:06.888 --> 00:02:08.606
they are different concepts that

00:02:08.606 --> 00:02:11.367
are closely related to each other.

00:02:11.367 --> 00:02:12.898
So, that's what a gram is.

00:02:12.898 --> 00:02:14.062
Now, another thing that you will see

00:02:14.062 --> 00:02:16.796
in the context of mass and measuring mass,

00:02:16.796 --> 00:02:18.509
is the kilogram.

00:02:22.093 --> 00:02:25.641
And all the kilogram is,
and you're going to see

00:02:25.641 --> 00:02:30.038
the prefix "kilo-" over and over again,

00:02:30.038 --> 00:02:33.507
in your, I guess your education careers,

00:02:33.507 --> 00:02:35.214
especially when you're
thinking about units.

00:02:35.214 --> 00:02:37.973
But "kilo-" is just a prefix for 1000.

00:02:37.973 --> 00:02:41.945
So that literally means 1000 grams.

00:02:41.945 --> 00:02:45.147
So instead of saying "2000 grams,"

00:02:45.147 --> 00:02:48.223
I could say "2 kilograms."

00:02:48.223 --> 00:02:49.188
So with that out of the way, let's see

00:02:49.188 --> 00:02:53.011
if we can get a sense for what one gram

00:02:53.011 --> 00:02:55.146
or different numbers
of grams of masses are

00:02:55.146 --> 00:02:58.121
and what might have a mass of one kilogram

00:02:58.121 --> 00:03:01.095
or 10 kilograms or 100 kilograms.

00:03:01.095 --> 00:03:05.749
And that's exactly why I got
all of these pictures prepared.

00:03:05.749 --> 00:03:07.649
So if, if you want to
get a sense of something

00:03:07.649 --> 00:03:12.649
that has a mass of one
gram, think of a paper clip.

00:03:13.386 --> 00:03:15.160
And I have this little
squiggly thing here,

00:03:15.160 --> 00:03:17.798
because that right there,
that means "roughly."

00:03:17.798 --> 00:03:20.540
Every paper clip isn't
going to be exactly 1 gram;

00:03:20.540 --> 00:03:22.351
it might be 1 1/2 grams, it might be

00:03:22.351 --> 00:03:24.760
small paper clips that are 1/2 gram.

00:03:24.760 --> 00:03:25.653
But it gives you the sense of

00:03:25.653 --> 00:03:27.917
what would have a mass of 1 gram.

00:03:27.917 --> 00:03:30.154
And if you want to know how
much something with a mass

00:03:30.154 --> 00:03:33.185
of 1 gram weighs on the
surface of the earth,

00:03:33.185 --> 00:03:36.242
well, you could just hold that
paper clip and kind of judge,

00:03:36.242 --> 00:03:40.163
feel how much it's
pulling down on your hand.

00:03:40.163 --> 00:03:42.191
Now, or how much the weight of it

00:03:42.191 --> 00:03:44.564
is pushing down on your
hand, I guess I could say.

00:03:44.564 --> 00:03:45.806
Now, if you want to get a sense of things

00:03:45.806 --> 00:03:47.466
that are little bit more massive,

00:03:47.466 --> 00:03:51.061
a pencil is five times as
massive as a paper clip.

00:03:51.061 --> 00:03:53.245
And hopefully you'll see that,
if you hold five paper clips

00:03:53.245 --> 00:03:55.254
on one side and a pencil
on the other side,

00:03:55.254 --> 00:03:57.390
you'll see that they
roughly have the same mass,

00:03:57.390 --> 00:04:02.288
and their weight will be roughly the same.

00:04:02.288 --> 00:04:03.914
Now, a little bit more massive,

00:04:03.914 --> 00:04:05.110
you could think of a field mouse.

00:04:05.110 --> 00:04:07.134
A field mouse is equivalent to the mass

00:04:07.134 --> 00:04:11.502
of roughly 4 pencils, or 20 paperclips.

00:04:11.502 --> 00:04:12.751
You want to get even more massive,

00:04:12.751 --> 00:04:14.713
you can think of a kitten.

00:04:14.713 --> 00:04:16.546
So, a kitten after a couple of weeks,

00:04:16.546 --> 00:04:19.209
it's still a kitten now, it's
still not a fully grown cat,

00:04:19.209 --> 00:04:23.189
it'll have a mass of
about 200 to 300 grams.

00:04:23.189 --> 00:04:26.449
And this makes a rough amount of sense,

00:04:26.449 --> 00:04:29.490
you can imagine this
thing having the same mass

00:04:29.490 --> 00:04:33.147
as maybe 10 or 15 of
these little field mice.

00:04:33.147 --> 00:04:37.628
You can imagine the
mouse is about that size,

00:04:37.628 --> 00:04:38.905
right over there.

00:04:41.811 --> 00:04:43.512
I could do a better job drawing a mouse.

00:04:43.512 --> 00:04:46.001
But that makes some sense.

00:04:46.001 --> 00:04:47.224
So what if we get even larger?

00:04:47.224 --> 00:04:50.399
What if we get into the
ranges of, say, 1000 grams?

00:04:50.399 --> 00:04:52.908
Well, now we can start
talking about kilograms.

00:04:52.908 --> 00:04:55.223
And just to be clear,
when I put a "g" here,

00:04:55.223 --> 00:04:57.164
that's just an abbreviation for "gram."

00:04:57.164 --> 00:05:01.694
So this is 1 gram, 5 grams, 20 grams.

00:05:01.694 --> 00:05:03.647
And now, if you want to go to 1000 grams,

00:05:03.647 --> 00:05:06.105
you could think of something like

00:05:06.105 --> 00:05:10.672
a hard-cover textbook like this one.

00:05:10.672 --> 00:05:13.324
And they're obviously,
some textbooks are huge,

00:05:13.324 --> 00:05:14.794
but the one that I'm showing here

00:05:14.794 --> 00:05:17.390
I didn't weigh or figure out
the mass of this exact one.

00:05:17.390 --> 00:05:19.359
But it would have a mass of roughly

00:05:19.359 --> 00:05:23.445
1 kilogram, or 1000 grams.

00:05:23.445 --> 00:05:27.332
So that's a kilogram right
over there, 1000 grams.

00:05:27.332 --> 00:05:31.476
A fully grown cat, that's
about 3 to 5 kilograms,

00:05:31.476 --> 00:05:33.264
depending on how fully grown it is

00:05:33.264 --> 00:05:35.882
or I guess how big of a cat it is.

00:05:35.882 --> 00:05:37.762
So, that's pretty
interesting, a fully grown cat

00:05:37.762 --> 00:05:42.762
is more than 10 times
the mass of a kitten.

00:05:43.857 --> 00:05:46.725
Actually that makes sense,
most fully grown humans

00:05:46.725 --> 00:05:49.487
are more than 10 times their mass

00:05:49.487 --> 00:05:51.751
when they were very, very young,

00:05:51.751 --> 00:05:53.625
or actually roughly 10
times the mass of when

00:05:53.625 --> 00:05:58.572
they were very young, so
it actually makes sense.

00:05:58.572 --> 00:06:00.763
Now, if you want to get even larger,

00:06:00.763 --> 00:06:04.169
you can go to a large
ape, like a human being,

00:06:04.169 --> 00:06:06.860
in particular, this one right over here.

00:06:06.860 --> 00:06:08.232
70 kilograms, and this is actually

00:06:08.232 --> 00:06:12.591
pretty close to my actual
mass, 70 kilograms.

00:06:12.591 --> 00:06:14.486
And you get a sense of
what that would mean

00:06:14.486 --> 00:06:16.869
if you're used to dealing with pounds.

00:06:16.869 --> 00:06:19.876
I weigh a little over 160 pounds,

00:06:19.876 --> 00:06:21.582
so that's about 70 kilograms.

00:06:21.582 --> 00:06:25.425
Or 1 kilogram is roughly 2.2 pounds

00:06:25.425 --> 00:06:27.773
if you are used to thinking
in terms of pounds.

00:06:27.773 --> 00:06:30.739
But, once again, pounds
are a unit of weight,

00:06:30.739 --> 00:06:34.046
while kilograms are a unit of mass.

00:06:34.046 --> 00:06:35.836
And then, if you want to get even heavier,

00:06:35.836 --> 00:06:38.809
you would get 10 times more massive

00:06:38.809 --> 00:06:41.259
than a human being, than a
normal-sized human being,

00:06:41.259 --> 00:06:43.999
you might get to something like this cow,

00:06:43.999 --> 00:06:47.463
and this cow would have
a mass of 750 kilograms.

00:06:47.463 --> 00:06:49.084
So, how many paper clips would you need

00:06:49.084 --> 00:06:51.312
to have the same mass as this cow?

00:06:51.312 --> 00:06:53.484
Well, you might be tempted to say 750,

00:06:53.484 --> 00:06:58.484
but remember this is 1
gram, this is 750 kilograms.

00:06:58.967 --> 00:07:03.967
So the mass of this cow
is actually 750,000 grams.

00:07:04.744 --> 00:07:08.300
So you would need about 750,000 of these

00:07:08.300 --> 00:07:12.555
paper clips to have the
same mass as this cow.


Marking data on line plots

WEBVTT
Kind: captions
Language: en

00:00:00.317 --> 00:00:01.454
- [Voiceover] Let's get a little practice

00:00:01.454 --> 00:00:02.905
doing the Khan Academy exercise

00:00:02.905 --> 00:00:05.833
Marking Data on Line Plots.

00:00:06.174 --> 00:00:07.234
So right over here it says,

00:00:07.234 --> 00:00:08.514
"Tessa measured the distance

00:00:08.514 --> 00:00:10.414
"from her house to several locations

00:00:10.414 --> 00:00:13.877
"to the nearest half kilometer."

00:00:14.297 --> 00:00:16.317
All right, and so they
have different locations

00:00:16.317 --> 00:00:19.137
and it tells us the
distance from her house

00:00:19.137 --> 00:00:20.457
to each of these locations.

00:00:20.457 --> 00:00:23.247
So from her house to the
school is 3 1/2 kilometers,

00:00:23.247 --> 00:00:25.148
from the library is 5 1/2 kilometers,

00:00:25.148 --> 00:00:26.630
to the park is two kilometers,

00:00:26.630 --> 00:00:28.588
the movie theater is four kilometers,

00:00:28.588 --> 00:00:30.998
to the post office, 5 1/2 kilometers.

00:00:30.998 --> 00:00:32.832
Then they ask us, "How many dots

00:00:32.832 --> 00:00:34.768
"should be on a line plot
of these measurements?"

00:00:34.768 --> 00:00:37.763
So a line plot, and
they have one down here,

00:00:38.071 --> 00:00:40.128
a line plot, this is
literally a number line

00:00:40.128 --> 00:00:42.078
and what we do is we put

00:00:42.078 --> 00:00:43.629
as many dots there are

00:00:43.629 --> 00:00:46.574
at a certain point in the line.

00:00:46.574 --> 00:00:48.546
So if we have two data points at 5 1/2

00:00:48.546 --> 00:00:51.044
we would put two dots right over there,

00:00:51.044 --> 00:00:52.488
but we'll get to that in a second.

00:00:52.965 --> 00:00:54.925
They're just asking us how many dots

00:00:54.925 --> 00:00:56.885
should be on the line plot
of these measurements?

00:00:56.885 --> 00:00:58.105
Well, we should make a dot

00:00:58.105 --> 00:00:59.505
for each of these data points

00:00:59.505 --> 00:01:01.825
and there are one, two, three, four,

00:01:01.825 --> 00:01:03.385
five data points here.

00:01:03.385 --> 00:01:05.005
So there's five data points

00:01:05.005 --> 00:01:07.406
that I'm going to want
to put on my line plot.

00:01:07.707 --> 00:01:10.027
Then they say, "Create a line plot

00:01:10.027 --> 00:01:11.827
"that shows all of the measurements.

00:01:11.827 --> 00:01:14.027
"Click above the marking
on the number line

00:01:14.027 --> 00:01:15.267
"that matches each measurement.

00:01:15.267 --> 00:01:18.307
"Click higher to add more
dots above the same marking.

00:01:18.307 --> 00:01:19.623
"Click lower to add fewer dots."

00:01:19.947 --> 00:01:23.010
So they say that the
distance from the school

00:01:23.010 --> 00:01:25.308
to Tessa's house is 3 1/2 kilometers

00:01:25.308 --> 00:01:29.208
so we would want to put a dot at 3 1/2

00:01:29.208 --> 00:01:30.968
and I do that just by clicking,

00:01:30.968 --> 00:01:33.528
clicking right over there, 3 1/2.

00:01:33.535 --> 00:01:35.900
So that's one data point at 3 1/2.

00:01:35.900 --> 00:01:37.631
Half way between three and four.

00:01:38.261 --> 00:01:39.704
Let's do it for the next ones.

00:01:40.155 --> 00:01:41.496
5 1/2, that's the distance

00:01:41.496 --> 00:01:43.132
from the library to her house.

00:01:43.963 --> 00:01:48.317
5 1/2, then we have two kilometers.

00:01:49.256 --> 00:01:51.656
Two kilometers, distance from ...

00:01:52.335 --> 00:01:53.677
Two kilometers was the distance

00:01:53.677 --> 00:01:54.944
from the park to her house.

00:01:55.275 --> 00:01:57.455
Then the movie theater if four kilometers.

00:01:57.877 --> 00:01:59.597
I wish you could see it all in one screen.

00:02:00.295 --> 00:02:01.133
Four kilometers.

00:02:01.755 --> 00:02:05.776
And then the post office
is also 5 1/2 kilometers.

00:02:05.776 --> 00:02:07.356
So you actually have two data points

00:02:07.356 --> 00:02:09.456
at 5 1/2 kilometers.

00:02:09.636 --> 00:02:11.526
So right here I've just
constructed a line plot.

00:02:11.526 --> 00:02:13.326
At different points on the number line

00:02:13.326 --> 00:02:15.386
I'm showing how many data points we have.

00:02:15.386 --> 00:02:17.806
So we have one data
point at two kilometers,

00:02:17.806 --> 00:02:20.873
one at 3 1/2 kilometers,
one at four kilometers,

00:02:20.873 --> 00:02:23.373
and two at 5 1/2 kilometers.

00:02:23.647 --> 00:02:26.439
Now notice, we said we
would have five dots

00:02:26.439 --> 00:02:29.091
when we have one, two, three, four, five.

00:02:29.091 --> 00:02:30.686
One for each data point.

00:02:31.211 --> 00:02:32.471
Let's do this one more time.

00:02:32.471 --> 00:02:34.154
Let's check and make sure we got it right.

00:02:34.154 --> 00:02:35.534
Now let's do one more.

00:02:36.474 --> 00:02:38.374
All right, measure the length ...

00:02:39.220 --> 00:02:41.094
Measure the length of each line

00:02:41.094 --> 00:02:43.254
to the nearest quarter
inch to collect data

00:02:43.254 --> 00:02:44.794
for the line plot below.

00:02:44.794 --> 00:02:45.874
To make your measurements,

00:02:45.874 --> 00:02:48.094
drag the ruler on top of the lines.

00:02:48.094 --> 00:02:49.715
Align the left end of the ruler

00:02:49.715 --> 00:02:51.075
with the left end of the lines

00:02:51.075 --> 00:02:53.285
then read the length of each line

00:02:53.285 --> 00:02:55.322
from the right end of the line.

00:02:55.856 --> 00:02:57.743
All right, so really I
just wanna measure ...

00:02:58.208 --> 00:02:59.564
I just wanna measure, oh this is ...

00:03:00.136 --> 00:03:01.554
I didn't mean to tilt it like that.

00:03:02.248 --> 00:03:04.558
I just wanna measure each of these lines.

00:03:05.142 --> 00:03:07.522
So this green line right over here

00:03:07.522 --> 00:03:11.562
looks like it is six
and this is split into

00:03:11.562 --> 00:03:12.962
one, two, three, four.

00:03:12.962 --> 00:03:15.795
It looks like it's 6 3/4ths.

00:03:16.242 --> 00:03:21.242
6 3/4ths, the green line
is 6 3/4ths inches long.

00:03:21.762 --> 00:03:24.794
So we have one data point at 6 3/4ths.

00:03:25.442 --> 00:03:27.262
So I'll put that right over there.

00:03:28.342 --> 00:03:31.242
Then we have the red line is ...

00:03:32.242 --> 00:03:36.168
Let's see, this is 7 2/4ths or 7 1/2.

00:03:36.822 --> 00:03:38.826
It's halfway between seven and eight.

00:03:39.382 --> 00:03:44.002
So 7 2/4ths or 7 1/2, so halfway between

00:03:44.002 --> 00:03:45.423
seven and eight is right over there.

00:03:46.302 --> 00:03:49.602
Then finally I have the purple line.

00:03:50.342 --> 00:03:52.194
The purple line, let's see.

00:03:53.042 --> 00:03:56.082
That is 8 3/4ths inches long.

00:03:56.102 --> 00:03:58.582
Notice I aligned it right at the beginning

00:03:58.582 --> 00:04:00.362
right over there and it goes all the way

00:04:00.362 --> 00:04:04.202
to eight and one, two,
three out of the four.

00:04:04.202 --> 00:04:09.202
8 3/4ths inches long, so let me, 8 3/4ths.

00:04:10.202 --> 00:04:11.102
So there you go.

00:04:11.102 --> 00:04:13.062
I have one line that was 6 3/4ths,

00:04:13.062 --> 00:04:16.434
one line that was 7 2/4ths or 7 1/2

00:04:16.434 --> 00:04:19.260
and then one line that's
8 3/4ths inches long.

00:04:19.768 --> 00:04:21.282
Let's check our answer.

00:04:23.412 --> 00:04:24.302
Let's do one more.

00:04:24.302 --> 00:04:25.243
This is actually a lot of fun.

00:04:25.842 --> 00:04:27.619
All right, we gotta measure things again.

00:04:27.988 --> 00:04:30.599
So let's see, the green line.

00:04:31.239 --> 00:04:34.380
Green line is exactly three inches long,

00:04:34.380 --> 00:04:36.376
three inches long so let's plot it

00:04:36.376 --> 00:04:37.094
right over there.

00:04:38.076 --> 00:04:41.256
Then we have the red line is exactly,

00:04:41.256 --> 00:04:43.616
let's see, 3 3/4ths inches long.

00:04:44.316 --> 00:04:47.076
3 3/4ths, let's put a
dot right over there.

00:04:47.806 --> 00:04:52.697
Then the purple line is
three inches long as well.

00:04:52.697 --> 00:04:55.377
So I have another data
point at three inches.

00:04:55.377 --> 00:04:57.717
So two of the lines were three inches

00:04:57.717 --> 00:04:59.838
and one of them is 3 3/4ths

00:04:59.838 --> 00:05:01.441
and we are ...

00:05:02.018 --> 00:05:03.428
And we're done.