Equivalent fractions with models

WEBVTT
Kind: captions
Language: en

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- [Instructor] So what we're
going to do in this video

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is think about equivalent fractions.

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So let's say we have the fraction 3/4,

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and I wanna think about

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what is an equivalent number of eighths?

00:00:12.790 --> 00:00:17.360
So 3/4 is equal to how many eighths?

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And to represent that how many,

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I could put a question mark there,

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but instead of a question
mark, I'll just put a letter.

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So what should y be?

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3/4 is equal to y/8.

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What does y need to be to make this true?

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And before I tell you
go go pause this video

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and try to work on it on your own,

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which I will do in a little bit,

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I'll give you a little bit of a hint.

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So let's try to represent 3/4,
or I'll represent it for you.

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So I will do it with this rectangle.

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So I'm going to divide it
into four equal sections.

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So let's see, that would be
dividing it roughly in half.

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I'm hand-drawing it, so it's not perfect,

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but these should be equal sections.

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The areas of each of these
sections should be equal.

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So there you go, this is my
hand-drawn version of that.

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And so three of those four,
and I will do that in purple.

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Three of those four, it could be one,

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two, and then just for kicks
I will do this one out here.

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So that is 3/4 right over there.

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So if I were to think about
this in terms of eighths.

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So I'm going to draw another whole.

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But this time instead of just
splitting them into fourths,

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I'm going to split it into eighths.

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So let's do the fourths first,

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just 'cause it's easy to
look at the one above that.

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So that's my fourths.

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And then I'll divide each
of the fourths into two.

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So that gives me eighths.

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All right, almost there.

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The drawing is really
the hardest part here.

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And so each of these is an eighth.

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It's hand-drawn.

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But imagine if there were
eight equal sections.

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So how many eighths is equal to 3/4?

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Pause the video and try to
work it out on your own.

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All right, well, we can
just look at this visually.

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So this first fourth, we
could say that's equivalent

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to filling out this
eighth and this eighth.

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So that first fourth is equal to 2/8.

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This second fourth is equal

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to another two of these eighths.

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And then this third fourth is equal

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to another two of the eighths.

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So how many eighths have I shaded in?

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Well, I have one, two,
three, four, five, 6/8.

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So I have six over eight.

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So 3/4 is equivalent to 6/8.

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So in this scenario, y is equal to 6/8,

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or we could say 6/8 is equal to y.

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Now let's do another example.

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So what we could see
here in this top circle

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is we've divided it
into six equal sections.

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So each of these are one of
the six equal sections or 1/6.

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And we can see that one, two, three, four

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of them are shaded in.

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So what we have represented
in that top circle,

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that is four out of six.

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So what I wanna think about is

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how many thirds are equivalent to 4/6?

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Pause this video and think about it.

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So once again, how many
thirds are equivalent to 4/6?

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Instead of just putting
a question mark there,

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I'll put the letter x.

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So what should x be for these
two things to be equivalent?

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Or another way to think about it is

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four over six is equal to

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x over three.

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4/6 is equal to how many thirds?

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All right, now let's do this together.

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And so one way you could think about it,

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let's see, for 1/3,

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for us to make this equal to 1/3,

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it looks like that is equivalent to

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what I am circling in the orange up here.

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And that also makes sense.

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If I were to divided 1/3 into two,

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so now I would have this would
be 1/6 and that would be 1/6.

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So you need 2/6 to make up 1/3,

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or each 1/3 is equivalent to 2/6.

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So this is 1/3 right over here.

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And that is equivalent
to two of these sixths.

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But we are not completely done yet.

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We have another two sixths.

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So we could say those
two sixths are equivalent

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to another third.

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And it's a little tricky
because they didn't put

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this sixth next to that sixth,

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but you could imagine if we were to move.

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Let's say we were to move this sixth.

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So I'm gonna color this one in white.

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So if I were to move that
sixth to right over here.

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So we're shading this one in instead.

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So then you can see that these
two sixths right over here,

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these two sixths are equivalent

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to this third right over there.

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So what you can see is, is that our 4/6,

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and remember, I moved this one over,

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so I'm not shading this one in anymore.

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But you can see the 4/6

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that I've shaded in is equivalent to 2/3.

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Or another way to say it
is x would be equal to two.

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X would be equal to two.

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4/6 is equal to x/3,
or 4/6 is equal to 2/3.


Subtracting mixed numbers with regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.250
- [Instructor] So let's
see how we could approach

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four and 1/4 minus two and 2/4.

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Pause this video and have a go at that

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before we work on this together.

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All right, so the first thing
that you might try to do

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is re-write this as four
and 1/4 minus two and 2/4.

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And the reason why it's
useful to write it this way

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is we could say look,
each of these mixed number

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have a whole number part

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and then they have a fractional part.

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And so I could try to
subtract the fractional part

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from the fractional part

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and then the whole number part
from the whole number part.

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And that will often work.

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But when we go to just
the fractional parts

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and we say hey, how do
I subtract 2/4 from 1/4,

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2/4 is larger than 1/4.

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You immediately might find yourself

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in something of a conundrum.

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So what do you do?

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Well, what comes to the rescue
is a notion called regrouping

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and that's that we're not
just subtracting 2/4 from 1/4.

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We're subtracting two and
2/4 from four and 1/4.

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So what we could do is we
could take some of the value

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that's in this four and
regroup it into the fraction.

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What do I mean by that?

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Well instead of four,

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if I view four as three and 4/4, 4/4.

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And so 4/4 plus 1/4 is 5/4, 5/4.

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So I could re-write this as three and 5/4

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minus two and 2/4.

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Once again, why is that useful?

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Because 5/4 is greater than 2/4.

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What is 5/4 minus 2/4?

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Well that's going to be 3/4.

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And then what is three minus two?

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Well that's going to be one.

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And we get one and 3/4.

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Another way we could of thought about it

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is this is the same
thing as four minus two

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plus 1/4, plus 1/4,

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minus 2/4, minus 2/4.

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And we have trouble
with the 1/4 minus 2/4.

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That's what we saw right over here.

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So what we do is we take some
of that value from the four.

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Instead of four we could
re-write that as three plus 4/4.

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That's the same thing as four.

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And of course you have minus
two plus 1/4 minus 2/4.

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And then if you add the 4/4 to the 1/4

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that's going to give us 5/4.

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So you're going to have three minus two.

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That's that part and that part.

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And then you're going to have

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if you add the 4/4 and the 1/4,

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so plus 5/4 minus 2/4.

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And that exactly what we have here.

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Three and 5/4, three and 5/4

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minus two, minus two, and minus 2/4.

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And so three minus two is one.

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5/4 minus 2/4 is 3/4.


Mixed number addition with regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:02.130
- [Narrator] Let's see if we can add

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five and 2/5 to three and 4/5.

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Pause this video and see if you
can figure out what this is.

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All right, now let's do this together.

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We've had a little bit of practice

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adding mixed numbers in the past.

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And so one way to think about it is,

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you could view five and
2/5 as five plus 2/5.

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And then to that we're going
to be adding three and 4/5,

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which you could view as three plus 4/5.

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And then you could just change the order

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with which you are adding,

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and say, all right,

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well, I could say five plus three,

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so that's five plus three,

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and then to that I could add 2/5 and 4/5,

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so, plus 2/5, plus 4/5.

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Now what is that going to get me?

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Well five plus three is
going to be equal to eight.

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Eight plus, and then if I have 2/5,

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and I add four more fifths to that,

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well now I'm going to have 6/5.

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Two of something plus
four of that something

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is going to be six of that something,

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and the something in this case are fifths.

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So now I'm going to have 6/5.

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So some of you might be tempted to say,

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"Hey, isn't this just going to be equal to

00:01:15.123 --> 00:01:19.040
eight and 6/5?"

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And you wouldn't be completely
wrong if you said that.

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But pause this video and think about

00:01:24.300 --> 00:01:26.655
why this feels a little bit off.

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Well, the reason why this isn't standard,

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is the fractional part
of this mixed number 6/5

00:01:33.020 --> 00:01:34.830
is greater than one.

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So there's a whole inside of this 6/5.

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So the standard way to do this is

00:01:40.060 --> 00:01:41.537
see if we can break out that whole.

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Now what do I mean by that?

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Well I could rewrite eight plus 6/5

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as eight plus 6/5 is the same thing

00:01:50.550 --> 00:01:55.550
as a whole, or 5/5 plus 1/5.

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And why is this useful?

00:01:57.930 --> 00:02:01.030
Well 5/5 is the same thing as one.

00:02:01.030 --> 00:02:03.140
And so now I can say
this is going to be equal

00:02:03.140 --> 00:02:06.040
to eight plus one whole is nine,

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that's eight plus the 5/5,

00:02:07.690 --> 00:02:10.350
and then what I have leftover is 1/5.

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So nine and 1/5.

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And so this is the direction

00:02:14.990 --> 00:02:17.600
that people will traditionally go in.

00:02:17.600 --> 00:02:19.240
Now there's another way
that you could approach it,

00:02:19.240 --> 00:02:20.920
which is really the same idea,

00:02:20.920 --> 00:02:23.140
we're just writing things
a little bit differently.

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We could write this as five and 2/5

00:02:31.518 --> 00:02:32.351
plus three and 4/5,

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and notice the way that I wrote it.

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I put all the fractions
in the fraction column,

00:02:41.270 --> 00:02:42.210
I guess you could call it that way,

00:02:42.210 --> 00:02:45.270
and I put all of our whole
numbers underneath each other,

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and if I had multiple digits here,

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I would align them
according to place value.

00:02:49.490 --> 00:02:51.030
And then what we could do is we could say,

00:02:51.030 --> 00:02:55.270
okay 2/5 plus 4/5 is going to be 6/5.

00:02:55.270 --> 00:02:57.230
We could write 6/5 there.

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But we say, hey, there's
something a little bit

00:02:58.750 --> 00:03:01.270
fishy about 6/5.

00:03:01.270 --> 00:03:04.862
That's really the same
thing as 5/5 plus 1/5,

00:03:04.862 --> 00:03:08.030
or you could say that's the
same thing as one and 1/5.

00:03:08.030 --> 00:03:11.410
6/5 is equal to one and 1/5,

00:03:11.410 --> 00:03:13.100
and so what you could do is,

00:03:13.100 --> 00:03:16.020
you could write the 1/5 part
in the fractions column,

00:03:16.020 --> 00:03:19.340
and then the one well now you're
going to be regrouping that

00:03:19.340 --> 00:03:20.560
into our whole numbers,

00:03:20.560 --> 00:03:22.040
so you put a one right over there.

00:03:22.040 --> 00:03:25.940
Notice, 2/5 plus 4/5 is one and 1/5,

00:03:25.940 --> 00:03:27.570
which is the same thing as 6/5.

00:03:27.570 --> 00:03:29.510
And then you add the whole number part.

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One plus five plus three is nine.

00:03:32.370 --> 00:03:34.006
So you get nine and 1/5.

00:03:34.006 --> 00:03:35.330
But hopefully you realize

00:03:35.330 --> 00:03:36.940
that these are really the same idea,

00:03:36.940 --> 00:03:39.230
just different way of writing things


Adding mixed numbers with like denominators

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:01.510
- [Instructor] What we're
going to do in this video

00:00:01.510 --> 00:00:05.250
is to start thinking about
adding mixed numbers.

00:00:05.250 --> 00:00:07.640
Now just as a reminder of
what a mixed number is,

00:00:07.640 --> 00:00:11.900
it'd be something like three and 2/8.

00:00:11.900 --> 00:00:14.640
It's called mixed because part of the way

00:00:14.640 --> 00:00:16.650
we represent this number
is there's a whole number,

00:00:16.650 --> 00:00:19.720
the three, and then part
is as a fraction, 2/8.

00:00:19.720 --> 00:00:22.510
But the numbers three and 2/8ths.

00:00:22.510 --> 00:00:24.380
So let's think about how
we might be able to add

00:00:24.380 --> 00:00:29.380
three and 2/8ths to five and 3/8ths.

00:00:33.260 --> 00:00:36.670
So pause this video and see
if you can have a go at this

00:00:36.670 --> 00:00:39.050
before we work through this together.

00:00:39.050 --> 00:00:40.750
All right, now let's work
through this together.

00:00:40.750 --> 00:00:43.550
Now there's a couple of ways
that we could approach this.

00:00:43.550 --> 00:00:47.160
You might recognize that three
and 2/8ths is the same thing

00:00:47.160 --> 00:00:52.160
as three plus 2/8ths, and five and 3/8ths

00:00:52.620 --> 00:00:57.620
is the same thing as five plus 3/8ths.

00:00:57.940 --> 00:00:59.850
And so if we add these two things together

00:00:59.850 --> 00:01:03.750
we're just adding really
these four things together

00:01:03.750 --> 00:01:06.410
and the order in which
we add doesn't matter

00:01:06.410 --> 00:01:11.050
so you could view this as three plus five

00:01:12.190 --> 00:01:17.190
plus 2/8ths plus 2/8ths, plus 3/8ths,

00:01:19.100 --> 00:01:20.300
notice it's just the same thing,

00:01:20.300 --> 00:01:21.920
I'm just changing the order with which

00:01:21.920 --> 00:01:26.330
we are adding, and now
what is three plus five?

00:01:26.330 --> 00:01:28.200
Three wholes plus five wholes,

00:01:28.200 --> 00:01:30.140
we'll that's just going to be eight.

00:01:30.140 --> 00:01:34.530
And then what is 2/8ths plus 3/8ths?

00:01:34.530 --> 00:01:36.930
Well if I have two of
something, in this case eighths,

00:01:36.930 --> 00:01:39.250
and I add three more of that something,

00:01:39.250 --> 00:01:40.900
in this case eighths, I'm gonna have five

00:01:40.900 --> 00:01:43.700
of that something, in this case 5/8ths.

00:01:43.700 --> 00:01:46.920
So this is going to be eight plus 5/8ths.

00:01:46.920 --> 00:01:49.320
Which we can express as a mixed number

00:01:49.320 --> 00:01:54.320
as just being equal to eight and 5/8ths.

00:01:55.810 --> 00:01:57.770
Now there's other ways that you might see

00:01:57.770 --> 00:01:59.690
folks approaching this.

00:01:59.690 --> 00:02:01.390
You might see something like this

00:02:01.390 --> 00:02:06.340
where people will write three plus 2/8ths

00:02:06.340 --> 00:02:08.400
and then they will write right below that

00:02:08.400 --> 00:02:10.870
they'll write the five in the ones place,

00:02:10.870 --> 00:02:12.770
or in the whole number and
there's only a ones place

00:02:12.770 --> 00:02:14.803
in these whole numbers,
so you'd write the five

00:02:14.803 --> 00:02:17.870
right below the three and then plus

00:02:17.870 --> 00:02:20.850
and then the fraction you'd
write right below the fraction.

00:02:20.850 --> 00:02:21.723
3/8ths.

00:02:23.370 --> 00:02:26.720
And then we can add vertically like this

00:02:26.720 --> 00:02:29.400
where first we can look
at the fractional part,

00:02:29.400 --> 00:02:34.400
2/8ths plus 3/8ths is
going to be equal to 5/8ths

00:02:36.050 --> 00:02:40.990
and then three plus five is
going to be equal to eight

00:02:40.990 --> 00:02:42.620
and if you add these two together,

00:02:42.620 --> 00:02:45.740
eight and 5/8ths, you get eight and 5/8s.

00:02:45.740 --> 00:02:47.360
So different ways to approach it,

00:02:47.360 --> 00:02:50.050
but that will hopefully
start to get you comfortable

00:02:50.050 --> 00:02:52.153
with adding a mixed numbers.


Fraction multiplcation on the number line

WEBVTT
Kind: captions
Language: en

00:00:00.960 --> 00:00:01.793
- [Instructor] So, what
we're gonna think about

00:00:01.793 --> 00:00:04.890
in this video is multiplying fractions.

00:00:04.890 --> 00:00:07.870
So, let's say that we wanted to take 2/3

00:00:08.920 --> 00:00:12.150
and we want to multiply it by four,

00:00:12.150 --> 00:00:14.920
what is this going to be equal to?

00:00:14.920 --> 00:00:18.320
Pause this video and try to
think about it on your own.

00:00:18.320 --> 00:00:20.020
Alright, now let's work
through this together.

00:00:20.020 --> 00:00:23.480
And, to help us, I will use a number line,

00:00:23.480 --> 00:00:25.902
and let's say that each
of these hash marks

00:00:25.902 --> 00:00:27.240
represent a third.

00:00:27.240 --> 00:00:32.240
So, this is zero, this is 1/3, 2/3, 3/3,

00:00:32.971 --> 00:00:37.971
4/3, 5/3, 6/3, 7/3,

00:00:39.138 --> 00:00:44.138
8/3, and 9/3, and so
where is 2/3 times one?

00:00:46.320 --> 00:00:48.650
Well, 2/3 times one is
just going to be 2/3,

00:00:48.650 --> 00:00:53.290
we just take a jump of
2/3, so that is times 1.

00:00:53.290 --> 00:00:56.250
If we multiply by, or if
we take 2/3 times two,

00:00:56.250 --> 00:00:57.330
that'll be two jumps,

00:00:57.330 --> 00:01:02.330
so one 2/3, two 2/3, three
2/3, and then four 2/3.

00:01:06.830 --> 00:01:10.100
So, we just took four jumps of 2/3 each.

00:01:10.100 --> 00:01:15.100
You could view that as 2/3
plus 2/3 plus 2/3 plus 2/3,

00:01:19.640 --> 00:01:21.510
and where does that get us to?

00:01:21.510 --> 00:01:23.890
It got us to 8/3.

00:01:23.890 --> 00:01:28.890
So, notice, 2/3 times
four is equal to 8/3.

00:01:28.890 --> 00:01:30.110
Now, we could go the other way,

00:01:30.110 --> 00:01:31.520
we could look at a number line

00:01:31.520 --> 00:01:33.640
and think about what are ways to represent

00:01:33.640 --> 00:01:35.090
what the number line is showing us?

00:01:35.090 --> 00:01:37.080
And, on Khan Academy, we
have some example problems

00:01:37.080 --> 00:01:37.913
that do it that way,

00:01:37.913 --> 00:01:39.890
so I thought it would be good
to do an example like that.

00:01:39.890 --> 00:01:41.620
And, so, let's label this number line

00:01:41.620 --> 00:01:42.453
a little bit different.

00:01:42.453 --> 00:01:44.840
Instead of each of these
lines representing a third,

00:01:44.840 --> 00:01:46.740
let's say they represent a half,

00:01:46.740 --> 00:01:51.740
so zero, 1/2, 2/2, 3/2, 4/2, 5/2,

00:01:54.560 --> 00:01:57.270
why did I write 5/6, my
brain is going ahead,

00:01:57.270 --> 00:02:02.270
5/2, 6/2, 7/2, 8/2, and 9/2.

00:02:04.570 --> 00:02:08.420
And, let's say we were to
see something like this.

00:02:08.420 --> 00:02:10.804
So, if you were to just
see this representation,

00:02:10.804 --> 00:02:14.990
so I'm going to try to draw it like this,

00:02:14.990 --> 00:02:17.740
so if you were to just
see this representation,

00:02:17.740 --> 00:02:19.920
what is that trying to represent?

00:02:19.920 --> 00:02:24.138
What type of multiplication
is that trying to represent?

00:02:24.138 --> 00:02:28.646
Well, you could view that
as 3/2 plus another 3/2

00:02:28.646 --> 00:02:30.188
plus another 3/2, 'cause, notice,

00:02:30.188 --> 00:02:34.430
each of these jumps are three 1/2, or 3/2.

00:02:34.430 --> 00:02:39.430
So, you could view this
as 3/2 plus 3/2 plus 3/2,

00:02:41.220 --> 00:02:42.380
or another way of thinking about it

00:02:42.380 --> 00:02:45.110
is this is three jumps of 3/2.

00:02:45.110 --> 00:02:47.580
So, you can also view this
as doing the same thing

00:02:47.580 --> 00:02:52.340
as three times 3/2, and
what are these equal to?

00:02:52.340 --> 00:02:55.726
Well, 3/2 plus 3/2 plus
3/2, or three times 3/2,

00:02:55.726 --> 00:02:59.033
it gets you to 9/2.


Multiplying mixed numbers

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:01.740
- [Instructor] Let's
see if we can figure out

00:00:01.740 --> 00:00:06.000
what 2 1/4 times three is.

00:00:06.000 --> 00:00:08.563
Pause this video and see
if you can work that out.

00:00:09.540 --> 00:00:12.270
All right, now there's different ways

00:00:12.270 --> 00:00:14.520
that we could approach this.

00:00:14.520 --> 00:00:16.620
One way to approach this is to recognize

00:00:16.620 --> 00:00:18.410
that if I multiply anything times three,

00:00:18.410 --> 00:00:20.220
that means that I'm taking
three of these things

00:00:20.220 --> 00:00:21.240
and adding them together.

00:00:21.240 --> 00:00:23.210
So, this is going to be the same thing

00:00:23.210 --> 00:00:28.210
as 2 1/4 plus 2 1/4 plus 2 1/4.

00:00:30.800 --> 00:00:34.420
Now, I could also view
this as each of these 2 1/4

00:00:34.420 --> 00:00:39.420
is two plus 1/4 plus two
plus 1/4 plus two plus 1/4.

00:00:42.030 --> 00:00:44.550
All I did is I broke up each of the 2 1/4

00:00:44.550 --> 00:00:46.250
into a two plus 1/4.

00:00:46.250 --> 00:00:47.870
And then what I could do is

00:00:47.870 --> 00:00:49.310
I could add the whole number parts,

00:00:49.310 --> 00:00:54.310
two plus two plus two is
going to be equal to six.

00:00:55.630 --> 00:00:59.590
And then, if I were to
take 1/4 plus 1/4 plus 1/4,

00:00:59.590 --> 00:01:01.430
how many fourths do I now have?

00:01:01.430 --> 00:01:06.020
I have 3/4, so it's gonna be six plus 3/4,

00:01:06.020 --> 00:01:08.983
or I could write this as 6 3/4.

00:01:12.510 --> 00:01:16.030
Now, another way that we
could have approached this

00:01:16.030 --> 00:01:19.350
is we could've rewritten 2 1/4.

00:01:19.350 --> 00:01:24.350
I could have rewritten that as
the same thing as two is 8/4,

00:01:27.040 --> 00:01:29.140
8/4 is the same thing as two,

00:01:29.140 --> 00:01:31.720
plus 1/4 and then I'm multiplying

00:01:31.720 --> 00:01:34.290
that whole thing times three.

00:01:34.290 --> 00:01:35.970
And so, this is going to be the same thing

00:01:35.970 --> 00:01:40.970
as 9/4, 8/4 plus 1/4 is 9/4, times three.

00:01:42.810 --> 00:01:44.667
And so, we could say, "Hey,
that's just going to be equal

00:01:44.667 --> 00:01:49.667
"to 9/4 plus 9/4 plus 9/4,
which would be equal to what?"

00:01:52.460 --> 00:01:54.210
Well, I have,

00:01:54.210 --> 00:01:57.960
this is going to be nine
plus nine plus nine fourths,

00:01:57.960 --> 00:02:01.220
which is going to be equal to 27/4.

00:02:01.220 --> 00:02:02.107
Now, you might be saying,

00:02:02.107 --> 00:02:05.130
"Hey, these look like they are different",

00:02:05.130 --> 00:02:06.160
but you could check this.

00:02:06.160 --> 00:02:08.760
Six is equal to how many fourths?

00:02:08.760 --> 00:02:12.520
So, six is equal to 24/4,

00:02:12.520 --> 00:02:17.520
24/4 plus 3/4, which is
exactly equal to 27/4.

00:02:20.650 --> 00:02:22.263
And we are done.


Mixed number subtraction

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:02.110
- [Instructor] Let's say
that we want to figure out

00:00:02.110 --> 00:00:07.110
what is seven and 11/12
minus one and 6/12.

00:00:09.540 --> 00:00:12.363
Pause this video and see
if you can figure that out.

00:00:13.270 --> 00:00:15.750
All right now let's work on this together.

00:00:15.750 --> 00:00:18.350
So there's a couple of ways
that you could approach this.

00:00:18.350 --> 00:00:23.350
You can view this as the same
thing as seven plus 11/12

00:00:24.660 --> 00:00:29.480
and then minus one and you
might be tempted to say minus

00:00:29.480 --> 00:00:31.900
one and then plus 6/12 but
remember you're subtracting

00:00:31.900 --> 00:00:34.630
not just the one you're
subtracting one and you're

00:00:34.630 --> 00:00:39.520
subtracting 6/12 so minus one minus 6/12.

00:00:39.520 --> 00:00:42.240
Now why is that useful to
think about it that way?

00:00:42.240 --> 00:00:45.490
Well now you can think about
the whole numbers so you can

00:00:45.490 --> 00:00:50.490
say this is going to be
seven minus one plus 11/12,

00:00:50.790 --> 00:00:55.790
plus 11/12 minus 6/12, minus 6/12.

00:00:56.160 --> 00:01:01.160
And so seven minus one is six,
so it's going to be six plus

00:01:01.310 --> 00:01:04.090
now if I have 11 of
something in this case 12ths

00:01:04.090 --> 00:01:06.770
and I'm subtracting six of
them away six of the 12ths,

00:01:06.770 --> 00:01:10.620
I'm gonna be left with five of
those somethings, five 12ths.

00:01:10.620 --> 00:01:14.450
So it's going to be six plus
5/12 which is the same thing

00:01:14.450 --> 00:01:18.540
as six and 5/12.

00:01:18.540 --> 00:01:20.400
Now as you get more used
to this you could do some

00:01:20.400 --> 00:01:22.130
of this maybe even in your
head, you could say hey look,

00:01:22.130 --> 00:01:27.130
look at the whole number
parts, seven minus one is going

00:01:27.470 --> 00:01:32.470
to give me six and then
if I say 11/12 minus 6/12

00:01:33.080 --> 00:01:36.240
is going to give me 5/12 so
that takes up a little bit

00:01:36.240 --> 00:01:37.190
less space.

00:01:37.190 --> 00:01:39.385
Another way that you might
see this approached is you can

00:01:39.385 --> 00:01:44.385
rewrite this as seven and 11/12
minus, minus one and 6/12.

00:01:51.970 --> 00:01:55.080
Let me do the 6/12 in that
blue color that I'm using

00:01:55.080 --> 00:01:59.500
for the fraction parts,
6/12 and then I would first

00:01:59.500 --> 00:02:03.162
focus on the fractional parts
and I'd say 11/12 minus 6/12

00:02:03.162 --> 00:02:08.162
is 5/12 and seven minus one
is six and I got six and 5/12.

00:02:10.480 --> 00:02:12.830
So many different ways to
approach the same thing

00:02:12.830 --> 00:02:15.350
and at the end of the day
they really are the same

00:02:15.350 --> 00:02:17.563
underlying idea.


Reform in the Gilded Age

WEBVTT
Kind: captions
Language: en

00:00:00.863 --> 00:00:02.390
- [Instructor] In the year 2000

00:00:02.390 --> 00:00:05.660
a wealthy Bostonian named Julian West

00:00:05.660 --> 00:00:09.140
woke up from a very long nap.

00:00:09.140 --> 00:00:13.320
He had fallen asleep in the year 1887.

00:00:13.320 --> 00:00:15.270
The United States in the year 2000

00:00:15.270 --> 00:00:18.300
was very different from
the Gilded Age he knew.

00:00:18.300 --> 00:00:21.880
It was a utopian society
where there was no poverty

00:00:21.880 --> 00:00:24.840
no labor strikes, no pollution.

00:00:24.840 --> 00:00:27.160
His new friends in the
future explained to him

00:00:27.160 --> 00:00:28.950
how this society worked.

00:00:28.950 --> 00:00:31.740
There was no private
property and no money.

00:00:31.740 --> 00:00:34.570
Everyone worked at the job
they were most suited for

00:00:34.570 --> 00:00:37.860
and received their fair
share of the national wealth

00:00:37.860 --> 00:00:41.810
on a credit card that they
could use to buy necessities.

00:00:41.810 --> 00:00:44.110
This is the plot of Edward Bellamy's

00:00:44.110 --> 00:00:47.590
1888 novel "Looking Backward."

00:00:47.590 --> 00:00:50.570
It was a bestseller of its
day but today we remember it

00:00:50.570 --> 00:00:53.530
not only for its
predictions about the future

00:00:53.530 --> 00:00:56.740
but as an example of how
thinkers of the time period

00:00:56.740 --> 00:00:59.970
explored the burning
question of the Gilded Age.

00:00:59.970 --> 00:01:04.110
Was it possible to have a
modern industrial society

00:01:04.110 --> 00:01:06.880
without great inequalities of wealth?

00:01:06.880 --> 00:01:10.810
In the late 19th century the
rapid changes in American life

00:01:10.810 --> 00:01:13.750
stemming from the rise
of industrial capitalism

00:01:13.750 --> 00:01:16.410
caused a great deal of concern.

00:01:16.410 --> 00:01:18.890
The United States prided
itself on being different

00:01:18.890 --> 00:01:20.200
from the countries of Europe

00:01:20.200 --> 00:01:23.330
where the inequality
between the aristocracy

00:01:23.330 --> 00:01:27.230
and the working class caused
strife and revolution.

00:01:27.230 --> 00:01:29.670
But industrialization had brought

00:01:29.670 --> 00:01:34.200
both millionaires and
impoverished millions

00:01:34.200 --> 00:01:37.300
to the United States,
packing them in to cities

00:01:37.300 --> 00:01:41.540
where mansions sat side-by-side
with filthy tenements.

00:01:41.540 --> 00:01:46.540
Suddenly, the New World looked
a lot more like the old world

00:01:46.810 --> 00:01:48.590
and so people at the time wondered

00:01:48.590 --> 00:01:52.380
was inequality an inevitable by-product

00:01:52.380 --> 00:01:54.100
of advancing society?

00:01:54.100 --> 00:01:58.350
Those who answered yes were
called Social Darwinists

00:01:58.350 --> 00:02:00.480
who thought that the
survival of the fittest

00:02:00.480 --> 00:02:04.270
would weed out the weak and
improve society overall.

00:02:04.270 --> 00:02:06.630
We'll talk more about
Social Darwinism elsewhere

00:02:06.630 --> 00:02:08.820
but in this video I wanna concentrate

00:02:08.820 --> 00:02:12.920
on those who believe that it
was possible to have a society

00:02:12.920 --> 00:02:15.710
that was both modern and equitable.

00:02:15.710 --> 00:02:18.900
During the Gilded Age there
were a number of reformers

00:02:18.900 --> 00:02:22.990
and reform movements that
attempted to solve the problems

00:02:22.990 --> 00:02:26.650
posed by urban and industrial life.

00:02:26.650 --> 00:02:29.580
So let's talk about some
of the ways that reformers

00:02:29.580 --> 00:02:33.500
attempted to respond to the
inequalities of the Gilded Age.

00:02:33.500 --> 00:02:36.760
One was to suggest new economic systems

00:02:36.760 --> 00:02:38.110
for the United States.

00:02:38.110 --> 00:02:39.640
For example, you might have noticed

00:02:39.640 --> 00:02:43.120
that in Edward Bellamy's utopian society

00:02:43.120 --> 00:02:44.880
there was no private property

00:02:44.880 --> 00:02:48.260
and the national wealth
was equally shared.

00:02:48.260 --> 00:02:51.060
In fact, what he was
suggesting was socialism

00:02:51.060 --> 00:02:54.410
a system in which the government,
not private individuals

00:02:54.410 --> 00:02:56.630
owns economic enterprises.

00:02:56.630 --> 00:03:00.410
Bellamy carefully avoided
saying the word socialism

00:03:00.410 --> 00:03:02.740
which was associated with anarchists

00:03:02.740 --> 00:03:04.810
and immigrant radicals

00:03:04.810 --> 00:03:07.560
but he portrayed it as the ultimate remedy

00:03:07.560 --> 00:03:09.280
to all problems in the country.

00:03:09.280 --> 00:03:12.030
Bellamy's work influenced many reformers

00:03:12.030 --> 00:03:14.980
including the labor activist
and socialist leader

00:03:14.980 --> 00:03:17.620
Eugene V. Debs who ran for president

00:03:17.620 --> 00:03:20.520
on a socialist platform five times.

00:03:20.520 --> 00:03:23.080
Another popular suggestion of the time

00:03:23.080 --> 00:03:27.690
was the single tax which
was proposed by Henry George

00:03:27.690 --> 00:03:29.970
in his book "Progress and Poverty."

00:03:29.970 --> 00:03:32.400
George's solution to wealth inequality

00:03:32.400 --> 00:03:37.020
was to replace all other
taxes with a single high tax

00:03:37.020 --> 00:03:38.360
on the value of land.

00:03:38.360 --> 00:03:40.430
He believed that the revenue from this tax

00:03:40.430 --> 00:03:41.263
would be enough to pay

00:03:41.263 --> 00:03:44.110
for all necessary government services.

00:03:44.110 --> 00:03:47.730
One of George's many
admirers was Jacob Riis.

00:03:47.730 --> 00:03:50.260
He was a social reformer who published

00:03:50.260 --> 00:03:54.000
a photo expose of tenement
life in New York City

00:03:54.000 --> 00:03:56.300
called "How the Other Half Lives."

00:03:56.300 --> 00:03:58.710
Riis was one of the first muckrakers

00:03:58.710 --> 00:04:01.860
whose chosen method of
combating social problems

00:04:01.860 --> 00:04:03.690
was to shine a light on them.

00:04:03.690 --> 00:04:07.460
His images of dangerous
living conditions in tenements

00:04:07.460 --> 00:04:10.770
led to laws which
regulated building safety.

00:04:10.770 --> 00:04:14.720
Other muckrakers targeted
the corruption in industry

00:04:14.720 --> 00:04:18.040
like Ida Tarbell, who wrote a history

00:04:18.040 --> 00:04:19.650
of the Standard Oil Company

00:04:19.650 --> 00:04:22.660
that exposed its unscrupulous practices.

00:04:22.660 --> 00:04:27.120
Some focused on the unjust
treatment of racial minorities.

00:04:27.120 --> 00:04:31.080
In 1881, Helen Hunt
Jackson published a book

00:04:31.080 --> 00:04:33.640
called "A Century of Dishonor"

00:04:33.640 --> 00:04:36.240
which discussed the mistreatment, violence

00:04:36.240 --> 00:04:39.840
and broken treaties that
indigenous Americans had faced

00:04:39.840 --> 00:04:42.910
at the hands of the U.S.
government and white settlers.

00:04:42.910 --> 00:04:45.300
Journalist Ida B. Wells campaigned

00:04:45.300 --> 00:04:48.080
against the lynching of
black men in the South.

00:04:48.080 --> 00:04:49.500
In addition to campaigning

00:04:49.500 --> 00:04:52.520
against economic and social inequality

00:04:52.520 --> 00:04:56.370
many Gilded Age reformers
attempted to remedy the problems

00:04:56.370 --> 00:04:58.750
befalling cities and their residents.

00:04:58.750 --> 00:05:02.020
The most famous of these was
the settlement house movement.

00:05:02.020 --> 00:05:04.410
Settlement houses were community centers

00:05:04.410 --> 00:05:06.230
based in immigrant neighborhoods

00:05:06.230 --> 00:05:09.570
where newcomers could learn
English, get job skills

00:05:09.570 --> 00:05:13.240
attain childcare and
find a range of services

00:05:13.240 --> 00:05:16.300
that helped them adapt
to life in urban America.

00:05:16.300 --> 00:05:20.950
Jane Addams founded Hull
House in Chicago in 1889

00:05:20.950 --> 00:05:22.300
and many other reformers

00:05:22.300 --> 00:05:25.060
founded settlement houses in other cities.

00:05:25.060 --> 00:05:27.760
Similarly some churches of the time period

00:05:27.760 --> 00:05:30.300
began to emphasize that confronting

00:05:30.300 --> 00:05:33.940
contemporary social problems
and helping the poor

00:05:33.940 --> 00:05:37.500
were the embodiment of the
teachings of Christianity.

00:05:37.500 --> 00:05:40.830
This Social Gospel
movement as it was known

00:05:40.830 --> 00:05:44.190
led to the establishment
of missions in urban areas

00:05:44.190 --> 00:05:46.510
and churches opened libraries

00:05:46.510 --> 00:05:49.900
gymnasiums and classrooms for public use.

00:05:49.900 --> 00:05:52.020
Some reformers focused their energies

00:05:52.020 --> 00:05:53.940
on the physical setting of cities

00:05:53.940 --> 00:05:57.940
believing that the squalor of
dirty streets and tenements

00:05:57.940 --> 00:06:01.560
depressed people and
encouraged moral decay.

00:06:01.560 --> 00:06:02.831
The City Beautiful movement

00:06:02.831 --> 00:06:05.950
works to incorporate parks

00:06:05.950 --> 00:06:09.100
inspiring architecture and good design

00:06:09.100 --> 00:06:10.740
into American cities.

00:06:10.740 --> 00:06:14.580
Frederick Law Olmsted designed
Central Park in New York City

00:06:14.580 --> 00:06:16.930
to be a respite from the urban jungle

00:06:16.930 --> 00:06:20.110
and City Beautiful architects
like Daniel Burnham

00:06:20.110 --> 00:06:24.380
created monumental spaces and
buildings in Washington D.C.

00:06:24.380 --> 00:06:27.370
These spaces were supposed
to inspire harmony

00:06:27.370 --> 00:06:30.830
order and civic virtue in society.

00:06:30.830 --> 00:06:34.790
One aspect of Gilded Age reform
you may have noticed by now

00:06:34.790 --> 00:06:38.170
is that a large number
of reformers were women.

00:06:38.170 --> 00:06:42.260
Jane Addams, Ida B. Wells and Ida Tarbell

00:06:42.260 --> 00:06:44.960
all placed prominent
roles in the movement.

00:06:44.960 --> 00:06:48.990
After the Civil War a growing
number of middle class women

00:06:48.990 --> 00:06:52.280
went to college and these
active and educated women

00:06:52.280 --> 00:06:55.960
began looking for work and
meaning outside of the home.

00:06:55.960 --> 00:06:58.850
Many white, middle- and upper-class women

00:06:58.850 --> 00:07:01.670
joined clubs dedicated to social reforms

00:07:01.670 --> 00:07:05.180
and they argued that
women's traditional role

00:07:05.180 --> 00:07:07.070
of keeping their homes clean

00:07:07.070 --> 00:07:09.670
and the people within
them upstanding and moral

00:07:09.670 --> 00:07:12.080
also extended to their communities

00:07:12.080 --> 00:07:14.810
which they called municipal housekeeping.

00:07:14.810 --> 00:07:17.830
The Women's Christian
Temperance Union, for example

00:07:17.830 --> 00:07:21.300
became the Gilded Age's
largest female organization

00:07:21.300 --> 00:07:24.100
with more than 150,000 members.

00:07:24.100 --> 00:07:28.420
Led by Frances Willard, the
WCTU grew from its roots

00:07:28.420 --> 00:07:31.600
in opposing the sale and
consumption of alcohol

00:07:31.600 --> 00:07:35.100
to advocate for policy
solutions to social problems

00:07:35.100 --> 00:07:38.600
ranging from prison reform
to domestic violence.

00:07:38.600 --> 00:07:42.320
The WCTU also campaigned
for women's suffrage.

00:07:42.320 --> 00:07:46.530
In 1890 the two major women's
suffrage organizations

00:07:46.530 --> 00:07:48.600
which had been at odds with each other

00:07:48.600 --> 00:07:52.450
since the passage of the 15th
Amendment reunited to form

00:07:52.450 --> 00:07:56.410
the National American
Woman Suffrage Association.

00:07:56.410 --> 00:07:57.820
Their efforts would lead

00:07:57.820 --> 00:08:00.640
to the growth of woman
suffrage at the state level

00:08:00.640 --> 00:08:03.840
and later, with the help of
the National Woman's Party

00:08:03.840 --> 00:08:07.553
the passage of the 19th Amendment
to the U.S. Constitution.


Factoring quadratics with a common factor

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:02.020
- [Instructor] Averil was trying to factor

00:00:02.020 --> 00:00:05.270
six x squared minus 18x plus 12.

00:00:05.270 --> 00:00:08.550
She found that the greatest common factor

00:00:08.550 --> 00:00:13.240
of these terms was six
and made an area model.

00:00:13.240 --> 00:00:16.320
What is the width of Averil's area model?

00:00:16.320 --> 00:00:18.200
So pause this video and see
if you can figure that out,

00:00:18.200 --> 00:00:20.263
and then we'll work through this together.

00:00:21.120 --> 00:00:23.800
All right, so there's a couple
of ways to think about it.

00:00:23.800 --> 00:00:26.550
She's trying to factor six x squared

00:00:26.550 --> 00:00:29.160
minus 18x plus 12,

00:00:29.160 --> 00:00:32.230
and she figured out that the
greatest common factor was six.

00:00:32.230 --> 00:00:34.570
So one way you could think about it is

00:00:34.570 --> 00:00:39.250
this could be rewritten as
six times something else.

00:00:39.250 --> 00:00:40.690
And to help her think about it,

00:00:40.690 --> 00:00:42.320
she thought about an area model,

00:00:42.320 --> 00:00:45.320
where if you had a rectangle,

00:00:45.320 --> 00:00:47.520
if you had a rectangle like this,

00:00:47.520 --> 00:00:50.970
and if the height is six and the width,

00:00:50.970 --> 00:00:53.430
let's just call that the width for now,

00:00:53.430 --> 00:00:56.060
so this is the width right over here.

00:00:56.060 --> 00:00:58.440
If you multiply six times the width,

00:00:58.440 --> 00:01:00.800
maybe I could write width right over here,

00:01:00.800 --> 00:01:02.580
if you multiply six times the width,

00:01:02.580 --> 00:01:04.620
you multiply the height times the width,

00:01:04.620 --> 00:01:05.550
you're going to get the area.

00:01:05.550 --> 00:01:07.350
So imagine that the area of this rectangle

00:01:07.350 --> 00:01:08.920
was our original expression,

00:01:08.920 --> 00:01:12.920
six x squared minus 18x plus 12.

00:01:12.920 --> 00:01:15.230
And that's exactly what's drawn here.

00:01:15.230 --> 00:01:18.010
Now, what's interesting
is is that they broke up

00:01:18.010 --> 00:01:20.140
the area into three sections.

00:01:20.140 --> 00:01:24.080
This pink section is the six x squared,

00:01:24.080 --> 00:01:27.460
this blue section is the negative 18x,

00:01:27.460 --> 00:01:29.770
and this peach section is the 12.

00:01:29.770 --> 00:01:30.960
And, of course, these
aren't drawn to scale,

00:01:30.960 --> 00:01:33.270
'cause we don't even know
how wide each of these are

00:01:33.270 --> 00:01:34.500
'cause we don't know what x is.

00:01:34.500 --> 00:01:35.940
So this is all a little bit abstract,

00:01:35.940 --> 00:01:38.380
but it's to show that we
can break our bigger area

00:01:38.380 --> 00:01:40.340
into three smaller areas.

00:01:40.340 --> 00:01:43.230
And what's useful about
this is we could think about

00:01:43.230 --> 00:01:46.780
the width of each of these sub-areas,

00:01:46.780 --> 00:01:48.730
and then we can add them together

00:01:48.730 --> 00:01:50.300
to figure out the total width.

00:01:50.300 --> 00:01:54.120
So what is the width of this
pink section right over here?

00:01:54.120 --> 00:01:57.120
Well, six times what is six x squared?

00:01:57.120 --> 00:02:00.170
Well, six times x
squared is six x squared,

00:02:00.170 --> 00:02:02.400
so the width here is x squared.

00:02:02.400 --> 00:02:05.780
Now, what about this blue area?

00:02:05.780 --> 00:02:08.180
A height of six times what width

00:02:08.180 --> 00:02:10.440
is equal to negative 18x?

00:02:10.440 --> 00:02:13.560
So let's see, if I take
six times negative three,

00:02:13.560 --> 00:02:15.140
I get negative 18,

00:02:15.140 --> 00:02:17.710
then I have to multiply
it times an x as well

00:02:17.710 --> 00:02:18.990
to get negative 18x.

00:02:18.990 --> 00:02:22.950
So six times negative
three x is negative 18x.

00:02:22.950 --> 00:02:25.040
And then, last but not least,

00:02:25.040 --> 00:02:26.670
six, our height of six,

00:02:26.670 --> 00:02:29.090
times what is going to be equal to 12?

00:02:29.090 --> 00:02:31.690
Well, six times two is equal to 12.

00:02:31.690 --> 00:02:34.960
So we figured out the widths
of each of these subregions,

00:02:34.960 --> 00:02:36.840
and now we know what the total width is.

00:02:36.840 --> 00:02:39.550
The total width is going
to be our x squared

00:02:39.550 --> 00:02:43.420
plus our negative three x, plus our two.

00:02:43.420 --> 00:02:46.420
So the width is going to be x squared,

00:02:46.420 --> 00:02:47.580
and I can just write that as,

00:02:47.580 --> 00:02:51.430
minus three x, plus two.

00:02:51.430 --> 00:02:53.870
So we have answered the question.

00:02:53.870 --> 00:02:56.080
And you could substitute
that back in for this,

00:02:56.080 --> 00:02:57.880
and you could see, if
you multiplied six times

00:02:57.880 --> 00:02:59.760
all of this, if you distributed the six,

00:02:59.760 --> 00:03:03.573
you would indeed get six x
squared minus 18x plus 12.


Multiply monomials by polynomials: Area model

WEBVTT
Kind: captions
Language: en

00:00:00.120 --> 00:00:03.270
- [Instructor] We are told a
rectangle has a height of five

00:00:03.270 --> 00:00:08.220
and a width of 3x
squared minus x plus two.

00:00:08.220 --> 00:00:10.460
Then we're told express the area

00:00:10.460 --> 00:00:12.550
of the entire rectangle,

00:00:12.550 --> 00:00:14.930
and the expression should be expanded.

00:00:14.930 --> 00:00:18.900
So pause this video and see if
you could work through this.

00:00:18.900 --> 00:00:21.420
All right, now let's work
through this together.

00:00:21.420 --> 00:00:23.390
What this diagram is showing us

00:00:23.390 --> 00:00:25.910
is exactly an indicative rectangle,

00:00:25.910 --> 00:00:28.440
where its height is five

00:00:28.440 --> 00:00:33.100
and its width is 3x
squared minus x plus two.

00:00:33.100 --> 00:00:35.410
What this shows us is is the area

00:00:35.410 --> 00:00:38.440
of the entire rectangle can be broken down

00:00:38.440 --> 00:00:40.820
into three smaller areas.

00:00:40.820 --> 00:00:44.200
You have this blue area right over here.

00:00:44.200 --> 00:00:46.860
Where the width is 3x
squared, the height is five,

00:00:46.860 --> 00:00:48.463
so what's that area going to be?

00:00:48.463 --> 00:00:50.900
It's going to be the height times width.

00:00:50.900 --> 00:00:53.780
Five times 3x squared.

00:00:53.780 --> 00:00:57.220
That's the same thing as 15x squared.

00:00:57.220 --> 00:01:00.880
Now what about this purple
area right over here?

00:01:00.880 --> 00:01:02.930
It's going to be height times width again,

00:01:02.930 --> 00:01:05.870
so it'll be five times negative x,

00:01:05.870 --> 00:01:08.580
which is the same thing as negative 5x.

00:01:08.580 --> 00:01:09.570
I know what some of you are thinking,

00:01:09.570 --> 00:01:12.690
how can you have a width of negative x?

00:01:12.690 --> 00:01:14.070
Well, we don't know what x is,

00:01:14.070 --> 00:01:16.270
so this is all a little bit abstract.

00:01:16.270 --> 00:01:19.610
But you could imagine having
x be a negative value,

00:01:19.610 --> 00:01:22.770
in which case this actually
would be a positive width.

00:01:22.770 --> 00:01:24.080
Another thing you might be saying is,

00:01:24.080 --> 00:01:26.420
hey, this magenta area,
when I just eyeball it,

00:01:26.420 --> 00:01:28.540
looks like it has a larger
width than this blue area.

00:01:28.540 --> 00:01:29.373
How do we know that?

00:01:29.373 --> 00:01:30.206
We don't.

00:01:30.206 --> 00:01:32.960
They're just showing this
as an indicative way.

00:01:32.960 --> 00:01:35.100
They might have actually the same width.

00:01:35.100 --> 00:01:37.310
They might have, one might
be larger than the other,

00:01:37.310 --> 00:01:38.490
but this is just showing us that

00:01:38.490 --> 00:01:39.980
they're not necessarily the same.

00:01:39.980 --> 00:01:41.100
It's very abstract.

00:01:41.100 --> 00:01:42.450
It's definitely not drawn to scale

00:01:42.450 --> 00:01:44.940
'cause we don't know what x is.

00:01:44.940 --> 00:01:47.020
All right, and then this last area

00:01:47.020 --> 00:01:49.160
is going to be the height, which is five,

00:01:49.160 --> 00:01:50.920
times the width, which is two.

00:01:50.920 --> 00:01:53.050
So that's just going to be equal to 10.

00:01:53.050 --> 00:01:56.000
So what we just saw is that
the area of the whole thing

00:01:56.000 --> 00:01:58.610
is equal to the sum of these areas,

00:01:58.610 --> 00:02:03.440
and the sum of those areas is 15x squared

00:02:03.440 --> 00:02:06.860
plus negative 5x, or we could
just write that as minus 5x.

00:02:06.860 --> 00:02:09.480
And then we have plus 10.

00:02:09.480 --> 00:02:11.010
Now I know what some of you are thinking.

00:02:11.010 --> 00:02:12.610
If I know that the height is five

00:02:12.610 --> 00:02:14.640
and the width is this value,

00:02:14.640 --> 00:02:16.920
well couldn't I have
just multiplied height

00:02:16.920 --> 00:02:21.020
times the entire width, 3x
squared minus x plus two,

00:02:21.020 --> 00:02:24.410
and then I would've just
naturally distributed the five?

00:02:24.410 --> 00:02:26.630
And essentially that's
exactly what we did here.

00:02:26.630 --> 00:02:27.890
Area models, you might have first

00:02:27.890 --> 00:02:29.340
seen them in elementary school,

00:02:29.340 --> 00:02:32.260
really to understand the
distributive property.

00:02:32.260 --> 00:02:35.740
So if you distribute the
five you get 15x squared

00:02:35.740 --> 00:02:38.010
minus 5x plus 10.

00:02:38.010 --> 00:02:39.110
Same idea.

00:02:39.110 --> 00:02:40.360
Let's do another example.

00:02:41.780 --> 00:02:43.570
So here we're given another rectangle.

00:02:43.570 --> 00:02:44.960
It has a height of 2x.

00:02:44.960 --> 00:02:46.360
We see that right there.

00:02:46.360 --> 00:02:48.150
A width of 3x plus four.

00:02:48.150 --> 00:02:50.020
We see that over there.

00:02:50.020 --> 00:02:53.280
Express the area of the entire rectangle.

00:02:53.280 --> 00:02:54.860
So same drill.

00:02:54.860 --> 00:02:56.050
See if you can pause this video

00:02:56.050 --> 00:02:57.603
and work on this on your own.

00:02:58.640 --> 00:02:59.500
All right.

00:02:59.500 --> 00:03:02.900
Well, we can see that the height is 2x,

00:03:02.900 --> 00:03:07.060
the width in total is 3x plus four.

00:03:07.060 --> 00:03:09.620
But we can clearly see that
the area of the entire thing

00:03:09.620 --> 00:03:11.560
can be broken up into this blue section

00:03:11.560 --> 00:03:13.050
and this magenta section.

00:03:13.050 --> 00:03:16.230
The blue section's area is 3x times 2x.

00:03:16.230 --> 00:03:20.620
So 3x times 2x, which
is equal to 6x squared.

00:03:20.620 --> 00:03:22.270
The magenta section is equal

00:03:22.270 --> 00:03:25.270
to width times height, its area.

00:03:25.270 --> 00:03:27.510
So that going to be equal to 8x.

00:03:27.510 --> 00:03:29.080
So the area of the whole thing is going

00:03:29.080 --> 00:03:30.250
to be the sum of these areas,

00:03:30.250 --> 00:03:32.990
which is going to be 6x squared plus 8x.

00:03:32.990 --> 00:03:33.920
And we're done.

00:03:33.920 --> 00:03:36.340
Once again, we could've
just thought about it as

00:03:36.340 --> 00:03:39.760
the height is 2x, we're gonna
multiply it times the width,

00:03:39.760 --> 00:03:41.520
times 3x plus four.

00:03:41.520 --> 00:03:44.000
Then we just distribute the 2x.

00:03:44.000 --> 00:03:46.669
So 2x times 3x is 6x squared.

00:03:46.669 --> 00:03:49.370
2x times four is 8x.

00:03:49.370 --> 00:03:51.010
Same idea.

00:03:51.010 --> 00:03:53.600
It's really just to
make sure we understand

00:03:53.600 --> 00:03:57.670
that when we distribute a 2x
onto this 3x and this four,

00:03:57.670 --> 00:04:00.750
we can conceptualize it as figuring out

00:04:00.750 --> 00:04:05.153
the areas of the sub-parts
of the entire rectangle.


Interpret a quadratic graph

WEBVTT
Kind: captions
Language: en

00:00:00.110 --> 00:00:02.010
- [Instructor] Katie
throws a ball in the air

00:00:02.010 --> 00:00:03.910
for her dog to chase.

00:00:03.910 --> 00:00:08.260
The function f models the
height of the ball, in meters,

00:00:08.260 --> 00:00:11.870
as a function of time, in
seconds, after Katie threw it.

00:00:11.870 --> 00:00:13.620
And we could see that right over here.

00:00:13.620 --> 00:00:15.230
This is our function f.

00:00:15.230 --> 00:00:17.390
So at time t equals zero,

00:00:17.390 --> 00:00:19.760
the height looks like
it's a couple of meters.

00:00:19.760 --> 00:00:22.880
And then as we go forward in time,

00:00:22.880 --> 00:00:26.180
to a little under 2 1/2
seconds, the ball's going up.

00:00:26.180 --> 00:00:28.700
And then after a little
under 2 1/2 seconds,

00:00:28.700 --> 00:00:30.420
the ball starts going down.

00:00:30.420 --> 00:00:32.140
And after, by the time you get

00:00:32.140 --> 00:00:33.760
to five seconds or close to five seconds,

00:00:33.760 --> 00:00:35.600
it looks like the ball is on the ground.

00:00:35.600 --> 00:00:38.070
Its height is zero meters.

00:00:38.070 --> 00:00:40.510
So then they ask us

00:00:40.510 --> 00:00:42.640
which of these statements are true?

00:00:42.640 --> 00:00:44.350
Choose all that apply.

00:00:44.350 --> 00:00:47.643
So pause this video, and
see if you can work it out.

00:00:49.120 --> 00:00:50.680
All right, now let's
look through the choices.

00:00:50.680 --> 00:00:52.680
So the first one says Katie threw the ball

00:00:52.680 --> 00:00:54.260
from a height of five meters.

00:00:54.260 --> 00:00:56.910
So let's see if that bears fruit

00:00:56.910 --> 00:00:59.310
or (chuckles) see if that's true.

00:00:59.310 --> 00:01:01.360
So if she threw it from
a height of five meters,

00:01:01.360 --> 00:01:03.040
that means that the y-intercept

00:01:03.040 --> 00:01:04.230
would've been at five meters.

00:01:04.230 --> 00:01:05.260
At time t equals zero,

00:01:05.260 --> 00:01:06.990
we would've been at a
height of five meters.

00:01:06.990 --> 00:01:08.690
Clearly, that is not the y-intercept.

00:01:08.690 --> 00:01:09.920
It looks like she threw it from a height

00:01:09.920 --> 00:01:11.910
of maybe 1 1/2 or two meters.

00:01:11.910 --> 00:01:15.050
So I'm assuming that Katie
is not five meters tall.

00:01:15.050 --> 00:01:16.430
So she wasn't on a ladder or anything.

00:01:16.430 --> 00:01:18.260
She just threw it from her regular height.

00:01:18.260 --> 00:01:22.370
And so we can rule out this first choice.

00:01:22.370 --> 00:01:24.640
The second one says at its highest point,

00:01:24.640 --> 00:01:27.560
the ball was about 31
meters above the ground.

00:01:27.560 --> 00:01:29.530
Let's see if that is true.

00:01:29.530 --> 00:01:32.810
So the highest point is right over here.

00:01:32.810 --> 00:01:36.570
And yeah, that looks about 31 meters,

00:01:36.570 --> 00:01:38.460
so I like that choice.

00:01:38.460 --> 00:01:40.623
So I will select that one.

00:01:41.640 --> 00:01:45.420
The ball was in the air
for about 2 1/2 seconds.

00:01:45.420 --> 00:01:49.130
So we can clearly see
that that is not the case.

00:01:49.130 --> 00:01:50.500
The ball was going up

00:01:50.500 --> 00:01:52.830
for about 2 1/2 seconds,
but then it was going down

00:01:52.830 --> 00:01:55.280
for roughly another 2 1/2 seconds.

00:01:55.280 --> 00:01:58.760
And so it was actually in the
air for almost five seconds.

00:01:58.760 --> 00:02:03.083
So I would rule this one out.

00:02:04.250 --> 00:02:06.250
The ball reached its
highest point in the air

00:02:06.250 --> 00:02:09.600
about 2 1/2 seconds after Katie threw it.

00:02:09.600 --> 00:02:11.270
So let's see, after Katie,

00:02:11.270 --> 00:02:13.430
2 1/2 seconds after Katie threw the ball,

00:02:13.430 --> 00:02:15.600
it's right over there.

00:02:15.600 --> 00:02:17.210
It looks like it reached its highest point

00:02:17.210 --> 00:02:19.410
a little bit before that.

00:02:19.410 --> 00:02:22.420
But they said about, so they're
speaking in rough terms.

00:02:22.420 --> 00:02:24.320
So I think that statement can be true.

00:02:25.211 --> 00:02:26.780
It's about 2 1/2 seconds.

00:02:26.780 --> 00:02:28.340
If they said exactly 2 1/2 seconds,

00:02:28.340 --> 00:02:30.390
I wouldn't have selected it
'cause it seems like it happened

00:02:30.390 --> 00:02:33.070
at like 2.45 seconds
or something like that.

00:02:33.070 --> 00:02:36.293
But there you go, those are
the ones that seem true.


Factoring completely with a common factor

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:02.340
- [Instructor] So let's
see if we can try to factor

00:00:02.340 --> 00:00:05.820
the following expression completely.

00:00:05.820 --> 00:00:07.130
So factor this completely,

00:00:07.130 --> 00:00:09.143
pause the video and have a go at that.

00:00:10.160 --> 00:00:12.200
All right, now let's work
through this together.

00:00:12.200 --> 00:00:14.410
So the way that I like to think about it,

00:00:14.410 --> 00:00:17.240
I first try to see is
there any common factor

00:00:17.240 --> 00:00:19.820
to all the terms, and I
try to find the greatest

00:00:19.820 --> 00:00:20.900
of the common factor,

00:00:20.900 --> 00:00:23.500
possible common factors
to all of the terms.

00:00:23.500 --> 00:00:26.410
So let's see, they're
all divisible by two,

00:00:26.410 --> 00:00:27.960
so two would be a common factor,

00:00:27.960 --> 00:00:30.270
but let's see, they're
also all divisible by four,

00:00:30.270 --> 00:00:33.030
four is divisible by four,
eight is divisible by four,

00:00:33.030 --> 00:00:34.780
12 is divisible by four,

00:00:34.780 --> 00:00:37.390
and that looks like the
greatest common factor.

00:00:37.390 --> 00:00:38.820
They're not all divisible by x,

00:00:38.820 --> 00:00:40.540
so I can't throw an x in there.

00:00:40.540 --> 00:00:42.970
So what I wanna do is factor out a four.

00:00:42.970 --> 00:00:46.290
So I could re-write this as four times,

00:00:46.290 --> 00:00:48.320
now what would it be, four times what?

00:00:48.320 --> 00:00:51.150
Well if I factor a four
out of four x squared,

00:00:51.150 --> 00:00:53.910
I'm just going to be
left with an x squared.

00:00:53.910 --> 00:00:57.530
If I factor a four out
of negative eight x,

00:00:57.530 --> 00:01:00.420
negative eight x divided
by four is negative two,

00:01:00.420 --> 00:01:03.000
so I'm going to have negative two x.

00:01:03.000 --> 00:01:08.000
And if I factor a four out of negative 12,

00:01:08.070 --> 00:01:12.520
negative 12 divided by
four is negative three.

00:01:12.520 --> 00:01:14.200
Now am I done factoring?

00:01:14.200 --> 00:01:16.310
Well it looks like I
could factor this thing

00:01:16.310 --> 00:01:17.500
a little bit more.

00:01:17.500 --> 00:01:21.200
Can I think of two numbers
that add up to negative two,

00:01:21.200 --> 00:01:23.510
and when I multiply it
I get negative three,

00:01:23.510 --> 00:01:25.320
since when I multiply
I get a negative value,

00:01:25.320 --> 00:01:26.741
one of the 'em is going to be positive

00:01:26.741 --> 00:01:27.770
and one of 'em is going to be negative.

00:01:27.770 --> 00:01:29.350
I can think about it this way.

00:01:29.350 --> 00:01:32.050
A plus B is equal to negative two,

00:01:32.050 --> 00:01:35.070
A times B needs to be
equal to negative three.

00:01:35.070 --> 00:01:37.600
So let's see, A could be
equal to negative three

00:01:37.600 --> 00:01:39.670
and B could be equal to one

00:01:39.670 --> 00:01:43.450
because negative three
plus one is negative two,

00:01:43.450 --> 00:01:46.470
and negative three times
one is negative three.

00:01:46.470 --> 00:01:48.160
So I could re-write all of this

00:01:48.160 --> 00:01:53.160
as four times x plus negative three,

00:01:53.670 --> 00:01:56.090
or I could just write
that as x minus three,

00:01:56.090 --> 00:02:00.430
times x plus one, x plus one.

00:02:00.430 --> 00:02:05.280
And now I have actually
factored this completely.

00:02:05.280 --> 00:02:07.773
Let's do another example.

00:02:09.300 --> 00:02:12.200
So let's say that we had the expression

00:02:12.200 --> 00:02:17.200
negative three x squared
plus 21 x minus 30.

00:02:17.760 --> 00:02:21.370
Pause the video and see if you
can factor this completely.

00:02:21.370 --> 00:02:22.760
All right now let's do this together.

00:02:22.760 --> 00:02:24.830
So what would be the
greatest common factor?

00:02:24.830 --> 00:02:28.010
So let's see, they're
all divisible by three,

00:02:28.010 --> 00:02:29.690
so you could factor out a three.

00:02:29.690 --> 00:02:31.470
Let's see what happens if
you factor out a three.

00:02:31.470 --> 00:02:34.120
This is the same thing as three times,

00:02:34.120 --> 00:02:35.600
well negative three x squared divided

00:02:35.600 --> 00:02:38.110
by three is negative x squared,

00:02:38.110 --> 00:02:41.810
21 x divided by three is
seven x, so plus seven x,

00:02:41.810 --> 00:02:45.520
and then negative 30 divided
by three is negative 10.

00:02:45.520 --> 00:02:46.930
You could do it this way,

00:02:46.930 --> 00:02:49.970
but having this negative
out on the x squared term

00:02:49.970 --> 00:02:51.560
still makes it a little bit confusing

00:02:51.560 --> 00:02:53.240
on how you would factor this further.

00:02:53.240 --> 00:02:54.610
You can do it, but it still takes

00:02:54.610 --> 00:02:57.120
a little bit more of a mental load.

00:02:57.120 --> 00:02:58.590
So instead of just factoring out a three,

00:02:58.590 --> 00:03:00.120
let's factor out a negative three.

00:03:00.120 --> 00:03:01.050
So we could write it this way.

00:03:01.050 --> 00:03:04.070
If we factor out a negative
three, what does that become?

00:03:04.070 --> 00:03:06.090
Well then if you factor
out a negative three out

00:03:06.090 --> 00:03:08.730
of this term, you're just
left with an x squared.

00:03:08.730 --> 00:03:11.520
If you factor out a negative
three from this term,

00:03:11.520 --> 00:03:15.000
21 divided by negative
three is negative seven x.

00:03:15.000 --> 00:03:18.990
And if you factor out a negative
three out of negative 30,

00:03:18.990 --> 00:03:22.820
you're left with a
positive 10, positive 10.

00:03:22.820 --> 00:03:25.370
And now let's see if we
can factor this thing

00:03:25.370 --> 00:03:26.470
a little bit more.

00:03:26.470 --> 00:03:30.110
Can I think of two numbers
where if I were to add them

00:03:30.110 --> 00:03:32.320
I get to negative seven,

00:03:32.320 --> 00:03:35.790
and if I were to multiply
them, I get to 10?

00:03:35.790 --> 00:03:37.890
And let's see, they'd
have to have the same sign

00:03:37.890 --> 00:03:39.660
'cause their product is positive.

00:03:39.660 --> 00:03:42.580
So let's see A could be
equal to negative five,

00:03:42.580 --> 00:03:45.230
and then B is equal to negative two.

00:03:45.230 --> 00:03:47.490
So I can re-write this
whole thing as equal

00:03:47.490 --> 00:03:50.430
to negative three times
x plus negative five,

00:03:50.430 --> 00:03:52.400
which is the same thing as x minus five,

00:03:52.400 --> 00:03:53.910
times x plus negative two,

00:03:53.910 --> 00:03:56.810
which is the same thing as x minus two.

00:03:56.810 --> 00:03:59.233
And now we have factored completely.


Writing functions with exponential decay

WEBVTT
Kind: captions
Language: en

00:00:00.100 --> 00:00:03.250
- [Instructor] We are told
a phone sells for $600

00:00:03.250 --> 00:00:08.250
and loses 25% of its value per year.

00:00:08.530 --> 00:00:12.550
Write a function that gives
the phone's value, V of t,

00:00:12.550 --> 00:00:17.550
so value as a function of
time, t years after it is sold.

00:00:17.930 --> 00:00:20.210
So pause this video, and have a go at that

00:00:20.210 --> 00:00:22.480
before we work through it together.

00:00:22.480 --> 00:00:25.200
All right, so let's just
think about it a little bit.

00:00:25.200 --> 00:00:27.220
And I could even set up a table

00:00:27.220 --> 00:00:28.870
to think about what is going on.

00:00:28.870 --> 00:00:32.910
So this is t, and this
is the value of our phone

00:00:32.910 --> 00:00:34.990
as a function of t.

00:00:34.990 --> 00:00:37.130
So it sells for $600.

00:00:37.130 --> 00:00:40.670
At time t equals zero, what is V of zero?

00:00:40.670 --> 00:00:43.650
Well it's going to be equal to $600.

00:00:43.650 --> 00:00:46.500
That's what it sells for
at time t equals zero.

00:00:46.500 --> 00:00:49.200
Now at t equals one,
what's going to happen?

00:00:49.200 --> 00:00:52.090
Well it says that the phone loses 25%

00:00:52.090 --> 00:00:53.600
of its value per year.

00:00:53.600 --> 00:00:57.270
Another way to rewrite that
it loses 25% of its value

00:00:57.270 --> 00:01:01.010
per year is that it retains, it retains,

00:01:01.010 --> 00:01:04.130
100% minus 25% of its value per year,

00:01:04.130 --> 00:01:06.980
or it retains 75%

00:01:06.980 --> 00:01:11.980
of its value per year, per year.

00:01:13.330 --> 00:01:16.480
So how much is it going to
be worth after one year?

00:01:16.480 --> 00:01:19.617
Well it's going to be worth $600,

00:01:19.617 --> 00:01:24.160
$600 times 75%.

00:01:24.160 --> 00:01:25.550
Now what about year two?

00:01:25.550 --> 00:01:28.151
Well it's going to be worth
what it was in year one

00:01:28.151 --> 00:01:30.670
times 75% again.

00:01:30.670 --> 00:01:35.620
So it's going to be $600
times 75% times 75%.

00:01:35.620 --> 00:01:40.620
And so you could write
that as times 75% squared.

00:01:41.300 --> 00:01:42.830
And I think you see a pattern.

00:01:42.830 --> 00:01:47.200
In general, if we have gone,
let's just call it t years,

00:01:47.200 --> 00:01:48.890
well then the value of our phone,

00:01:48.890 --> 00:01:52.210
if we're saying it in dollars,
is just going to be $600

00:01:52.210 --> 00:01:54.575
times, and I could write it as a decimal,

00:01:54.575 --> 00:01:59.575
0.75, instead of 75%, to the t power.

00:02:00.540 --> 00:02:05.230
So V of t is going to be equal to 600

00:02:05.230 --> 00:02:10.230
times 0.75 to the t power, and we're done.

00:02:11.363 --> 00:02:12.963
Let's do another example.

00:02:14.080 --> 00:02:17.320
So here, we are told that
a biologist has a sample

00:02:17.320 --> 00:02:19.620
of 6,000 cells.

00:02:19.620 --> 00:02:22.650
The biologist introduces
a virus that kills 1/3

00:02:22.650 --> 00:02:25.190
of the cells every week.

00:02:25.190 --> 00:02:29.060
Write a function that gives
the number of cells remaining,

00:02:29.060 --> 00:02:32.140
which would be C of t, the
cells as a function of time,

00:02:32.140 --> 00:02:36.140
in the sample t weeks after
the virus is introduced.

00:02:36.140 --> 00:02:38.060
So again, pause this video

00:02:38.060 --> 00:02:39.810
and see if you can figure that out.

00:02:40.790 --> 00:02:43.710
All right, so I'll set
up another table again.

00:02:43.710 --> 00:02:47.400
So this is time, it's in weeks,

00:02:47.400 --> 00:02:49.200
and this is the number of cells, C.

00:02:49.200 --> 00:02:51.590
We could say it's a function of time.

00:02:51.590 --> 00:02:54.900
So time t equals zero, when
zero weeks have gone by,

00:02:54.900 --> 00:02:56.510
we have 6,000 cells.

00:02:56.510 --> 00:02:57.950
That's pretty clear.

00:02:57.950 --> 00:03:00.140
Now after one week, how
many cells do we have?

00:03:00.140 --> 00:03:01.750
What's C of one?

00:03:01.750 --> 00:03:04.320
Well it says that the virus kills 1/3

00:03:04.320 --> 00:03:07.030
of the cells every week,
which is another way of saying

00:03:07.030 --> 00:03:09.692
that 2/3 of the cells are able to live

00:03:09.692 --> 00:03:11.580
for the next week.

00:03:11.580 --> 00:03:16.543
And so after one week, we're
going to have 6,000 times 2/3.

00:03:17.720 --> 00:03:21.020
And then after two weeks,
or another week goes by,

00:03:21.020 --> 00:03:22.920
we're gonna have 2/3 of
the number that we had

00:03:22.920 --> 00:03:23.753
after one week.

00:03:23.753 --> 00:03:27.360
So we're gonna have 6,000
times 2/3 times 2/3,

00:03:27.360 --> 00:03:30.630
or we could just write
that as 2/3 squared.

00:03:30.630 --> 00:03:33.940
So once again, you are likely
seeing the pattern here.

00:03:33.940 --> 00:03:36.830
We are going to, at time t
equals zero, we have 6,000,

00:03:36.830 --> 00:03:39.120
and then we're going to multiply by 2/3

00:03:39.120 --> 00:03:42.440
however many weeks have gone by.

00:03:42.440 --> 00:03:45.710
So the cells as a function
of the weeks of t,

00:03:45.710 --> 00:03:48.680
which is in weeks, is going
to be our original amount,

00:03:48.680 --> 00:03:51.790
and then however many weeks have gone by,

00:03:51.790 --> 00:03:55.140
we're going to multiply
by 2/3 that many times,

00:03:55.140 --> 00:03:57.537
so times 2/3 to the t power.

00:03:57.537 --> 00:03:58.753
And we're done.


Analyzing mistakes when finding extrema example 2

WEBVTT
Kind: captions
Language: en

00:00:00.310 --> 00:00:02.500
- [Instructor] Erin was
asked to find if f of x

00:00:02.500 --> 00:00:05.620
is equal to x squared
minus one to the 2/3 power

00:00:05.620 --> 00:00:07.890
has a relative maximum.

00:00:07.890 --> 00:00:09.820
This is her solution.

00:00:09.820 --> 00:00:12.333
And then they give us her
steps, and at the end they say,

00:00:12.333 --> 00:00:13.970
is Erin's work correct?

00:00:13.970 --> 00:00:16.230
If not, what's her mistake?

00:00:16.230 --> 00:00:17.790
So pause this video and see if you can

00:00:17.790 --> 00:00:18.630
figure it out yourself.

00:00:18.630 --> 00:00:20.270
Is Erin correct, or did
you she make a mistake,

00:00:20.270 --> 00:00:21.510
and where was that mistake?

00:00:21.510 --> 00:00:23.400
All right, now let's just do it together.

00:00:23.400 --> 00:00:25.190
So she says that this is the derivative.

00:00:25.190 --> 00:00:28.470
I'm just going to reevaluate
here to the right of her work.

00:00:28.470 --> 00:00:30.900
So let's see, f prime
of x is just going to be

00:00:30.900 --> 00:00:31.980
the chain rule.

00:00:31.980 --> 00:00:33.420
I'm gonna take the
derivative of the outside

00:00:33.420 --> 00:00:34.510
with respect to the inside.

00:00:34.510 --> 00:00:38.980
So this is going to be 2/3
times x squared minus one

00:00:38.980 --> 00:00:43.350
to the 2/3 minus one, so
to the negative 1/3 power,

00:00:43.350 --> 00:00:46.410
times the derivative of the
inside with respect to x.

00:00:46.410 --> 00:00:48.614
So the derivative of x squared
minus one with respect to x

00:00:48.614 --> 00:00:51.310
is two x.

00:00:51.310 --> 00:00:52.143
(siren ringing)

00:00:52.143 --> 00:00:54.278
There's a fire hydrant, a fire (laughing),

00:00:54.278 --> 00:00:55.990
not a hydrant, that
would be a noisy hydrant.

00:00:55.990 --> 00:00:58.440
There's a fire truck outside, but okay,

00:00:58.440 --> 00:00:59.780
I think it's passed.

00:00:59.780 --> 00:01:02.040
But this looks like what
she got for the derivative.

00:01:02.040 --> 00:01:05.230
Because if you multiply two times two x,

00:01:05.230 --> 00:01:07.260
you do indeed get four x.

00:01:07.260 --> 00:01:10.510
You have this three right
over here in the denominator.

00:01:10.510 --> 00:01:13.150
And x squared minus one
to the negative 1/3,

00:01:13.150 --> 00:01:15.155
that's the same thing
as x squared minus one

00:01:15.155 --> 00:01:16.790
to the 1/3 in the denominator,

00:01:16.790 --> 00:01:18.410
which is the same thing as the cubed root

00:01:18.410 --> 00:01:20.100
of x squared minus one.

00:01:20.100 --> 00:01:22.380
So all of this is looking good.

00:01:22.380 --> 00:01:24.700
That is indeed the derivative.

00:01:24.700 --> 00:01:28.430
Step two, the critical
point is x equals zero.

00:01:28.430 --> 00:01:32.480
So let's see, a critical point
is where our first derivative

00:01:32.480 --> 00:01:36.340
is either equal to zero
or it is undefined.

00:01:36.340 --> 00:01:40.770
And so it does indeed
seem that f prime of zero

00:01:40.770 --> 00:01:41.970
is going to be four times zero,

00:01:41.970 --> 00:01:45.295
it's gonna be zero over
three times the cubed root

00:01:45.295 --> 00:01:48.130
of zero minus one, of negative one.

00:01:48.130 --> 00:01:50.540
And so this is three times negative one,

00:01:50.540 --> 00:01:51.580
or zero over negative three,

00:01:51.580 --> 00:01:53.760
so this is indeed equal to zero.

00:01:53.760 --> 00:01:55.090
So this is true.

00:01:55.090 --> 00:01:57.970
A critical point is at x equals zero.

00:01:57.970 --> 00:02:00.404
But a question is, is this
the only critical point?

00:02:00.404 --> 00:02:02.640
Well as we've mentioned, a critical point

00:02:02.640 --> 00:02:06.060
is where a function's derivative
is either equal to zero

00:02:06.060 --> 00:02:07.650
or it's undefined.

00:02:07.650 --> 00:02:10.610
This is the only one where the
derivative's equal to zero,

00:02:10.610 --> 00:02:12.270
but can you find some x-values

00:02:12.270 --> 00:02:13.970
where the derivative is undefined?

00:02:13.970 --> 00:02:15.750
Well what if we make the derivative,

00:02:15.750 --> 00:02:18.150
what would make the
denominator of the derivative

00:02:18.150 --> 00:02:19.790
equal to zero?

00:02:19.790 --> 00:02:21.497
Well if x squared minus
one is equal to zero,

00:02:21.497 --> 00:02:23.310
you take the cube root of zero,

00:02:23.310 --> 00:02:25.280
you're gonna get zero in the denominator.

00:02:25.280 --> 00:02:28.460
So what would make x squared
minus one equal to zero?

00:02:28.460 --> 00:02:30.890
Well x is equal to plus or minus one.

00:02:30.890 --> 00:02:33.560
These are also critical
points because they make

00:02:33.560 --> 00:02:37.043
f prime of x undefined.

00:02:38.050 --> 00:02:40.950
So I'm not feeling good about step two.

00:02:40.950 --> 00:02:43.620
It is true that a critical
point is x equals zero,

00:02:43.620 --> 00:02:46.010
but it is not the only critical point.

00:02:46.010 --> 00:02:47.740
So I would put that there.

00:02:47.740 --> 00:02:49.150
And the reason why it's important,

00:02:49.150 --> 00:02:51.047
you might say, "Well what's
the harm in not noticing

00:02:51.047 --> 00:02:52.001
"these other critical points?

00:02:52.001 --> 00:02:53.447
"She identified one,

00:02:53.447 --> 00:02:55.960
"maybe this is the
relative maximum point."

00:02:55.960 --> 00:02:57.790
But as we talked about in other videos,

00:02:57.790 --> 00:03:01.100
in order to use the first
derivative test, so to speak,

00:03:01.100 --> 00:03:03.710
and find this place where
the first derivative is zero,

00:03:03.710 --> 00:03:07.250
in order to test whether it
is a maximum or minimum point,

00:03:07.250 --> 00:03:09.790
is you have to sample
values on either side of it

00:03:09.790 --> 00:03:11.670
to make sure that you have a change,

00:03:11.670 --> 00:03:14.420
a change in sign of the derivative.

00:03:14.420 --> 00:03:15.760
But you have to make
sure that when you test

00:03:15.760 --> 00:03:17.372
on either side that
you're not going beyond

00:03:17.372 --> 00:03:18.610
another critical point.

00:03:18.610 --> 00:03:20.330
Because critical points are places

00:03:20.330 --> 00:03:22.600
where you can change direction.

00:03:22.600 --> 00:03:24.530
And so let's see what
she does in step three

00:03:24.530 --> 00:03:25.820
right over here.

00:03:25.820 --> 00:03:28.600
Well it is indeed in step
three that's she's testing,

00:03:28.600 --> 00:03:30.788
she's trying to test values on either side

00:03:30.788 --> 00:03:33.910
of the critical point that she,

00:03:33.910 --> 00:03:36.380
that the one critical
point that she identified.

00:03:36.380 --> 00:03:39.461
But the problem here, the reason
why this is a little shady,

00:03:39.461 --> 00:03:43.250
is this is beyond another critical point

00:03:43.250 --> 00:03:44.710
that is less than zero,

00:03:44.710 --> 00:03:46.330
and this is beyond, this is greater

00:03:46.330 --> 00:03:48.820
than another critical point
that is greater than zero.

00:03:48.820 --> 00:03:50.970
This is larger than
the critical point one,

00:03:50.970 --> 00:03:54.660
and this is less than the
critical point negative one.

00:03:54.660 --> 00:03:58.540
What she should've tried is x equals 0.5

00:03:58.540 --> 00:04:00.950
and x equals negative 0.5.

00:04:00.950 --> 00:04:02.260
So this is what she shoulda done

00:04:02.260 --> 00:04:07.260
is try maybe negative two, negative one,

00:04:07.500 --> 00:04:10.410
negative 1/2, zero,

00:04:10.410 --> 00:04:14.900
1/2, and then one we know is undefined,

00:04:14.900 --> 00:04:16.390
and then positive two.

00:04:16.390 --> 00:04:18.240
Because this is a candidate extremum,

00:04:19.716 --> 00:04:21.630
this is a candidate extremum,

00:04:21.630 --> 00:04:24.200
and this is a candidate
extremum right over here.

00:04:24.200 --> 00:04:27.160
And so you wanna see in
which of these situations

00:04:27.160 --> 00:04:29.260
you have a sign change of the derivative.

00:04:29.260 --> 00:04:30.580
And you just wanna test in the intervals

00:04:30.580 --> 00:04:32.590
between the extremum points.

00:04:32.590 --> 00:04:36.430
So I would say that really
the main mistake she made

00:04:36.430 --> 00:04:38.545
is in step two is not identifying

00:04:38.545 --> 00:04:40.833
all of the critical points.


Geometric series convergence and divergence examples

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:02.640
- [Instructor] So here we
have three different series.

00:00:02.640 --> 00:00:04.880
And what I would like you
to do is pause this video,

00:00:04.880 --> 00:00:09.880
and think about whether each
of them converges or diverges.

00:00:11.100 --> 00:00:15.090
All right, now let's
work on this together.

00:00:15.090 --> 00:00:17.390
So, just as a refresher, converge means

00:00:17.390 --> 00:00:18.640
that even though you're summing up

00:00:18.640 --> 00:00:21.610
an infinite number of terms
in all of these cases,

00:00:21.610 --> 00:00:23.070
if they converge, that
means you actually get

00:00:23.070 --> 00:00:26.230
a finite value for that infinite sum,

00:00:26.230 --> 00:00:28.330
or that infinite number
of terms being summed up,

00:00:28.330 --> 00:00:30.330
which I always find somewhat amazing.

00:00:30.330 --> 00:00:32.810
And diverging means that
you're not going to get

00:00:32.810 --> 00:00:36.020
an actual finite value for the sum

00:00:36.020 --> 00:00:38.290
of all of the infinite terms.

00:00:38.290 --> 00:00:40.040
So how do we think about that?

00:00:40.040 --> 00:00:42.950
Well, we already know something
about geometric series,

00:00:42.950 --> 00:00:45.310
and these look kind of
like geometric series.

00:00:45.310 --> 00:00:47.600
So let's just remind ourselves
what we already know.

00:00:47.600 --> 00:00:49.190
We know that a geometric series,

00:00:49.190 --> 00:00:50.830
the standard way of writing it is

00:00:50.830 --> 00:00:53.270
we're starting n equals,

00:00:53.270 --> 00:00:55.440
typical you'll often
see n is equal to zero,

00:00:55.440 --> 00:00:58.090
but let's say we're
starting at some constant.

00:00:58.090 --> 00:00:59.090
And then you're going to have,

00:00:59.090 --> 00:01:03.100
you're gonna go to infinity
of a times r to the n,

00:01:03.100 --> 00:01:04.830
where r is our common ratio,

00:01:04.830 --> 00:01:07.300
we've talked about that
in depth in other videos.

00:01:07.300 --> 00:01:08.930
We know, this is the standard way

00:01:08.930 --> 00:01:10.100
to write a geometric series.

00:01:10.100 --> 00:01:14.200
We know that if the absolute value of r

00:01:14.200 --> 00:01:19.200
is between zero, is between zero and one,

00:01:19.540 --> 00:01:24.540
then this thing is going
to converge, converge.

00:01:24.680 --> 00:01:27.770
And if it doesn't, I'll
just write it else,

00:01:27.770 --> 00:01:30.520
it will diverge.

00:01:30.520 --> 00:01:32.750
So maybe a good path would be,

00:01:32.750 --> 00:01:35.140
hey, can we rewrite these expressions

00:01:35.140 --> 00:01:36.890
that we're trying to
take, that are defining

00:01:36.890 --> 00:01:39.140
each of our terms as we increment n,

00:01:39.140 --> 00:01:41.010
if we can rewrite it in this form,

00:01:41.010 --> 00:01:43.300
then we can identify the common ratio

00:01:43.300 --> 00:01:45.950
and think about whether
it converges or diverges.

00:01:45.950 --> 00:01:50.390
So I'm going to focus on
this part right over here.

00:01:50.390 --> 00:01:51.780
Let's see if I can rewrite that.

00:01:51.780 --> 00:01:54.210
So let's see, can I rewrite, let's see,

00:01:54.210 --> 00:01:56.630
five to the n minus
one, I can rewrite that

00:01:56.630 --> 00:02:01.630
as five to the n times
five to the negative one,

00:02:02.280 --> 00:02:07.280
and then that's gonna
be times 9/10 to the n.

00:02:07.290 --> 00:02:10.710
And let's see, this is
going to be equal to,

00:02:10.710 --> 00:02:11.960
going to be, I can just write this

00:02:11.960 --> 00:02:13.945
as this part right over here.

00:02:13.945 --> 00:02:17.380
I'll write it as 1/5,
that's the same thing

00:02:17.380 --> 00:02:19.160
as five to the negative one

00:02:19.160 --> 00:02:23.280
times, and then five to
the n and 9/10 to the n,

00:02:23.280 --> 00:02:26.770
well, I have the same exponent,
so I can rewrite that as,

00:02:26.770 --> 00:02:28.160
and we're multiplying all this stuff,

00:02:28.160 --> 00:02:29.620
I'm just switching the order,

00:02:29.620 --> 00:02:34.620
this is the same thing as five
times 9/10 to the nth power.

00:02:36.310 --> 00:02:40.260
And so this is going to
be equal to 1/5 times,

00:02:40.260 --> 00:02:45.260
well five times nine is 45 divided by 10

00:02:45.450 --> 00:02:50.450
is going to be 4.5, so
times 4.5 to the nth power.

00:02:50.810 --> 00:02:54.440
So that original series I can rewrite as,

00:02:54.440 --> 00:02:56.810
just for good measure, I'm
starting at n equals two,

00:02:56.810 --> 00:02:59.780
I'm going to infinity,
and this can be rewritten

00:02:59.780 --> 00:03:04.780
as 1/5 times 4.5 to the n.

00:03:05.060 --> 00:03:08.270
So what's our common
ratio, what's our r here?

00:03:08.270 --> 00:03:10.403
Well, you can see very clearly, it is 4.5.

00:03:11.590 --> 00:03:14.290
The absolute value of 4.5 is clearly

00:03:14.290 --> 00:03:16.780
not between zero and one.

00:03:16.780 --> 00:03:21.780
So this is a situation where
we are going to diverge.

00:03:22.010 --> 00:03:23.370
Now if you found that inspiring,

00:03:23.370 --> 00:03:24.990
and if you weren't able
to do it the first time

00:03:24.990 --> 00:03:25.980
I asked you to pause the video,

00:03:25.980 --> 00:03:28.100
try to pause the video again
and try to work these out now,

00:03:28.100 --> 00:03:29.700
now that you've seen an example.

00:03:30.640 --> 00:03:32.990
All right, let's jump into it.

00:03:32.990 --> 00:03:35.376
So I'm just gonna try
algebraically manipulate this part

00:03:35.376 --> 00:03:37.740
to get it into this form.

00:03:37.740 --> 00:03:38.940
So let's do that.

00:03:38.940 --> 00:03:40.290
So I can rewrite this, let's see,

00:03:40.290 --> 00:03:42.550
if I can get some things
just to the nth power,

00:03:42.550 --> 00:03:47.550
so I can rewrite it as
3/2 to the nth power,

00:03:47.650 --> 00:03:52.650
and I could write this
part right over here as

00:03:52.650 --> 00:03:57.650
times one over nine to the
nth times nine squared.

00:03:58.670 --> 00:04:01.080
This is going to be equal to

00:04:02.090 --> 00:04:04.880
3/2 to the n.

00:04:04.880 --> 00:04:07.410
And let's see, I could
factor out or bring out

00:04:07.410 --> 00:04:09.410
the one over nine squared,
so let me do that.

00:04:09.410 --> 00:04:12.350
So I'll write that as one over
81, I'll write it out there,

00:04:12.350 --> 00:04:15.110
so that's this part right over there.

00:04:15.110 --> 00:04:17.753
1/81 times 3/2 to the n

00:04:17.753 --> 00:04:21.900
times one over nine to the n.

00:04:21.900 --> 00:04:24.335
But one over nine to the
n, that is the same thing

00:04:24.335 --> 00:04:29.335
as one over nine all of
that to the nth power.

00:04:29.520 --> 00:04:31.560
And the reason why I did
that is now I have both

00:04:31.560 --> 00:04:32.790
of these things to the nth power,

00:04:32.790 --> 00:04:34.690
and I can do just what
I did over here before.

00:04:34.690 --> 00:04:38.650
So this is all going to
be equal to one over 81

00:04:38.650 --> 00:04:43.490
times 3/2 times 1/9

00:04:43.490 --> 00:04:45.760
to the nth power.

00:04:45.760 --> 00:04:47.380
These are just exponent properties

00:04:47.380 --> 00:04:50.060
that I am applying right over here.

00:04:50.060 --> 00:04:53.260
And so this is going to
be equal to one over 81

00:04:53.260 --> 00:04:57.920
times, let's see, 3/2 times 1/9 is

00:04:57.920 --> 00:05:01.220
3/18 which is the same thing as 1/6,

00:05:01.220 --> 00:05:04.660
times 1/6 to the nth power.

00:05:04.660 --> 00:05:06.650
If I were to rewrite the original series,

00:05:06.650 --> 00:05:11.333
it's the sum from n equals
five to infinity of,

00:05:12.610 --> 00:05:15.970
of, now I can rewrite it as one over 81

00:05:15.970 --> 00:05:19.830
times 1/6 to the nth power.

00:05:19.830 --> 00:05:23.940
This is our common
ratio, 1/6, very clearly.

00:05:23.940 --> 00:05:27.610
I'll do that in this
light blue color, 1/6.

00:05:27.610 --> 00:05:31.730
That absolute value is
clearly between zero and one,

00:05:31.730 --> 00:05:34.693
so this is a situation where
we will converge, converge.

00:05:37.321 --> 00:05:38.970
Now let's do this last example.

00:05:38.970 --> 00:05:40.890
I'll do this one a little bit faster.

00:05:40.890 --> 00:05:42.920
So let's see, I could, if I'm just trying

00:05:42.920 --> 00:05:44.770
to algebraically
manipulate that part there,

00:05:44.770 --> 00:05:48.420
that's going to be the same
thing as two to the nth power

00:05:48.420 --> 00:05:52.800
times one over three to the n

00:05:52.800 --> 00:05:55.746
times three to the negative one.

00:05:55.746 --> 00:05:59.440
Well, this is going to be the
same thing as two to the n

00:05:59.440 --> 00:06:03.367
times one over, why did I write equals?

00:06:03.367 --> 00:06:07.710
Times one over three to the n.

00:06:07.710 --> 00:06:10.760
And one over three to the negative one,

00:06:10.760 --> 00:06:12.780
that's the same thing as one over 1/3,

00:06:12.780 --> 00:06:16.170
that's just going to be
equal to three times three.

00:06:16.170 --> 00:06:17.820
Well, that's going to be equal to,

00:06:17.820 --> 00:06:19.884
I'll give myself some
space, we'll start out here.

00:06:19.884 --> 00:06:22.720
That's equal to, I'll
put the three out front,

00:06:22.720 --> 00:06:25.900
three times two to the n times,

00:06:25.900 --> 00:06:27.950
and one over three to
the n is the same thing

00:06:27.950 --> 00:06:31.600
as 1/3 to the nth power.

00:06:31.600 --> 00:06:35.762
And so this is going to
be equal to three times

00:06:35.762 --> 00:06:40.080
two times 1/3, all that to the nth power.

00:06:40.080 --> 00:06:44.240
And so that's going to
be equal to three times

00:06:44.240 --> 00:06:47.180
2/3 to the nth power.

00:06:47.180 --> 00:06:49.740
So we just simplify this
part right over here

00:06:49.740 --> 00:06:52.300
to three times 2/3 to the nth power.

00:06:52.300 --> 00:06:55.290
We can see that our common ratio is 2/3,

00:06:55.290 --> 00:06:57.540
so the absolute value of 2/3 is clearly

00:06:57.540 --> 00:06:59.250
between zero and one.

00:06:59.250 --> 00:07:03.503
So once again, we are going
to converge, converge.

00:07:05.700 --> 00:07:06.693
And we're done.


Finding derivative with fundamental theorem of calculus

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.560
- [Instructor] Let's say that
we have the function g of x,

00:00:03.560 --> 00:00:08.560
and it is equal to the
definite integral from 19 to x

00:00:08.700 --> 00:00:12.370
of the cube root of t dt.

00:00:12.370 --> 00:00:14.500
And what I'm curious about finding

00:00:14.500 --> 00:00:19.500
or trying to figure out
is, what is g prime of 27?

00:00:19.700 --> 00:00:20.580
What is that equal to?

00:00:20.580 --> 00:00:22.070
Pause this video and
try to think about it,

00:00:22.070 --> 00:00:23.910
and I'll give you a little bit of a hint.

00:00:23.910 --> 00:00:28.400
Think about the second
fundamental theorem of calculus.

00:00:28.400 --> 00:00:30.550
All right, now let's
work on this together.

00:00:30.550 --> 00:00:32.670
So we wanna figure out what g prime,

00:00:32.670 --> 00:00:34.340
we could try to figure
out what g prime of x is,

00:00:34.340 --> 00:00:35.900
and then evaluate that at 27,

00:00:35.900 --> 00:00:37.850
and the best way that I
can think about doing that

00:00:37.850 --> 00:00:41.090
is by taking the derivative of
both sides of this equation.

00:00:41.090 --> 00:00:44.150
So let's take the derivative
of both sides of that equation.

00:00:44.150 --> 00:00:46.150
So the left-hand side,
we'll take the derivative

00:00:46.150 --> 00:00:47.900
with respect to x of g of x,

00:00:47.900 --> 00:00:51.430
and the right-hand side, the
derivative with respect to x

00:00:51.430 --> 00:00:53.550
of all of this business.

00:00:53.550 --> 00:00:55.570
Now, the left-hand side is
pretty straight forward.

00:00:55.570 --> 00:00:57.640
The derivative with
respect to x of g of x,

00:00:57.640 --> 00:01:01.270
that's just going to be g prime of x,

00:01:01.270 --> 00:01:04.400
but what is the right-hand
side going to be equal to?

00:01:04.400 --> 00:01:06.610
Well, that's where the
second fundamental theorem

00:01:06.610 --> 00:01:08.520
of calculus is useful.

00:01:08.520 --> 00:01:09.640
I'll write it right over here.

00:01:09.640 --> 00:01:14.233
Second fundamental, I'll
abbreviate a little bit,

00:01:16.620 --> 00:01:19.760
theorem of calculus.

00:01:19.760 --> 00:01:24.710
It tells us, let's say we have
some function capital F of x,

00:01:24.710 --> 00:01:28.930
and it's equal to the
definite integral from a,

00:01:28.930 --> 00:01:33.930
sum constant a to x of
lowercase f of t dt.

00:01:35.210 --> 00:01:37.940
The second fundamental
theorem of calculus tells us

00:01:37.940 --> 00:01:41.180
that if our lowercase f,

00:01:41.180 --> 00:01:46.180
if lowercase f is continuous
on the interval from a to x,

00:01:47.590 --> 00:01:48.560
so I'll write it this way,

00:01:48.560 --> 00:01:52.023
on the closed interval from a to x,

00:01:53.860 --> 00:01:57.660
then the derivative of our capital f of x,

00:01:57.660 --> 00:02:01.920
so capital F prime of x
is just going to be equal

00:02:01.920 --> 00:02:06.090
to our inner function f
evaluated at x instead of t

00:02:06.090 --> 00:02:09.020
is going to become lowercase f of x.

00:02:09.020 --> 00:02:10.390
Now, I know when you first saw this,

00:02:10.390 --> 00:02:12.207
you thought that, "Hey, this
might be some cryptic thing

00:02:12.207 --> 00:02:14.240
"that you might not use too often."

00:02:14.240 --> 00:02:16.650
Well, we're gonna see that
it's actually very, very useful

00:02:16.650 --> 00:02:19.300
and even in the future, and
some of you might already know,

00:02:19.300 --> 00:02:20.730
there's multiple ways to try

00:02:20.730 --> 00:02:22.430
to think about a definite
integral like this,

00:02:22.430 --> 00:02:24.030
and you'll learn it in the future.

00:02:24.030 --> 00:02:26.090
But this can be extremely simplifying,

00:02:26.090 --> 00:02:30.430
especially if you have a hairy
definite integral like this,

00:02:30.430 --> 00:02:32.760
and so this just tells us,
hey, look, the derivative

00:02:32.760 --> 00:02:35.020
with respect to x of all of this business,

00:02:35.020 --> 00:02:37.610
first we have to check
that our inner function,

00:02:37.610 --> 00:02:40.120
which would be analogous
to our lowercase f here,

00:02:40.120 --> 00:02:44.620
is this continuous on the
interval from 19 to x?

00:02:44.620 --> 00:02:46.800
Well, no matter what x is,

00:02:46.800 --> 00:02:50.180
this is going to be
continuous over that interval,

00:02:50.180 --> 00:02:53.290
because this is continuous for all x's,

00:02:53.290 --> 00:02:57.110
and so we meet this first
condition or our major condition,

00:02:57.110 --> 00:02:58.850
and so then we can just say, all right,

00:02:58.850 --> 00:03:01.170
then the derivative of all of this

00:03:01.170 --> 00:03:05.020
is just going to be this inner
function replacing t with x.

00:03:05.020 --> 00:03:06.490
So we're going to get the cube root,

00:03:06.490 --> 00:03:07.800
instead of the cube root of t,

00:03:07.800 --> 00:03:09.800
you're gonna get the cube root of x.

00:03:09.800 --> 00:03:12.620
And so we can go back to
our original question,

00:03:12.620 --> 00:03:16.230
what is g prime of 27
going to be equal to?

00:03:16.230 --> 00:03:19.490
Well, it's going to be equal
to the cube root of 27,

00:03:19.490 --> 00:03:22.563
which is of course equal
to three, and we're done.