What to think about when taking over an existing business?

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- Every now and then I get
a friend or a family member

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saying that they're interested in buying

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some type of a business.

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So this wouldn't be about
starting a business,

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but there might be an existing business.

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Maybe it's a convenience store

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or maybe it's some type
of health care practice

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or whatever it might be, and they say,

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hey, I'm going to buy this
from the existing owner

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and I think it's going
to be a great business.

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And they'll often say it
generates this much money

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and I only have to put this much in

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and it feels risk free.

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And what I tend to tell them,

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and I don't wanna be one
of these naysayer people,

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I say, possibly.

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But what you always have to think about

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is why are they selling it?

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And sometimes there's a good
reason why they're selling it.

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Maybe they're retiring and they don't have

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someone to pass the business onto

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and they see in you kind of
the person that they would

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love to see running that
business going forward.

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That could be a very legitimate thing.

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But oftentimes, there might be situations

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where they might be pumping
up some of the numbers,

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they might be showing you business

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in the busy season or on the busy day.

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And so I wouldn't take, especially
if it's a small business

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that doesn't have auditing,

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that doesn't have really
strict accounting practices.

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I would observe the business
for a very long period of time

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and I would take your own data

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as to whether the numbers
you're hearing about are real.

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And I also wouldn't
discount how much effort

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and work the existing
owner is putting into it

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and what happens to the business

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if they're no longer there.

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If all of that works out

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and if it really does make sense,

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then, yeah, you might be onto something.


What advice do you have for someone wanting to be an entrepreneur?

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- So what advice would I have for someone

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who wants to be an entrepreneur?

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So everyone's path is different,

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so take anything I have to say

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with a grain of salt.

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But a lot of folks think
of entrepreneurship as,

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"Hey, I have a new idea for a business,"

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whether it's a restaurant,
or a new technology company,

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a new social network,
whatever it might be,

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and they put a lot of
emphasis on the idea itself.

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And I think most
entrepreneurs would tell you

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that a lot more depends
not only on the idea

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but the execution of the idea,

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and really just how ready is the world,

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is the market, for your idea?

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And so my advice would be, especially

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if you're early in your career,

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try to build as many skills as you can,

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and to be an entrepreneur,

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you have to be a little bit
of a jack of all trades.

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So know your finance.

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Know your accounting.

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Know your marketing,

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And you don't have to know
it necessarily academically,

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but know it intuitively.

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Pay attention to how
other people are doing it.

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Think about your strategy.

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And wherever your area of interest is,

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if it's to start a restaurant,

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get to know that business
really, really, really well.

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Work at a restaurant at all levels.

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If your goal is to work in tech,

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work at some tech companies.

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Get to know that really, really well.

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Get to know not just the technology,

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not just the product management,

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not just the design, but
the sales aspect of it.

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Get to know the management aspect of it.

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And the whole time that
you're building yourself up,

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that you're working in
industry and learning

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these various skills,
make sure that you have

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some time to fiddle.

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The best entrepreneurial ideas
that I've seen are the ones

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where folks get to know a space well

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through their craft, and
then they have an idea,

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and then they're able to have a little bit

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of time to test out that idea,

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not only build it, not only prototype it,

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but put it out there somehow

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and see how it the market responds.

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And if the market is
responding very favorably,

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if people are saying,
"Wow, I want more of this.

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"How can I get it?"

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Then I think you're on to something.

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And all of those skills
that you would have built up

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over those many years,
not only will they help

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you execute on the idea,

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but they'll help you convince other people

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that you're ready to execute on the idea,

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which is maybe just as important,

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potential investors, people that you need

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to co-found your venture with,

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and also, my last point is
know what your weak points are.

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As you experience all of these dimensions

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of starting a business, if you know that,

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hey maybe the marketing
isn't my thing, know that.

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And then you could find
someone who's good at it,

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who can complement you, if you're like,

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"Hey, I like the entrepreneurial side,

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"but scaling up a business
isn't my cup of tea."

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Find great operators who
can help you scale it up.

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So that's my two cents.

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Take it all with a grain of salt.


How to handle naysayers when starting a business?

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- So one of the hardest things

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when you are starting as an entrepreneur,

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when you're trying to start
some type of anything,

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whether it's a business or a
nonprofit or an organization,

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is you're going to wanna share your idea

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and what you're working
on with a lot of folks,

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and you're going to get
different reactions.

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And it's not unlikely that you're going

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to get a lot of folks who
will tell you things like,

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"Well, this is already done before.

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"Why do you think you're
going to be able to do it?"

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or "This has never been done before.

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"Why do you think you're
going to be able to do it?"

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And it's very hard to judge whether,

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okay, are they just being
naysayers and just being negative

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and I should just power through it,

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or should I listen seriously
to what they have to say?

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And like all things, take
this all with a grain of salt,

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it's hard to know for sure,
but I like to think about

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whether people are coming
from a point of view

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of that they really want you to succeed.

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And if there are people in your life

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that you know at the end of
the day have got your back

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and really want you to succeed,

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I would listen to what they're saying.

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Now, it doesn't mean what they're
saying is absolutely true.

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It's completely possible
that they might have

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a different tolerance
for risk than you do,

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but it's worth processing.

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It's worth at least listening.

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And sometimes when people say,

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"Oh, well, this has been tried before.

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"Why do you think you
doing it is going to work?"

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Well, the answer to that is sometimes

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how you do it matters a lot
more than the idea itself.

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Khan Academy is an example of that.

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I wasn't the first person to think

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that online could be a way
to disseminate information.

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I wasn't the first person to think

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that software could personalize
education for folks.

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I wasn't the first person
to make content on YouTube

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or to help teach people through video.

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But there was something about
how it all came together

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and also the time and space

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and how accessible it was
that things caught on.

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And that would've been
very hard to predict

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had I not at least tried.

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And the way that I protected myself

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from both the positive and negative,

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positive and negative feedback
(laughs) was I told myself,

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"You know what?

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"This is all good information.

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"I'm going to listen to
what people have to say,

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"but I'm also going to try it.

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"And I'm gonna try it in a
way that I can start to learn

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"what is actually working
and what's not working."

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And if you can work on
your day job, so to speak,

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while you start to
fiddle and you can start

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to build those data points
and points of confirmation

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that you might be on to something,

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well, then it makes you a
little bit more resilient

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to what folks might tell you.

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And you should keep listening,

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but you shouldn't rest everything

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on necessarily what everyone tells you.


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- How do you know if a business
idea is worth pursuing?

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So my take on that is

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if you think about what
you are going to uniquely

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bring to the table.

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Sometimes I've been at a
restaurant with a friend

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and they're like, "Oh, this
restaurant's doing so well.

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"I think I could do this,"

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and they start thinking about maybe I'll

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open up another restaurant just like that

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across the street.

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Well, I think my friends in that situation

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are probably underestimating how much work

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that the restaurant
we're in had put into it,

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how much thought they put into it.

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And also if you open up
an identical restaurant

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across the street, well
that's just going to hurt

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both of your businesses and neither of you

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might have a viable business.

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Another thing is you
could look at a business

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that someone else
started and you might see

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the cash register running and
there's a lot of customers,

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but you don't realize
everything, all of the costs,

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all of the risks that that person took on.

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Their rent might be a lot.

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You might just be going
through during the busy time,

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and if you go the next day during lunch,

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there might not be anyone there.

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And so, it's very easy
to always see things

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that the grass is greener
on the other side.

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Most entrepreneurs will tell you

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that even the most successful ventures

00:01:07.280 --> 00:01:08.960
went through their dark periods

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and went through a period

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where it looked like it was going to fail.

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It didn't look like it
was going to work out.

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And so what I would say if
you have a business idea,

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is one, think about what you're going to

00:01:21.920 --> 00:01:23.060
uniquely bring to the table.

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Maybe it is an idea that no
one's ever thought about before.

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But maybe it is an idea that people

00:01:28.460 --> 00:01:29.320
have thought about before,

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but you have a unique
way of approaching it,

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or you have unique
skills, or a unique team

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that can approach it right,

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or you kind of see the nexus
of several of these things.

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And I would try to test the
idea as well as possible.

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Can you put out some
advertising for your business

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before it even exists and
see if people respond?

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Can you build a prototype of a product

00:01:52.850 --> 00:01:55.320
and see if people want it?

00:01:55.320 --> 00:01:57.620
And the more that you
can test it in that way

00:01:57.620 --> 00:02:00.387
and say, "Wow, there's actually
a product market fit here,

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"or a service market fit here;

00:02:01.677 --> 00:02:05.790
"people really want what this idea is,"

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then I think you might be on to something.

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Obviously the ideal is you
even start the business

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on a small scale without
necessarily going all in yet

00:02:13.630 --> 00:02:17.210
and seeing that wow, there's
a lot of demand for this.

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So once again, no right
answer here always.

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When you're starting a business,

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you really just have to
prepare yourself mentally

00:02:24.360 --> 00:02:26.910
that there will be rough patches.

00:02:26.910 --> 00:02:29.190
But if you can come up with something

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and have a unique take on it and test it,

00:02:31.880 --> 00:02:35.000
and it's actually getting
traction with folks,

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and you're able to get the capital,

00:02:37.050 --> 00:02:39.900
and you yourself have the
capital you've saved up, maybe,

00:02:39.900 --> 00:02:43.943
if things get a little bit rough
as you start your business,

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I think you might be in a
good position to start it.


How do you prepare yourself mentally to be an entrepreneur?

WEBVTT
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- So how do you prepare yourself mentally

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to be an entrepreneur?

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So what I will say is,
maybe borrowing a little bit

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from Buddhism or philosophical Hinduism,

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but it's really this notion

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to try to not get attached to the outcome.

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Obviously you're going
into entrepreneurship

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because you're imagining

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that one, you could be your own boss,

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you'll be able to apply
your creative ideas,

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you'll be able to do a
service for the world.

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But it's good to go into it
with a bit of a Zen philosophy

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where you're gonna do what's right,

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you're gonna do what's in
service of the customer

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or the user or whomever you are serving.

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But you're not going to define yourself

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based on how successful
you are or you're not.

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Because what typically
happens in entrepreneurship

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is that you start with this optimism

00:00:52.580 --> 00:00:53.860
and that's where I started too,

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and it's almost delusional
or it sometimes can be,

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and you might quit your
job and then you go into it

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and you quickly realize

00:01:01.414 --> 00:01:04.620
that the world doesn't immediately

00:01:04.620 --> 00:01:06.823
come to recognize the brilliance

00:01:06.823 --> 00:01:10.730
of what you might be working on.

00:01:10.730 --> 00:01:12.500
And so there will likely be a period

00:01:12.500 --> 00:01:14.241
where you'll really question yourself,

00:01:14.241 --> 00:01:16.490
you'll be looking at your finances,

00:01:16.490 --> 00:01:18.480
you'll be depleting your finances,

00:01:18.480 --> 00:01:19.930
and it's all that much worse

00:01:19.930 --> 00:01:23.160
if your ego is heavily
invested in that as well.

00:01:23.160 --> 00:01:24.727
And if you say, "What
will people think of me?

00:01:24.727 --> 00:01:26.667
"What is going to happen to my career?

00:01:26.667 --> 00:01:27.917
"I'm not going to be able to afford

00:01:27.917 --> 00:01:30.181
"the lifestyle that I'm used to."

00:01:30.181 --> 00:01:33.620
But if you suppress your ego a little bit

00:01:33.620 --> 00:01:36.780
and say, "I'm doing this
for the good of the world,"

00:01:36.780 --> 00:01:38.597
whether it's for-profit,
nonprofit, whatever it is,

00:01:38.597 --> 00:01:40.997
"I'm doing it in service to others,

00:01:40.997 --> 00:01:42.960
"and I'm doing what's right,"

00:01:42.960 --> 00:01:45.570
and hopefully you will have a good outcome

00:01:45.570 --> 00:01:47.270
but you're not super attached to it,

00:01:47.270 --> 00:01:49.350
then it's going to
lower the anxiety level.

00:01:49.350 --> 00:01:50.807
I heard a quote recently that,

00:01:50.807 --> 00:01:53.326
"Anxiety is misplaced imagination."

00:01:53.326 --> 00:01:57.900
And I think most entrepreneurs,
or aspiring entrepreneurs,

00:01:57.900 --> 00:02:00.330
the one thing they have a
surplus of is imagination

00:02:00.330 --> 00:02:01.940
and that's a great thing,

00:02:01.940 --> 00:02:04.280
but sometimes your imagination
can get into cycles

00:02:04.280 --> 00:02:06.760
of all of the bad things that'll happen

00:02:06.760 --> 00:02:09.360
or how people will think of
you et cetera, et cetera.

00:02:09.360 --> 00:02:12.290
So I would try not to go there.

00:02:12.290 --> 00:02:15.240
And if you do that I think
you'll just be more resilient

00:02:15.240 --> 00:02:17.185
because a lot of entrepreneurial ideas

00:02:17.185 --> 00:02:20.710
it's not a matter of whether
it's a good idea or not,

00:02:20.710 --> 00:02:22.939
it's not a matter of even

00:02:22.939 --> 00:02:25.090
whether you're executing well or not,

00:02:25.090 --> 00:02:26.710
you do need to execute well,

00:02:26.710 --> 00:02:28.520
but sometimes it's a matter of timing,

00:02:28.520 --> 00:02:31.187
and the more resilient
you can make yourself,

00:02:31.187 --> 00:02:34.650
also the more resilient
you can make your business.

00:02:34.650 --> 00:02:36.310
So making yourself resilient

00:02:36.310 --> 00:02:37.580
and we're talking about mental resilience,

00:02:37.580 --> 00:02:39.403
you can make yourself
financially resilient,

00:02:39.403 --> 00:02:42.541
by living below your means,
having a lot of savings,

00:02:42.541 --> 00:02:46.600
and you can make your
business financially resilient

00:02:46.600 --> 00:02:48.790
so that it can hit the
timing at the right time

00:02:48.790 --> 00:02:51.285
or have more time to be able to,

00:02:51.285 --> 00:02:54.240
for the market to recognize its value,

00:02:54.240 --> 00:02:56.500
if you do it in a very lean way

00:02:56.500 --> 00:02:59.350
and if you're always
taking new data points

00:02:59.350 --> 00:03:00.380
and processing it.

00:03:00.380 --> 00:03:01.240
And that's another place

00:03:01.240 --> 00:03:02.490
where it's good to suppress your ego

00:03:02.490 --> 00:03:05.630
'cause sometimes early on you
get so enamored with an idea,

00:03:05.630 --> 00:03:08.600
that even if some data
points, even if some research,

00:03:08.600 --> 00:03:10.320
even if some customers are telling you

00:03:10.320 --> 00:03:12.590
that aah, it's not exactly what they want,

00:03:12.590 --> 00:03:14.257
you're confirmation bias is almost like,

00:03:14.257 --> 00:03:15.377
"Well I'm talking to the wrong people,

00:03:15.377 --> 00:03:16.780
"or that data might not be right,"

00:03:16.780 --> 00:03:17.963
and maybe that's the case,

00:03:17.963 --> 00:03:20.430
but I think the less
that you have your ego

00:03:20.430 --> 00:03:22.490
super invested in your idea,

00:03:22.490 --> 00:03:23.930
that it's your idea,

00:03:23.930 --> 00:03:25.420
the more that you'll be willing to pivot,

00:03:25.420 --> 00:03:27.470
and the more that you're
willing to change.

00:03:27.470 --> 00:03:29.290
Famously in the early days of Khan Academy

00:03:29.290 --> 00:03:30.600
when I was working on all of the software

00:03:30.600 --> 00:03:31.470
I showed it to a friend

00:03:31.470 --> 00:03:33.280
and he suggested that I make videos

00:03:33.280 --> 00:03:34.430
to supplement the software

00:03:34.430 --> 00:03:36.580
and I immediately thought
it was a horrible idea,

00:03:36.580 --> 00:03:38.520
but it took a little bit
of my ego suppression

00:03:38.520 --> 00:03:41.050
to realize that just
because I didn't think of it

00:03:41.050 --> 00:03:42.320
doesn't mean it's a bad idea,

00:03:42.320 --> 00:03:43.750
and just because other people had done it

00:03:43.750 --> 00:03:45.640
doesn't mean it was a bad idea,

00:03:45.640 --> 00:03:47.010
so I said, "Let me try it out,"

00:03:47.010 --> 00:03:49.910
and there was a unique
take on how I was doing it

00:03:49.910 --> 00:03:51.150
to complement the software

00:03:51.150 --> 00:03:54.130
that that very nascent
Khan Academy was doing

00:03:54.130 --> 00:03:58.441
that really helped us get off the ground.

00:03:58.441 --> 00:04:01.280
So what I would say is become resilient,

00:04:01.280 --> 00:04:04.020
a look that is being aware of your ego,

00:04:04.020 --> 00:04:05.780
you don't have to destroy your ego,

00:04:05.780 --> 00:04:07.280
but know it's there,

00:04:07.280 --> 00:04:10.250
and you have to put it in
it's place every now and then.

00:04:10.250 --> 00:04:11.720
Always be open to new data,

00:04:11.720 --> 00:04:14.470
make yourself financially
and mentally resilient

00:04:14.470 --> 00:04:15.900
and do the same for your organization

00:04:15.900 --> 00:04:18.323
or your nascent organization
that you're starting.


Fundamental theorem to evaluate derivative

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:01.163
- [Instructor] Let's say that I were

00:00:01.163 --> 00:00:02.480
to walk up to you on the street

00:00:02.480 --> 00:00:05.420
and said alright, I have
this function g of x,

00:00:05.420 --> 00:00:08.060
which I'm going to define
as the definite integral

00:00:08.060 --> 00:00:12.120
from 19 to x of the cube root

00:00:12.120 --> 00:00:15.140
of t dt.

00:00:15.140 --> 00:00:19.740
And then I were to ask
you what is the derivative

00:00:19.740 --> 00:00:23.810
of g evaluated at 27?

00:00:23.810 --> 00:00:24.720
Pause this video,

00:00:24.720 --> 00:00:27.540
and think about how you
would approach that.

00:00:27.540 --> 00:00:30.120
Well, the most obvious thing would be

00:00:30.960 --> 00:00:33.170
g-prime of x, and I'll switch colors here

00:00:33.170 --> 00:00:34.830
just for a little bit of contrast,

00:00:34.830 --> 00:00:38.510
g-prime of x is going to be the derivative

00:00:38.510 --> 00:00:42.210
with respect to x of all of this business,

00:00:42.210 --> 00:00:45.500
of all of, I'll put it in brackets here.

00:00:45.500 --> 00:00:50.500
The integral from 19 to x
of the cube root of t dt.

00:00:52.400 --> 00:00:55.610
And you might be tempted to
take the antiderivative here

00:00:55.610 --> 00:00:58.710
and evaluated it x, evaluated it at 19,

00:00:58.710 --> 00:01:01.080
find the difference and
then take the derivative

00:01:01.080 --> 00:01:05.250
and you could do that, and
then evaluate that at 27.

00:01:05.250 --> 00:01:07.470
But we're going to see a short cut.

00:01:07.470 --> 00:01:12.040
You could just use the
Fundamental Theorem of Calculus.

00:01:12.040 --> 00:01:14.837
If you're taking the
derivative with respect to x

00:01:14.837 --> 00:01:17.430
of a function that's defined

00:01:17.430 --> 00:01:20.360
by a definite integral like this

00:01:20.360 --> 00:01:22.540
where our upper boundary is x,

00:01:22.540 --> 00:01:26.320
this is just going to be
equal to our inner function

00:01:26.320 --> 00:01:29.120
right over here, with instead
of t being the variable,

00:01:29.120 --> 00:01:30.750
it would now be x.

00:01:30.750 --> 00:01:35.350
So this is just going to be
equal to the cube root of x.

00:01:35.350 --> 00:01:37.010
And that saved us some time there,

00:01:37.010 --> 00:01:39.710
but it can be really
useful if this function

00:01:39.710 --> 00:01:42.440
inside the integral is really hard

00:01:42.440 --> 00:01:44.950
to take the antiderivative of.

00:01:44.950 --> 00:01:47.060
It's really hard to
evaluate this integral.

00:01:47.060 --> 00:01:49.920
The Fundamental Theorem of
Calculus can be very useful.

00:01:49.920 --> 00:01:52.330
And so going back to
our original question,

00:01:52.330 --> 00:01:55.730
g-prime of 27, well this is just going

00:01:55.730 --> 00:01:58.850
to be equal to the cube root of 27,

00:01:58.850 --> 00:02:02.163
which, of course, is equal
to three, and we're done.


Finding derivative with fundamental theorem of calculus: chain rule

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:01.950
- [Instructor] Let's say
that we have the function

00:00:01.950 --> 00:00:03.850
capital F of x,

00:00:03.850 --> 00:00:06.270
which we're going to define
as the definite integral

00:00:06.270 --> 00:00:08.760
from one to sine of x,

00:00:08.760 --> 00:00:11.830
so that's an interesting
upper bound right over there,

00:00:11.830 --> 00:00:14.930
of two t minus one,

00:00:14.930 --> 00:00:16.650
and of course, dt,

00:00:16.650 --> 00:00:18.770
and what we are curious about

00:00:18.770 --> 00:00:23.710
is trying to figure out
what is F prime of x going

00:00:23.710 --> 00:00:24.543
to be equal to?

00:00:24.543 --> 00:00:27.230
So pause this video and see
if you can figure that out.

00:00:28.200 --> 00:00:29.320
All right.

00:00:29.320 --> 00:00:31.310
So some of you might have
been a little bit challenged

00:00:31.310 --> 00:00:32.690
by this notion of hey,

00:00:32.690 --> 00:00:34.350
instead of an x on this upper bound,

00:00:34.350 --> 00:00:35.740
I now have a sine of x.

00:00:35.740 --> 00:00:36.770
If it was just an x,

00:00:36.770 --> 00:00:39.160
I could have used the
fundamental theorem of calculus.

00:00:39.160 --> 00:00:40.240
Just to review that,

00:00:40.240 --> 00:00:42.920
if I had a function,
let me call it h of x,

00:00:42.920 --> 00:00:46.710
if I have h of x that was
defined as the definite integral

00:00:46.710 --> 00:00:51.710
from one to x of two t minus one dt,

00:00:52.290 --> 00:00:55.200
we know from the fundamental
theorem of calculus

00:00:55.200 --> 00:00:57.170
that h prime of x

00:00:57.170 --> 00:00:59.390
would be simply this inner function

00:00:59.390 --> 00:01:01.160
with the t replaced by the x.

00:01:01.160 --> 00:01:04.720
It would just be two x minus
one, pretty straightforward.

00:01:04.720 --> 00:01:06.680
But this one isn't quite
as straightforward.

00:01:06.680 --> 00:01:08.150
Instead of having an x up here,

00:01:08.150 --> 00:01:10.840
our upper bound is a sine of x.

00:01:10.840 --> 00:01:15.800
So one way to think about it
is if we were to define g of x

00:01:15.800 --> 00:01:19.260
as being equal to sine of x,

00:01:19.260 --> 00:01:20.710
equal to sine of x,

00:01:20.710 --> 00:01:25.710
our capital F of x can be
expressed as capital F of x

00:01:25.710 --> 00:01:28.970
is the same thing as h of,

00:01:28.970 --> 00:01:31.580
h of, instead of an x,

00:01:31.580 --> 00:01:32.680
everywhere we see an x,

00:01:32.680 --> 00:01:34.530
we're replacing it with a sine of x,

00:01:34.530 --> 00:01:38.740
so it's h of g of x, g of x.

00:01:38.740 --> 00:01:42.490
You can see the g of x right over there.

00:01:42.490 --> 00:01:44.770
So you replace x with g of x

00:01:44.770 --> 00:01:46.700
for where, in this expression,

00:01:46.700 --> 00:01:47.990
you get h of g of x

00:01:47.990 --> 00:01:49.920
and that is capital F of x.

00:01:49.920 --> 00:01:52.030
Now why am I doing all of that?

00:01:52.030 --> 00:01:54.360
Well, this might start making you think

00:01:54.360 --> 00:01:55.850
about the chain rule.

00:01:55.850 --> 00:01:57.200
Because if this is true,

00:01:57.200 --> 00:02:00.970
then that means that capital F prime of x

00:02:00.970 --> 00:02:05.970
is going to be equal to h prime of g of x,

00:02:08.210 --> 00:02:11.010
h prime of g of x

00:02:11.010 --> 00:02:14.300
times g prime of x.

00:02:14.300 --> 00:02:15.910
And so what would that be?

00:02:15.910 --> 00:02:18.500
Well, we already know
what h prime of x is,

00:02:18.500 --> 00:02:20.570
so I'll need to do this in another color.

00:02:20.570 --> 00:02:22.930
This part right over
here is going to be equal

00:02:22.930 --> 00:02:25.210
to everywhere we see an x here,

00:02:25.210 --> 00:02:26.630
we'll replace with a g of x,

00:02:26.630 --> 00:02:29.510
so it's going to be two,

00:02:29.510 --> 00:02:32.440
two times sine of x.

00:02:32.440 --> 00:02:36.140
Two sine of x,

00:02:36.140 --> 00:02:39.720
and then minus one, minus one.

00:02:39.720 --> 00:02:42.650
This is this right over here,

00:02:42.650 --> 00:02:45.120
and then what's g prime of x?

00:02:45.120 --> 00:02:47.210
G prime of x,

00:02:47.210 --> 00:02:50.360
well g prime of x is just, of course,

00:02:50.360 --> 00:02:53.490
the derivative of sine
of x is cosine of x,

00:02:53.490 --> 00:02:54.400
is cosine of x.

00:02:54.400 --> 00:02:56.040
So this part right over here

00:02:56.040 --> 00:02:59.350
is going to be cosine of x.

00:02:59.350 --> 00:03:00.580
And we could keep going.

00:03:00.580 --> 00:03:01.970
We could try to,

00:03:01.970 --> 00:03:04.140
we could try to simplify this a little bit

00:03:04.140 --> 00:03:05.530
or rewrite it in different ways,

00:03:05.530 --> 00:03:06.860
but there you have it.


Introducing Khan Academy Learnstorm 2019!

WEBVTT
Kind: captions
Language: en

00:00:00.100 --> 00:00:01.260
- Hello teachers,

00:00:01.260 --> 00:00:04.800
I'm Sal Khan, founder of the
not-for-profit Khan Academy,

00:00:04.800 --> 00:00:07.470
and I'm here to announce a nationwide

00:00:07.470 --> 00:00:11.518
back to school learning
challenge called LearnStorm.

00:00:11.518 --> 00:00:14.300
LearnStorm is an exciting
way to jump start

00:00:14.300 --> 00:00:17.410
your school year around
learning activities.

00:00:17.410 --> 00:00:19.610
It's designed to motivate your students

00:00:19.610 --> 00:00:22.420
and to help build their growth mindset.

00:00:22.420 --> 00:00:24.490
- This year was amazing with LearnStorm,

00:00:24.490 --> 00:00:26.610
I had a group of students
that couldn't add,

00:00:26.610 --> 00:00:29.090
subtract, multiply, or divide fractions.

00:00:29.090 --> 00:00:31.380
They avoided it at all costs,

00:00:31.380 --> 00:00:33.180
but since it was an assignment,

00:00:33.180 --> 00:00:36.670
they would try it and they
would eventually do much better.

00:00:36.670 --> 00:00:40.100
And what was amazing was that
the lower achieving students

00:00:40.100 --> 00:00:44.300
were able to see that they
were part of something huge.

00:00:44.300 --> 00:00:46.080
They were able to contribute,
(cheering)

00:00:46.080 --> 00:00:49.400
and that, I thought, boosted
their self-confidence,

00:00:49.400 --> 00:00:51.480
their self-esteem, and I didn't realize

00:00:51.480 --> 00:00:53.948
that would happen after
doing a LearnStorm.

00:00:53.948 --> 00:00:54.781
(cheering)
(clapping)

00:00:54.781 --> 00:00:57.781
(warm bouncy music)

00:00:59.009 --> 00:01:03.070
- LearnStorm is a good
way to differentiate

00:01:03.070 --> 00:01:05.300
and as well as motivate.

00:01:05.300 --> 00:01:07.510
And with the growth mindset pieces,

00:01:07.510 --> 00:01:10.580
you can also educate them about

00:01:10.580 --> 00:01:11.910
why it's important to learn,

00:01:11.910 --> 00:01:15.150
how they learn, how their
body is set up to learn,

00:01:15.150 --> 00:01:17.470
what strategies help them learn.

00:01:17.470 --> 00:01:20.035
It's amazing, it's amazing and it's free!

00:01:20.035 --> 00:01:21.168
(laughing)

00:01:21.168 --> 00:01:22.740
It's free!

00:01:22.740 --> 00:01:25.180
- We think this is really
going to motivate your students

00:01:25.180 --> 00:01:27.900
to understand, to
appreciate that they truly

00:01:27.900 --> 00:01:29.770
can learn learn anything.

00:01:29.770 --> 00:01:31.890
The activities have been designed

00:01:31.890 --> 00:01:35.210
to have measurable gains
in learning attitudes.

00:01:35.210 --> 00:01:37.860
- One thing I learned from
the growth mindset activities

00:01:37.860 --> 00:01:40.350
is that your intelligence can grow

00:01:40.350 --> 00:01:41.700
just like your muscles can grow

00:01:41.700 --> 00:01:44.000
and that's something that
really stuck with me.

00:01:44.000 --> 00:01:46.480
'Cause I didn't know that before.

00:01:46.480 --> 00:01:49.200
- I did over 300 lessons in one week.

00:01:49.200 --> 00:01:52.240
I felt proud of myself, getting
all of these good grades.

00:01:52.240 --> 00:01:54.730
- It helps the students tremendously,

00:01:54.730 --> 00:01:56.260
their confidence is higher,

00:01:56.260 --> 00:01:58.370
they're able to do it
and it warms my heart

00:01:58.370 --> 00:01:59.980
that they think that way now.

00:01:59.980 --> 00:02:02.110
And they know that they can keep going

00:02:02.110 --> 00:02:05.860
and keep growing and nothing
will stand in their way.

00:02:05.860 --> 00:02:08.570
- So together, as we jump
start the school year,

00:02:08.570 --> 00:02:12.450
I look forward for all of us
living the LearnStorm motto,

00:02:12.450 --> 00:02:15.547
to keep going and to keep growing.

00:02:15.547 --> 00:02:19.464
(all chanting "Go and cheat!")


Harmonic series and 𝑝-series

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:01.810
- [Instructor] For many hundreds of years,

00:00:01.810 --> 00:00:04.020
mathematicians have been fascinated

00:00:04.020 --> 00:00:07.280
by the infinite sum, which
we would call a series,

00:00:07.280 --> 00:00:12.280
of one plus 1/2 plus 1/3 plus 1/4,

00:00:12.540 --> 00:00:15.600
and you just keep adding
on and on and on forever.

00:00:15.600 --> 00:00:17.460
And this is interesting on many layers.

00:00:17.460 --> 00:00:18.293
One, it just feels like something

00:00:18.293 --> 00:00:20.070
that would be interesting to explore.

00:00:20.070 --> 00:00:23.300
It's one over one plus one
over two plus one over three,

00:00:23.300 --> 00:00:25.600
that each of these terms are
getting smaller and smaller.

00:00:25.600 --> 00:00:27.270
They're approaching zero,

00:00:27.270 --> 00:00:28.780
but when you add them all together,

00:00:28.780 --> 00:00:30.480
these infinite number of terms,

00:00:30.480 --> 00:00:31.940
do you get a finite number

00:00:31.940 --> 00:00:35.020
or does it diverge, do
not get a finite number?

00:00:35.020 --> 00:00:37.390
This also shows up in music

00:00:37.390 --> 00:00:38.610
and this actually might have been one

00:00:38.610 --> 00:00:42.140
of the early motivations
for studying this series.

00:00:42.140 --> 00:00:45.660
Where if you have a fundamental note,

00:00:45.660 --> 00:00:48.590
a fundamental frequency in music,

00:00:48.590 --> 00:00:50.650
and the point of this video
isn't to teach you too much

00:00:50.650 --> 00:00:53.860
about music, but if you
have a fundamental note,

00:00:53.860 --> 00:00:56.280
that might be a pure A
or something like that.

00:00:56.280 --> 00:00:57.780
I'm just showing you
one of its wavelengths.

00:00:57.780 --> 00:00:59.570
Obviously, you would keep going like that

00:00:59.570 --> 00:01:02.010
and hit is a hand-drawn
version, so it's not perfect.

00:01:02.010 --> 00:01:06.840
The harmonics are the
frequencies, the overtones,

00:01:06.840 --> 00:01:09.710
that at least to our
ear, reinforce that A,

00:01:09.710 --> 00:01:12.430
and what's true about the harmonics are

00:01:12.430 --> 00:01:16.060
that they will be 1/2
of the wavelength of A.

00:01:16.060 --> 00:01:19.290
In which case, it might
look something like this.

00:01:19.290 --> 00:01:23.870
So this would be a harmonic of A.

00:01:23.870 --> 00:01:26.700
It has half of the
wavelength of A and notice,

00:01:26.700 --> 00:01:31.110
it gets, when it finishes
its second full wave form,

00:01:31.110 --> 00:01:33.350
it ends again right at the same time

00:01:33.350 --> 00:01:35.060
that the wavelength of A ends.

00:01:35.060 --> 00:01:36.730
And then it would be another harmonic

00:01:36.730 --> 00:01:39.347
where it'd be something that
has 1/3 the wavelength of an A

00:01:39.347 --> 00:01:40.780
and a 1/4 of a wavelength of A,

00:01:40.780 --> 00:01:42.490
and if you look at a lot
of musical instruments

00:01:42.490 --> 00:01:44.180
or what sounds good to our ears,

00:01:44.180 --> 00:01:46.070
they're playing not
just a fundamental tone,

00:01:46.070 --> 00:01:47.000
but a lot of the harmonics.

00:01:47.000 --> 00:01:49.020
But anyway, that was a long-winded way

00:01:49.020 --> 00:01:54.020
of justifying why this is
called the harmonic series.

00:01:54.500 --> 00:01:58.600
Harmonic, harmonic series.

00:01:58.600 --> 00:02:00.970
And in a future video,

00:02:00.970 --> 00:02:04.110
we will prove that, and I don't
want to ruin the punchline,

00:02:04.110 --> 00:02:05.570
but this actually diverges,

00:02:05.570 --> 00:02:07.250
and I will come up with general rules

00:02:07.250 --> 00:02:09.110
for when things that look like this

00:02:09.110 --> 00:02:10.420
might converge or diverge,

00:02:10.420 --> 00:02:13.490
but the harmonic series
in particular diverges.

00:02:13.490 --> 00:02:15.410
So if we were to write it,

00:02:15.410 --> 00:02:17.570
so in sigma form,

00:02:17.570 --> 00:02:18.910
we would write it like this.

00:02:18.910 --> 00:02:23.910
We're going from n equals one
to infinity of one over n.

00:02:24.540 --> 00:02:26.190
Now another interesting thing

00:02:26.190 --> 00:02:28.800
is well, what if we were to
throw in some exponents here?

00:02:28.800 --> 00:02:30.670
So we already said, and
I'll just rewrite it.

00:02:30.670 --> 00:02:33.720
Doesn't hurt to rewrite it
and get more familiar with it.

00:02:33.720 --> 00:02:35.700
This right over here
is the harmonic series.

00:02:35.700 --> 00:02:38.010
One over one, which is just one

00:02:38.010 --> 00:02:40.810
plus one over two plus one over three,

00:02:40.810 --> 00:02:42.550
so on and so forth,

00:02:42.550 --> 00:02:45.840
but what if we were to raise
each of these denominators

00:02:45.840 --> 00:02:49.000
to say, the second power?

00:02:49.000 --> 00:02:51.750
So you might have something
that looks like this,

00:02:51.750 --> 00:02:54.740
where you have from n
equals one to infinity

00:02:54.740 --> 00:02:58.400
of one over n to the second power.

00:02:58.400 --> 00:02:59.470
Well, then it would look like this.

00:02:59.470 --> 00:03:02.387
It'd be one over one
squared, which is one,

00:03:02.387 --> 00:03:05.330
and we can just write
that first term as one,

00:03:05.330 --> 00:03:08.210
plus one over two squared,
which would be 1/4,

00:03:08.210 --> 00:03:11.820
plus one over three squared, which is 1/9,

00:03:11.820 --> 00:03:16.350
and then you could go on and on forever.

00:03:16.350 --> 00:03:17.280
Forever,

00:03:17.280 --> 00:03:18.940
and then you could generalize it.

00:03:18.940 --> 00:03:20.510
You could say hey, all right,

00:03:20.510 --> 00:03:23.800
what if we wanted to have
a general class of series

00:03:23.800 --> 00:03:25.940
that we were to describe like this?

00:03:25.940 --> 00:03:28.690
Going from n equals one to infinity

00:03:28.690 --> 00:03:32.170
of one over n to the p,

00:03:32.170 --> 00:03:35.630
where p could be any exponent.

00:03:35.630 --> 00:03:37.410
So for example,

00:03:37.410 --> 00:03:38.870
well the way this would play out

00:03:38.870 --> 00:03:42.620
is this would be one plus
one over two to the p

00:03:42.620 --> 00:03:45.060
plus one over three to the p

00:03:45.060 --> 00:03:47.320
plus one over four to the p,

00:03:47.320 --> 00:03:49.450
and it doesn't just have
to be an integer value.

00:03:49.450 --> 00:03:51.553
It could be, some, p could be 1/2,

00:03:52.400 --> 00:03:53.580
in which case, you would have one

00:03:53.580 --> 00:03:54.860
plus one over the square root of two

00:03:54.860 --> 00:03:56.670
plus one of the square root of three.

00:03:56.670 --> 00:03:58.890
This entire class of series

00:03:58.890 --> 00:04:01.240
and of course, harmonic
series is a special case

00:04:01.240 --> 00:04:02.980
where p is equal to one,

00:04:02.980 --> 00:04:06.280
this is known as p series.

00:04:06.280 --> 00:04:09.060
So these are known as p series

00:04:09.060 --> 00:04:10.420
and I try to remember it

00:04:10.420 --> 00:04:13.220
'cause it's p for the power

00:04:13.220 --> 00:04:15.410
that you are raising this denominator to.

00:04:15.410 --> 00:04:16.243
You could also view it

00:04:16.243 --> 00:04:17.440
as you're raising the
whole expression to it

00:04:17.440 --> 00:04:21.240
because one to any exponent
is still going to be one.

00:04:21.240 --> 00:04:22.570
But I hinted a little bit

00:04:22.570 --> 00:04:25.920
that maybe some of these converge
and some of these diverge,

00:04:25.920 --> 00:04:28.340
and we're going to prove
it in future videos,

00:04:28.340 --> 00:04:30.680
but the general principle is

00:04:30.680 --> 00:04:34.380
if p is greater than one,

00:04:34.380 --> 00:04:38.230
then we are going to converge.

00:04:38.230 --> 00:04:39.950
And that makes sense intuitively

00:04:39.950 --> 00:04:43.130
because that means that the
terms are getting smaller

00:04:43.130 --> 00:04:44.580
and smaller fast enough

00:04:44.580 --> 00:04:47.410
because the larger the
exponent for that denominator,

00:04:47.410 --> 00:04:50.630
that means that the denominator's
going to get bigger faster

00:04:50.630 --> 00:04:53.750
which means that the fraction
is going to get smaller faster

00:04:53.750 --> 00:04:57.230
and if p is less than or equal to one,

00:04:57.230 --> 00:04:58.680
and of course, when p is equal to one,

00:04:58.680 --> 00:05:01.440
we're dealing with the
famous harmonic series,

00:05:01.440 --> 00:05:05.360
that's a situation in which we diverge

00:05:05.360 --> 00:05:08.743
and we will prove these
things in future videos.


Time past between two clock faces

WEBVTT
Kind: captions
Language: en

00:00:00.220 --> 00:00:02.907
- [Instructor] We are asked,
"How much time has passed

00:00:02.907 --> 00:00:05.597
"from the time on the
left," so right over here,

00:00:05.597 --> 00:00:07.000
"to the time on the right?"

00:00:07.000 --> 00:00:07.960
Which we see right over here.

00:00:07.960 --> 00:00:12.450
And they say that, "The time
passed is less than 12 hours."

00:00:12.450 --> 00:00:14.630
So like always, pause this video

00:00:14.630 --> 00:00:16.300
and see if you can answer that on your own

00:00:16.300 --> 00:00:18.920
before we work on this together.

00:00:18.920 --> 00:00:20.190
All right.

00:00:20.190 --> 00:00:22.150
So first I just wanna figure out

00:00:22.150 --> 00:00:26.400
what time is being told by
each of these clock faces.

00:00:26.400 --> 00:00:27.920
So let's go here on the left.

00:00:27.920 --> 00:00:31.120
So the first thing I'm gonna
do is look at the hour hand.

00:00:31.120 --> 00:00:33.360
And we can see that the hour hand

00:00:33.360 --> 00:00:36.190
has gone a little bit past six.

00:00:36.190 --> 00:00:39.140
So it's going to be six-something.

00:00:39.140 --> 00:00:40.570
Six-something.

00:00:40.570 --> 00:00:43.320
And then to figure out
what the minutes are

00:00:43.320 --> 00:00:45.950
I can look at the minute
hand right over here.

00:00:45.950 --> 00:00:50.460
And say, "Well, how far
has it gone in minutes?"

00:00:50.460 --> 00:00:52.780
And I like to count by fives

00:00:52.780 --> 00:00:56.180
because we know when we go
from 12 on the clock to one

00:00:56.180 --> 00:00:58.120
that actually one, two, three,

00:00:58.120 --> 00:00:59.940
four, five minutes have gone by.

00:00:59.940 --> 00:01:02.790
And in fact, between any two
numbers, it's five minutes.

00:01:02.790 --> 00:01:07.430
So we go five, 10, 15, 20,

00:01:07.430 --> 00:01:11.430
25, 30, 35, 40.

00:01:11.430 --> 00:01:15.150
And then 41, 42, 43.

00:01:15.150 --> 00:01:16.410
That's where it's pointing.

00:01:16.410 --> 00:01:20.250
So it's 6:43, is where we're starting.

00:01:20.250 --> 00:01:22.420
And then where we're ending, let's see.

00:01:22.420 --> 00:01:25.240
The hour hand is a little bit past nine.

00:01:25.240 --> 00:01:27.000
You can see it's past nine.

00:01:27.000 --> 00:01:28.220
And so it's not necessarily

00:01:28.220 --> 00:01:30.160
what it's closest to, it's what it's past.

00:01:30.160 --> 00:01:32.810
So even if the hour hand was like there,

00:01:32.810 --> 00:01:36.470
even if it was closer to
10 but not quite at 10,

00:01:36.470 --> 00:01:38.650
you would say, "Okay, it's
gonna be nine-something."

00:01:38.650 --> 00:01:40.113
It's passed nine.

00:01:41.051 --> 00:01:42.940
And only if it got to 10 or past 10,

00:01:42.940 --> 00:01:44.070
then you would say, "It's 10-something."

00:01:44.070 --> 00:01:46.130
So this is gonna be nine-something.

00:01:46.130 --> 00:01:47.950
And to figure out what that something is,

00:01:47.950 --> 00:01:50.710
once again, we start here,
we can count by fives.

00:01:50.710 --> 00:01:54.300
Five minutes, 10 minutes,
and then we have one more.

00:01:54.300 --> 00:01:55.250
We get to 11 minutes.

00:01:55.250 --> 00:01:57.810
So it's 9:11.

00:01:57.810 --> 00:02:00.610
And they tell us that this
is less than 12 hours apart.

00:02:00.610 --> 00:02:04.250
So it's not like 6:43 in
one day, maybe at night,

00:02:04.250 --> 00:02:06.170
and then 9:11 the next morning.

00:02:06.170 --> 00:02:09.150
This is maybe the same
morning or the same evening.

00:02:09.150 --> 00:02:12.150
And so we can set up a timeline here.

00:02:12.150 --> 00:02:13.970
So let's set up a timeline.

00:02:13.970 --> 00:02:15.143
And so, let's see.

00:02:17.934 --> 00:02:19.840
The sixth hour is going to be important,

00:02:19.840 --> 00:02:20.970
so that's six o'clock.

00:02:20.970 --> 00:02:24.670
And let's see, maybe we wanna
go all the way to 10 o'clock.

00:02:24.670 --> 00:02:27.040
So that we go beyond 9:11.

00:02:27.040 --> 00:02:28.820
And so then we could say, let's go,

00:02:28.820 --> 00:02:30.425
have to do seven, eight, and nine.

00:02:30.425 --> 00:02:32.160
I just wanna get my spacing right.

00:02:32.160 --> 00:02:36.360
So let's say that this is
seven o'clock right over here.

00:02:36.360 --> 00:02:40.370
This is eight o'clock right over there.

00:02:40.370 --> 00:02:43.570
And then that is nine
o'clock right over there.

00:02:43.570 --> 00:02:45.510
It's not gonna be a perfect drawing,

00:02:45.510 --> 00:02:46.700
but this is pretty close.

00:02:46.700 --> 00:02:47.533
And so let's see.

00:02:47.533 --> 00:02:49.530
We are starting at 6:43.

00:02:49.530 --> 00:02:53.000
Which is going to be, let's
see, it's going to be roughly,

00:02:53.000 --> 00:02:54.380
and I just have to say roughly,

00:02:54.380 --> 00:02:55.980
it's going to be right over there.

00:02:55.980 --> 00:02:57.430
That's where we are starting.

00:02:57.430 --> 00:03:01.720
And then 9:11 is going to
be roughly right over there.

00:03:01.720 --> 00:03:04.260
So that is when we are ending.

00:03:04.260 --> 00:03:07.590
So how much time has gone by?

00:03:07.590 --> 00:03:11.430
Well, to go from 6:43 to seven o'clock,

00:03:11.430 --> 00:03:12.973
how many minutes is that?

00:03:14.030 --> 00:03:17.360
Well, to go from 43 minutes to 60 minutes,

00:03:17.360 --> 00:03:19.460
you have to add 17 minutes.

00:03:19.460 --> 00:03:24.410
So this right over here,
that is 17 minutes.

00:03:24.410 --> 00:03:26.060
Just write it out.

00:03:26.060 --> 00:03:29.340
And then we can see, we
have another two hours

00:03:29.340 --> 00:03:32.120
that goes by from seven
o'clock to nine o'clock.

00:03:32.120 --> 00:03:34.180
So then that's two hours.

00:03:34.180 --> 00:03:37.610
And then to go from nine o'clock
to 9:11, what is going on?

00:03:37.610 --> 00:03:40.190
Well, that's going to
be another 11 minutes.

00:03:40.190 --> 00:03:42.630
11 minutes.

00:03:42.630 --> 00:03:43.940
So how much time has happened?

00:03:43.940 --> 00:03:48.030
It's going to be 17 minutes
plus two hours plus 11 minutes.

00:03:48.030 --> 00:03:49.100
Well, I find it a little bit easier

00:03:49.100 --> 00:03:50.450
just to think about the minutes first.

00:03:50.450 --> 00:03:55.450
So if I add 17 minutes to 11
minutes, 17 plus 11 is 28.

00:03:57.770 --> 00:04:00.720
I can do that on the side
if you want to see it.

00:04:00.720 --> 00:04:02.350
Seven plus one is eight.

00:04:02.350 --> 00:04:04.310
One plus one is two.

00:04:04.310 --> 00:04:08.810
So we are going to have a total
of two hours and 28 minutes.

00:04:08.810 --> 00:04:10.300
And I could write it that way.

00:04:10.300 --> 00:04:15.300
Two hours, two hours and 28 minutes.

00:04:17.270 --> 00:04:19.283
And we are done.


Finding the end time for a movie in 24 hour time

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.010
- [Instructor] We are told
that Andre goes to a movie

00:00:03.010 --> 00:00:06.410
that starts at 19 hours 45 minutes,

00:00:06.410 --> 00:00:08.310
or 19:45,

00:00:08.310 --> 00:00:11.080
and is 90 minutes long.

00:00:11.080 --> 00:00:13.870
What time is the movie finished?

00:00:13.870 --> 00:00:16.160
So pause this video, and
see if you can answer that

00:00:16.160 --> 00:00:18.560
before we work through it together.

00:00:18.560 --> 00:00:21.900
All right, so what I like to do is set up

00:00:21.900 --> 00:00:24.110
what we could call a timeline.

00:00:24.110 --> 00:00:28.280
So if this is our timeline,

00:00:28.280 --> 00:00:32.570
and let's say that this is
19 hours right over here,

00:00:32.570 --> 00:00:33.670
19 hours,

00:00:33.670 --> 00:00:37.930
maybe this is 20 hours right over there,

00:00:37.930 --> 00:00:42.690
and then this would be 21
hours right over there.

00:00:42.690 --> 00:00:44.120
Now, what time does the movie start?

00:00:44.120 --> 00:00:47.060
It starts at 19 hours 45 minutes.

00:00:47.060 --> 00:00:50.370
So this would be 19 hours 30 minutes,

00:00:50.370 --> 00:00:53.240
which would be halfway
between 19 hours and 20 hours,

00:00:53.240 --> 00:00:54.950
and 19 hours 45 minutes

00:00:54.950 --> 00:00:58.510
would be halfway between 19:30 and 20.

00:00:58.510 --> 00:01:02.050
So 19 hours and 45 would
be right over there.

00:01:02.050 --> 00:01:03.050
So let me write that down.

00:01:03.050 --> 00:01:06.020
That is 19 hours and 45 minutes,

00:01:06.020 --> 00:01:09.920
and that is when this
movie is going to start.

00:01:09.920 --> 00:01:13.390
Now, it's going to
continue for 90 minutes.

00:01:13.390 --> 00:01:15.180
So when does it end?

00:01:15.180 --> 00:01:17.130
So there's a couple of ways
you could think about it.

00:01:17.130 --> 00:01:20.800
The way I think about it is
it will take us 15 minutes

00:01:20.800 --> 00:01:24.780
to get to 20 hours, so that's 15 minutes.

00:01:24.780 --> 00:01:28.570
And so 15 minutes of the movie
has happened by 20 hours.

00:01:28.570 --> 00:01:30.340
And then how much is left?

00:01:30.340 --> 00:01:34.010
Well, 19 minus 15 is 75,

00:01:34.010 --> 00:01:37.400
so then there's another 75 minutes left.

00:01:37.400 --> 00:01:41.730
Well, 60 of those 75 are gonna
get you to the next hour,

00:01:41.730 --> 00:01:45.870
so then it's going to be like this.

00:01:45.870 --> 00:01:49.690
So then you're going to
have another 60 minutes

00:01:49.690 --> 00:01:50.720
to get to the next hour.

00:01:50.720 --> 00:01:52.550
So, so far we've used 75,

00:01:52.550 --> 00:01:54.480
and then you're going to
need another 15 minutes

00:01:54.480 --> 00:01:56.600
to have a total of 90 minutes.

00:01:56.600 --> 00:02:00.870
So then you're going to
have another 15 minutes.

00:02:00.870 --> 00:02:03.200
And so when does the movie end?

00:02:03.200 --> 00:02:05.740
Well, as you can see, we started at 19:45,

00:02:05.740 --> 00:02:07.120
15 minutes got us to 20,

00:02:07.120 --> 00:02:09.290
another 60 minutes got us to 21,

00:02:09.290 --> 00:02:10.940
so now we have a total of 75.

00:02:10.940 --> 00:02:14.720
And then another 15 would
give us a total of 90,

00:02:14.720 --> 00:02:17.530
so we are finishing 15 minutes

00:02:17.530 --> 00:02:19.740
after the 21st hour

00:02:19.740 --> 00:02:22.700
or after hour 21 or however
you wanna think about it.

00:02:22.700 --> 00:02:24.920
So this time right over here is going

00:02:24.920 --> 00:02:28.100
to be 21 hours and 15 minutes,

00:02:28.100 --> 00:02:29.790
or 21:15.

00:02:29.790 --> 00:02:34.123
So that's the time that
the movie is finished.


Length of a trip in 24 hour time

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:01.073
- [Instructor] Hello.

00:00:01.073 --> 00:00:04.330
So we're told Colette rides her bike home

00:00:04.330 --> 00:00:06.610
from school every afternoon.

00:00:06.610 --> 00:00:11.610
She leaves school at 14:55
and arrives home at 15:25.

00:00:15.140 --> 00:00:18.470
How long does she ride her bike?

00:00:18.470 --> 00:00:20.440
So pause this video, like always,

00:00:20.440 --> 00:00:21.610
and see if you can answer that question.

00:00:21.610 --> 00:00:24.740
How long is her bike ride home?

00:00:24.740 --> 00:00:25.730
All right,

00:00:25.730 --> 00:00:27.670
now to help us answer that question,

00:00:27.670 --> 00:00:30.370
I'm going to set up a timeline.

00:00:30.370 --> 00:00:33.260
So that's my timeline there,

00:00:33.260 --> 00:00:34.760
and let's say that, let's see.

00:00:34.760 --> 00:00:37.520
Everything is going on between 14 hours

00:00:37.520 --> 00:00:39.800
and it looks like between
actually past 15 hours

00:00:39.800 --> 00:00:41.490
so I'll go to 16 hours,

00:00:41.490 --> 00:00:44.960
so that is 14 hours

00:00:44.960 --> 00:00:49.180
and this is 15 hours

00:00:49.180 --> 00:00:54.180
and then 16 hours would be
maybe right over here, 16 hours,

00:00:54.870 --> 00:00:58.200
and this would be 14 and a half,

00:00:58.200 --> 00:00:59.890
just to make things a little bit clearer.

00:00:59.890 --> 00:01:03.670
This would be 15 and a
half, just like that.

00:01:03.670 --> 00:01:05.030
Actually, it'd be a little bit,

00:01:05.030 --> 00:01:06.340
it would be something like that,

00:01:06.340 --> 00:01:07.460
and so let's see.

00:01:07.460 --> 00:01:10.650
She leaves school at 14:55.

00:01:10.650 --> 00:01:13.230
Where is that on this timeline?

00:01:13.230 --> 00:01:17.703
Well, 14:55 is five minutes before 15,

00:01:18.680 --> 00:01:23.680
so it's going to be
roughly right over here.

00:01:24.350 --> 00:01:26.350
So that is when Colette leaves,

00:01:26.350 --> 00:01:30.010
so that is 14:55,

00:01:30.010 --> 00:01:33.220
and then when she arrives, 15:25,

00:01:33.220 --> 00:01:36.660
that's 25 minutes after 15 hours,

00:01:36.660 --> 00:01:39.280
so that is right over there,

00:01:39.280 --> 00:01:44.280
so this right over there is
when she arrives home, 15:25,

00:01:44.370 --> 00:01:46.190
and so how long is that?

00:01:46.190 --> 00:01:48.240
Well the way I think about it is

00:01:48.240 --> 00:01:51.750
if we go from 14:55 to 15,

00:01:51.750 --> 00:01:53.680
that is five minutes,

00:01:53.680 --> 00:01:56.160
so this part right over there is five,

00:01:56.160 --> 00:01:59.090
and then to go from 15 to 15:25,

00:01:59.090 --> 00:02:01.133
that's another 25 minutes.

00:02:02.150 --> 00:02:03.550
So the total,

00:02:03.550 --> 00:02:06.770
the total length of her trip in time

00:02:06.770 --> 00:02:10.140
is going to be five
minutes plus 25 minutes.

00:02:10.140 --> 00:02:15.140
Five plus 25 is 30, 30 minutes,

00:02:16.500 --> 00:02:18.463
and we are done.


The rise of industrial capitalism

WEBVTT
Kind: captions
Language: en

00:00:00.130 --> 00:00:03.010
- [Instructor] The period
from the end of the Civil War

00:00:03.010 --> 00:00:06.210
to the start of the
20th Century was one of

00:00:06.210 --> 00:00:09.920
incredible economic transformation
in the United States.

00:00:09.920 --> 00:00:13.400
In 1865 the United States
was the 4th largest

00:00:13.400 --> 00:00:15.300
industrial economy in the world.

00:00:15.300 --> 00:00:18.930
By the 1890s, it had leapt to 1st place.

00:00:18.930 --> 00:00:23.480
At the same time, where people
worked, how people worked,

00:00:23.480 --> 00:00:27.180
and how much money they made,
all changed drastically.

00:00:27.180 --> 00:00:29.530
During the Gilded Age, the
United States went from

00:00:29.530 --> 00:00:34.090
being a nation of farmers to
a nation of factory workers.

00:00:34.090 --> 00:00:36.750
The nature of work itself also changed

00:00:36.750 --> 00:00:39.470
as large corporations began to implement

00:00:39.470 --> 00:00:41.340
management techniques aimed at

00:00:41.340 --> 00:00:44.000
increasing efficiency and profit.

00:00:44.000 --> 00:00:47.240
The gap between rich
and poor also increased

00:00:47.240 --> 00:00:49.100
considerably during this era.

00:00:49.100 --> 00:00:52.440
So what caused this
economic transformation?

00:00:52.440 --> 00:00:55.460
In this video I want to
explore some of the factors

00:00:55.460 --> 00:00:59.320
that contributed to these
changes in work and the economy:

00:00:59.320 --> 00:01:03.490
technological advancements,
new business strategies,

00:01:03.490 --> 00:01:07.830
business consolidation, and
pro-growth government policies.

00:01:07.830 --> 00:01:11.090
So let's dive a little
deeper into each of these.

00:01:11.090 --> 00:01:13.680
One of the biggest factors
contributing to the rise

00:01:13.680 --> 00:01:16.470
of industrial capitalism was technology.

00:01:16.470 --> 00:01:19.260
The late 19th Century
was an era of innovation.

00:01:19.260 --> 00:01:22.330
Nearly half a million patents were issued

00:01:22.330 --> 00:01:24.810
between 1860 and 1900.

00:01:24.810 --> 00:01:28.360
Improvements in machinery
and manufacturing processes,

00:01:28.360 --> 00:01:30.580
like the Bessemer process to make steel,

00:01:30.580 --> 00:01:32.100
increased productivity.

00:01:32.100 --> 00:01:34.990
And there were new technologies
that helped business:

00:01:34.990 --> 00:01:37.730
the telephone to coordinate
transactions over

00:01:37.730 --> 00:01:41.750
long distances, the typewriter
to speed up record keeping,

00:01:41.750 --> 00:01:44.190
and electricity which made it possible

00:01:44.190 --> 00:01:46.340
to work safely after dark.

00:01:46.340 --> 00:01:48.620
And the expansion of the
railroad, made it easy

00:01:48.620 --> 00:01:50.505
to get raw materials to factories

00:01:50.505 --> 00:01:53.070
and finished goods to markets.

00:01:53.070 --> 00:01:56.000
Corporations also devised new strategies

00:01:56.000 --> 00:01:59.190
to cope with doing business
at a national scale.

00:01:59.190 --> 00:02:02.190
In this era the first
national brands emerged.

00:02:02.190 --> 00:02:05.640
Companies like Coca-Cola
and Kellogg's Corn Flakes

00:02:05.640 --> 00:02:08.370
began advertising to national audiences.

00:02:08.370 --> 00:02:12.320
And mail order catalogs like
Montgomery Ward and Sears

00:02:12.320 --> 00:02:14.310
sold products across the country.

00:02:14.310 --> 00:02:19.140
An integrated nation-wide
system of business and shipping

00:02:19.140 --> 00:02:22.300
made it easy for customers
and companies to connect.

00:02:22.300 --> 00:02:26.400
During the Gilded Age
coordinating supplies and workers,

00:02:26.400 --> 00:02:29.910
time tables and sales, became its own

00:02:29.910 --> 00:02:32.510
full time job called management.

00:02:32.510 --> 00:02:37.510
Managers worked to increase
efficiency and cut costs.

00:02:37.590 --> 00:02:39.400
They did this in a number of ways:

00:02:39.400 --> 00:02:41.637
by replacing workers with machines,

00:02:41.637 --> 00:02:46.520
increasing working hours, and
decreasing wages for laborers.

00:02:46.520 --> 00:02:48.870
The titans of industry used other measures

00:02:48.870 --> 00:02:50.500
to maximize profits as well.

00:02:50.500 --> 00:02:54.720
The Gilded Age was an era of
ruthless business competition

00:02:54.720 --> 00:02:57.330
and the magnates of each industry set out

00:02:57.330 --> 00:02:59.150
to crush their enemies.

00:02:59.150 --> 00:03:01.760
Many of the men who made fabulous fortunes

00:03:01.760 --> 00:03:04.740
during the Gilded Age, started
out in the railroad industry

00:03:04.740 --> 00:03:08.470
taking advantage of government
subsidies and land grants.

00:03:08.470 --> 00:03:11.080
The U.S. Government took a laissez faire,

00:03:11.080 --> 00:03:14.440
or hands off, approach to
regulating business at this time.

00:03:14.440 --> 00:03:17.350
And there were no
corporate or income taxes

00:03:17.350 --> 00:03:20.870
so it was possible for a few
individuals and companies

00:03:20.870 --> 00:03:22.830
to amass enormous wealth.

00:03:22.830 --> 00:03:25.950
They did so by consolidating
their businesses,

00:03:25.950 --> 00:03:29.460
reducing competition,
and controlling markets.

00:03:29.460 --> 00:03:32.600
Steel baron Andrew Carnegie
was one of the first

00:03:32.600 --> 00:03:37.030
businessmen to employ vertical
integration in his companies.

00:03:37.030 --> 00:03:39.770
The goal of vertical
integration is to control

00:03:39.770 --> 00:03:42.610
every part of the supply
chain for a product.

00:03:42.610 --> 00:03:46.340
For example, Carnegie
owned not just steel mills,

00:03:46.340 --> 00:03:49.520
but the mines that produced
the iron ore and coal

00:03:49.520 --> 00:03:51.300
necessary for making steel,

00:03:51.300 --> 00:03:53.790
and the ships and
railroads that transported

00:03:53.790 --> 00:03:55.730
raw materials to the factories,

00:03:55.730 --> 00:03:58.290
and finished steel from the factories.

00:03:58.290 --> 00:04:00.920
This cut out middlemen and ensured that

00:04:00.920 --> 00:04:03.510
Carnegie never had to
wait for other companies

00:04:03.510 --> 00:04:05.210
to send him supplies.

00:04:05.210 --> 00:04:08.010
Big businesses in the
Gilded Age also reduced

00:04:08.010 --> 00:04:12.330
competition through holding
companies, trusts, and pools.

00:04:12.330 --> 00:04:15.610
Holding companies and
trusts allowed mergers

00:04:15.610 --> 00:04:18.320
that put many companies under the control

00:04:18.320 --> 00:04:19.910
of one parent company.

00:04:19.910 --> 00:04:22.580
Using these tactics, John D. Rockefeller

00:04:22.580 --> 00:04:26.400
who owned Standard Oil, controlled 95%

00:04:26.400 --> 00:04:29.940
of the country's oil supply by
the end of the 19th Century.

00:04:29.940 --> 00:04:32.800
Standard Oil become the nation's first

00:04:32.800 --> 00:04:34.700
billion dollar company.

00:04:34.700 --> 00:04:37.360
Some companies realized that cooperation

00:04:37.360 --> 00:04:40.300
was better than competition
and simply agreed

00:04:40.300 --> 00:04:43.070
to divide markets and
profits between them.

00:04:43.070 --> 00:04:47.100
These groups of supposedly
competing business entities

00:04:47.100 --> 00:04:49.090
were known as pools.

00:04:49.090 --> 00:04:53.160
The rise of industrial
capitalism had major consequences

00:04:53.160 --> 00:04:56.650
on American life, politics,
and foreign policy.

00:04:56.650 --> 00:04:59.150
For some, this new economy meant a higher

00:04:59.150 --> 00:05:01.210
standard of living than ever before,

00:05:01.210 --> 00:05:04.340
with cheap and plentiful
material comforts.

00:05:04.340 --> 00:05:07.280
But this new way of doing
business came at the expense

00:05:07.280 --> 00:05:09.360
of wages and working conditions,

00:05:09.360 --> 00:05:11.635
leading workers to begin organizing unions

00:05:11.635 --> 00:05:16.350
and advocating for political
solutions to economic problems.

00:05:16.350 --> 00:05:19.590
And as the United States
produced more and more,

00:05:19.590 --> 00:05:22.750
it would begin looking
abroad for new markets

00:05:22.750 --> 00:05:25.290
to sell its goods and consequently,

00:05:25.290 --> 00:05:27.353
for greater influence in the world.


Celebrate 10 years of Khan Academy! 🎂

WEBVTT
Kind: captions
Language: en

00:00:00.156 --> 00:00:01.976
- Hello teachers, Sal here,

00:00:01.976 --> 00:00:04.166
from the not-for-profit Khan Academy,

00:00:04.166 --> 00:00:05.796
and I just wanted to thank you

00:00:05.796 --> 00:00:09.456
for all of your partnership
over the last 10 years.

00:00:09.456 --> 00:00:12.166
All of us here at Khan Academy,
it's much more than me now,

00:00:12.166 --> 00:00:14.796
we're over 200 folks,
including researchers,

00:00:14.796 --> 00:00:17.946
and teachers, and designers,
and we all recognize

00:00:17.946 --> 00:00:19.796
that the most important actor

00:00:19.796 --> 00:00:23.446
in a student's learning
experience is you the teacher.

00:00:23.446 --> 00:00:26.696
And that's why we've been so
excited to partner with you.

00:00:26.696 --> 00:00:31.036
Over 700,000 teachers have
registered for Khan Academy

00:00:31.036 --> 00:00:35.976
and helped us reach over
74 million students.

00:00:35.976 --> 00:00:38.576
There's over 1200 translators

00:00:38.576 --> 00:00:40.926
who have helped translate Khan Academy

00:00:40.926 --> 00:00:43.656
into over 40 languages that are being used

00:00:43.656 --> 00:00:46.786
pretty much around the
world in over 190 countries.

00:00:46.786 --> 00:00:49.356
And a lot of people think
of math when they think

00:00:49.356 --> 00:00:51.656
of Khan Academy, and we
do have a lot of math,

00:00:51.656 --> 00:00:54.176
but we've been able to
expand well beyond that

00:00:54.176 --> 00:00:58.546
into the sciences, the humanities,
computing, and the arts.

00:00:58.546 --> 00:00:59.936
So I really wanna thank you

00:00:59.936 --> 00:01:02.266
for going on this journey with us.

00:01:02.266 --> 00:01:05.616
I've often said if I had to
pick between an amazing teacher

00:01:05.616 --> 00:01:07.066
and an amazing technology,

00:01:07.066 --> 00:01:10.286
I would pick the amazing
teacher every time.

00:01:10.286 --> 00:01:13.266
But the good thing is we don't
have to make that trade off.

00:01:13.266 --> 00:01:16.416
Our goal is to try to
empower you, the teacher,

00:01:16.416 --> 00:01:19.756
with the tools you need to
best meet your students,

00:01:19.756 --> 00:01:23.116
to save you time, and improve outcomes

00:01:23.116 --> 00:01:24.576
so that your students learn more

00:01:24.576 --> 00:01:27.606
and become empowered
citizens for the world.

00:01:27.606 --> 00:01:30.186
So thank you for being
on this journey with us,

00:01:30.186 --> 00:01:32.206
and I really look forward
to seeing what we can do

00:01:32.206 --> 00:01:33.959
in the next decade together.


Fractions in context

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:01.360
- [Instructor] In this video

00:00:01.360 --> 00:00:04.250
we're going to think about
how fractions can be used

00:00:04.250 --> 00:00:07.240
to represent things in the real world.

00:00:07.240 --> 00:00:11.000
So here we're told that
on the Sharks dive team

00:00:11.000 --> 00:00:13.810
there are three divers in third grade.

00:00:13.810 --> 00:00:17.000
There are eight total divers on the team.

00:00:17.000 --> 00:00:22.000
What fraction of Sharks
dive team is in third grade?

00:00:22.210 --> 00:00:25.383
So pause this video and see
if you can figure that out.

00:00:26.220 --> 00:00:28.540
All right, so first of all they tell us

00:00:28.540 --> 00:00:31.170
there are eight total divers on the team.

00:00:31.170 --> 00:00:32.800
So maybe I'll represent each diver

00:00:32.800 --> 00:00:34.500
with a little circle like this,

00:00:34.500 --> 00:00:36.120
I'll try to make it look
kind of like a diver,

00:00:36.120 --> 00:00:37.890
so that's not quite a
circle but you get the idea,

00:00:37.890 --> 00:00:39.410
it looks like something
kind of diving down.

00:00:39.410 --> 00:00:44.410
So one, two, three, four,
five, six, seven, and eight.

00:00:47.830 --> 00:00:51.350
Now if we were to talk
about just one diver here,

00:00:51.350 --> 00:00:54.720
just like that, that would
be one out of the eight.

00:00:54.720 --> 00:00:59.240
Or we would often call that one eighth,

00:00:59.240 --> 00:01:01.210
so eight with a h at the end.

00:01:01.210 --> 00:01:03.120
That is one eighth right over there,

00:01:03.120 --> 00:01:04.580
or I could represent it like this,

00:01:04.580 --> 00:01:09.443
I could say this is an eighth, an eighth,

00:01:09.443 --> 00:01:14.030
or I could say that this is equal to 1/8.

00:01:14.030 --> 00:01:19.030
This is one of the eight
members of our dive team.

00:01:19.480 --> 00:01:24.480
Now they tell us that there are
three divers in third grade,

00:01:24.480 --> 00:01:28.410
what fraction of Sharks
dive team is in third grade?

00:01:28.410 --> 00:01:30.760
So that is, let's say it these three,

00:01:30.760 --> 00:01:32.946
so there's three out of the eight.

00:01:32.946 --> 00:01:35.640
So if you wanted to
represent that as a fraction

00:01:35.640 --> 00:01:39.773
you could represent it as 3/8 like this,

00:01:40.670 --> 00:01:42.900
or you could represent it as,

00:01:42.900 --> 00:01:44.830
if you wanted to write it out as a word,

00:01:44.830 --> 00:01:47.570
three, instead of having
it three over eight,

00:01:47.570 --> 00:01:52.560
you could write three eighths like that.

00:01:52.560 --> 00:01:53.610
If you were doing this on Khan Academy

00:01:53.610 --> 00:01:55.030
there'd be some choices out there

00:01:55.030 --> 00:01:57.980
where you'd pick one
of the correct choice,

00:01:57.980 --> 00:02:00.310
but you could represent the
fraction of Sharks dive team

00:02:00.310 --> 00:02:03.676
that is in third grade either
as three over eight, 3/8,

00:02:03.676 --> 00:02:07.260
or three and then spell
out the word eighths.

00:02:07.260 --> 00:02:08.510
Let's do another example.

00:02:10.450 --> 00:02:13.800
Here we are told Yuma divided his clay

00:02:14.900 --> 00:02:17.410
into four equal parts.

00:02:17.410 --> 00:02:22.410
He made clay animals out
of three of the parts.

00:02:23.470 --> 00:02:27.100
What fraction of the clay did
Yuma use to make clay animals?

00:02:27.100 --> 00:02:30.250
So one again, pause this
video, and think about it.

00:02:30.250 --> 00:02:32.510
All right, so let's just imagine

00:02:32.510 --> 00:02:35.940
that this is his clay, initially,

00:02:35.940 --> 00:02:38.660
and he divides it into four equal parts.

00:02:38.660 --> 00:02:41.490
And so let's say he divides it like this.

00:02:41.490 --> 00:02:44.480
And let's say that I've divided
it into four equal parts,

00:02:44.480 --> 00:02:47.260
that these all have the
exact amount of clay in it,

00:02:47.260 --> 00:02:49.250
it's hand drawn so it's
not going to be perfect

00:02:49.250 --> 00:02:50.540
the way I drew it but
let's assume they all have

00:02:50.540 --> 00:02:52.363
the exact same amount of clay.

00:02:52.363 --> 00:02:54.690
Now it says that he made clay animals

00:02:54.690 --> 00:02:56.923
out of three of the parts.

00:02:56.923 --> 00:02:59.530
So maybe, this part right over here,

00:02:59.530 --> 00:03:01.468
he was able to make a clay animal out of,

00:03:01.468 --> 00:03:04.404
this part right over here,
he made a clay animal out of,

00:03:04.404 --> 00:03:06.520
and then that part, right over there,

00:03:06.520 --> 00:03:08.264
he made a clay animal out of.

00:03:08.264 --> 00:03:09.910
So what fraction of the clay

00:03:09.910 --> 00:03:12.392
did he use to make clay animals?

00:03:12.392 --> 00:03:15.730
So what would you call
each of the equal parts?

00:03:15.730 --> 00:03:18.080
So if I were to just focus
on that right over there,

00:03:18.080 --> 00:03:21.870
you would call that a fourth, a fourth.

00:03:21.870 --> 00:03:26.700
You could also represent it
as one fourth, like that,

00:03:26.700 --> 00:03:28.420
or you could represent it as 1/4.

00:03:28.420 --> 00:03:31.440
That's if you were to just
circle one of these equal parts,

00:03:31.440 --> 00:03:35.074
that one, or that one,
or that one, or that one.

00:03:35.074 --> 00:03:37.490
Now if you're talking
about all of his clay,

00:03:37.490 --> 00:03:38.440
what are you talking about?

00:03:38.440 --> 00:03:42.810
Well you could view it as
four fourths, four fourths,

00:03:42.810 --> 00:03:46.450
or four over four, this would
also be read four fourths.

00:03:46.450 --> 00:03:47.283
That would be referring to

00:03:47.283 --> 00:03:50.370
all one, two, three, four of his clay.

00:03:50.370 --> 00:03:52.460
Now if you wanted to say
what fraction of the clay

00:03:52.460 --> 00:03:54.620
did Yuma use to make clay animals,

00:03:54.620 --> 00:03:57.710
we can see that three of the fourths,

00:03:57.710 --> 00:04:00.080
were used to make clay animals.

00:04:00.080 --> 00:04:01.210
So to answer that question,

00:04:01.210 --> 00:04:04.240
we would say three of the fourths,

00:04:04.240 --> 00:04:09.080
so three fourths were
used to make clay animals.

00:04:09.080 --> 00:04:12.080
You can also express that as a fraction.

00:04:12.080 --> 00:04:17.020
You could also write
that as 3/4 like this.

00:04:17.020 --> 00:04:18.550
You would read these the same,

00:04:18.550 --> 00:04:21.520
three fourths, or three fourths.

00:04:21.520 --> 00:04:24.274
Three out of the four
equal sections of clay

00:04:24.274 --> 00:04:27.653
were used to make the clay animals.


Relating fractions to 1

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.647
- [Instructor] We are told,
"Select the two fractions

00:00:03.647 --> 00:00:06.120
"that are greater than one."

00:00:06.120 --> 00:00:08.540
So pause this video and
see if you can figure out

00:00:08.540 --> 00:00:12.163
which two of these fractions
are greater than one.

00:00:13.000 --> 00:00:15.540
All right, now let's
work on this together.

00:00:15.540 --> 00:00:18.750
And so, the main realization here,

00:00:18.750 --> 00:00:22.821
the main thing to pay attention
to is how the numerator

00:00:22.821 --> 00:00:25.330
relates to the denominator.

00:00:25.330 --> 00:00:28.190
When we are at one, they're
equal to each other.

00:00:28.190 --> 00:00:32.290
For example, one is equal to one whole.

00:00:32.290 --> 00:00:34.015
I guess you could write
it that way, one oneths.

00:00:34.015 --> 00:00:36.630
Or you could also write it as two halves,

00:00:36.630 --> 00:00:38.768
is also equal to three thirds,

00:00:38.768 --> 00:00:41.030
is also equal to four fourths.

00:00:41.030 --> 00:00:41.930
We could go on and on.

00:00:41.930 --> 00:00:43.450
If you want it in terms of sixths,

00:00:43.450 --> 00:00:46.250
one would be equal to six sixths.

00:00:46.250 --> 00:00:49.410
And so, if whatever we have up here

00:00:49.410 --> 00:00:51.445
is larger than the denominator,

00:00:51.445 --> 00:00:53.790
so if we have seven sixths

00:00:53.790 --> 00:00:58.630
or if we were to have maybe five fourths.

00:00:58.630 --> 00:01:01.080
Notice, the numerator is
larger than denominator.

00:01:01.080 --> 00:01:05.740
These are all situations
when we are greater than one.

00:01:05.740 --> 00:01:07.660
And all the situations where the numerator

00:01:07.660 --> 00:01:11.710
is less than the denominator,
so example one half

00:01:11.710 --> 00:01:16.710
or nine 11ths, or 10 11ths,

00:01:18.560 --> 00:01:22.230
these are all situations
where we are less than one.

00:01:22.230 --> 00:01:23.330
All of this is less than one.

00:01:23.330 --> 00:01:24.770
So let's look over here.

00:01:24.770 --> 00:01:27.540
Four is less than six.

00:01:27.540 --> 00:01:29.850
Four sixths is less than six sixths.

00:01:29.850 --> 00:01:32.310
Remember is the same thing as six sixths.

00:01:32.310 --> 00:01:34.040
So this is not greater than one,

00:01:34.040 --> 00:01:35.780
so I would not select that.

00:01:35.780 --> 00:01:38.870
Nine fourths, well
that's definitely larger

00:01:38.870 --> 00:01:40.220
than four fourths.

00:01:40.220 --> 00:01:41.750
That is greater than one

00:01:41.750 --> 00:01:45.030
and once again our numerator
is larger than our denominator,

00:01:45.030 --> 00:01:46.720
so we know that we are greater than one.

00:01:46.720 --> 00:01:48.720
So I will select that one.

00:01:48.720 --> 00:01:51.505
And then we see again five halves.

00:01:51.505 --> 00:01:53.582
Two halves is equal to one,

00:01:53.582 --> 00:01:56.660
so five halves is
definitely greater than one.

00:01:56.660 --> 00:01:58.310
So I like that one as well.

00:01:58.310 --> 00:02:00.161
And notice, five is greater than two

00:02:00.161 --> 00:02:03.480
and then we are already know
we picked our two choices,

00:02:03.480 --> 00:02:04.530
but we can look at the other ones.

00:02:04.530 --> 00:02:05.754
Seven is less than eight,

00:02:05.754 --> 00:02:08.360
so seven eighths is
less than eight eighths,

00:02:08.360 --> 00:02:09.850
so this is less than one.

00:02:09.850 --> 00:02:12.220
And three thirds, we've
already talked about it,

00:02:12.220 --> 00:02:13.320
that's equal to one.

00:02:13.320 --> 00:02:15.390
So I like those two choices.

00:02:15.390 --> 00:02:18.040
Let's do another example that tackles it

00:02:18.040 --> 00:02:19.328
a little bit of a different way.

00:02:19.328 --> 00:02:22.657
It says, "What, which
fraction could represent

00:02:22.657 --> 00:02:24.450
"Point A on the number line?"

00:02:24.450 --> 00:02:25.320
And Point A is here.

00:02:25.320 --> 00:02:27.470
They don't tell us a lot about Point A,

00:02:27.470 --> 00:02:30.030
but all they do tell us by
looking at the number line

00:02:30.030 --> 00:02:33.680
is that Point A is less than one.

00:02:33.680 --> 00:02:36.100
So which of these, another
way to think about it,

00:02:36.100 --> 00:02:38.183
is less than one?

00:02:39.530 --> 00:02:41.790
All right, in order for
it to be less than one

00:02:41.790 --> 00:02:43.944
the numerator has to be
less than the denominator.

00:02:43.944 --> 00:02:47.330
Here our numerator is
greater than the denominator.

00:02:47.330 --> 00:02:50.830
Seven fourths is definitely
larger than four fourths.

00:02:50.830 --> 00:02:53.200
Remember, four fourths is equal to one.

00:02:53.200 --> 00:02:56.080
This right over here I could
rewrite as four fourths.

00:02:56.080 --> 00:02:59.220
And so, seven fourths is
going to be someplace,

00:02:59.220 --> 00:03:01.420
seven fourths is going to
be someplace over there.

00:03:01.420 --> 00:03:03.820
So this is definitely not our choice.

00:03:03.820 --> 00:03:07.070
Two halves, well once again
that's the same thing as one.

00:03:07.070 --> 00:03:09.000
That's right over there.

00:03:09.000 --> 00:03:10.590
So that's not our choice.

00:03:10.590 --> 00:03:12.083
What about five eighths?

00:03:13.550 --> 00:03:14.383
Well, five eighths,

00:03:14.383 --> 00:03:17.039
the numerator is less
than the denominator.

00:03:17.039 --> 00:03:20.020
Eight eighths is equal to one,

00:03:20.020 --> 00:03:21.340
so that's that point again.

00:03:21.340 --> 00:03:23.720
So five eighths would
be to the left of that.

00:03:23.720 --> 00:03:25.434
So that could be Point A.

00:03:25.434 --> 00:03:27.963
So I like that choice.


Writing whole numbers as fractions

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:03.110
- [Instructor] We're told that
each rectangle is a whole.

00:00:03.110 --> 00:00:06.210
So this is a whole right over there,

00:00:06.210 --> 00:00:10.920
that's one whole, and
so this is two wholes.

00:00:10.920 --> 00:00:15.040
Which expressions describe the
shaded part of the picture?

00:00:15.040 --> 00:00:16.520
And they've shaded in everything,

00:00:16.520 --> 00:00:18.020
and they say choose two answers.

00:00:18.020 --> 00:00:19.130
So pause this video and see

00:00:19.130 --> 00:00:22.770
if you can answer that before
we work on this together.

00:00:22.770 --> 00:00:24.130
So before I even look at the choices,

00:00:24.130 --> 00:00:26.270
there's different ways
I could represent this.

00:00:26.270 --> 00:00:27.559
I have, I've already talked about

00:00:27.559 --> 00:00:29.770
that if each of these
rectangles is a whole,

00:00:29.770 --> 00:00:33.890
we have two of 'em, so I
could say this is two wholes.

00:00:33.890 --> 00:00:35.630
That's one way to represent it.

00:00:35.630 --> 00:00:37.540
Another way to represent it is,

00:00:37.540 --> 00:00:39.046
let's see each of the wholes is divided

00:00:39.046 --> 00:00:41.740
into two equal sections so this

00:00:41.740 --> 00:00:46.740
is a half right over there
and this is another half.

00:00:47.110 --> 00:00:51.590
This is a half and this is another half.

00:00:51.590 --> 00:00:53.460
So total shaded in we have one,

00:00:53.460 --> 00:00:56.520
two, three, four halves shaded in.

00:00:56.520 --> 00:00:59.465
So we could also represent
it as four halves.

00:00:59.465 --> 00:01:02.367
Or instead of writing half out like that,

00:01:02.367 --> 00:01:05.940
you could also represent
a half like that 1/2,

00:01:05.940 --> 00:01:10.350
that's another 1/2, that's
another 1/2, that's another 1/2.

00:01:10.350 --> 00:01:11.760
And how many of these halves do we have,

00:01:11.760 --> 00:01:13.710
we have one, two, three, four.

00:01:13.710 --> 00:01:15.244
So how can we write four halves,

00:01:15.244 --> 00:01:18.312
well you can write it as 4/2.

00:01:18.312 --> 00:01:19.943
So which expressions describe

00:01:19.943 --> 00:01:21.770
the shaded part of the picture?

00:01:21.770 --> 00:01:23.220
Well it's definitely not just one whole,

00:01:23.220 --> 00:01:26.140
we have two wholes here
so I'll rule that one out.

00:01:26.140 --> 00:01:29.640
4/2, yep we already
thought about that one.

00:01:29.640 --> 00:01:31.210
Four halves, yeah we thought

00:01:31.210 --> 00:01:33.553
about that one as well so we're done.


Comparing unit fractions

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.480
- [Instructor] So which
of the following numbers

00:00:03.480 --> 00:00:08.480
is greater, 1/3 or 1/5?

00:00:10.020 --> 00:00:13.290
Pause this video and try to answer that.

00:00:13.290 --> 00:00:16.470
All right, now let's
think about this together.

00:00:16.470 --> 00:00:19.160
And the way that I can best think about it

00:00:19.160 --> 00:00:21.070
is by visualizing them.

00:00:21.070 --> 00:00:23.460
So let's imagine a whole.

00:00:23.460 --> 00:00:28.460
So this is a whole right over there.

00:00:28.720 --> 00:00:31.300
And then let's say that
this is another whole

00:00:31.300 --> 00:00:33.720
right over there, I'm gonna
try to make these rectangles

00:00:33.720 --> 00:00:35.770
about looking about the same.

00:00:35.770 --> 00:00:38.380
And now, how would I represent 1/3?

00:00:38.380 --> 00:00:41.310
Well, I would divide this whole
into three equal sections.

00:00:41.310 --> 00:00:44.740
And so I'm gonna try to divide
it into three equal sections.

00:00:44.740 --> 00:00:47.180
So that's three equal
sections right over there.

00:00:47.180 --> 00:00:49.340
Now they're supposed to
be three equal sections,

00:00:49.340 --> 00:00:52.300
these are hand-drawn so
give me a little slack.

00:00:52.300 --> 00:00:56.090
But one of these three equal
sections, well that's 1/3.

00:00:56.090 --> 00:00:59.780
So that is 1/3 right over there.

00:00:59.780 --> 00:01:01.690
Now what about 1/5?

00:01:01.690 --> 00:01:06.160
Well then I would try to divide
this in five equal sections.

00:01:06.160 --> 00:01:09.091
So one,

00:01:09.091 --> 00:01:11.190
two,

00:01:11.190 --> 00:01:12.630
three,

00:01:12.630 --> 00:01:13.670
four,

00:01:13.670 --> 00:01:15.530
and five equal sections.

00:01:15.530 --> 00:01:19.320
And so 1/5 would be just
one of these fifths.

00:01:19.320 --> 00:01:22.930
So it would be that right over there.

00:01:22.930 --> 00:01:27.180
So when you compare it like
this, what's larger 1/3 or 1/5?

00:01:27.180 --> 00:01:29.310
And if it isn't obvious just yet,

00:01:29.310 --> 00:01:32.440
I could drag this one over,

00:01:32.440 --> 00:01:34.750
so that we can compare them,

00:01:34.750 --> 00:01:37.230
so that we can compare them directly.

00:01:37.230 --> 00:01:40.560
And you can see very clearly that 1/3

00:01:40.560 --> 00:01:43.210
covers more of the whole,
it's a larger fraction

00:01:43.210 --> 00:01:45.660
of the whole than 1/5 is.

00:01:45.660 --> 00:01:49.050
So 1/3 is greater than 1/5.

00:01:49.050 --> 00:01:51.550
And so you might have noticed
an interesting pattern

00:01:51.550 --> 00:01:54.300
or might start thinking about a pattern.

00:01:54.300 --> 00:01:56.570
You might have been tempted
when you saw the five here,

00:01:56.570 --> 00:02:01.570
five is larger than three,
but 1/5 is less than 1/3

00:02:01.720 --> 00:02:04.150
or 1/3 is greater than 1/5.

00:02:04.150 --> 00:02:06.900
And that is generally true,

00:02:06.900 --> 00:02:08.940
that the larger the denominator,

00:02:08.940 --> 00:02:11.560
the smaller the fraction is going to be.

00:02:11.560 --> 00:02:12.590
Why is that?

00:02:12.590 --> 00:02:16.490
Because you're dividing your
whole into more equal chunks.

00:02:16.490 --> 00:02:18.360
So if you're only dividing into three,

00:02:18.360 --> 00:02:21.760
if it's one of three of the
whole, 1/3 of the whole,

00:02:21.760 --> 00:02:24.140
or if it's one of three
equal chunks of the whole,

00:02:24.140 --> 00:02:26.150
it's gonna be bigger than one of five

00:02:26.150 --> 00:02:27.910
equal chunks of the whole.

00:02:27.910 --> 00:02:29.718
And so based on this,

00:02:29.718 --> 00:02:33.220
how would you compare these two numbers?

00:02:33.220 --> 00:02:38.220
How would you compare 2/3 to 2/5?

00:02:40.460 --> 00:02:42.975
Well, same idea here.

00:02:42.975 --> 00:02:45.370
1/3 is bigger than 1/5,

00:02:45.370 --> 00:02:48.980
so 2/3 is definitely going
to be bigger than 2/5,

00:02:48.980 --> 00:02:49.813
and you can see it here.

00:02:49.813 --> 00:02:52.433
2/3 is that,

00:02:53.380 --> 00:02:58.380
while 2/5 is that right over there.

00:02:58.390 --> 00:03:00.000
And I can do another example

00:03:00.000 --> 00:03:02.320
where I haven't even drawn it out.

00:03:02.320 --> 00:03:06.400
How would you compare four over six

00:03:06.400 --> 00:03:09.970
to four over eight?

00:03:09.970 --> 00:03:12.403
So 4/6 versus 4/8?

00:03:13.890 --> 00:03:15.550
Well, same idea.

00:03:15.550 --> 00:03:18.330
1/6 is larger than 1/8.

00:03:18.330 --> 00:03:21.970
1/6 is greater than 1/8,

00:03:21.970 --> 00:03:24.570
because the denominator here is smaller.

00:03:24.570 --> 00:03:27.420
We have the same numerator,
but the denominator is smaller.

00:03:27.420 --> 00:03:30.070
So four of the bigger
things is going to be larger

00:03:30.070 --> 00:03:31.630
than four of the smaller things.

00:03:31.630 --> 00:03:35.093
So 4/6 is greater than 4/8.


Fractions greater than 1 on the number line

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.188
- [Instructor] We're asked
to move the dot to 7/6

00:00:04.188 --> 00:00:06.130
on the number line.

00:00:06.130 --> 00:00:07.380
So pause this video.

00:00:07.380 --> 00:00:09.410
I can move this dot right over here,

00:00:09.410 --> 00:00:11.140
but I encourage you, pause the video,

00:00:11.140 --> 00:00:13.700
and put your finger on where 7/6 would be

00:00:13.700 --> 00:00:15.290
on the number line.

00:00:15.290 --> 00:00:16.730
All right, now let's
work on this together.

00:00:16.730 --> 00:00:18.120
So what they're saying is,

00:00:18.120 --> 00:00:21.720
is from zero to this
point on the number line

00:00:21.720 --> 00:00:24.270
right over there, that gets us to 1/6.

00:00:24.270 --> 00:00:27.010
So each of these spaces are a sixth.

00:00:27.010 --> 00:00:30.705
So we go zero, 1/6, 2/6,

00:00:30.705 --> 00:00:35.674
3/6, 4/6, 5/6,

00:00:35.674 --> 00:00:36.507
6/6, 7/6.

00:00:39.088 --> 00:00:40.140
Let me make sure I got that.

00:00:40.140 --> 00:00:41.690
So each of these are a sixth.

00:00:41.690 --> 00:00:44.810
So we have one, two, three,

00:00:44.810 --> 00:00:47.193
four, five, six, 7/6.

00:00:49.640 --> 00:00:52.670
So that's 7/6 on that number line.

00:00:52.670 --> 00:00:56.930
Now they have other ways of
getting at the same idea.

00:00:56.930 --> 00:01:00.200
For example, they say
which point is at 9/4

00:01:00.200 --> 00:01:01.270
on the number line?

00:01:01.270 --> 00:01:03.930
And they ask us to choose one answer,

00:01:03.930 --> 00:01:05.960
and we can look at the choices here.

00:01:05.960 --> 00:01:10.570
So which choice shows
9/4 on the number line?

00:01:10.570 --> 00:01:12.870
Pause this video and see
if you can pick that.

00:01:13.780 --> 00:01:15.960
All right, now let's
look at each of these.

00:01:15.960 --> 00:01:19.026
So it looks like in choice,
in this first choice,

00:01:19.026 --> 00:01:22.114
the space between zero and one

00:01:22.114 --> 00:01:26.890
is split into one, two,
three, four equal spaces.

00:01:26.890 --> 00:01:29.640
So as we go from zero to this next line,

00:01:29.640 --> 00:01:30.480
that's a fourth.

00:01:30.480 --> 00:01:32.520
And that seems like it keeps going.

00:01:32.520 --> 00:01:36.580
So this is 1/4, 2/4, 3/4, 4/4,

00:01:36.580 --> 00:01:40.310
5/4, 6/4, 7/4,

00:01:40.310 --> 00:01:44.510
8/4, 9/4 is here.

00:01:44.510 --> 00:01:45.540
That's what we're looking for.

00:01:45.540 --> 00:01:47.360
But the dot is not at 9/4.

00:01:47.360 --> 00:01:50.500
It's at 10/4, 11/4, it's at 12/4.

00:01:50.500 --> 00:01:52.860
So I don't like choice A.

00:01:52.860 --> 00:01:55.910
Let's see, choice B.

00:01:55.910 --> 00:01:57.190
Let's see what is, let's see,

00:01:57.190 --> 00:01:59.700
we have divided the space
between zero and one

00:01:59.700 --> 00:02:01.830
into one, two, three,

00:02:01.830 --> 00:02:04.444
four, five, six equal spaces.

00:02:04.444 --> 00:02:06.487
So each of these are a sixth.

00:02:06.487 --> 00:02:08.430
So to go from zero to one,

00:02:08.430 --> 00:02:13.430
you've already gone 6/6,
and then 7/6, 8/6, 9/6.

00:02:14.980 --> 00:02:18.800
So this is 9/6, not 9/4.

00:02:18.800 --> 00:02:20.710
And so let's look at this last choice.

00:02:20.710 --> 00:02:23.280
I'm already feeling like it
should be the answer, but.

00:02:23.280 --> 00:02:25.450
We can see that the spaces are the same

00:02:25.450 --> 00:02:28.000
as in our first choice,
so these are each fourths.

00:02:28.000 --> 00:02:29.640
Once again I know that because the space

00:02:29.640 --> 00:02:32.650
between zero and one or
any two whole numbers

00:02:32.650 --> 00:02:35.080
is divided into four equal spaces.

00:02:35.080 --> 00:02:38.320
So to go from zero to one, you go 4/4,

00:02:38.320 --> 00:02:40.830
and then 5/4, 6/4,

00:02:40.830 --> 00:02:45.630
7/4, 8/4, and 9/4.

00:02:45.630 --> 00:02:48.655
So choice C is definitely looking good.

00:02:48.655 --> 00:02:50.536
Let's do one more example.

00:02:50.536 --> 00:02:53.920
So here they say, what fraction is located

00:02:53.920 --> 00:02:56.750
at Point A on the number line?

00:02:56.750 --> 00:02:59.313
Pause this video and see
if you can answer that.

00:02:59.313 --> 00:03:00.880
All right.

00:03:00.880 --> 00:03:02.990
So between the whole numbers,

00:03:02.990 --> 00:03:05.740
how many spaces, equal spaces, do we have?

00:03:05.740 --> 00:03:08.730
It looks like we have one, two, three,

00:03:08.730 --> 00:03:12.210
four, five, six equal spaces.

00:03:12.210 --> 00:03:14.860
So things are divided into sixths.

00:03:14.860 --> 00:03:17.350
So 1/6, 2/6, 3/6,

00:03:17.350 --> 00:03:21.570
4/6, 5/6, 6/6,

00:03:21.570 --> 00:03:23.170
which is equal to one,

00:03:23.170 --> 00:03:25.510
and then 7/6.

00:03:25.510 --> 00:03:30.510
So this is seven over six, just like that,

00:03:30.680 --> 00:03:31.863
and we are done.