Khan Academy Classrooms has a new mastery system that makes personalized learning easier than ever!

WEBVTT
Kind: captions
Language: en

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- Hello, teachers, I'm Sal Khan,

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founder of the
not-for-profit Khan Academy,

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with the goal of helping you

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accelerate outcomes in your classroom.

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And I have an exciting announcement.

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What we're launching is a new

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mastery framework on Khan Academy.

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We have some of the most
significant features ever

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around personalization
and mastery learning,

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where your students can get
as much practice as they need,

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as much feedback as they
need, and the most important

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part of a mastery learning
system is the teacher,

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is the human being in the classroom.

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So it's explicitly designed to
increase human interactions,

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to give you, the teacher,
insights on which students

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might need help, which students might need

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a little bit of motivation.

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We've designed dashboards so
that teachers can identify

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gaps and misconceptions in
their students in real time.

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We've significantly added
to the amount of content,

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not just in math, where we've
always had tens of thousands

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of exercises from
kindergarten through calculus

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and statistics, but also in
high school and college-level

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sciences and the humanities.

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And, most importantly, it's
flexible to meet your needs.

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This can be done in the
classroom, it could be used

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as part of in-class practice,
it could be part of review,

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or it can be done as part of homework.

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And we are seeing the
classrooms that use Khan Academy

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for the classroom, at
least 30 minutes per week,

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are seeing incredible gains
in their student outcomes.

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- I look at my reports a lot.

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I use data a lot to drive my instruction

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and to drive what I need to reteach.

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- They tell me exactly
where that student is.

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It tells me how many
attempts that they have done

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and specifically, so there's
real data telling me,

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hey, this student is getting it wrong.

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- They're not doing 30 problems
and then realizing, uh,

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I don't know what I'm
doing and I just made

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some really bad habits for myself.

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- So thank you, teachers.

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All of us at Khan Academy
are incredibly inspired

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by the work that you
do day in and day out.

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Anything we can do to
make your lives easier

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or to help accelerate the
learning for your students,

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we are excited to do with you.

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(upbeat music)


Technology and presidential communication

WEBVTT
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Language: en

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- [Tutor] In this video, we're
going to talk a little bit

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about how modern technology
like social media

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has enhanced the communication
power of the presidency.

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Now, being president
has a lot of advantages,

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but politically one of
those advantages is that

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as president, you're
automatically newsworthy.

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So cameras and shows
are gonna be interested

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in talking to you, hearing
what you have to say.

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Now, this has always been
the case for presidents.

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If they wanted to publish
an editorial in a newspaper

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back in the 1790s, they
could have done that.

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But one major advantage
that presidents have today

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is advances in communication technology.

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And by that, I mean not only
the wealth of media choices

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that are out there in terms of channels

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and methods of distribution like YouTube,

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but also social media

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like Facebook or Twitter.

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So why would presidents be interested

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in using social media in the first place?

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Well, check out this graph
from the Pew Research Center

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that shows how much the
following of politicians

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has grown in just four
years, from 2010 to 2014.

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You can see that it doesn't
matter how old someone is

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or whether they're Republican or Democrat.

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Across the board, the
percent of registered voters

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who follow candidates for office

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has doubled in just this short time.

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So what does this mean for
the president in particular?

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Well, one thing it does
is make it much easier

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for the president to do
something called going public.

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And going public is when

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the president goes
directly to the electorate,

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speaks directly to the people

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to gain support for their policies.

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So they're bypassing members of Congress

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and speaking directly to the electorate

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to try to drum up support

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for whatever policy they'd like to enact.

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And the range of diverse media outlets

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that serve different demographics

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means that it's pretty
easy for the president

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to speak directly to whichever demographic

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he or she's interested in talking to.

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So they can craft a message

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that appeals to one specific demographic.

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For example, let's take a look at

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an appearance that former
president Barack Obama

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made on the internet
show Between Two Ferns.

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- Have you heard of the
Affordable Care Act?

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- Oh yeah, I heard about that.

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That's the thing that doesn't work.

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Why would you get the
guy that created the Zune

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to make your website?

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- Healthcare.gov works great now,

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and millions of Americans
have already gotten

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health insurance plans.

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And what we want is for people to know

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that you can get affordable healthcare.

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And most young Americans,
right now they're not covered.

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And the truth is is that
they can get coverage,

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all for what it costs to
pay your cell phone bill.

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- [Tutor] So you'll
notice that in that clip,

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Obama was speaking
directly to young people,

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going through a show that young
people were likely to watch

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and even using language that
was geared toward young people,

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like saying that healthcare costs

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about as much as your cell phone bill,

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all in service of trying
to promote the policy

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of the Affordable Care Act.

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Barack Obama was also the first president

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to create an official presidential
Twitter handle @POTUS,

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President of the United States,

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which then in the transfer of power

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from the Obama administration
to the Trump administration

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now is the official Twitter
handle of Donald Trump.

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And Trump is particularly
well known for using Twitter

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to reach a broad audience.

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Check out this tweet

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from July 5th, 2018, which talks about

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Trump's position on immigration laws.

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So let's read these.

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Congress must pass smart, fast

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and reasonable immigration laws now.

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Law enforcement at the
border is doing a great job,

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but the laws they are forced
to work with are insane.

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When people, with or without
children, enter our country,

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they must be told to
leave without our country

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being forced to endure
a long and costly trial.

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Tell the people "Out" and they must leave,

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just as they would if they were
standing on your front lawn.

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Hiring thousands of judges does not work

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and is not acceptable.

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Only country in the world that does this.

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So you can see what Trump is doing here

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is speaking directly to the people

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and asking them to influence Congress

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in support of his policy stance.

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So using Twitter allows Trump to speak to

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more than 50 million followers,

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and it allows the president
to have a rapid response

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to events as they unfold.

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So instead of having
to wait for a newspaper

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to print an editorial
tomorrow or on Sunday,

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it's possible to speak to the electorate

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within minutes of an
important event happening.

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So overall, modern
technology and social media

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have really increased the
impact of the president's power

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to communicate with the electorate,

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to get out policy initiatives
and to advertise them,

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and to speak quickly and
directly to the people.


Voting rights

WEBVTT
Kind: captions
Language: en

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- [Instructor] In this video,

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we're gonna do a brief overview

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of how amendments to the Constitution

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and federal legislation

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have increased voting rights over time.

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Now, why does this matter?

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Apart from just the innate value

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of voting rights in a democracy,

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it matter because who participates

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in the political process at any one time

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affects what political outcomes are.

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So, as we go through
these amendments and laws

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which increase voting rights,

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think about how each new group of voters

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might affect the overall opinion
of the voting population.

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The first major expansion of voting rights

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happened in 1870 when the
15th Amendment was ratified.

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And the 15th Amendment extended suffrage,

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the right to vote, to African American men

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and men in particular, at the time

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women were also hoping
to get the right to vote

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but there really was
only the political will

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to grand suffrage to African American men.

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Now, this happened immediately
after the Civil War,

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which ended in 1865,
and before the Civil War

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really only white men
had the right to vote.

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Now, the Civil War ended slavery

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but it didn't clear up the question

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of what the legal citizenship status

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of formerly enslaved
men and women would be

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and a lot of states in the south

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passed laws after the Civil War

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specifically denying the right to vote

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to African American men.

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So, in 1870, Congress kind of fought back

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against this and then they
passed the 15th Amendment

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which was then ratified by the states.

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The next notable
expansion of voting rights

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didn't concern so much who got to vote

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as who people got to vote for.

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In 1913 the states
ratified the 17th Amendment

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which provided for the
popular election of senators.

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So, before 1913, senators
were actually appointed

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by state legislatures

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instead of directly elected by citizens

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like members of the
House of Representatives.

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And switching to popular
election of senators

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is important because it
gives citizens more say

00:02:34.280 --> 00:02:35.813
over the national government.

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So, more say over who goes into Congress

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and therefore more say
over what kinds of laws

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that Congress passes.

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And in 1920, women finally
got the right to vote

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in the 19th Amendment, which doubled

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the voting eligible population.

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Now let's talk about some more recent

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voting rights amendments and laws.

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In 1964 and 1965 there were two amendments

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and pieces of federal legislation

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that had a major impact

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on African American voting rights.

00:03:17.810 --> 00:03:20.500
Now, although African American men

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technically got the right to vote in 1870,

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African American women
technically got the right to vote

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in 1920, there were a number of measures

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of voter suppression
enacted in southern states

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that effectively prevented
them from voting.

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One of these was the use of poll taxes

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which is effectively paying
money for the right to vote.

00:03:45.340 --> 00:03:48.070
The 24th Amendment outlawed that

00:03:48.070 --> 00:03:52.150
and then the Voting Rights
Act took that a step further

00:03:52.150 --> 00:03:54.760
by adding a number of other bans

00:03:54.760 --> 00:03:58.840
on voter suppression
tactics in southern states.

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So, in addition to poll taxes,

00:04:01.270 --> 00:04:04.430
the Voting Rights Act
banned literacy tests

00:04:04.430 --> 00:04:08.570
and forms of voter suppression
like grandfather clauses

00:04:10.350 --> 00:04:13.680
which were laws that
said you could only vote

00:04:13.680 --> 00:04:15.860
if your grandfather had voted.

00:04:15.860 --> 00:04:18.050
And while you might
think, oh, literacy tests

00:04:18.050 --> 00:04:21.240
don't sound that bad,
seems like a good idea

00:04:21.240 --> 00:04:23.400
to have a literate voting population,

00:04:23.400 --> 00:04:25.530
literacy tests really weren't fair

00:04:25.530 --> 00:04:29.440
because they were administered
by local voting boards

00:04:29.440 --> 00:04:32.740
who got to decide who
was and was not literate,

00:04:32.740 --> 00:04:35.040
so if you were an African American man

00:04:35.040 --> 00:04:37.860
you could have been a
Pulitzer Prize winning author

00:04:37.860 --> 00:04:40.130
and still had your local board say,

00:04:40.130 --> 00:04:43.080
oh, you're not literate soy
you're not eligible to vote.

00:04:43.080 --> 00:04:46.770
So, things like poll
taxes and literacy tests

00:04:46.770 --> 00:04:48.543
and grandfather clauses,

00:04:50.080 --> 00:04:52.660
nowhere in the language
of these voting laws

00:04:52.660 --> 00:04:55.060
did they say you can't
vote if you're black

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but they were very carefully crafted

00:04:57.410 --> 00:05:00.710
to specifically single out and suppress

00:05:00.710 --> 00:05:02.770
the votes of African Americans.

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The Voting Rights Act also
provided for federal examiners

00:05:09.070 --> 00:05:12.713
who had the power to
register people to vote,

00:05:14.410 --> 00:05:17.380
to monitor elections in jurisdictions

00:05:17.380 --> 00:05:20.360
that had large black populations

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but few registered black voters.

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And this effort was enormously successful.

00:05:27.080 --> 00:05:28.210
By the end of 1965,

00:05:29.647 --> 00:05:34.647
250,000 new African American voters

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had been registered in southern states.

00:05:37.640 --> 00:05:41.070
So, imagine how all of
these new participants

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might change legislative outcomes.

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Okay, there are two more
things I'd like to discuss.

00:05:47.230 --> 00:05:49.410
Now, one aspect of voting rights

00:05:49.410 --> 00:05:51.160
you might not have heard of

00:05:51.160 --> 00:05:54.910
is that until 1971,

00:05:54.910 --> 00:05:57.310
when the 26th Amendment was ratified,

00:05:57.310 --> 00:06:00.130
you had to be 21 years old to vote.

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Now, you can kinda guess at the
reason behind this amendment

00:06:03.920 --> 00:06:05.400
based on its year.

00:06:05.400 --> 00:06:07.270
In 1971 the United States

00:06:07.270 --> 00:06:09.190
was involved in the Vietnam war

00:06:09.190 --> 00:06:11.350
and young men were being drafted.

00:06:11.350 --> 00:06:13.480
So, it would have been possible

00:06:13.480 --> 00:06:16.980
to be drafted for the U.S. Army at age 18

00:06:16.980 --> 00:06:19.270
without ever having cast a vote.

00:06:19.270 --> 00:06:21.380
It's amazing to think
that the right to vote

00:06:21.380 --> 00:06:25.170
for 18 year olds is
less than 50 years old.

00:06:25.170 --> 00:06:28.510
The last piece of legislation
that I want to talk about

00:06:28.510 --> 00:06:32.460
is the Motor Voter Act
which was passed in 1993.

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The official name of this was the

00:06:34.200 --> 00:06:38.210
National Voter Registration Act of 1993

00:06:38.210 --> 00:06:39.890
but it's called the Motor Voter Act

00:06:39.890 --> 00:06:44.610
because it requires states
to provide individuals

00:06:44.610 --> 00:06:47.730
with the opportunity to register to vote

00:06:47.730 --> 00:06:51.303
when they apply for or renew
their driver's license.

00:06:54.150 --> 00:06:56.350
So, this makes it easier
to register to vote

00:06:56.350 --> 00:06:58.910
in the first place and it makes it easier

00:06:58.910 --> 00:07:00.980
to maintain your voter registration

00:07:00.980 --> 00:07:02.720
if you move to a new state

00:07:02.720 --> 00:07:05.170
because you'll need to
get a new driver's license

00:07:05.170 --> 00:07:08.720
so you can quickly make sure
you get on the polls to vote

00:07:08.720 --> 00:07:09.750
in your new state.

00:07:09.750 --> 00:07:12.450
The Motor Voter Act also requires states

00:07:12.450 --> 00:07:15.480
to allow individuals to register by mail

00:07:16.940 --> 00:07:19.650
30 days before a federal election

00:07:19.650 --> 00:07:21.980
and to allow individuals
to register to vote

00:07:21.980 --> 00:07:24.160
if they're going to an office

00:07:24.160 --> 00:07:25.460
providing services to people

00:07:25.460 --> 00:07:27.943
with disabilities or public assistance.

00:07:30.060 --> 00:07:33.040
So, in general, the Motor Voter Act

00:07:33.040 --> 00:07:34.450
makes it easier

00:07:35.440 --> 00:07:37.700
for people to get registered to vote

00:07:37.700 --> 00:07:39.490
and to stay registered to vote.

00:07:39.490 --> 00:07:43.940
So, it kinda removes barriers
to political participation.

00:07:43.940 --> 00:07:46.290
So, I know there's a
lot of information here,

00:07:46.290 --> 00:07:48.450
it's a lot to take in,
but the biggest thing

00:07:48.450 --> 00:07:50.250
to take away from this, again,

00:07:50.250 --> 00:07:53.640
is how participation affects outcomes.

00:07:53.640 --> 00:07:57.160
Who gets to vote, and more
importantly who does vote

00:07:57.160 --> 00:08:00.290
in any given election,
has an enormous impact

00:08:00.290 --> 00:08:03.370
on what kinds of politicians we elect

00:08:03.370 --> 00:08:05.753
and what kind of policies they enact.


Definite integrals: reverse power rule

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:02.430
- [Instructor] Let's evaluate
the definite integral

00:00:02.430 --> 00:00:05.370
from negative three to five

00:00:05.370 --> 00:00:08.290
of four dx.

00:00:08.290 --> 00:00:10.030
What is this going to be equal to?

00:00:10.030 --> 00:00:11.700
And I encourage you to pause the video

00:00:11.700 --> 00:00:13.603
and try to figure it out on your own.

00:00:14.580 --> 00:00:16.810
All right, so in order to evaluate this,

00:00:16.810 --> 00:00:20.670
we need to remember the
fundamental theorem of calculus,

00:00:20.670 --> 00:00:22.470
which connects the notion

00:00:22.470 --> 00:00:25.920
of a definite integral and antiderivative.

00:00:25.920 --> 00:00:28.490
So the fundamental theorem
of calculus tells us

00:00:28.490 --> 00:00:32.540
that our definite integral
from a to b of f of x dx

00:00:33.860 --> 00:00:36.840
is going to be equal to the antiderivative

00:00:36.840 --> 00:00:40.700
of our function f, which we
denote with the capital F,

00:00:40.700 --> 00:00:43.240
evaluated at the upper bound,

00:00:43.240 --> 00:00:44.520
minus

00:00:44.520 --> 00:00:46.450
our antiderivative,

00:00:46.450 --> 00:00:49.130
evaluated at the lower bound.

00:00:49.130 --> 00:00:51.570
So we just have to do
that right over here.

00:00:51.570 --> 00:00:54.190
So this is going to be equal to,

00:00:54.190 --> 00:00:58.400
well, what is the antiderivative of four?

00:00:58.400 --> 00:00:59.880
Well, you might immediately say,

00:00:59.880 --> 00:01:01.380
well, that's just going to be four x.

00:01:01.380 --> 00:01:04.370
You could even think of it in
terms of reverse power rule.

00:01:04.370 --> 00:01:07.350
Four is the same thing
as four x to the zero.

00:01:07.350 --> 00:01:09.390
So you increase zero by one.

00:01:09.390 --> 00:01:11.060
So it's going to be four x to the first,

00:01:11.060 --> 00:01:12.980
and then you divide by that new exponent.

00:01:12.980 --> 00:01:14.980
Four x to the first divided by one,

00:01:14.980 --> 00:01:16.870
well, that's just going to be four x.

00:01:16.870 --> 00:01:19.610
So the antiderivative is four x.

00:01:19.610 --> 00:01:22.400
This is, you could say,
our capital F of x,

00:01:22.400 --> 00:01:24.110
and we're going to evaluate that

00:01:24.110 --> 00:01:26.370
at five and at negative three.

00:01:26.370 --> 00:01:28.500
We're gonna find the
difference between these two.

00:01:28.500 --> 00:01:30.860
So what we have right over here,

00:01:30.860 --> 00:01:34.160
evaluating the antiderivative
at our upper bound,

00:01:34.160 --> 00:01:37.740
that is going to be four times five.

00:01:37.740 --> 00:01:40.700
And then from that,
we're going to subtract,

00:01:40.700 --> 00:01:44.850
evaluating our antiderivative
at the lower bound.

00:01:44.850 --> 00:01:47.000
So that's four times negative three.

00:01:47.000 --> 00:01:49.810
Four times negative three.

00:01:49.810 --> 00:01:52.290
And what is that going to be equal to?

00:01:52.290 --> 00:01:54.240
So this is

00:01:54.240 --> 00:01:57.730
20 and then minus negative 12.

00:01:57.730 --> 00:01:59.930
So this is going to be plus 12,

00:01:59.930 --> 00:02:03.630
which is going to be equal to 32.

00:02:03.630 --> 00:02:05.760
Let's do another example

00:02:05.760 --> 00:02:08.660
where we're going to do
the reverse power rule.

00:02:08.660 --> 00:02:09.980
So let's say

00:02:09.980 --> 00:02:12.090
that we want to find

00:02:12.090 --> 00:02:16.350
the indefinite or we want to
find the definite integral

00:02:16.350 --> 00:02:19.030
going from negative one

00:02:19.030 --> 00:02:20.740
to three

00:02:20.740 --> 00:02:23.210
of seven

00:02:23.210 --> 00:02:25.100
x squared

00:02:25.100 --> 00:02:25.933
dx.

00:02:26.840 --> 00:02:28.890
What is this going to be equal to?

00:02:28.890 --> 00:02:30.430
Well, what we want to do is

00:02:30.430 --> 00:02:33.360
evaluate what is the
antiderivative of this?

00:02:33.360 --> 00:02:35.790
Or you could say what is,
if this is lowercase f of x,

00:02:35.790 --> 00:02:37.420
what is capital F of x?

00:02:37.420 --> 00:02:38.780
Well, the reverse power rule,

00:02:38.780 --> 00:02:41.450
we increase this exponent by one.

00:02:41.450 --> 00:02:45.660
So we're going to have
seven times x to the third,

00:02:45.660 --> 00:02:49.050
and then we divide by
that increased exponent.

00:02:49.050 --> 00:02:51.200
So seven x to the third divided by three,

00:02:51.200 --> 00:02:54.300
and we want to evaluate
that at our upper bound

00:02:54.300 --> 00:02:55.482
and then subtract from that

00:02:55.482 --> 00:02:57.610
and it evaluate it at our lower bound.

00:02:57.610 --> 00:03:00.090
So this is going to be equal to,

00:03:00.090 --> 00:03:03.390
so evaluating it at our upper bound,

00:03:03.390 --> 00:03:07.390
it's going to be seven
times three to the third.

00:03:07.390 --> 00:03:10.710
I'll just write that three
to the third over three.

00:03:10.710 --> 00:03:14.410
And then from that, we
are going to subtract

00:03:14.410 --> 00:03:16.080
this capital F of x,

00:03:16.080 --> 00:03:19.160
the antiderivative evaluated
at the lower bound.

00:03:19.160 --> 00:03:21.480
So that is going to be

00:03:21.480 --> 00:03:24.960
seven times negative one to the third,

00:03:24.960 --> 00:03:26.940
all of that over three.

00:03:26.940 --> 00:03:28.950
And so this first expression, let's see,

00:03:28.950 --> 00:03:30.080
this is going to be seven

00:03:30.080 --> 00:03:31.940
times three to the third over three.

00:03:31.940 --> 00:03:33.510
This is 27

00:03:33.510 --> 00:03:34.670
over three.

00:03:34.670 --> 00:03:37.330
This is going to be the same
thing as seven times nine.

00:03:37.330 --> 00:03:39.730
So this is going to be 63.

00:03:39.730 --> 00:03:40.950
And this over here,

00:03:40.950 --> 00:03:44.240
negative one to the third
power is negative one.

00:03:44.240 --> 00:03:46.880
But then we were subtracting a negative,

00:03:46.880 --> 00:03:49.038
so this is just gonna be adding.

00:03:49.038 --> 00:03:53.200
And so this is just going
to be plus seven over three.

00:03:53.200 --> 00:03:55.540
Plus seven

00:03:55.540 --> 00:03:56.373
over three,

00:03:56.373 --> 00:03:59.050
if we wanted to express
this as a mixed number,

00:03:59.050 --> 00:04:01.833
seven over three is the
same thing as 2 1/3.

00:04:03.400 --> 00:04:05.290
So when we add everything together,

00:04:05.290 --> 00:04:10.290
we are going to get 65 1/3.

00:04:10.900 --> 00:04:12.223
And we are done.


Analyzing related rates problems: equations (Pythagoras)

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:02.150
- [Instructor] Two cars are driving

00:00:02.150 --> 00:00:06.290
towards an intersection from
perpendicular directions.

00:00:06.290 --> 00:00:08.370
The first car's velocity is 50 km/h

00:00:09.520 --> 00:00:13.100
and the second car's velocity is 90 km/h.

00:00:13.100 --> 00:00:15.330
At a certain instant, T sub zero,

00:00:15.330 --> 00:00:19.080
the first car is a
distance, X sub T sub zero

00:00:19.080 --> 00:00:22.490
or X of T sub zero, of half a kilometer

00:00:22.490 --> 00:00:24.220
from the intersection and the second car

00:00:24.220 --> 00:00:27.080
is a distance, Y of T sub zero,

00:00:27.080 --> 00:00:29.970
of 1.2 kilometers from the intersection.

00:00:29.970 --> 00:00:32.330
What is the rate of
change of the distance,

00:00:32.330 --> 00:00:35.220
D of T, between the cars at that instant?

00:00:35.220 --> 00:00:36.930
So, T sub zero.

00:00:36.930 --> 00:00:39.940
Which equation should be
used to solve the problem?

00:00:39.940 --> 00:00:43.150
And they give us a
choice of four equations,

00:00:43.150 --> 00:00:45.530
right over here, so you
could pause the video

00:00:45.530 --> 00:00:47.100
and try to work through it on your own

00:00:47.100 --> 00:00:49.790
but I'm about to do it, as well.

00:00:49.790 --> 00:00:51.440
So, let's just draw what's going on.

00:00:51.440 --> 00:00:53.100
That's always a healthy thing to do.

00:00:53.100 --> 00:00:55.040
So, two cars are driving towards

00:00:55.040 --> 00:00:57.710
an intersection from
perpendicular directions.

00:00:57.710 --> 00:01:01.280
So, let's say that this
is one car right over here

00:01:02.140 --> 00:01:05.100
and it is moving in the X direction

00:01:05.100 --> 00:01:07.760
towards that intersection,
which is right over there.

00:01:07.760 --> 00:01:09.130
And then you have another car

00:01:09.130 --> 00:01:11.150
that is moving in the Y direction.

00:01:11.150 --> 00:01:13.470
So, let's say it's moving like this.

00:01:13.470 --> 00:01:15.150
So, this is the other car.

00:01:15.150 --> 00:01:17.300
I should have maybe done a top view.

00:01:17.300 --> 00:01:18.580
Well, here we go.

00:01:18.580 --> 00:01:20.250
This square represents the car

00:01:20.250 --> 00:01:23.020
and it is moving in that direction.

00:01:23.020 --> 00:01:25.220
Now, they say at a certain instance.

00:01:25.220 --> 00:01:27.430
T sub zero, so let's draw that instant.

00:01:27.430 --> 00:01:29.750
So, the first car is a distance,

00:01:29.750 --> 00:01:33.960
X of T sub zero, of 0.5 kilometers.

00:01:33.960 --> 00:01:36.690
So, this distance, right over here,

00:01:36.690 --> 00:01:40.360
let's just call this X of T and let's call

00:01:40.360 --> 00:01:44.830
this distance, right over here, Y of T.

00:01:44.830 --> 00:01:47.160
Now, how does the distance between the

00:01:47.160 --> 00:01:50.840
cars relate to X of T and Y of T?

00:01:50.840 --> 00:01:52.830
Well, we could just use
the distance formula,

00:01:52.830 --> 00:01:55.240
which is essentially just
the Pythagorean theorem,

00:01:55.240 --> 00:01:57.110
to say, well, the
distance between the cars

00:01:57.110 --> 00:02:00.180
would be the hypotenuse
of this right triangle.

00:02:00.180 --> 00:02:03.040
Remember, they're traveling
from perpendicular directions.

00:02:03.040 --> 00:02:04.760
So, that's a right triangle, there.

00:02:04.760 --> 00:02:09.760
So, this distance, right over
here, would be X of T squared

00:02:11.300 --> 00:02:16.300
plus Y of T squared, and
the square root of that,

00:02:16.710 --> 00:02:19.070
and that's just the Pythagorean
theorem, right over here.

00:02:19.070 --> 00:02:23.170
This would be D of T, or we could say

00:02:23.170 --> 00:02:27.240
that D of T squared is equal to

00:02:29.273 --> 00:02:34.273
X of T squared plus Y,
too many parentheses.

00:02:34.900 --> 00:02:38.300
Plus Y of T squared.

00:02:38.300 --> 00:02:39.610
So, that's the relationship between

00:02:39.610 --> 00:02:43.870
D of T, X of T, and Y
of T, and it's useful

00:02:43.870 --> 00:02:45.740
for solving this problem because, now,

00:02:45.740 --> 00:02:47.680
we can take the derivative of both sides

00:02:47.680 --> 00:02:49.510
of this equation with respect to T.

00:02:49.510 --> 00:02:52.000
We'd be using various derivative rules,

00:02:52.000 --> 00:02:54.400
including the chain
rule, in order to do it,

00:02:54.400 --> 00:02:56.610
and then that would give us a relationship

00:02:56.610 --> 00:02:59.620
between the rate of change of D of T,

00:02:59.620 --> 00:03:01.370
which would be D prime of T,

00:03:01.370 --> 00:03:04.480
and the rate of change of X of T, Y of T,

00:03:04.480 --> 00:03:07.450
and X of T and Y of T, themselves.

00:03:07.450 --> 00:03:10.423
And so, if we look at these
choices, right over here,

00:03:11.766 --> 00:03:14.830
we indeed see that D sets up that exact

00:03:14.830 --> 00:03:17.940
same relationship that
we just did, ourselves.

00:03:17.940 --> 00:03:20.230
That it shows that the distance

00:03:20.230 --> 00:03:22.430
squared between the cars is equal to

00:03:22.430 --> 00:03:25.370
that X distance from the
intersection squared,

00:03:25.370 --> 00:03:27.880
plus the Y distance from
the intersection squared,

00:03:27.880 --> 00:03:30.440
and then we can take the
derivative of both sides

00:03:30.440 --> 00:03:33.383
to actually figure out this
related, rates question.


Comparing constants of proportionality

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:02.390
- [Instructor] Betty's Bakery calculates

00:00:02.390 --> 00:00:07.080
the total price D in
dollars for C cupcakes

00:00:07.080 --> 00:00:12.080
using the equation D is
equal to two times C.

00:00:12.230 --> 00:00:15.170
What does two mean in this situation?

00:00:15.170 --> 00:00:18.370
So pause this video and
see if you can answer that.

00:00:18.370 --> 00:00:20.030
Alright, before I even
look at the choices,

00:00:20.030 --> 00:00:21.150
let me just interpret this.

00:00:21.150 --> 00:00:25.880
This says D equals 2C tells us

00:00:25.880 --> 00:00:28.830
that however many cupcakes someone buys,

00:00:28.830 --> 00:00:33.440
we multiply that times two
to get the amount of dollars,

00:00:33.440 --> 00:00:35.950
the price that they need to pay.

00:00:35.950 --> 00:00:39.420
So this must mean that each
cupcake is two dollars.

00:00:39.420 --> 00:00:41.220
Or that it's two dollars per cupcake.

00:00:41.220 --> 00:00:43.030
Because however many cupcakes we get,

00:00:43.030 --> 00:00:44.810
we multiply that by two dollars

00:00:44.810 --> 00:00:48.010
per cupcake to get the actual price.

00:00:48.010 --> 00:00:49.150
So let's see, choice A says

00:00:49.150 --> 00:00:51.800
the bakery charges two
dollars for each cupcake.

00:00:51.800 --> 00:00:53.910
Yeah, that's pretty close
to what I just said.

00:00:53.910 --> 00:00:55.620
So I'll go with that one.

00:00:55.620 --> 00:01:00.180
The bakery sells two
cupcakes for a dollar.

00:01:00.180 --> 00:01:02.190
No, that would not be the case.

00:01:02.190 --> 00:01:03.890
And you could even try it out.

00:01:03.890 --> 00:01:05.590
If we had one cupcake.

00:01:05.590 --> 00:01:08.430
So if C is one, what is D going to be?

00:01:08.430 --> 00:01:09.960
And actually, let me just do that for you

00:01:09.960 --> 00:01:11.030
'cause it's interesting.

00:01:11.030 --> 00:01:12.350
C and D.

00:01:12.350 --> 00:01:15.240
So if you get one cupcake,
you're gonna multiply it by two.

00:01:15.240 --> 00:01:16.870
It's going to be two dollars.

00:01:16.870 --> 00:01:18.410
Two cupcakes, we're gonna multiply

00:01:18.410 --> 00:01:20.330
it by two and be four dollars.

00:01:20.330 --> 00:01:22.400
It's consistent with this first choice,

00:01:22.400 --> 00:01:23.690
but to sell two cupcakes,

00:01:23.690 --> 00:01:24.880
it's not gonna be four dollars,

00:01:24.880 --> 00:01:27.140
it's gonna be for four dollars.

00:01:27.140 --> 00:01:29.150
The bakery sells two types of cupcakes.

00:01:29.150 --> 00:01:30.410
Well, they don't say anything about that,

00:01:30.410 --> 00:01:32.350
so I'll rule that out, as well.

00:01:32.350 --> 00:01:33.400
Let's do another one.

00:01:34.420 --> 00:01:37.410
Here, we are told, select the store with

00:01:37.410 --> 00:01:40.840
the least expensive ice cream per scoop.

00:01:40.840 --> 00:01:43.640
There's definitely a dessert
theme going on over here.

00:01:43.640 --> 00:01:45.960
Alright, so pause this video
and see if you can work it out.

00:01:45.960 --> 00:01:49.173
Is it choice A, choice B or choice C.

00:01:50.600 --> 00:01:52.940
Alright, now let's go
through these together.

00:01:52.940 --> 00:01:56.360
Choice A calculates the total price D

00:01:56.360 --> 00:02:00.750
in dollars of ice cream
with S scoops using

00:02:00.750 --> 00:02:04.393
the equation D is equal to 0.75S.

00:02:05.530 --> 00:02:08.210
So whatever the number of scoops are,

00:02:08.210 --> 00:02:11.420
we're gonna multiply that times 75 cents

00:02:11.420 --> 00:02:14.950
or 75 hundredth of a
dollar to get the price.

00:02:14.950 --> 00:02:18.780
And so based on the logic we
just used in that last example,

00:02:18.780 --> 00:02:23.100
in store A, it is 75 cents,

00:02:23.100 --> 00:02:26.780
75 cents per, per scoop.

00:02:26.780 --> 00:02:28.190
So we know the price there.

00:02:28.190 --> 00:02:29.629
And anything like this,
when you're comparing,

00:02:29.629 --> 00:02:32.450
you wanna put it all in the same terms.

00:02:32.450 --> 00:02:34.230
Okay, here it's 75 cents per scoop.

00:02:34.230 --> 00:02:36.550
Let's think about how many,
how much per scoop it is at B.

00:02:36.550 --> 00:02:40.040
And how many, how much
per scoop it is for C.

00:02:40.040 --> 00:02:42.270
Alright, now store B.

00:02:42.270 --> 00:02:44.640
So, when I get three scoops,

00:02:44.640 --> 00:02:47.820
I multiply that times
one to get three dollars.

00:02:47.820 --> 00:02:48.800
When I get eight scoops,

00:02:48.800 --> 00:02:51.450
I multiply it times one
to get eight dollars.

00:02:51.450 --> 00:02:52.530
When I have 12 scoops,

00:02:52.530 --> 00:02:55.440
I multiply it by one to get 12 dollars.

00:02:55.440 --> 00:02:57.940
So the equation that store B must use

00:02:57.940 --> 00:02:59.900
is that the dollar is D.

00:02:59.900 --> 00:03:04.810
That's going to be equal to
one times the number of scoops.

00:03:04.810 --> 00:03:06.930
Or you could view this as, hey,

00:03:06.930 --> 00:03:09.220
it's a dollar per scoop at store B.

00:03:09.220 --> 00:03:11.230
So one dollar,

00:03:11.230 --> 00:03:14.620
one dollar per scoop.

00:03:14.620 --> 00:03:17.880
So we already know that store
A is cheaper than store B.

00:03:17.880 --> 00:03:19.690
'Cause 75 cents per scoop is cheaper

00:03:19.690 --> 00:03:21.550
than one dollar per scoop.

00:03:21.550 --> 00:03:23.960
Store C, alright, so here this,

00:03:23.960 --> 00:03:26.950
this relationship is
described with a graph but we

00:03:26.950 --> 00:03:29.690
can put it in the same
forms that we saw before.

00:03:29.690 --> 00:03:31.770
So for store C.

00:03:31.770 --> 00:03:33.530
Make a little table here.

00:03:33.530 --> 00:03:37.790
And so if I have the scoops
and I have the dollars.

00:03:37.790 --> 00:03:38.780
So let's see.

00:03:38.780 --> 00:03:41.970
When I get two scoops, it looks like,

00:03:41.970 --> 00:03:43.390
and I'm just picking values where it looks

00:03:43.390 --> 00:03:45.330
like I can read the graph easily.

00:03:45.330 --> 00:03:48.050
Two scoops looks like three dollars.

00:03:48.050 --> 00:03:50.130
Two scoops, three dollars.

00:03:50.130 --> 00:03:53.850
Four scoops, it is six dollars.

00:03:53.850 --> 00:03:56.650
Four scoops, it's six dollars.

00:03:56.650 --> 00:03:59.602
So it looks like I'm multiplying
times one point half,

00:03:59.602 --> 00:04:02.130
one and a half, I was
gonna say one point half.

00:04:02.130 --> 00:04:06.120
1.5 or one and a half to
go from scoops to dollars.

00:04:06.120 --> 00:04:08.000
Or another way you can think about it is

00:04:08.000 --> 00:04:13.000
the dollars is equal to
1.5 times the scoops.

00:04:13.630 --> 00:04:14.860
Or another way to think about it,

00:04:14.860 --> 00:04:19.860
at store C, they're charging
a 1.50, 1.50 per, per scoop.

00:04:21.670 --> 00:04:23.910
So store C is the most expensive followed

00:04:23.910 --> 00:04:26.130
by B and then store A is the cheapest

00:04:26.130 --> 00:04:27.080
and that's what they're asking us.

00:04:27.080 --> 00:04:30.653
The least expensive ice
cream per scoop is store A.


Interpreting graphs of proportional relationships

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:02.670
- [Instructor] We are told
the proportional relationship

00:00:02.670 --> 00:00:05.710
between the number of
hours a business operates

00:00:05.710 --> 00:00:09.660
and its total cost of electricity is shown

00:00:09.660 --> 00:00:13.030
in the following graph, all right.

00:00:13.030 --> 00:00:15.840
Which statements about the graph are true?

00:00:15.840 --> 00:00:18.580
Choose all answers that apply.

00:00:18.580 --> 00:00:22.130
So pause this video and see
if you can figure this out.

00:00:22.130 --> 00:00:23.680
All right, now let's do this together.

00:00:23.680 --> 00:00:26.590
And before I even look at the choices,

00:00:26.590 --> 00:00:28.670
let me analyze this a little bit.

00:00:28.670 --> 00:00:30.980
It is a proportional relationship.

00:00:30.980 --> 00:00:33.740
So we know that our total
cost, let me write it here.

00:00:33.740 --> 00:00:37.180
Our total cost

00:00:37.180 --> 00:00:41.260
is going to be equal to some
constant of proportionality

00:00:41.260 --> 00:00:45.520
times our number of hours.

00:00:45.520 --> 00:00:46.850
And we can even figure out

00:00:46.850 --> 00:00:50.050
what that constant of
proportionality is going to be,

00:00:50.050 --> 00:00:52.260
because they give us this point A.

00:00:52.260 --> 00:00:55.620
We know that when our hours are four,

00:00:55.620 --> 00:00:58.160
so when this is four right over here,

00:00:58.160 --> 00:01:02.627
our total cost is $120.

00:01:02.627 --> 00:01:04.380
$120.

00:01:04.380 --> 00:01:08.720
So what times four is equal to 120?

00:01:08.720 --> 00:01:10.400
Well, we know that this k must be 30,

00:01:10.400 --> 00:01:14.480
'cause 30 times four is 120.

00:01:14.480 --> 00:01:16.810
So we can write that
proportional relationship

00:01:16.810 --> 00:01:20.410
where our total cost is going to be equal

00:01:20.410 --> 00:01:23.030
to our constant of proportionality, 30,

00:01:23.030 --> 00:01:26.430
times our number of hours.

00:01:26.430 --> 00:01:29.440
Number of hours.

00:01:29.440 --> 00:01:30.980
So let's see which of these choices,

00:01:30.980 --> 00:01:32.620
and it might be more than one,

00:01:32.620 --> 00:01:36.720
say this or describe what's going on here.

00:01:36.720 --> 00:01:39.670
So choice A, the y-coordinate of point A,

00:01:39.670 --> 00:01:43.510
so point A is at the point four comma 120,

00:01:43.510 --> 00:01:45.570
so the y-coordinate is the 120.

00:01:45.570 --> 00:01:47.570
That's the total cost

00:01:47.570 --> 00:01:49.730
when you run your business for four hours.

00:01:49.730 --> 00:01:52.490
The y-coordinate of point
A represents the total cost

00:01:52.490 --> 00:01:55.300
of electricity when the business
operates for four hours.

00:01:55.300 --> 00:01:57.860
Yes, that is exactly
or very close (laughs)

00:01:57.860 --> 00:02:01.050
to what I just said, so I like this one.

00:02:01.050 --> 00:02:04.070
The total cost of electricity is $35

00:02:04.070 --> 00:02:06.800
when operating the business for one hour.

00:02:06.800 --> 00:02:08.810
So let's go to one hour here.

00:02:08.810 --> 00:02:11.370
This is going to be the total cost.

00:02:11.370 --> 00:02:15.720
Now, you might say, hey, this
looks kinda close to $35,

00:02:15.720 --> 00:02:17.310
but that's why it was useful for us

00:02:17.310 --> 00:02:19.690
to write this relationship
right over here,

00:02:19.690 --> 00:02:22.130
because what we see is that
our total cost is going

00:02:22.130 --> 00:02:25.550
to be 30 times our number of hours.

00:02:25.550 --> 00:02:28.690
Our total cost here is actually
going to be 30, not 35.

00:02:28.690 --> 00:02:30.260
And it actually does look smack dab

00:02:30.260 --> 00:02:33.960
in between 20 and 40 versus
a little bit closer to 40.

00:02:33.960 --> 00:02:36.580
So this one is not going to be true.

00:02:36.580 --> 00:02:38.060
And we're not gonna
select none of the above,

00:02:38.060 --> 00:02:40.230
'cause we actually did
select one of the above.

00:02:40.230 --> 00:02:41.063
And we're done.


Identifying proportional relationships from graphs

WEBVTT
Kind: captions
Language: en

00:00:00.230 --> 00:00:02.690
- [Narrator] We are asked
how many proportional

00:00:02.690 --> 00:00:06.340
relationships are shown in
the coordinate plane below?

00:00:06.340 --> 00:00:09.010
And we have the choices,
but let's actually look

00:00:09.010 --> 00:00:12.040
at the coordinate plane below,

00:00:12.040 --> 00:00:14.780
to think about how many
proportional relationships

00:00:14.780 --> 00:00:15.613
are depicted here.

00:00:15.613 --> 00:00:18.343
So pause this video and try
to answer that yourself.

00:00:19.640 --> 00:00:21.760
So let's do this together.

00:00:21.760 --> 00:00:24.520
So, if we're thinking about
a proportional relationship,

00:00:24.520 --> 00:00:27.080
or the graph of a
proportional relationship,

00:00:27.080 --> 00:00:29.390
there should be two things
that we're looking for.

00:00:29.390 --> 00:00:32.090
One, it should be a line.

00:00:32.090 --> 00:00:33.830
It should be a linear relationship

00:00:33.830 --> 00:00:35.580
between the two variables.

00:00:35.580 --> 00:00:38.170
Y should be some constant,

00:00:38.170 --> 00:00:42.050
some proportionality constant, times X.

00:00:42.050 --> 00:00:46.240
So you immediately would rule
out our green curve, here

00:00:46.240 --> 00:00:48.060
because this is not a line.

00:00:48.060 --> 00:00:50.120
You don't have a constant relationship

00:00:50.120 --> 00:00:53.970
between Y being some
proportionality constant times X.

00:00:57.070 --> 00:01:01.240
And for the same reason you
would rule out this blue curve.

00:01:01.240 --> 00:01:03.120
Now what about this purple line?

00:01:03.120 --> 00:01:06.600
This might be tempting
because it is a line,

00:01:06.600 --> 00:01:09.520
but it does not go through the origin.

00:01:09.520 --> 00:01:13.610
When X is two, Y is zero times X.

00:01:13.610 --> 00:01:18.240
While, when X is four, Y is one times X.

00:01:18.240 --> 00:01:23.240
And when X is six, Y looks
to be, 1 and 1/3 times X.

00:01:25.100 --> 00:01:27.600
So you don't have the same
proportionality constant

00:01:27.600 --> 00:01:28.780
the entire time.

00:01:28.780 --> 00:01:32.720
So, we have zero proportional
relationships depicted here.

00:01:32.720 --> 00:01:35.560
So I would pick zero there.

00:01:35.560 --> 00:01:37.433
Let's do one more example.

00:01:38.440 --> 00:01:40.890
Natalie is an expert archer.

00:01:40.890 --> 00:01:43.460
The following table
shows her scores, points,

00:01:43.460 --> 00:01:45.890
based on the number of targets she hits.

00:01:45.890 --> 00:01:49.150
All right, targets hit
and then points she gets.

00:01:49.150 --> 00:01:51.430
Plot the ordered pairs from the table.

00:01:51.430 --> 00:01:55.130
All right, so the first one is 1, 3.

00:01:55.130 --> 00:01:56.930
So here I'm doing it on Khan Academy.

00:01:56.930 --> 00:01:59.660
My horizontal axis is targets hit,

00:01:59.660 --> 00:02:02.400
and my vertical axis is points.

00:02:02.400 --> 00:02:05.240
So, one target hit, three points.

00:02:05.240 --> 00:02:08.400
So this is going to be one target hit,

00:02:08.400 --> 00:02:10.283
this is going to be three points.

00:02:11.280 --> 00:02:14.840
Then I have two targets hit, six points.

00:02:14.840 --> 00:02:19.510
So two targets hit, and I have six points.

00:02:19.510 --> 00:02:24.053
And then I'm gonna have
five targets hit, 15 points.

00:02:24.940 --> 00:02:29.940
So then I'm going to
have five targets hit,

00:02:31.200 --> 00:02:35.240
and that is going to be 15 points.

00:02:35.240 --> 00:02:38.210
And so this is looking like
a proportional relationship.

00:02:38.210 --> 00:02:42.000
In every situation my point is equal to

00:02:42.000 --> 00:02:44.460
three times the targets hit.

00:02:44.460 --> 00:02:46.820
So my proportionality constant is three.

00:02:46.820 --> 00:02:48.920
And you can see if you
try to connect these dots

00:02:48.920 --> 00:02:51.090
with a line, it will be a line.

00:02:51.090 --> 00:02:53.400
A line can go through all three of these,

00:02:53.400 --> 00:02:55.273
and it will go through the origin.

00:02:56.330 --> 00:02:59.460
So are Natalie's points
proportional to the number

00:02:59.460 --> 00:03:01.210
of targets she hit?

00:03:01.210 --> 00:03:02.783
Yes, absolutely.


Generalizabilty of survey results example

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:02.140
- [Instructor] Nkechi took a random sample

00:00:02.140 --> 00:00:04.340
of 10 countries to study fertility rate.

00:00:04.340 --> 00:00:06.160
And life expectancy.

00:00:06.160 --> 00:00:08.710
She noticed a strong
negative linear relationship

00:00:08.710 --> 00:00:10.980
between those variables
in the sample data.

00:00:10.980 --> 00:00:12.570
Here is computer output

00:00:12.570 --> 00:00:14.360
from a least-squares regression analysis

00:00:14.360 --> 00:00:16.260
for using fertility rate

00:00:16.260 --> 00:00:18.460
to predict life expectancy.

00:00:18.460 --> 00:00:21.480
Use this model to predict the
life expectancy of a country

00:00:21.480 --> 00:00:24.120
whose fertility rate is
two babies per woman.

00:00:24.120 --> 00:00:25.360
And you can round your answer

00:00:25.360 --> 00:00:26.750
to the nearest whole number of years.

00:00:26.750 --> 00:00:27.583
So pause this number

00:00:27.583 --> 00:00:28.416
and see if you can do it,

00:00:28.416 --> 00:00:30.420
you might need to use a calculator.

00:00:30.420 --> 00:00:32.320
All right now let's do this together.

00:00:32.320 --> 00:00:33.440
So in general,

00:00:33.440 --> 00:00:36.010
this computer output is actually
giving us a lot of data,

00:00:36.010 --> 00:00:38.640
more than we need actually,
to do this prediction.

00:00:38.640 --> 00:00:40.420
But it's giving us the data we need

00:00:40.420 --> 00:00:42.740
to know the equation
for a regression line.

00:00:42.740 --> 00:00:44.540
So the general form of a regression line,

00:00:44.540 --> 00:00:46.440
a linear regression line would be,

00:00:46.440 --> 00:00:47.590
our estimate,

00:00:47.590 --> 00:00:50.880
and that little hat means
we're estimating our y value,

00:00:50.880 --> 00:00:53.250
would be equal to our y-intercept

00:00:53.250 --> 00:00:56.420
plus our slope, times our x value.

00:00:56.420 --> 00:00:58.010
Now in this situation,

00:00:58.010 --> 00:01:02.700
we're using fertility to
predict life expectancy.

00:01:02.700 --> 00:01:05.100
Or let me circle all of life expectancy.

00:01:05.100 --> 00:01:06.920
So the thing that we're trying to predict,

00:01:06.920 --> 00:01:09.130
that is y, life expectancy.

00:01:09.130 --> 00:01:10.810
And fertility, is the thing

00:01:10.810 --> 00:01:12.060
that we're using to predict that.

00:01:12.060 --> 00:01:14.750
So that is going to be
our x, right over there.

00:01:14.750 --> 00:01:16.420
Now what are a and b?

00:01:16.420 --> 00:01:18.540
Well, our computer output gives us that.

00:01:18.540 --> 00:01:21.110
It's these numbers right over here.

00:01:21.110 --> 00:01:24.460
Our constant coefficient
right over here, this is a.

00:01:24.460 --> 00:01:28.743
And our slope, is going
to be negative 5.97.

00:01:29.710 --> 00:01:32.120
You could view it as the
coefficient on fertility.

00:01:32.120 --> 00:01:34.800
Remember, this right
over here, is fertility.

00:01:34.800 --> 00:01:35.800
You could even write,

00:01:35.800 --> 00:01:37.570
rewrite this as

00:01:37.570 --> 00:01:40.563
our estimated life expectancy,

00:01:41.560 --> 00:01:43.340
estimated life expectancy.

00:01:43.340 --> 00:01:44.370
I could put a little hat on it

00:01:44.370 --> 00:01:46.780
to show this is estimated life expectancy,

00:01:46.780 --> 00:01:51.750
is going to be equal to 89.70

00:01:51.750 --> 00:01:54.890
minus 5.97

00:01:54.890 --> 00:01:56.650
times fertility,

00:01:56.650 --> 00:01:58.560
times fertility rate.

00:01:58.560 --> 00:02:00.300
I'll just call it, say fert.

00:02:00.300 --> 00:02:01.990
And period, right over there.

00:02:01.990 --> 00:02:04.770
Notice, this is the
coefficient on fertility,

00:02:04.770 --> 00:02:06.860
and then this is the constant coefficient.

00:02:06.860 --> 00:02:08.970
We could do that right over there.

00:02:08.970 --> 00:02:13.520
And now, we can use this to
estimate the life expectancy

00:02:13.520 --> 00:02:16.900
of a country whose fertility
rate is two babies per woman.

00:02:16.900 --> 00:02:18.870
For fertility, you just put a two here.

00:02:18.870 --> 00:02:21.460
And then you get your
estimated life expectancy.

00:02:21.460 --> 00:02:22.570
So what's that going to be?

00:02:22.570 --> 00:02:23.970
We can get out a calculator.

00:02:25.270 --> 00:02:27.220
So we can say,

00:02:27.220 --> 00:02:29.330
5.97

00:02:29.330 --> 00:02:31.910
times two is equal to that

00:02:31.910 --> 00:02:34.120
and then we wanna subtract that from,

00:02:34.120 --> 00:02:35.660
so put in a negative there,

00:02:35.660 --> 00:02:39.700
and add that to 89.7

00:02:39.700 --> 00:02:40.760
is equal to,

00:02:40.760 --> 00:02:42.810
and we wanna round to the
nearest whole number of years,

00:02:42.810 --> 00:02:45.880
so that's approximately 78 years.

00:02:45.880 --> 00:02:47.570
So this is approximately

00:02:47.570 --> 00:02:49.750
78 years.

00:02:49.750 --> 00:02:50.970
And we're done.

00:02:50.970 --> 00:02:53.680
And just to be clear
what even happened here,

00:02:53.680 --> 00:02:56.253
is that Nkechi, she did a regression,

00:02:57.090 --> 00:03:00.800
on the x-axis with fertility,

00:03:00.800 --> 00:03:01.633
fertility,

00:03:01.633 --> 00:03:03.100
on the y-axis is

00:03:03.100 --> 00:03:06.080
let's call it l period dot e period.

00:03:06.080 --> 00:03:07.600
That's our y-axis.

00:03:07.600 --> 00:03:09.000
Took 10 data points,

00:03:09.000 --> 00:03:13.070
one, two, three, four, five,
six, seven, eight, nine, 10.

00:03:13.070 --> 00:03:14.320
Put a regression line on,

00:03:14.320 --> 00:03:15.153
try to fit

00:03:15.153 --> 00:03:17.740
try to fit a regression line.

00:03:17.740 --> 00:03:20.200
Saw a negative linear relationship,

00:03:20.200 --> 00:03:22.950
and then using this
regression line to estimate,

00:03:22.950 --> 00:03:24.560
hey, if fertility is,

00:03:24.560 --> 00:03:26.750
let's say this is two right over here,

00:03:26.750 --> 00:03:29.620
what is the estimated life expectancy?

00:03:29.620 --> 00:03:32.503
And we just saw that that
would be roughly 78 years.


Visually assessing standard deviation

WEBVTT
Kind: captions
Language: en

00:00:00.270 --> 00:00:01.790
- [Instructor] Each dot plot below

00:00:01.790 --> 00:00:03.920
represents a different set of data.

00:00:03.920 --> 00:00:05.160
We see that here.

00:00:05.160 --> 00:00:08.420
Order the dot plots from
largest standard deviation,

00:00:08.420 --> 00:00:12.960
top, to smallest standard
deviation, bottom.

00:00:12.960 --> 00:00:14.810
So, pause this video and
see if you can do that

00:00:14.810 --> 00:00:16.360
or at least if you could rank these

00:00:16.360 --> 00:00:18.750
from largest standard deviation

00:00:18.750 --> 00:00:20.543
to smallest standard deviation.

00:00:21.400 --> 00:00:22.780
All right, now, let's
work through this together

00:00:22.780 --> 00:00:24.150
and I'm doing this on Khan Academy

00:00:24.150 --> 00:00:26.550
where I can move these
around to order them,

00:00:26.550 --> 00:00:28.060
but let's just remind ourselves

00:00:28.060 --> 00:00:29.650
what the standard deviation is

00:00:29.650 --> 00:00:31.210
or how we can perceive it.

00:00:31.210 --> 00:00:33.760
You could view the standard
deviation as a measure

00:00:33.760 --> 00:00:35.450
of the typical distance

00:00:35.450 --> 00:00:39.020
from each of the data points to the mean.

00:00:39.020 --> 00:00:41.880
So, the largest standard deviation,

00:00:41.880 --> 00:00:43.230
which you want to put on top,

00:00:43.230 --> 00:00:46.320
would be the one where
typically our data points

00:00:46.320 --> 00:00:47.990
are further from the mean

00:00:47.990 --> 00:00:49.910
and our smallest standard deviation

00:00:49.910 --> 00:00:52.100
would be the ones where it feels like,

00:00:52.100 --> 00:00:54.881
on average, our data points
are closer to the mean.

00:00:54.881 --> 00:00:56.950
Well, in all of these examples,

00:00:56.950 --> 00:01:00.210
our mean looks to be right in the center,

00:01:00.210 --> 00:01:03.900
right between 50 and
100, so right around 75.

00:01:03.900 --> 00:01:08.200
So, it's really about how
spread apart they are from that.

00:01:08.200 --> 00:01:09.790
And if you look at this first one,

00:01:09.790 --> 00:01:12.920
it has these two data points,

00:01:12.920 --> 00:01:14.550
one on the left and one on the right,

00:01:14.550 --> 00:01:16.130
that are pretty far, and then you have

00:01:16.130 --> 00:01:17.420
these two that are a little bit closer,

00:01:17.420 --> 00:01:19.860
and then these two that are inside.

00:01:19.860 --> 00:01:21.570
This one right over here,

00:01:21.570 --> 00:01:24.100
to get from this top
one to this middle one

00:01:24.100 --> 00:01:26.140
you essentially are taking this data point

00:01:26.140 --> 00:01:27.880
and making it go further

00:01:27.880 --> 00:01:31.110
and taking this data point
and making it go further

00:01:31.110 --> 00:01:34.500
and so this one is going to
have a higher standard deviation

00:01:34.500 --> 00:01:37.730
than that one, so let me
put it just like that.

00:01:37.730 --> 00:01:39.580
And I just want to make it very clear,

00:01:39.580 --> 00:01:41.460
keep track of what's the difference

00:01:41.460 --> 00:01:43.510
between these two things.

00:01:43.510 --> 00:01:46.360
Here, you have this data
point and this data point

00:01:46.360 --> 00:01:49.680
that was closer in and then
if you move it further,

00:01:49.680 --> 00:01:51.540
that's going to make your typical distance

00:01:51.540 --> 00:01:55.410
from the middle more, which is
exactly what happened there.

00:01:55.410 --> 00:01:57.110
Now, what about this one?

00:01:57.110 --> 00:02:01.310
Well, this one is starting here

00:02:01.310 --> 00:02:04.270
and then taking this point
and taking this point

00:02:04.270 --> 00:02:06.900
and moving it closer
and so that would make

00:02:06.900 --> 00:02:09.080
our typical distance from the middle,

00:02:09.080 --> 00:02:10.870
from the mean, shorter,

00:02:10.870 --> 00:02:13.200
so this would have the
smallest standard deviation

00:02:13.200 --> 00:02:15.200
and this would have the largest.

00:02:15.200 --> 00:02:16.963
Let's do another example.

00:02:18.280 --> 00:02:20.490
So, same idea, order the dot plots

00:02:20.490 --> 00:02:22.420
from largest standard deviation on the top

00:02:22.420 --> 00:02:24.300
to smallest standard
deviation on the bottom.

00:02:24.300 --> 00:02:27.330
Pause this video and see
if you can figure that out.

00:02:27.330 --> 00:02:28.310
So, this is interesting

00:02:28.310 --> 00:02:30.130
because these all have different means.

00:02:30.130 --> 00:02:32.420
Just eyeballing it, the
mean for this first one

00:02:32.420 --> 00:02:34.800
is right around here, the
mean for the second one

00:02:34.800 --> 00:02:36.560
is right around here, at around 10,

00:02:36.560 --> 00:02:38.200
and the mean for the third one,

00:02:38.200 --> 00:02:40.890
it looks like the same
mean as this top one.

00:02:40.890 --> 00:02:42.860
And so, pause this video.

00:02:42.860 --> 00:02:44.110
How would you order them?

00:02:45.560 --> 00:02:49.030
All right, so just eyeballing it,

00:02:49.030 --> 00:02:51.800
these, this middle one right over here,

00:02:51.800 --> 00:02:54.660
your typical data point seems furthest

00:02:54.660 --> 00:02:56.570
from the mean, you definitely have,

00:02:56.570 --> 00:03:00.150
if the mean is here,
you have this data point

00:03:00.150 --> 00:03:01.990
and this data point that are quite far

00:03:01.990 --> 00:03:04.630
from that mean, and even this data point

00:03:04.630 --> 00:03:07.030
and this data point are at least as far

00:03:07.030 --> 00:03:08.810
as any of the data points that we have

00:03:08.810 --> 00:03:09.900
in the top or the bottom one,

00:03:09.900 --> 00:03:13.100
so, I would say this has the
largest standard deviation

00:03:13.100 --> 00:03:17.063
and if I were to compare
between these two,

00:03:18.040 --> 00:03:20.530
if you think about how you
would get the difference

00:03:20.530 --> 00:03:23.650
between these two, is if
you took this data point

00:03:23.650 --> 00:03:28.320
and moved it to the mean and
if you took this data point

00:03:28.320 --> 00:03:30.140
and you moved it to the mean,

00:03:30.140 --> 00:03:32.720
you would get this third situation.

00:03:32.720 --> 00:03:36.830
And so, this third situation
you have the fewest data points

00:03:36.830 --> 00:03:38.610
that are sitting away from the mean

00:03:38.610 --> 00:03:39.890
relative to this one.

00:03:39.890 --> 00:03:41.880
And so, I actually like this ordering

00:03:41.880 --> 00:03:44.250
that this top one has the
largest standard deviation

00:03:44.250 --> 00:03:47.983
and this bottom one has the
smallest standard deviation.


Estimating mean and median in data displays

WEBVTT
Kind: captions
Language: en

00:00:00.160 --> 00:00:03.750
- [Instructor] We are told
researchers scored 31 athletes

00:00:03.750 --> 00:00:05.340
on an agility test.

00:00:05.340 --> 00:00:07.160
Here are their scores.

00:00:07.160 --> 00:00:08.860
It's in this histogram.

00:00:08.860 --> 00:00:10.710
And what I'm going to ask you is,

00:00:10.710 --> 00:00:14.249
which of these intervals,
interval A, B, or C,

00:00:14.249 --> 00:00:18.520
which one contains the
median of the scores,

00:00:18.520 --> 00:00:21.890
and which one, or give an
estimate which one contains

00:00:21.890 --> 00:00:24.250
the mean of the scores.

00:00:24.250 --> 00:00:27.310
Pause this video and see
if you can figure that out.

00:00:27.310 --> 00:00:29.870
So let's just start with the median.

00:00:29.870 --> 00:00:33.140
Remember, the median you could
view as the middle number,

00:00:33.140 --> 00:00:35.750
or if you have an even
number of data points,

00:00:35.750 --> 00:00:37.970
it would be the average of the middle two.

00:00:37.970 --> 00:00:40.330
Here we have an odd number of data points,

00:00:40.330 --> 00:00:42.440
so it would be the middle number.

00:00:42.440 --> 00:00:44.060
So what would be the middle number

00:00:44.060 --> 00:00:46.897
if you were to order them
from least to greatest?

00:00:46.897 --> 00:00:50.540
Well, it would be the one
that has 15 on either side,

00:00:50.540 --> 00:00:54.490
so it would be the 16th data point.

00:00:54.490 --> 00:00:58.110
16th data point.

00:00:58.110 --> 00:01:00.350
And so we could just think
about which interval here

00:01:00.350 --> 00:01:02.640
contains the 16th data point.

00:01:02.640 --> 00:01:04.590
You could view it for
the 16th from the highest

00:01:04.590 --> 00:01:06.100
or the 16th from the lowest.

00:01:06.100 --> 00:01:07.830
It is the middle one.

00:01:07.830 --> 00:01:09.350
All right, so let's
start from the highest.

00:01:09.350 --> 00:01:14.350
So this interval C contains
the 13 highest data points

00:01:15.860 --> 00:01:20.860
and then interval B goes
from the 14th highest

00:01:22.210 --> 00:01:25.160
all the way to the 18th highest.

00:01:25.160 --> 00:01:27.860
So this B contains the median.

00:01:27.860 --> 00:01:30.780
It contains the 16th highest data point,

00:01:30.780 --> 00:01:32.040
or if you started from the left,

00:01:32.040 --> 00:01:34.370
it would also be the
16th lowest data point.

00:01:34.370 --> 00:01:36.680
So that's where the median is.

00:01:36.680 --> 00:01:38.260
The median.

00:01:38.260 --> 00:01:41.590
Now what about an estimate for the mean?

00:01:41.590 --> 00:01:44.300
Well, you have calculated
the mean in the past,

00:01:44.300 --> 00:01:46.590
but when you're looking at
a distribution like this,

00:01:46.590 --> 00:01:48.400
when you're looking at a histogram,

00:01:48.400 --> 00:01:50.470
one way to think about the mean is

00:01:50.470 --> 00:01:52.230
it would be the balancing point.

00:01:52.230 --> 00:01:54.460
If you imagine that this
histogram was made out of

00:01:54.460 --> 00:01:58.000
some material of, let's
say, uniform density,

00:01:58.000 --> 00:02:00.876
where would you put a fulcrum
in order to balance it?

00:02:00.876 --> 00:02:03.337
If you put the fulcrum right over here,

00:02:03.337 --> 00:02:08.214
it feels like you would
tip over to the left

00:02:08.214 --> 00:02:11.320
because this is a
left-skewed distribution.

00:02:11.320 --> 00:02:13.890
You have this long tail to the left.

00:02:13.890 --> 00:02:15.960
If you really wanted to balance it out,

00:02:15.960 --> 00:02:18.631
it seems like you would
have to move your fulcrum

00:02:18.631 --> 00:02:22.090
in that direction of that left skew,

00:02:22.090 --> 00:02:23.720
in the direction of the tail.

00:02:23.720 --> 00:02:25.900
And so I would estimate to balance it out,

00:02:25.900 --> 00:02:27.920
it would actually be closer to that,

00:02:27.920 --> 00:02:29.840
which would be interval A.

00:02:29.840 --> 00:02:32.934
Interval A would contain the mean.

00:02:32.934 --> 00:02:35.040
The intention of this type of exercise

00:02:35.040 --> 00:02:37.900
isn't for you to try to
calculate every data point,

00:02:37.900 --> 00:02:40.150
in fact, they don't give you
all of the information here,

00:02:40.150 --> 00:02:42.760
and add them all up and then divide by 31.

00:02:42.760 --> 00:02:46.126
It's really to estimate and
to also get the intuition

00:02:46.126 --> 00:02:49.460
that when you have a left-skewed
distribution like this,

00:02:49.460 --> 00:02:53.070
you will often see a
situation where your mean

00:02:53.070 --> 00:02:55.430
is to the left of the median.

00:02:55.430 --> 00:02:57.050
If you have a right-skewed distribution,

00:02:57.050 --> 00:02:58.240
it would be the other way around.

00:02:58.240 --> 00:03:01.219
And as we will see, when we
see a symmetric distribution,

00:03:01.219 --> 00:03:04.670
the mean and the median will
be awfully close to each other,

00:03:04.670 --> 00:03:06.600
or when you have a roughly
symmetric distribution.

00:03:06.600 --> 00:03:08.440
If you have a perfectly
symmetric distribution,

00:03:08.440 --> 00:03:11.260
they might be exactly in the same place.

00:03:11.260 --> 00:03:13.155
So let's do another example.

00:03:13.155 --> 00:03:18.155
So here it says we have
the ages of 14 coworkers,

00:03:18.918 --> 00:03:23.243
and I want you to do is say
roughly where is the mean

00:03:23.243 --> 00:03:25.540
and roughly where is the median?

00:03:25.540 --> 00:03:28.410
Is it roughly at A, is it roughly at B,

00:03:28.410 --> 00:03:29.560
or its it roughly at C?

00:03:29.560 --> 00:03:31.660
Pause this video and try to figure it out.

00:03:32.700 --> 00:03:36.093
So let's first start off with the median.

00:03:36.093 --> 00:03:38.710
We have 14 data points.

00:03:38.710 --> 00:03:42.250
So this would be the average
of the middle two data points.

00:03:42.250 --> 00:03:45.210
It would be the average of the seventh

00:03:45.210 --> 00:03:46.880
and eighth data point.

00:03:46.880 --> 00:03:50.020
Well, you could say one, two,
three, four, five, six, seven,

00:03:50.020 --> 00:03:51.290
and then the eighth one is here.

00:03:51.290 --> 00:03:53.670
So the seventh data point is a 30.

00:03:53.670 --> 00:03:56.250
The eighth one is in the 31 bucket.

00:03:56.250 --> 00:03:59.420
So the average of the
two would get you to B.

00:03:59.420 --> 00:04:01.301
Another way that you
could think about it is

00:04:01.301 --> 00:04:03.670
you can just eyeball it and see you have

00:04:03.670 --> 00:04:06.200
just as many data points below B

00:04:06.200 --> 00:04:10.270
as you do have above B,
and so that also gives you

00:04:10.270 --> 00:04:14.150
a good indication that B
would be where the median is.

00:04:14.150 --> 00:04:16.340
So that is where the median is.

00:04:16.340 --> 00:04:17.773
Now what about the mean?

00:04:18.790 --> 00:04:21.719
Well, this is a perfectly
symmetric distribution.

00:04:21.719 --> 00:04:23.370
If I wanted to balance it,

00:04:23.370 --> 00:04:26.328
I would put the fulcrum
right in the middle.

00:04:26.328 --> 00:04:29.317
So I would say that the mean would also,

00:04:29.317 --> 00:04:33.973
the mean would also be
at B, and we are done.


Median in a histogram

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:02.660
- [Narrator] Miguel tracked
how much sleep he got

00:00:02.660 --> 00:00:07.660
for 50 consecutive days and
made a histogram of the results.

00:00:07.690 --> 00:00:11.730
Which interval contains
the median sleep amount?

00:00:11.730 --> 00:00:14.400
And so they're saying is it
this interval on the histogram

00:00:14.400 --> 00:00:18.530
from six to 6.5, or this one
or this one, or any of these.

00:00:18.530 --> 00:00:20.680
Which of these intervals
contain the median.

00:00:20.680 --> 00:00:23.570
Pause this video and see
if you can figure that out.

00:00:23.570 --> 00:00:24.970
All right now let's work
through this together.

00:00:24.970 --> 00:00:28.050
And let's just remind ourselves
how we find the median.

00:00:28.050 --> 00:00:33.050
If I had the data points 11,
nine, seven, three, and two,

00:00:33.790 --> 00:00:35.860
the way that we find the
median is we can order it

00:00:35.860 --> 00:00:37.680
from least to greatest or
actually you could do it

00:00:37.680 --> 00:00:39.640
from greatest to least, but
let's do least to greatest.

00:00:39.640 --> 00:00:43.170
So two, three, seven, nine, 11.

00:00:43.170 --> 00:00:44.790
And the median would be the middle number.

00:00:44.790 --> 00:00:46.030
And I have a clear middle number

00:00:46.030 --> 00:00:48.550
because I have five data points.

00:00:48.550 --> 00:00:51.080
If I have an even number of data points,

00:00:51.080 --> 00:00:53.300
I still would want to order
them from least to greatest,

00:00:53.300 --> 00:00:58.300
so let's say that I have a
one, one, three, and a seven.

00:00:58.800 --> 00:01:00.750
But here, you don't have a clear middle.

00:01:00.750 --> 00:01:04.900
So the median would be the
mean of the middle two numbers.

00:01:04.900 --> 00:01:06.230
So in this situation,

00:01:06.230 --> 00:01:08.900
Miguel has an even number of data points.

00:01:08.900 --> 00:01:10.900
So the median would be the mean

00:01:10.900 --> 00:01:15.900
of the 25th and 26th data point.

00:01:16.130 --> 00:01:18.460
These would be the middle two data points.

00:01:18.460 --> 00:01:20.220
So which interval here contains

00:01:20.220 --> 00:01:23.540
the 25th and the 26th data point?

00:01:23.540 --> 00:01:25.360
Well, we can start at the bottom.

00:01:25.360 --> 00:01:28.110
So we have, actually let's
just look at each interval

00:01:28.110 --> 00:01:29.750
and think about how many
data points they have in it.

00:01:29.750 --> 00:01:33.450
This one has two, this one has nine,

00:01:33.450 --> 00:01:36.980
this one has 12, and I'm just reading out

00:01:36.980 --> 00:01:39.300
how many data points there are
in each of these intervals.

00:01:39.300 --> 00:01:44.300
This one has 12, this one
has 11, see that there.

00:01:45.380 --> 00:01:47.343
This one has two and this one has two.

00:01:48.280 --> 00:01:50.630
So if we look at just this,
we have the two lowest,

00:01:50.630 --> 00:01:52.830
if we look at the two
bottom intervals combined

00:01:52.830 --> 00:01:54.790
we have the 11 lowest.

00:01:54.790 --> 00:01:57.970
If we look at the three bottom intervals,

00:01:57.970 --> 00:02:02.900
we have the 11 plus, 12,
you have the 23 lowest.

00:02:02.900 --> 00:02:07.600
So this is the 23 lowest data points.

00:02:07.600 --> 00:02:12.600
And so the 24th, 25th, 26th, the next 12

00:02:13.580 --> 00:02:15.410
data points starting from the bottom,

00:02:15.410 --> 00:02:16.850
starting from the lowest are going to be

00:02:16.850 --> 00:02:18.640
in this next interval here.

00:02:18.640 --> 00:02:21.250
And we care about the 25th and the 26th,

00:02:21.250 --> 00:02:23.660
so they're definitely going
to be in this interval

00:02:23.660 --> 00:02:27.563
from 7.5 hours of sleep
to eight hours of sleep.


Constant of proportionality from graph

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:01.870
- [Instructor] The following graph

00:00:01.870 --> 00:00:04.220
shows a proportional relationship.

00:00:04.220 --> 00:00:07.110
What is the constant of proportionality

00:00:07.110 --> 00:00:10.730
between y and x in the graph?

00:00:10.730 --> 00:00:13.750
Pause this video, and see
if you can figure that out.

00:00:13.750 --> 00:00:15.210
All right, now let's do this together,

00:00:15.210 --> 00:00:16.500
and let's remind ourselves

00:00:16.500 --> 00:00:20.630
what a constant of
proportionality even is.

00:00:20.630 --> 00:00:23.150
If we know that there is a
proportional relationship

00:00:23.150 --> 00:00:26.600
between y and x, then
there will be a constant

00:00:26.600 --> 00:00:28.950
of proportionality
between these variables,

00:00:28.950 --> 00:00:32.270
and what this is, is it is a number

00:00:32.270 --> 00:00:36.860
that I would have to
multiply x by to get to y.

00:00:36.860 --> 00:00:38.600
So I could make a little table here,

00:00:38.600 --> 00:00:42.080
as we often do when we describe
proportional relationships,

00:00:42.080 --> 00:00:43.470
x and y.

00:00:43.470 --> 00:00:46.053
We know that when x is zero, y is zero.

00:00:47.440 --> 00:00:50.660
But if I multiply zero by
anything, I'm gonna get zero.

00:00:50.660 --> 00:00:52.950
But then when x is one, what is y?

00:00:52.950 --> 00:00:55.980
When x is one, y is three.

00:00:55.980 --> 00:00:57.940
They mark it right over there.

00:00:57.940 --> 00:01:00.570
When x is two, what is y?

00:01:00.570 --> 00:01:02.070
X is two,

00:01:02.070 --> 00:01:05.470
we see that y is six.

00:01:05.470 --> 00:01:07.480
So our constant of proportionality is

00:01:07.480 --> 00:01:10.283
what are we multiplying x by to get to y?

00:01:11.130 --> 00:01:12.900
Well, let's see, to go from one to three,

00:01:12.900 --> 00:01:14.860
I have to multiply by three.

00:01:14.860 --> 00:01:19.540
To go from two to six, I
have to multiply by three.

00:01:19.540 --> 00:01:21.140
Another way to think about it is

00:01:21.140 --> 00:01:25.150
we could write the equation y is equal to

00:01:25.150 --> 00:01:27.190
something times x.

00:01:27.190 --> 00:01:30.530
The number that we multiply x by to get y

00:01:30.530 --> 00:01:33.340
is our constant of proportionality.

00:01:33.340 --> 00:01:35.730
And we've seen, in all
of these situations,

00:01:35.730 --> 00:01:38.360
this should be true for
any point on this line.

00:01:38.360 --> 00:01:41.660
You give the x, you multiply
it by three, you get your y.

00:01:41.660 --> 00:01:45.620
So the relationship here
is y is equal to three x.

00:01:45.620 --> 00:01:48.863
So three is our constant
of proportionality.


Categorical grants, mandates, and the Commerce Clause

WEBVTT
Kind: captions
Language: en

00:00:00.430 --> 00:00:01.590
- [Instructor] In a previous video,

00:00:01.590 --> 00:00:05.240
we've introduced ourselves
to the idea of federalism

00:00:05.240 --> 00:00:06.890
in the United States.

00:00:06.890 --> 00:00:09.950
At a high level, you could
view it as a contract

00:00:09.950 --> 00:00:11.790
between a national government

00:00:11.790 --> 00:00:14.390
and the states of which it is made,

00:00:14.390 --> 00:00:17.710
but you could also view it as
a layered form of government

00:00:17.710 --> 00:00:19.600
where you have your local government

00:00:19.600 --> 00:00:21.810
and then layered on top of
that your state government

00:00:21.810 --> 00:00:24.080
and then layered on top of
that your national government,

00:00:24.080 --> 00:00:26.400
often referred to as
the federal government.

00:00:26.400 --> 00:00:29.060
And we looked at the
example of a layered cake

00:00:29.060 --> 00:00:31.322
but have said that over the
course of American history,

00:00:31.322 --> 00:00:33.730
the layers have gotten
more and more mixed,

00:00:33.730 --> 00:00:35.580
more like a marbled cake.

00:00:35.580 --> 00:00:39.760
And it's been mixed more in
the favor of the national

00:00:39.760 --> 00:00:41.580
or the federal government,

00:00:41.580 --> 00:00:46.140
even though certain powers were
historically more associated

00:00:46.140 --> 00:00:49.720
with the states, as we will
see, there are several levers

00:00:49.720 --> 00:00:51.630
that the federal government has used

00:00:51.630 --> 00:00:54.950
in order to extend its
power into the domain

00:00:54.950 --> 00:00:58.400
of what used to be associated
with primarily the states.

00:00:58.400 --> 00:01:02.580
So one is the notion
of categorical grants.

00:01:02.580 --> 00:01:06.320
So these are grants for a specific purpose

00:01:06.320 --> 00:01:08.780
where the federal government
says, hey, states,

00:01:08.780 --> 00:01:10.260
we're gonna give you some money,

00:01:10.260 --> 00:01:11.660
but you've gotta use this money

00:01:11.660 --> 00:01:14.070
in exactly the way that we're telling you.

00:01:14.070 --> 00:01:17.840
Now, to be clear, not all
grants are categorical grants.

00:01:17.840 --> 00:01:20.060
You have things like block grants

00:01:20.060 --> 00:01:22.830
where the federal
government can give a grant

00:01:22.830 --> 00:01:24.020
to a state and say, hey, use this

00:01:24.020 --> 00:01:26.960
to generally improve the
safety of your citizens.

00:01:26.960 --> 00:01:29.690
That would still give a
lotta leeway to the states.

00:01:29.690 --> 00:01:32.600
But in categorical
grants, it's very specific

00:01:32.600 --> 00:01:36.140
in terms of how the states
are to use that money,

00:01:36.140 --> 00:01:38.010
even if historically it was something

00:01:38.010 --> 00:01:40.140
where the states had the powers.

00:01:40.140 --> 00:01:43.010
So an example of this would
be the federal program,

00:01:43.010 --> 00:01:45.870
the Special Supplemental
Nutrition Assistance Program

00:01:45.870 --> 00:01:48.557
for Women, Infants, and Children, or WIC.

00:01:49.750 --> 00:01:52.680
And to get a idea of
how prescriptive it is,

00:01:52.680 --> 00:01:56.170
here is an outline of the
program on the USDA website,

00:01:56.170 --> 00:01:58.580
a federal government agency.

00:01:58.580 --> 00:02:01.650
And if we go down here,
you can even see things

00:02:01.650 --> 00:02:03.280
like income requirements,

00:02:03.280 --> 00:02:06.070
and they'll be income
eligibility guidelines.

00:02:06.070 --> 00:02:09.903
These are set by the federal
government, not by the states.

00:02:11.060 --> 00:02:15.520
Along those lines, you also have mandates.

00:02:15.520 --> 00:02:19.130
So a mandate is the federal
government tying funding

00:02:19.130 --> 00:02:24.130
to one thing based on state
compliance with another thing.

00:02:24.190 --> 00:02:27.590
For example, the National
Minimum Drinking Age Act,

00:02:27.590 --> 00:02:32.590
which was passed in 1984,
ties federal highway funds

00:02:32.770 --> 00:02:37.600
to states raising their
minimum drinking age to 21.

00:02:37.600 --> 00:02:40.280
And I had direct experience with this act

00:02:40.280 --> 00:02:42.270
when I was growing up in Louisiana.

00:02:42.270 --> 00:02:44.170
Louisiana decided not to comply

00:02:44.170 --> 00:02:46.900
by the National Minimum Drinking Age Act,

00:02:46.900 --> 00:02:50.260
so the drinking age was 18,
but because they didn't comply,

00:02:50.260 --> 00:02:52.830
they weren't getting as much
federal funding for highways,

00:02:52.830 --> 00:02:55.610
and the highways weren't
as good as in other states.

00:02:55.610 --> 00:02:57.510
So even though the
drinking age is something

00:02:57.510 --> 00:02:59.740
that might be considered a state power,

00:02:59.740 --> 00:03:02.410
the federal government
was able to exercise a lot

00:03:02.410 --> 00:03:04.080
of influence on most states

00:03:04.080 --> 00:03:06.970
by tying what the
federal government wanted

00:03:06.970 --> 00:03:09.280
to highway funds.

00:03:09.280 --> 00:03:11.380
Now, outside of these examples

00:03:11.380 --> 00:03:14.820
of the federal government
tying state funds

00:03:14.820 --> 00:03:18.520
to the states doing what the
federal government wants,

00:03:18.520 --> 00:03:20.690
the federal government has also made use

00:03:20.690 --> 00:03:24.990
of the U.S. Constitution in
order to broaden its powers.

00:03:24.990 --> 00:03:29.170
In particular, the Commerce
Clause, Article I, Section 8.

00:03:29.170 --> 00:03:30.500
You might remember, that's the part

00:03:30.500 --> 00:03:33.250
where they say the
Congress shall have power,

00:03:33.250 --> 00:03:36.320
and then they list a bunch of powers,

00:03:36.320 --> 00:03:37.860
but one of them, the Commerce Clause,

00:03:37.860 --> 00:03:40.410
is to regulate Commerce
with foreign Nations,

00:03:40.410 --> 00:03:44.380
and among the several States,
and with the Indian Tribes.

00:03:44.380 --> 00:03:47.110
And the key part of the Commerce Clause is

00:03:47.110 --> 00:03:49.470
among the several States.

00:03:49.470 --> 00:03:51.270
Over the course of American history,

00:03:51.270 --> 00:03:54.890
this ability to regulate
interstate commerce,

00:03:54.890 --> 00:03:56.510
commerce between states,

00:03:56.510 --> 00:03:59.000
the federal government has used that

00:03:59.000 --> 00:04:01.580
to justify regulations and laws

00:04:01.580 --> 00:04:04.750
that focus on issues that
may at first be perceived

00:04:04.750 --> 00:04:08.580
as a state power but use the argument

00:04:08.580 --> 00:04:12.320
that it affects interstate
commerce in order to regulate it.

00:04:12.320 --> 00:04:13.620
And as you can imagine,

00:04:13.620 --> 00:04:16.180
when you have free-flowing
commerce between states,

00:04:16.180 --> 00:04:17.500
you have the same currency,

00:04:17.500 --> 00:04:19.300
you don't have tariffs between states,

00:04:19.300 --> 00:04:21.580
many things that you
would traditionally view

00:04:21.580 --> 00:04:25.470
as the power of the
state, one could argue,

00:04:25.470 --> 00:04:29.700
would have some influence
on interstate commerce.

00:04:29.700 --> 00:04:33.890
One notable example of this
would be federal drug laws

00:04:33.890 --> 00:04:38.130
where a state could decide
to, say, legalize marijuana,

00:04:38.130 --> 00:04:40.870
but the federal government
can make it pretty difficult

00:04:40.870 --> 00:04:44.330
by regulating how is that
marijuana transported?

00:04:44.330 --> 00:04:47.940
Or where does the cash for
that marijuana get deposited?

00:04:47.940 --> 00:04:49.940
Does it get deposited in a bank

00:04:49.940 --> 00:04:52.370
that has associations
with the Federal Reserve

00:04:52.370 --> 00:04:55.420
that needs to transfer that
money across state lines?

00:04:55.420 --> 00:04:58.590
So the Interstate Commerce
Clause has more influence

00:04:58.590 --> 00:05:01.853
on state affairs than you
might initially think.


Executive orders

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.294
- [Lecturer] What we're going
to discuss in this video

00:00:02.294 --> 00:00:05.370
is Executive Orders.

00:00:05.370 --> 00:00:07.850
And these are directives being issued

00:00:07.850 --> 00:00:10.170
by the President of the United States

00:00:10.170 --> 00:00:12.620
that can have the force of law.

00:00:12.620 --> 00:00:14.200
And I know what you're thinking,

00:00:14.200 --> 00:00:16.590
isn't Congress our legislative body,

00:00:16.590 --> 00:00:19.260
the body that actually creates the laws?

00:00:19.260 --> 00:00:22.110
And is it the job of the President to be

00:00:22.110 --> 00:00:26.340
the head of the Executive
Branch to execute on those laws?

00:00:26.340 --> 00:00:28.680
And if you are thinking
that, you are correct.

00:00:28.680 --> 00:00:31.410
But going all the way
back to George Washington,

00:00:31.410 --> 00:00:34.220
Presidents have issued Executive Orders.

00:00:34.220 --> 00:00:36.130
Some are fairly lightweight.

00:00:36.130 --> 00:00:38.970
They might be a directive
for something to be done

00:00:38.970 --> 00:00:41.860
in a certain way, or a small regulation,

00:00:41.860 --> 00:00:44.540
or even appointing someone to a job.

00:00:44.540 --> 00:00:46.590
But sometimes these Executive Orders

00:00:46.590 --> 00:00:48.350
can be quite significant.

00:00:48.350 --> 00:00:52.060
And the constitutional
justification for these Orders

00:00:52.060 --> 00:00:56.370
come mainly from two different
statements in Article II

00:00:56.370 --> 00:00:58.560
of the U.S. Constitution.

00:00:58.560 --> 00:01:00.747
In Section One, it starts off saying,

00:01:00.747 --> 00:01:03.747
"The Executive Power shall
be vested in a President

00:01:03.747 --> 00:01:05.970
"of the United States of America."

00:01:05.970 --> 00:01:07.727
And at the end of Section
Three, it says that,

00:01:07.727 --> 00:01:11.017
"the President shall
take care that the laws

00:01:11.017 --> 00:01:13.370
"be faithfully executed."

00:01:13.370 --> 00:01:16.550
And so, the clearest justification
for an Executive Order

00:01:16.550 --> 00:01:20.550
are times when a regulation
is needed or a directive

00:01:20.550 --> 00:01:23.990
is needed in order to
faithfully execute the laws.

00:01:23.990 --> 00:01:27.000
But as we will see,
Presidents throughout history

00:01:27.000 --> 00:01:29.760
have really pushed the boundary
here and have definitely

00:01:29.760 --> 00:01:32.470
gone into territory that you might expect

00:01:32.470 --> 00:01:35.450
to be the area of Congress.

00:01:35.450 --> 00:01:38.010
Now in terms of appreciating how frequent

00:01:38.010 --> 00:01:39.930
these Executive Orders are,

00:01:39.930 --> 00:01:43.400
here is a chart that gives
the average Executive Orders

00:01:43.400 --> 00:01:46.860
per year for the last few Presidents.

00:01:46.860 --> 00:01:48.260
And you can see at the low end,

00:01:48.260 --> 00:01:51.870
you have President Obama
and President George W. Bush

00:01:51.870 --> 00:01:55.870
averaging about 35 or 36
Executive Orders per year.

00:01:55.870 --> 00:01:57.510
And then all the way at the high end,

00:01:57.510 --> 00:02:00.110
you get to Gerald Ford and Jimmy Carter.

00:02:00.110 --> 00:02:02.160
Now if you were to go even
further back in history,

00:02:02.160 --> 00:02:05.190
Presidents like Franklin
Roosevelt had far more

00:02:05.190 --> 00:02:07.670
Executive Orders than even this.

00:02:07.670 --> 00:02:10.390
And just to appreciate some
of the most significant

00:02:10.390 --> 00:02:12.113
Executive Orders ever made,

00:02:13.210 --> 00:02:16.720
Lincoln's Emancipation
Proclamation in 1863

00:02:16.720 --> 00:02:19.420
which was delivered during the Civil War,

00:02:19.420 --> 00:02:22.420
it freed nearly three million slaves

00:02:22.420 --> 00:02:24.510
in the Confederate states.

00:02:24.510 --> 00:02:28.560
In 1942 a few months
after Japan's bombardment

00:02:28.560 --> 00:02:31.250
of Pearl Harbor, Franklin Roosevelt issues

00:02:31.250 --> 00:02:35.920
Executive Order 9066 which
called for the internment

00:02:35.920 --> 00:02:38.950
of Americans of Japanese ancestry.

00:02:38.950 --> 00:02:42.000
Historians have really frowned
on this Executive Order

00:02:42.000 --> 00:02:45.050
because it was a blatant
disregard for the rights

00:02:45.050 --> 00:02:46.410
of these Americans.

00:02:46.410 --> 00:02:49.030
But at the time, Franklin
Roosevelt justified it

00:02:49.030 --> 00:02:51.080
as a national security issue.

00:02:51.080 --> 00:02:54.710
And then in 1952 during the Korean war,

00:02:54.710 --> 00:02:57.480
steel workers are
threatening to go on strike.

00:02:57.480 --> 00:03:01.110
And so, President Truman
issues Executive Order 10340

00:03:03.140 --> 00:03:05.560
that puts steel mills under the control

00:03:05.560 --> 00:03:07.590
of the Commerce Secretary.

00:03:07.590 --> 00:03:09.870
And the justification
here was that they were

00:03:09.870 --> 00:03:11.930
in the middle of the war and that steel

00:03:11.930 --> 00:03:15.370
is an essential material
for conducting the war.

00:03:15.370 --> 00:03:18.480
Now what's really interesting
about this Executive Order,

00:03:18.480 --> 00:03:21.720
is that the owners of the
steel mills did not like this

00:03:21.720 --> 00:03:24.320
and they take the
government to Federal court.

00:03:24.320 --> 00:03:27.360
It eventually gets to
the U.S. Supreme Court.

00:03:27.360 --> 00:03:30.950
It's knows as the case,
Youngstown Sheet and Tube Company

00:03:30.950 --> 00:03:34.980
versus Sawyer where Sawyer
was the Commerce Secretary.

00:03:34.980 --> 00:03:39.270
And the Supreme Court rules
against President Truman

00:03:39.270 --> 00:03:43.030
saying that this Executive
Order went beyond the bounds

00:03:43.030 --> 00:03:46.160
of even the President's implied powers.

00:03:46.160 --> 00:03:48.650
So the big picture here
is that Executive Orders

00:03:48.650 --> 00:03:51.890
are a method by which Presidents
over time have been able

00:03:51.890 --> 00:03:55.930
to expand their power beyond
what is explicitly listed

00:03:55.930 --> 00:03:58.740
in Article II of the Constitution.

00:03:58.740 --> 00:04:02.260
Now like in all things, there
is a check on that power.

00:04:02.260 --> 00:04:05.240
And much of that comes from
the Supreme Court's ability

00:04:05.240 --> 00:04:08.530
to rule Executive Orders unconstitutional.

00:04:08.530 --> 00:04:11.890
But there still aren't clear
boundaries on what makes

00:04:11.890 --> 00:04:14.440
an Executive Order constitutional or not.

00:04:14.440 --> 00:04:17.400
And they are likely to continue
to be an area of debate

00:04:17.400 --> 00:04:19.943
when it comes to powers of the President.