Increasing political battles over slavery in mid 1800s

WEBVTT
Kind: captions
Language: en

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- [Voiceover] So Kim, in the last video

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we started talking about how slavery

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has been an issue in the United States

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really since its founding
with the Revolutionary War.

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And we see some images here of slavery.

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This was a particularly famous image

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that really kind of
helped spark indignation

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in the North among Abolitionists

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'cause you can see how this
person was beaten or whipped.

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- [Voiceover] Yes absolutely.

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- [Voiceover] So it's kind of, you know,

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and today we all think it's
morally reprehensible thing

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but you go back not too far
in the whole scope of history.

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We're going to the early 1800s,

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and this was something that was debated.

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It was slavery was allowed

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in a large chunk of the United States.

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And in the last video, we talked about

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that this issue of slavery
only got exacerbated

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the more territory that was added.

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You have the Louisiana Purchase in 1803

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that adds all of this territory.

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You start having states getting carved.

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Once you get a critical population,

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a critical mass of
people in a certain state

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they can apply to be, or
in a certain territory,

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they can apply to be a state.

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And as each of these states are added,

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and they want to get representation,

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it's a political issue.

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You mentioned how the
North, they didn't like it

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on moral grounds and if
it was a slave state.

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And they also didn't like
it on economic grounds

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'cause it's hard to compete
economically with slavery.

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While the South was afraid
of losing its political power

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if more free states
were to join the Union.

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And the Compromise of 1820, you mentioned

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that well that compromise,
Missouri is a slave state.

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Maine gets carved out of Massachusetts,

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becomes a free state.

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But that didn't solve the problem.

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That problem only
continues because we only

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add more territory.

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- [Voiceover] Yeah, I
mean most of the history

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of the 1800s, when it comes to slavery,

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is a history of putting off the problem.

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Henry Clay becomes this
very famous legislator

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because he's good at compromising.

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He's called the "Great Compromiser".

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So, instead of trying to actually solve

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the issue of slavery, which many people

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think of as an unsolvable problem.

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They're just saying,
"All right, well how can

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"we put off a conflict over
this a little bit longer?"

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- [Voiceover] And they
just keep doing that.

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You know, in 1836 the
territory expands further,

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or it starts to I guess.

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The expansion is catalyzed further

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by you have the Texas Revolution.

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Texas get its independence from Mexico.

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Which itself got independence
only a few decades

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before that from Spain.

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But so Texas, for a brief amount of time,

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is its own independent country.

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But then, it joins the US,
it's annexed by the US in 1845.

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So it's more territory for the US,

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and that was slave territory.

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- [Voiceover] Right, yeah so again,

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this is an area in the South where

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most of the reasons that
slavery existed in the South

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is because it's a very fertile
agricultural region, right?

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Where the things that you grow, crops,

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are very labor-intensive.

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So they figure, how are we going

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to find enough people to grow these crops?

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And the answer is that they've
been importing African slaves

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and forcing them to work.

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- [Voiceover] And the annexation of Texas,

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there's war disputes with Mexico

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which leads to even, another
conflict with Mexico.

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This time with the US and Mexico.

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This is the Mexican-American
War from 1846 to 1848.

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But in the context of this conversation,

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the reason why it helped
bring the slavery issue

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even more to a head, is
that when the US wins it,

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it gets even more territory.

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It gets all of this area in
the West, right over here.

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- [Voiceover] Right, it's more territory.

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And it's more sort of Southern territory

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which means that it has a high probability

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of becoming slave states.

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- [Voiceover] And so, this map that we

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looked at earlier in the other video,

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this is kind of showing
what the US looked like

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as we exit out the Mexican-American War.

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We're getting to about 1850.

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And so, what happens then?

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I mean, is it just one compromise

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after another at this point?

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- [Voiceover] Well, this is
the point where compromise

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begins to break down.

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Henry Clay again, is the architect

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of what's called the Compromise of 1850.

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As soon as the US goes to war with Mexico,

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people in Congress are wondering,

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"All right, if we get this territory

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"that we're trying to get,
what's going to happen?

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"Are they going to be free
states or slave states?"

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And this compromise over
free and slave states

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has been going on for 30 years

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since the Missouri Compromise.

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And even longer, if you
take it back to 1776.

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So, they're doing exactly the same thing.

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They're saying, "All right, well let's try

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"to keep a balance of power

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"between free states and slave states."

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Except, they add in a couple of provises

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that make people really
angry in the 1850s.

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- [Voiceover] And who gets angry?

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- [Voiceover] Both I
would say, slaveowners

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and Abolitionists and anti-slavery
activists in the North.

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- [Voiceover] So, I guess that's
what makes it a compromise.

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A little something to
make everybody angry.

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(laughter)

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What made the slave owners angry

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about the Compromise of 1850?

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- [Voiceover] One of the parts
of the Compromise of 1850,

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apart from deciding whether
these new territories

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were going to be slave or free

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was a part of the Act,

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was called the Fugitive Slave Act.

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And the Fugitive Slave
Act said that it was

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a federal offense not

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to help slave owners recover--

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- [Voiceover] So this is
something the Abolitionists angry?

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- [Voiceover] Yes.

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- [Voiceover] Or the anti-slavery angry?

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- [Voiceover] Yeah, absolutely both.

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So for example, if you're
maybe living in Massachusetts,

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you're a white middle-class person

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living in Massachusetts, you
don't think slavery is great.

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You think it's morally wrong,

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but it hasn't really directly
affected your life. Right?

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You're hundreds of miles away
from the nearest slave state.

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But now, the federal government says

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that if there is a person
who has escaped from slavery

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who has come to your town,

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it is a federal offense
for you not to help

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return that person to slavery.

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- [Voiceover] So this Fugitive Slave,

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this is part of the compromise?

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This is part of the Compromise of 1850.

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Is that I could be
sitting in Massachusetts,

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I could be anti-slavery or I could even be

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ambivalent about it.

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But now, I have to be complicit in it.

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If there's a slave, I can't.

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Or if there's someone who's escapes

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from the South who is a former slave,

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I have to actively, I
can't in any way help them.

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If I do, I could go to jail.

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If I'm a law officer, I
have to capture that person,

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and I have to bring them back.

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So it's kind of forcing people who are

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already not happy about slavery,

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it's kind of bringing it close to them.

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They have to partake in it.

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- [Voiceover] Yeah, and these are people

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who have very strong
religious convictions.

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This is the mid-19th century is a time

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when people feel their
religion very strongly.

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And so, there are people in
the North who are Quakers.

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Who are otherwise religiously
opposed to slavery.

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Who have maybe prayed
for the souls of slaves,

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but it's never been their job

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to try to keep someone in slavery before.

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So that is really infuriating.

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One of the things you get out this

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is a really strong backlash
of Abolitionist sentiment

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in the North.

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For example, Harriet Beecher Stowe,

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she's the daughter of a
reverend who's against slavery.

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She writes the book Uncle Tom's Cabin

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which becomes this smash hit.

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- [Voiceover] And when was this roughly?

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- [Voiceover] This was about 1852.

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- [Voiceover] OK, so this is
after the Compromise of 1850.

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So, people are getting, it
really is coming to a head.

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The people in the North, they're having

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to partake in this because
of the Fugitive Slave Act.

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You have Harriet Beecher Stowe
writes Uncle Tom's Cabin.

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It makes people even more upset

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about the realities of slavery.

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- [Voiceover] Right, and in the South,

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they've kind of gotten away,
white slave owners in this time

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with people saying, "Ya know,

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"slavery's not really my problem.

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"I don't like it."

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But now they're seeing
a concentrated attack,

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a moral and social attack
against slavery in the North.

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And their response is to become even more

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violently in favor of slavery.

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They pose the idea that
slavery is not just something

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that we could turn our eyes away from,

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but it's necessary.

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But it's a positive good.

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Slavery is actually going to actually

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make the country better.

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If it weren't for slavery, all
of these enslaved Africans,

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African-Americans, their lives
would be worse without us.

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- [Voiceover] And so, just to be clear

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on the Compromise of 1850.

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And there was a bunch of things,

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we went into a whole video
on the Compromise of 1850,

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all of the different facets of it.

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But it's one of its, in
terms of this conversation,

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one of its outcomes is because
of the Fugitive Slave Law.

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It infuriated many of the
anti-slavery Abolitionists

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in the North.

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They became more entrenched
in their positions

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which made the Southerners more entrenched

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in their positions.

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The Southerners didn't necessarily,

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they liked the Fugitive
Slave Law, the Southerners.

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The Compromise of 1850 sounds like it was,

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it was a compromise,
not everyone was happy.

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But it sounds like it made

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the anti-slavery folks more unhappy

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than the slavery folks.

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- [Voiceover] Yeah absolutely.

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So it makes the anti-slavery
folks super unhappy,

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but it also means that now
white slave owners in the South,

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they feel like there's
a target on their heads.

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And so, they're going
to dig in even further

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to make sure that their interests
in slavery are protected.

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- [Voiceover] And this gets
us to the election of 1860

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which I guess in some ways was the straw

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that breaks the camel's back,

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I guess from a Southern perspective.

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And why is that?

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- [Voiceover] Well, during the 1850s,

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you have all of these
political battles over slavery.

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In fact, it kind of breaks
the major political party

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of the 1850s, the Whig Party.

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So, in 1860, some of these leftover Whigs,

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they reorganize as the Republican Party.

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And this is the first election
with a Republican Party

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that we know today.

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But obviously, in 1860, the things

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that they're interested
in, their goals and aims

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are completely different.

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And the Republican Party
is an anti-slavery party.

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They're deliberately and
publicly against slavery.

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So, they nominate as their

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political candidate for 1860,

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Abraham Lincoln who is
well-known in the country

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for having been an anti-slavery agitator.

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He's given many speeches where he has

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made very eloquent
arguments against slavery.

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- [Voiceover] He's, we talked
about in the previous video,

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he's against it, I guess on moral grounds.

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But perhaps even more, his old father

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wasn't able to be a successful farmer

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because he had to compete
with slave owners.

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- [Voiceover] Right, so he's
brought up to hate slavery

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because it's a big
business that has harmed

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his own family's economic future.

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But, he comes, I think,
to his own conclusion

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that slavery is morally wrong.

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What he doesn't think that
he can do as president

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is legally or constitutionally,
get rid of slavery.

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He doesn't think that the
Constitution allows it.

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But, he is seen as enough
of a threat by Southerners

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that in many cases, on the
presidential ballots in 1860,

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you couldn't even vote for Abraham Lincoln

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if you wanted to.

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He didn't show up on the
ballot in Southern states.

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Nonetheless, he still gets
enough electoral votes

00:11:32.690 --> 00:11:34.670
that he's elected.

00:11:34.670 --> 00:11:36.760
- [Voiceover] And then
that's what takes us into

00:11:36.760 --> 00:11:41.760
the real meat of, you know,
catalyzes the Civil War.

00:11:42.050 --> 00:11:45.560
- [Voiceover] Yeah, stay
tuned, Abraham Lincoln.


Slavery and Missouri Compromise in early 1800s

WEBVTT
Kind: captions
Language: en

00:00:00.133 --> 00:00:02.223
- [Voiceover] This is Sal
here, and I'm with Kim Kutz,

00:00:02.223 --> 00:00:05.079
who is Khan Academy's American
History Content Fellow,

00:00:05.079 --> 00:00:07.273
and what I'm curious about is...

00:00:08.555 --> 00:00:10.900
In school, you learn about the Civil War,

00:00:10.900 --> 00:00:13.969
you learn about slavery, that slavery was

00:00:13.969 --> 00:00:17.712
a cause of the Civil War,
but at least for myself,

00:00:17.712 --> 00:00:19.732
I never got a full context of what were

00:00:19.732 --> 00:00:21.854
all the dynamics that
led to the Civil War?

00:00:21.854 --> 00:00:23.990
Was it just something
that happened overnight?

00:00:23.990 --> 00:00:25.137
- [Kim] Oh, defnitely not.

00:00:25.137 --> 00:00:27.876
You know, I think the
seeds of the Civil War

00:00:27.876 --> 00:00:31.786
were really with the United
States at its creation.

00:00:31.786 --> 00:00:34.967
I think there's sort of
an essential contradiction

00:00:34.967 --> 00:00:36.987
in the United States as it's born.

00:00:36.987 --> 00:00:40.284
We're this country where
all men are created equal,

00:00:40.284 --> 00:00:42.978
except that most of
the states in the South

00:00:42.978 --> 00:00:44.765
have slavery, where people are clearly

00:00:44.765 --> 00:00:48.132
not created equal, so they couldn't win

00:00:48.132 --> 00:00:51.940
the Revolutionary War without
including those states

00:00:51.940 --> 00:00:53.635
and kind of giving them what they wanted

00:00:53.635 --> 00:00:57.772
in retaining slavery,
but it means that the US

00:00:57.772 --> 00:01:02.345
is born with both free
states and slave states,

00:01:02.345 --> 00:01:06.292
and they're gonna continue
to try to figure out

00:01:06.292 --> 00:01:09.125
how to balance those for
the rest of the 1800s.

00:01:09.125 --> 00:01:10.348
- [Sal] We have this map here.

00:01:10.348 --> 00:01:13.635
This map is a later
period, but it shows...

00:01:14.477 --> 00:01:16.481
This is actually closer to the Civil War,

00:01:16.481 --> 00:01:18.674
but if we even look at
the original 13 colonies,

00:01:18.674 --> 00:01:20.276
you can see which ones were free states

00:01:20.276 --> 00:01:23.178
and which ones were slave states,

00:01:23.178 --> 00:01:24.622
and then you obviously
have these other states

00:01:24.622 --> 00:01:26.788
that come in later,
which we'll talk about.

00:01:26.788 --> 00:01:29.497
What you're saying is, at
the founding of the country,

00:01:29.497 --> 00:01:31.337
this was already an issue.

00:01:31.337 --> 00:01:33.044
There were people in the
North who weren't fans

00:01:33.044 --> 00:01:35.807
of slavery, and people
knew that at some point

00:01:35.807 --> 00:01:38.067
this would be irreconcilable,
or maybe they hoped

00:01:38.067 --> 00:01:40.951
that it would be a (laughs)
reconcilable difference.

00:01:40.951 --> 00:01:43.157
They said, "No, we gotta
unify against Great Britain."

00:01:43.157 --> 00:01:44.318
- [Kim] Exactly.
- [Sal] So they said,

00:01:44.318 --> 00:01:45.580
"Let's just become a country and do it."

00:01:45.580 --> 00:01:47.717
- [Kim] You know, even Thomas Jefferson,

00:01:47.717 --> 00:01:49.690
the author of the
Declaration of Independence,

00:01:49.690 --> 00:01:51.780
he knew that slavery was a contradiction.

00:01:51.780 --> 00:01:55.123
He called the issue of having slavery

00:01:55.123 --> 00:01:57.012
"like holding a wolf by the ears."

00:01:57.012 --> 00:01:59.671
You can't hold onto it,
but you can't let it go

00:01:59.671 --> 00:02:02.457
because so many of the wealthy elites

00:02:02.457 --> 00:02:04.152
who were going to end up in Congress

00:02:04.152 --> 00:02:05.662
in the South are slaveowners--

00:02:05.662 --> 00:02:07.412
- [Sal] Including himself.
- [Kim] Exactly,

00:02:07.412 --> 00:02:09.541
so they wanna protect their interests.

00:02:09.541 --> 00:02:13.543
- [Sal] So, the issue
is there from the moment

00:02:13.543 --> 00:02:15.261
that the country is
founded, and then we get

00:02:15.261 --> 00:02:17.398
into the 1800s, which
is really the run-up.

00:02:17.398 --> 00:02:20.936
The Civil War doesn't start
until we get into 1860

00:02:20.936 --> 00:02:25.623
or shortly thereafter, or actually, 1860.

00:02:28.383 --> 00:02:32.239
What's the big picture
that really leads up to it?

00:02:32.239 --> 00:02:34.421
- [Kim] Well, I think
what we're looking at

00:02:34.421 --> 00:02:38.517
when we get into the issues
that lead to the Civil War,

00:02:38.517 --> 00:02:42.638
it's really about how the US
handles getting new territory.

00:02:43.614 --> 00:02:45.183
- [Sal] And the US was getting
a lot of new territory.

00:02:45.183 --> 00:02:46.483
We have a map here.

00:02:46.483 --> 00:02:48.552
I guess the first really big chunk is

00:02:48.552 --> 00:02:51.223
you have the Louisiana Purchase in 1803,

00:02:51.223 --> 00:02:52.294
and so you get all of...

00:02:53.125 --> 00:02:54.188
Let me shade it in.

00:02:54.188 --> 00:02:58.010
You get roughly all of
this stuff right over here,

00:02:58.010 --> 00:03:01.447
so that's new areas that settlers can go,

00:03:01.447 --> 00:03:05.471
and it becomes officially
a part of the US,

00:03:05.471 --> 00:03:06.772
and what else happens?

00:03:06.772 --> 00:03:10.092
- [Kim] So, as we get
these new territories,

00:03:10.092 --> 00:03:12.553
out of them you're gonna get new states,

00:03:12.553 --> 00:03:14.991
and when new states come into the Union,

00:03:14.991 --> 00:03:17.267
they're going to come
in as either free states

00:03:17.267 --> 00:03:21.686
or slave states, so we've
balanced the interests

00:03:21.686 --> 00:03:24.482
of the North and South
up until this point,

00:03:24.482 --> 00:03:27.129
from the Revolutionary
War, so that there's

00:03:27.129 --> 00:03:29.382
equal representation in Congress

00:03:29.382 --> 00:03:32.052
between free states and slave states.

00:03:32.052 --> 00:03:33.840
- [Sal] Well, why does someone care?

00:03:33.840 --> 00:03:36.069
If I'm someone in Massachusetts,

00:03:36.069 --> 00:03:38.855
why do I care whether
the new state of Missouri

00:03:38.855 --> 00:03:41.223
is going to be a free
state or a slave state?

00:03:41.223 --> 00:03:42.918
- [Kim] Well, I think
there are two reasons

00:03:42.918 --> 00:03:44.427
why you might care.

00:03:44.427 --> 00:03:46.424
First, if you're an abolitionist,

00:03:46.424 --> 00:03:49.094
and these are the people
who we know very well,

00:03:49.094 --> 00:03:52.624
like Frederick Douglass,
or William Lloyd Garrison,

00:03:52.624 --> 00:03:55.456
who was the editor of the
newspaper The Liberator.

00:03:55.456 --> 00:03:57.268
These are the people who feel that,

00:03:57.268 --> 00:04:00.472
correctly, slavery is morally wrong.

00:04:00.472 --> 00:04:03.414
Slavery is a corruption

00:04:03.414 --> 00:04:05.875
of the essential principles

00:04:05.875 --> 00:04:08.623
on which the country was founded.

00:04:08.623 --> 00:04:13.151
It's something that destroys
lives, destroys families,

00:04:13.151 --> 00:04:15.978
but another reason if
you're, say, in Massachusetts

00:04:15.978 --> 00:04:18.904
or Pennsylvania, why
you might care whether

00:04:18.904 --> 00:04:23.153
a new state is a slave
state, is you're worried

00:04:23.153 --> 00:04:26.961
about opportunities for
yourself out in the West.

00:04:26.961 --> 00:04:29.097
We know that Horace Greeley,

00:04:29.097 --> 00:04:31.442
this famous newspaper editor, he says,

00:04:31.442 --> 00:04:34.391
"What do you do if you're
a young man in New York,

00:04:34.391 --> 00:04:37.223
a young white man, who doesn't
know how to get ahead?"

00:04:37.223 --> 00:04:39.174
He says, "Go West, young man.

00:04:39.174 --> 00:04:41.240
"You can go out there,
you can get some land,

00:04:41.240 --> 00:04:44.166
"you can start a farm,
but if you go out there

00:04:44.166 --> 00:04:47.045
"and you find that all of
the land has been bought up

00:04:47.045 --> 00:04:50.133
"by rich slaveholders from the South,

00:04:50.133 --> 00:04:51.944
"you might not be able to get any land,

00:04:51.944 --> 00:04:53.662
"and you certainly might not be able to,

00:04:53.662 --> 00:04:57.261
"for example, sell your
corn at a rate low enough

00:04:57.261 --> 00:05:00.512
"that you could beat
somebody who has free labor."

00:05:00.512 --> 00:05:02.952
- [Sal] So, a lot of times
there's a lot of focus

00:05:02.952 --> 00:05:05.413
on the moral argument, which
is a very strong argument,

00:05:05.413 --> 00:05:07.410
but there's also this
interesting economic argument

00:05:07.410 --> 00:05:08.757
which you just talked about,

00:05:08.757 --> 00:05:11.055
which is it's hard to
compete with slavery.

00:05:11.055 --> 00:05:12.852
I mean, you're literally
talking about labor

00:05:12.852 --> 00:05:15.197
that does not need traditional wages.

00:05:15.197 --> 00:05:16.939
- [Kim] Right.
- [Sal] That is literally

00:05:16.939 --> 00:05:18.781
slave labor, and so if you were having

00:05:18.781 --> 00:05:21.428
your own farm and you don't own slaves,

00:05:21.428 --> 00:05:23.100
how are you going to compete with that?

00:05:23.100 --> 00:05:25.152
So, that was the reason for
some folks in the North,

00:05:25.152 --> 00:05:26.684
an economic argument.

00:05:26.684 --> 00:05:28.913
Now, would these people be
considered abolitionists?

00:05:28.913 --> 00:05:31.453
- [Kim] No, the way that
we think about those,

00:05:31.453 --> 00:05:33.333
we call them "anti-slavery."

00:05:33.333 --> 00:05:37.745
So, anti-slavery
advocates, they don't think

00:05:37.745 --> 00:05:40.238
that they can get rid
of slavery in the South.

00:05:40.238 --> 00:05:42.491
Even if they don't like
slavery in the South,

00:05:42.491 --> 00:05:44.627
they don't even see how
it would be possible

00:05:44.627 --> 00:05:47.622
to get rid of it, but they do
think that as these new states

00:05:47.622 --> 00:05:51.778
are coming into the Union,
they could prevent them

00:05:51.778 --> 00:05:55.447
from becoming slave states,
so that it's possible

00:05:55.447 --> 00:05:58.422
for the Western lands to remain free.

00:05:59.914 --> 00:06:01.214
Abraham Lincoln, I think,

00:06:01.214 --> 00:06:03.939
is a really good poster child for this.

00:06:03.939 --> 00:06:06.633
I think we'll talk about
him a little bit more later,

00:06:06.633 --> 00:06:09.233
but Lincoln is born in Kentucky,

00:06:09.233 --> 00:06:11.532
one of these new, Western states.

00:06:11.532 --> 00:06:14.922
His father is a small, white farmer,

00:06:14.922 --> 00:06:17.662
and slaveowners move into Kentucky.

00:06:17.662 --> 00:06:20.123
Later it becomes a slave state,

00:06:20.123 --> 00:06:21.888
and his father can't find work,

00:06:21.888 --> 00:06:24.395
his father can't find land, so he ends up

00:06:24.395 --> 00:06:26.508
first having to move to Indiana,

00:06:26.508 --> 00:06:29.875
then moving to Illinois,
so this is literally a case

00:06:29.875 --> 00:06:31.895
of one of these poor,
white farmers who just

00:06:31.895 --> 00:06:34.078
can't compete with slavery,
which is one reason

00:06:34.078 --> 00:06:36.841
why Lincoln himself is
later gonna come out

00:06:36.841 --> 00:06:39.093
so strongly in favor of making sure

00:06:39.093 --> 00:06:40.672
there's no slavery in the West.

00:06:40.672 --> 00:06:45.185
- [Sal] So, abolitionists:
slavery is amoral,

00:06:45.185 --> 00:06:48.134
it needs to be removed from,
definitely the United States,

00:06:48.134 --> 00:06:49.759
possibly the world.
- [Kim] Yeah, absolutely.

00:06:49.759 --> 00:06:52.429
- [Sal] Anti-slavery: they
also think slavery is bad,

00:06:52.429 --> 00:06:54.690
they don't like it, but they think it's,

00:06:54.690 --> 00:06:57.113
"Well, but I'm not gonna
fight that fight to remove it.

00:06:57.113 --> 00:06:59.180
"Maybe that's hard to do, or impossible,

00:06:59.180 --> 00:07:00.619
"but it shouldn't spread.

00:07:00.619 --> 00:07:02.129
"It's not fair.

00:07:02.129 --> 00:07:03.382
"It's the reason my dad wasn't able

00:07:03.382 --> 00:07:05.751
"to be able to run his farm."

00:07:05.751 --> 00:07:07.245
- [Kim] Absolutely.

00:07:07.245 --> 00:07:10.124
- [Sal] So, you have
the Louisiana Purchase,

00:07:10.124 --> 00:07:12.013
and in other videos we
talk about that famously

00:07:12.013 --> 00:07:13.514
Napoleon sold it for quite cheap,

00:07:13.514 --> 00:07:14.744
because frankly, he couldn't defend it

00:07:14.744 --> 00:07:16.223
(Kim laughs)
because he was fighting

00:07:16.223 --> 00:07:17.386
these wars in Europe.

00:07:17.386 --> 00:07:19.151
That's the first chunk of land,

00:07:19.151 --> 00:07:20.242
so you have all of these states,

00:07:20.242 --> 00:07:21.542
and they need to figure out whether

00:07:21.542 --> 00:07:23.191
they're slave states or free states,

00:07:23.191 --> 00:07:24.198
but why would...

00:07:24.198 --> 00:07:26.018
I mean, I talked about why
would a Northerner care

00:07:26.018 --> 00:07:27.318
whether a slave or a free state.

00:07:27.318 --> 00:07:28.549
Why would a Southerner care?

00:07:28.549 --> 00:07:31.219
If I'm a slaveowner, I own a plantation

00:07:31.219 --> 00:07:33.842
in South Carolina or
Georgia, why do I care

00:07:33.842 --> 00:07:36.698
if Missouri is a slave
state or a free state?

00:07:36.698 --> 00:07:38.370
- [Kim] Well I think, just as their

00:07:38.370 --> 00:07:40.971
political interests
are tied up in slavery,

00:07:40.971 --> 00:07:43.432
all of their money is tied up in slavery.

00:07:43.432 --> 00:07:47.625
In 1860, the most valuable thing

00:07:47.625 --> 00:07:52.482
that anyone owns in the
United States is slaves.

00:07:52.482 --> 00:07:55.985
You can't compete with that kind of money,

00:07:55.985 --> 00:07:59.561
so they wanna make sure
that if a new state comes

00:07:59.561 --> 00:08:03.253
into the Union, that
state isn't a free state

00:08:03.253 --> 00:08:05.552
because then the free states might have

00:08:05.552 --> 00:08:07.827
more representation in Congress,

00:08:07.827 --> 00:08:10.219
and then they can vote to outlaw slavery.

00:08:10.219 --> 00:08:13.051
So, if your whole fortune
is built on slavery,

00:08:13.051 --> 00:08:16.302
if you're a white
slaveowner, they outlaw that,

00:08:16.302 --> 00:08:17.858
then you're left with nothing.

00:08:17.858 --> 00:08:20.458
- [Sal] I see, so the North,
there's the moral argument,

00:08:20.458 --> 00:08:21.851
there's the economic argument,

00:08:21.851 --> 00:08:23.593
slavery is hard to compete with,

00:08:23.593 --> 00:08:25.891
and in the South, "Hey,
if we have too many

00:08:25.891 --> 00:08:27.982
"of these free states, at
some point they're gonna have

00:08:27.982 --> 00:08:31.008
"enough of a voting
power in the government

00:08:31.008 --> 00:08:34.909
"to maybe abolish slavery one day,"

00:08:34.909 --> 00:08:36.644
which would completely undermine,

00:08:36.644 --> 00:08:38.039
if I'm a slaveowner,

00:08:38.039 --> 00:08:42.172
my economics of my reality.

00:08:42.172 --> 00:08:45.445
- [Kim] Right, and they are
sort of essentially amoral.

00:08:45.445 --> 00:08:47.953
Even someone like Jefferson who knows

00:08:47.953 --> 00:08:50.345
that slavery is wrong, his whole wealth,

00:08:50.345 --> 00:08:53.456
his whole fortune, his
whole political dynasty

00:08:53.456 --> 00:08:56.323
is built on the fortune of owning slaves.

00:08:57.306 --> 00:09:00.191
- [Sal] You know, one of the first points

00:09:00.191 --> 00:09:02.885
where this really gets
balanced, this issue,

00:09:02.885 --> 00:09:05.392
is we have the Louisiana Purchase in 1803,

00:09:05.392 --> 00:09:07.337
then starting to carve out
the Louisiana Purchase,

00:09:07.337 --> 00:09:08.962
you have states like Missouri.

00:09:08.962 --> 00:09:10.913
They get to their critical mass of people,

00:09:10.913 --> 00:09:13.211
of population, so that
they can become a state,

00:09:13.211 --> 00:09:14.651
and so what was the Missouri Compromise

00:09:14.651 --> 00:09:16.160
all about in 1820?

00:09:16.160 --> 00:09:17.832
- [Kim] So, the Missouri Compromise

00:09:17.832 --> 00:09:21.315
is when we have enough
people living in Missouri.

00:09:21.315 --> 00:09:24.158
These are white people, generally,

00:09:24.158 --> 00:09:27.082
who have come from the Eastern states,

00:09:27.082 --> 00:09:28.986
and they apply for statehood.

00:09:28.986 --> 00:09:31.493
You've got an equal number of slave states

00:09:31.493 --> 00:09:33.815
and free states already in Congress,

00:09:33.815 --> 00:09:37.205
so if Missouri comes in and
they wanna be a slave state,

00:09:37.205 --> 00:09:38.831
they're going to upset the apple cart.

00:09:38.831 --> 00:09:40.084
They're gonna upset the balance,

00:09:40.084 --> 00:09:42.221
so there will be more representatives

00:09:42.221 --> 00:09:44.705
for the South than
there will be for North,

00:09:44.705 --> 00:09:46.493
and everything they've done so far

00:09:46.493 --> 00:09:49.418
has been predicated on
this tenuous balance

00:09:49.418 --> 00:09:52.971
between free states and slave states.

00:09:52.971 --> 00:09:57.971
So, they debate this
in Congress for months,

00:09:57.994 --> 00:10:00.139
and eventually what they do is say,

00:10:00.139 --> 00:10:01.881
"All right, well, we can't decide,

00:10:01.881 --> 00:10:04.110
"so what we're going to do is admit

00:10:04.110 --> 00:10:07.291
"the state of Maine at the same time."

00:10:07.291 --> 00:10:09.706
- [Sal] And Maine, the territory of Maine

00:10:09.706 --> 00:10:11.907
was already part of the United States.

00:10:11.907 --> 00:10:14.358
How was it not already a state?

00:10:14.358 --> 00:10:16.216
- [Kim] It was part of Massachusetts,

00:10:16.216 --> 00:10:19.350
but as you can see, it's
really only tenuously

00:10:19.350 --> 00:10:22.461
connected to Massachusetts, so they divide

00:10:22.461 --> 00:10:24.296
this territory up so that it can have

00:10:24.296 --> 00:10:27.314
its own representation
in Congress, so they say,

00:10:27.314 --> 00:10:29.752
"All right, well, we
can't solve this problem

00:10:29.752 --> 00:10:31.981
"of the balance of power
between free states

00:10:31.981 --> 00:10:34.489
"and slave states right
now, so what we're gonna do

00:10:34.489 --> 00:10:36.416
"is just kind of extend our balance.

00:10:36.416 --> 00:10:38.599
"We're gonna keep this compromise going

00:10:38.599 --> 00:10:40.456
"to make sure that there
are the same number

00:10:40.456 --> 00:10:44.615
"of free and slave states,
so we'll let Missouri in

00:10:44.615 --> 00:10:46.479
"as a slave state at the same time

00:10:46.479 --> 00:10:49.219
"we let Maine in as a free state."

00:10:49.219 --> 00:10:50.303
- [Sal] Fascinating.

00:10:51.313 --> 00:10:52.983
I see where this is
going, that you have these

00:10:52.983 --> 00:10:55.130
very tenuous compromises while more

00:10:55.130 --> 00:10:57.066
and more territories are being added.

00:10:57.066 --> 00:10:59.127
It's exciting to see
where all of this goes.


Intuition for area with fractional side lengths

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:01.750
- [Voiceover] So I have a square here

00:00:01.750 --> 00:00:05.990
and let's say that its
height is one meter.

00:00:05.990 --> 00:00:08.250
So this is height right over here.

00:00:08.250 --> 00:00:10.990
That is one meter.

00:00:10.990 --> 00:00:13.290
And let's say its width is also one meter.

00:00:13.290 --> 00:00:15.450
So I'm talking about the
dimensions of the entire square,

00:00:15.450 --> 00:00:17.250
not just the shaded region.

00:00:17.250 --> 00:00:21.320
So this is also, that right
over there is also one meter.

00:00:21.320 --> 00:00:23.990
What's the area of the
entire square going to be,

00:00:23.990 --> 00:00:26.490
not just the shaded, the entire square?

00:00:26.490 --> 00:00:28.620
Well, the total area,

00:00:28.620 --> 00:00:32.350
total area is going to be equal to

00:00:32.350 --> 00:00:34.320
the height times the width.

00:00:34.320 --> 00:00:37.320
So one meter times one meter.

00:00:37.320 --> 00:00:38.920
One times one is of course one

00:00:38.920 --> 00:00:40.450
and meters times meters

00:00:40.450 --> 00:00:42.050
we could write that as a square meter

00:00:42.050 --> 00:00:44.690
or meter squared, however
you want to think about it.

00:00:44.690 --> 00:00:46.280
Now with that out of the way

00:00:46.280 --> 00:00:49.150
now let's focus on the shaded area.

00:00:49.150 --> 00:00:50.790
Let's think about what that is.

00:00:50.790 --> 00:00:54.090
So the shaded, shaded,

00:00:54.090 --> 00:00:56.650
shaded area is equal to what,

00:00:56.650 --> 00:00:58.010
and I encourage you to pause the video

00:00:58.010 --> 00:01:00.450
and try to figure that out.

00:01:00.450 --> 00:01:02.020
Well the one thing that
might jump out of you

00:01:02.020 --> 00:01:04.690
is that our entire area, our entire square

00:01:04.690 --> 00:01:09.050
is divided into these equally
equal, equal rectangles.

00:01:09.050 --> 00:01:10.590
So one way to think about it is well,

00:01:10.590 --> 00:01:13.720
what is the area of each
of these equal rectangles?

00:01:13.720 --> 00:01:15.390
For example what is the area

00:01:15.390 --> 00:01:17.710
of that rectangle right over there?

00:01:17.710 --> 00:01:18.590
And to figure it out we can say

00:01:18.590 --> 00:01:21.090
well, what fraction is that of the whole?

00:01:21.090 --> 00:01:22.850
And to figure that out
we have to figure out

00:01:22.850 --> 00:01:24.550
how many of these rectangles

00:01:24.550 --> 00:01:27.230
has our whole been divided into?

00:01:27.230 --> 00:01:29.420
We could try to count them out

00:01:29.420 --> 00:01:31.570
or we could say let's see, I have

00:01:31.570 --> 00:01:34.220
one, two, three, four, five,

00:01:34.220 --> 00:01:37.650
six, seven, eight, nine, 10 columns,

00:01:37.650 --> 00:01:41.890
and each columns has
one, two, three, four,

00:01:41.890 --> 00:01:44.150
five, six, seven.

00:01:44.150 --> 00:01:46.050
So we have 10 columns of seven

00:01:46.050 --> 00:01:48.450
or we have 70 of these rectangles,

00:01:48.450 --> 00:01:51.850
that our entire whole is
divided into 70 equal sections

00:01:51.850 --> 00:01:54.770
that we see these
rectangles right over here.

00:01:54.770 --> 00:01:57.650
This character, this
character right over there

00:01:57.650 --> 00:02:00.220
that is 1/70 of the entire area.

00:02:00.220 --> 00:02:02.850
So 1/70 of one square meter

00:02:02.850 --> 00:02:07.850
which is of course just going
to be 1/70 of a square meter.

00:02:08.250 --> 00:02:09.920
That's just one of these.

00:02:09.920 --> 00:02:12.020
That's just one of these rectangles.

00:02:12.020 --> 00:02:13.410
Now if we cared about the shaded area

00:02:13.410 --> 00:02:15.820
we can just count how many
of these rectangles there are

00:02:15.820 --> 00:02:20.510
and we see that there are
one, two, three, four,

00:02:20.510 --> 00:02:23.620
five, six, seven, eight, nine columns

00:02:23.620 --> 00:02:26.220
of one, two, three.

00:02:26.220 --> 00:02:30.320
So there's 27 of these rectangles,

00:02:30.320 --> 00:02:32.550
of these equal rectangles
in the shaded area.

00:02:32.550 --> 00:02:35.150
The shaded area is going to be,

00:02:35.150 --> 00:02:40.150
we have 27 of these
rectangles and each of them,

00:02:40.590 --> 00:02:42.250
and then each of them have an area

00:02:42.250 --> 00:02:44.750
of 1/70 of a square meter.

00:02:44.750 --> 00:02:47.250
1/70 of a square meter

00:02:47.250 --> 00:02:48.650
and what does that give us?

00:02:48.650 --> 00:02:50.920
Well that gives us the area of the shaded

00:02:50.920 --> 00:02:54.550
or the shaded area is going to be 27/70.

00:02:54.550 --> 00:02:58.820
27 times 1/70 is going to be 27/70.

00:02:58.820 --> 00:03:03.820
27/70 square meters

00:03:04.290 --> 00:03:05.320
and we're done.

00:03:05.320 --> 00:03:06.520
But what I want to appreciate now

00:03:06.520 --> 00:03:07.790
is that there's multiple ways

00:03:07.790 --> 00:03:09.350
that we could have tackled this.

00:03:09.350 --> 00:03:11.420
Another way we could have tackled it

00:03:11.420 --> 00:03:14.120
is to figure out what the dimensions,

00:03:14.120 --> 00:03:16.850
what the dimensions
are of the shaded area.

00:03:16.850 --> 00:03:18.820
So for example.

00:03:18.820 --> 00:03:23.050
For example, what is the height

00:03:23.050 --> 00:03:24.920
of just the shaded area?

00:03:24.920 --> 00:03:26.790
So just that height right over there

00:03:26.790 --> 00:03:27.820
and I encourage you to pause the video

00:03:27.820 --> 00:03:28.850
and try to think about what it is

00:03:28.850 --> 00:03:30.840
and it's going to be a fraction.

00:03:31.650 --> 00:03:34.420
Well, we see if we're going
in the vertical direction

00:03:34.420 --> 00:03:36.720
we've divided this one meter.

00:03:36.720 --> 00:03:41.550
We've divided it into one,
two, three, four, five.

00:03:41.550 --> 00:03:43.150
So let me do it a little bit differently.

00:03:43.150 --> 00:03:47.690
We've divided it into one, two,

00:03:47.690 --> 00:03:52.220
three, four, five, six,

00:03:52.220 --> 00:03:54.350
seven equal sections.

00:03:54.350 --> 00:03:55.720
That might have been
a little bit confusing

00:03:55.720 --> 00:03:56.820
the way I just drew it.

00:03:56.820 --> 00:03:58.610
So you can see it when
you look at the actual...

00:03:58.610 --> 00:04:01.370
Actually let me do it
in a more vibrant color.

00:04:02.790 --> 00:04:04.290
We have...

00:04:04.290 --> 00:04:06.420
I'm having trouble picking colors.

00:04:06.420 --> 00:04:07.750
All right, here we go.

00:04:07.750 --> 00:04:11.290
We have one, that's that right over there.

00:04:11.290 --> 00:04:16.120
Two, three, four, five, six,

00:04:16.120 --> 00:04:20.550
seven equal sections that we've
divided this one meter in.

00:04:20.550 --> 00:04:24.010
And the height of the shaded
area is three of them.

00:04:24.010 --> 00:04:26.250
So this height right over here,

00:04:26.250 --> 00:04:30.450
this height right over
here is 3/7 of the whole

00:04:30.450 --> 00:04:31.620
and the whole is a meter.

00:04:31.620 --> 00:04:35.090
So it's 3/7 of a meter.

00:04:35.090 --> 00:04:38.550
Now by that same logic what
is the width going to be?

00:04:38.550 --> 00:04:40.950
What is the width going to be?

00:04:40.950 --> 00:04:43.620
Well we can see that the entire meter

00:04:43.620 --> 00:04:47.420
has been divided into one,
two, three, four, five,

00:04:47.420 --> 00:04:50.720
six, seven, eight, nine, 10 equal sections

00:04:50.720 --> 00:04:54.920
so going from here to here
is going to be a tenth.

00:04:54.920 --> 00:04:57.220
So this distance right over
here is going to be a tenth.

00:04:57.220 --> 00:04:59.320
Let me do that in a
color that's different.

00:05:00.950 --> 00:05:04.090
So this distance right over
here is going to be a tenth

00:05:04.090 --> 00:05:06.150
and so how many tenths represent

00:05:06.150 --> 00:05:08.350
the width of the green area?

00:05:08.350 --> 00:05:11.650
Let's see, we get, have 1/10, 2/10,

00:05:11.650 --> 00:05:14.190
3/10, 4/10, 5/10,

00:05:14.190 --> 00:05:17.720
6/10, 7/10, 8/10, 9/10.

00:05:17.720 --> 00:05:21.720
So this width is 9/10 of this whole length

00:05:21.720 --> 00:05:25.590
which is a meter so it's 9/10 of a meter.

00:05:25.590 --> 00:05:27.350
Now to find the area

00:05:27.350 --> 00:05:29.920
we can multiply the width times the height

00:05:29.920 --> 00:05:31.650
or the height times the width.

00:05:31.650 --> 00:05:34.620
So we could say, so I'll write this again.

00:05:34.620 --> 00:05:36.390
The shaded area.

00:05:36.390 --> 00:05:40.250
Shaded area instead of doing it this way

00:05:40.250 --> 00:05:44.320
we could say I have a
height of 3/7 of a meter.

00:05:44.320 --> 00:05:47.950
So 3/7 meters

00:05:47.950 --> 00:05:50.250
and then I can multiply
that, times our width

00:05:50.250 --> 00:05:51.620
for just the shaded area

00:05:51.620 --> 00:05:53.650
which is 9/10 of a meter.

00:05:53.650 --> 00:05:56.390
9/10 of a meter.

00:05:56.390 --> 00:05:58.320
And now what is this going to get us?

00:05:58.320 --> 00:05:59.950
Well, this is going to be equal to

00:05:59.950 --> 00:06:01.450
the meters times the
meters is going to get

00:06:01.450 --> 00:06:03.350
square meters which is what we want

00:06:03.350 --> 00:06:04.720
and then we could multiple the numerators

00:06:04.720 --> 00:06:06.350
and multiply the denominators.

00:06:06.350 --> 00:06:10.750
Three times nine is going to give us 27

00:06:10.750 --> 00:06:13.650
and seven times 10 is going to give us 70.

00:06:13.650 --> 00:06:15.350
Exactly what we had before.

00:06:15.350 --> 00:06:18.150
27/70 square.

00:06:18.150 --> 00:06:19.390
Let me write that a little bit neater.

00:06:19.390 --> 00:06:21.320
27.

00:06:21.320 --> 00:06:25.090
27/70

00:06:25.090 --> 00:06:27.390
of a square meter.

00:06:27.390 --> 00:06:29.890
And you could think about
why did this work out

00:06:29.890 --> 00:06:31.020
regardless of how we did it?

00:06:31.020 --> 00:06:33.620
Notice, three and nine are the numerators.

00:06:33.620 --> 00:06:36.650
That was how many rows and columns we had

00:06:36.650 --> 00:06:38.750
of these little rectangles.

00:06:38.750 --> 00:06:40.150
And then the seven and 10

00:06:40.150 --> 00:06:43.490
that's to figure out how many
rectangles we actually had.

00:06:43.490 --> 00:06:44.520
So this is saying okay,

00:06:44.520 --> 00:06:47.620
the three times nine is how
many rectangles we have?

00:06:47.620 --> 00:06:49.750
And then the seven times 10 is

00:06:49.750 --> 00:06:51.120
what fraction of the whole

00:06:51.120 --> 00:06:52.820
each of those rectangles represent

00:06:52.820 --> 00:06:54.650
and that's essentially
what we did up here.

00:06:54.650 --> 00:06:56.790
So either way, you're going
to get the right answer

00:06:56.790 --> 00:06:59.800
but I really want you to
think about why this was.


Coordinates for triangle example

WEBVTT
Kind: captions
Language: en

00:00:00.341 --> 00:00:01.911
- [Voiceover] "Click on
the graph below to draw

00:00:01.911 --> 00:00:03.801
"a right triangle with right angle

00:00:03.801 --> 00:00:06.121
"at x equals eight, y equals six,

00:00:06.121 --> 00:00:09.128
"and hypotenuse extending
from zero comma 14

00:00:09.128 --> 00:00:11.751
"to negative 10 comma negative 12."

00:00:11.751 --> 00:00:14.027
Well, looks like they've
given us the three vertices

00:00:14.027 --> 00:00:16.105
of the triangle so we
could just plot them.

00:00:16.105 --> 00:00:17.985
So let's just, it says,
"Click to add vertices."

00:00:17.985 --> 00:00:18.728
So let's do that.

00:00:18.728 --> 00:00:20.145
We have one at eight comma six,

00:00:20.145 --> 00:00:24.036
x equals eight, y is equal to six.

00:00:24.036 --> 00:00:26.879
Then we have one at zero comma 14.

00:00:26.879 --> 00:00:30.433
So x equals zero, y is equal to 14

00:00:30.433 --> 00:00:31.919
and it drew the line there.

00:00:31.919 --> 00:00:34.473
And then we go from that point

00:00:34.473 --> 00:00:37.267
to negative 10 comma negative 12.

00:00:37.267 --> 00:00:38.931
Negative 10, wrong side,

00:00:38.931 --> 00:00:41.940
x is negative 10,

00:00:41.940 --> 00:00:43.467
y is negative 12.

00:00:43.467 --> 00:00:45.292
So we go to that point right over there

00:00:45.292 --> 00:00:47.196
and then we can just
connect everything back

00:00:47.196 --> 00:00:48.775
by going to our original point

00:00:48.775 --> 00:00:49.530
and there we have it.

00:00:49.530 --> 00:00:51.773
It even drew the nice right
angle right over there.

00:00:51.773 --> 00:00:52.269
There we have it.

00:00:52.269 --> 00:00:53.627
We have our right triangle.

00:00:53.627 --> 00:00:55.298
We'd really just have to plot the points

00:00:55.298 --> 00:00:56.457
in order actually.

00:00:56.457 --> 00:00:57.644
Any order would have worked

00:00:57.644 --> 00:00:59.605
because there's only three points here.

00:00:59.605 --> 00:01:01.922
So yeah, that was it.


Example finding area with fractional sides

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:01.984
- [Voiceover] So we've
got a rectangle here,

00:00:01.984 --> 00:00:03.711
it's five-ninths of a meter tall,

00:00:03.711 --> 00:00:05.438
and seven-eighths of a meter wide.

00:00:05.438 --> 00:00:06.597
What is its area?

00:00:06.597 --> 00:00:07.571
And I encourage you to pause the video

00:00:07.571 --> 00:00:09.340
to think about that.

00:00:09.340 --> 00:00:10.436
Well one way to think about it,

00:00:10.436 --> 00:00:12.368
is you can say our area,

00:00:12.368 --> 00:00:16.061
our area is just going to be
the width times the height.

00:00:16.061 --> 00:00:18.235
We're just going to multiply
these two dimensions.

00:00:18.235 --> 00:00:21.708
And so the width is
seven-eighths of a meter.

00:00:21.708 --> 00:00:25.568
So it's going to be
seven-eighths of a meter

00:00:25.568 --> 00:00:29.267
times the height, times the height

00:00:29.267 --> 00:00:32.680
which is five-ninths of a meter.

00:00:32.680 --> 00:00:35.178
Times five-ninths of a meter.

00:00:35.178 --> 00:00:36.621
And what's that going to get us?

00:00:36.621 --> 00:00:38.835
Well, that's just going
to be equal to the meters

00:00:38.835 --> 00:00:43.630
times the meters give us square
meters, so meters squared.

00:00:43.630 --> 00:00:44.911
We could write it like that.

00:00:44.911 --> 00:00:47.287
And then we're going have,
and then we're going to have

00:00:47.287 --> 00:00:50.497
seven times, this in a new color,

00:00:50.497 --> 00:00:55.497
we're going to have seven
times five in the numerator

00:00:55.578 --> 00:00:59.702
to get us 35, and then in the denominator,

00:00:59.702 --> 00:01:04.702
in the denominator we are
going to have eight times nine

00:01:04.780 --> 00:01:06.710
to give us 72.

00:01:06.710 --> 00:01:07.503
And we'd be done.

00:01:07.503 --> 00:01:09.494
This is the area of this rectangle here.

00:01:09.494 --> 00:01:13.497
It's 35-72nds of a square meter.

00:01:13.497 --> 00:01:15.102
What I want to do now is
think a little bit deeper

00:01:15.102 --> 00:01:16.910
about why that actually makes sense.

00:01:16.910 --> 00:01:19.044
Or just really another
way of thinking about it.

00:01:19.044 --> 00:01:20.770
And to do that, what I'm going to do

00:01:20.770 --> 00:01:25.770
is I'm going to split this
region into equal rectangles.

00:01:26.337 --> 00:01:28.612
So let's split it into equal rectangles.

00:01:28.612 --> 00:01:32.980
And we see that we have seven,
if we go in the horizontal

00:01:32.980 --> 00:01:35.479
direction we have one,
two, three, four, five,

00:01:35.479 --> 00:01:37.674
six, seven or you could say
in each row we have seven

00:01:37.674 --> 00:01:39.137
of these rectangles.

00:01:39.137 --> 00:01:42.367
In each column you have
one, two, three, four, five

00:01:42.367 --> 00:01:43.810
of these rectangles.

00:01:43.810 --> 00:01:46.512
So you can see we have
five times one, two, three,

00:01:46.512 --> 00:01:48.605
four, five, six, seven.

00:01:48.605 --> 00:01:51.470
So we have five times
seven of these rectangles.

00:01:51.470 --> 00:01:56.470
So, we have--so 35, we have 35 rectangles.

00:01:58.925 --> 00:02:00.996
I'll just write this, 35 rectangles.

00:02:00.996 --> 00:02:04.354
And what's the area of
each of those rectangles?

00:02:04.354 --> 00:02:07.707
Well, if this is
seven-eighths meters wide,

00:02:07.707 --> 00:02:10.165
and this is divided into
seven equal sections

00:02:10.165 --> 00:02:14.614
in the horizontal direction,
that means that each of these

00:02:14.614 --> 00:02:18.515
is exactly one-eighth of a meter wide.

00:02:18.515 --> 00:02:21.645
And by that same logic, each of these,

00:02:21.645 --> 00:02:23.371
if this whole thing is five-ninths,

00:02:23.371 --> 00:02:26.033
and the height of each
of these is one-fifth

00:02:26.033 --> 00:02:29.873
because we have five
rectangles per column,

00:02:29.873 --> 00:02:32.107
then the height of each
of these is going to be

00:02:32.107 --> 00:02:34.179
one-ninth of a meter.

00:02:34.179 --> 00:02:39.077
So what's the area of just
this character right over here?

00:02:39.077 --> 00:02:40.661
Well, it's going to be
one-ninth of a meter

00:02:40.661 --> 00:02:42.814
times one-eighth of a meter.

00:02:42.814 --> 00:02:46.879
So this area, this area right
over there is just going to be

00:02:46.879 --> 00:02:51.836
one-ninth of a meter times
one-eighth of a meter

00:02:51.836 --> 00:02:54.396
which is equal to one times one is one,

00:02:54.396 --> 00:02:57.443
nine times eight is 72,
and meters times meters

00:02:57.443 --> 00:02:59.088
is square meters.

00:02:59.088 --> 00:03:01.079
So the area of each of these 35

00:03:01.079 --> 00:03:03.092
is one-72nd of a square meter.

00:03:03.092 --> 00:03:05.753
So, if I say 35, so the
area of all of them combined

00:03:05.753 --> 00:03:08.637
is going to be 35 times
the area of each of them.

00:03:08.637 --> 00:03:11.706
35 times one-72nd of a square meter.

00:03:11.706 --> 00:03:13.066
And what's that going to be?

00:03:13.066 --> 00:03:15.667
Well, that's going to be
exactly what we got up here.

00:03:15.667 --> 00:03:19.182
35 times one-72nd of a
square meter is going to be

00:03:19.182 --> 00:03:24.182
35, 35-72nds of a square meter.

00:03:27.472 --> 00:03:31.474
And this 35 is the same
one that we had in yellow.

00:03:31.474 --> 00:03:33.302
That's this one right over there.

00:03:33.302 --> 00:03:35.293
So once again, you can
just multiply five-ninths

00:03:35.293 --> 00:03:37.792
times seven-eighths to
get what we have got here.

00:03:37.792 --> 00:03:39.947
But hopefully when we
thought about the area

00:03:39.947 --> 00:03:41.511
of each of these rectangles,

00:03:41.511 --> 00:03:43.095
it might make a little
bit more intuitive sense

00:03:43.095 --> 00:03:45.478
where this number came from.


Visually understanding multiplying fractions and whole numbers

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:03.596
- [Voiceover] Four unit
squares each have 4/5

00:00:03.596 --> 00:00:06.146
of their area shaded,
and we see that here.

00:00:06.146 --> 00:00:08.772
If we had this whole thing
shaded in, that would be

00:00:08.772 --> 00:00:12.169
a whole, but we only
have 4/5 of it shaded in.

00:00:12.169 --> 00:00:14.464
It's split up into five equal sections,

00:00:14.464 --> 00:00:17.242
and 4/5 are shaded in.

00:00:17.242 --> 00:00:19.090
So this is a fifth, that's another fifth,

00:00:19.090 --> 00:00:21.191
that's another fifth,
and that's another fifth,

00:00:21.191 --> 00:00:24.163
so we have 4/5 shaded in.

00:00:24.163 --> 00:00:27.416
And we have these 4/5 four times,

00:00:27.416 --> 00:00:29.578
so you could view this entire shaded area

00:00:29.578 --> 00:00:32.431
as four times 4/5.

00:00:32.431 --> 00:00:36.207
So all of this we could view

00:00:36.207 --> 00:00:39.494
as four times 4/5.

00:00:39.494 --> 00:00:40.564
Well, what is this going to be?

00:00:40.564 --> 00:00:43.175
What is the total area that is shaded?

00:00:43.175 --> 00:00:44.765
And I encourage you to
pause the video and try to

00:00:44.765 --> 00:00:46.315
think about it.

00:00:46.315 --> 00:00:49.251
Well, each of these bars,
this right over there,

00:00:49.251 --> 00:00:51.697
that is a fifth.

00:00:51.697 --> 00:00:52.585
We've already seen that.

00:00:52.585 --> 00:00:53.954
If you took a whole and you split it into

00:00:53.954 --> 00:00:55.419
five equal sections,

00:00:55.419 --> 00:00:57.499
one of those sections
is going to be a fifth.

00:00:57.499 --> 00:00:58.904
That's a fifth right over there.

00:00:58.904 --> 00:01:02.374
So how many fifths are shaded in?

00:01:02.374 --> 00:01:05.112
Well, that's, let me do
this in a color you can see.

00:01:05.112 --> 00:01:06.754
This is one.

00:01:06.754 --> 00:01:08.247
I didn't switch the colors.

00:01:08.247 --> 00:01:12.919
This is one, 2/5, 3/5, 4/5,

00:01:12.919 --> 00:01:16.948
5/5, 6/5, 7/5, 8/5,

00:01:16.948 --> 00:01:21.674
9/5, 10/5, 11/5, 12/5,

00:01:21.674 --> 00:01:24.150
13/5, 14/5,

00:01:24.150 --> 00:01:27.236
15/5,16/5.

00:01:27.236 --> 00:01:30.559
We have 16/5 shaded in.

00:01:30.559 --> 00:01:32.554
So what is the total area that is shaded?

00:01:32.554 --> 00:01:37.174
We can say 16/5

00:01:37.174 --> 00:01:41.205
are shaded in or 16/5 square units,

00:01:41.205 --> 00:01:43.534
where whole would be one square unit,

00:01:43.534 --> 00:01:46.802
but we have 16/5 of the square units

00:01:46.802 --> 00:01:48.698
actually shaded in.

00:01:48.698 --> 00:01:50.869
Let's do another example.

00:01:51.369 --> 00:01:54.522
What is 2/3 of six?

00:01:54.522 --> 00:01:56.335
Well, we have six triangles here,

00:01:56.335 --> 00:01:58.531
and you can see that they're
giving us a little hint,

00:01:58.531 --> 00:02:00.877
and they've divided it into thirds.

00:02:00.877 --> 00:02:02.772
They've divided it into
three equal groups.

00:02:02.772 --> 00:02:06.978
Notice that's one equal group,
that is two equal groups,

00:02:06.978 --> 00:02:09.200
and that is three equal groups.

00:02:09.200 --> 00:02:12.011
So each third, so each of these thirds,

00:02:12.011 --> 00:02:14.350
this is 1/3 of six.

00:02:14.350 --> 00:02:16.898
We see that 1/3 of six is two,

00:02:16.898 --> 00:02:20.547
but we want 2/3 of six,
so that's going to be

00:02:20.547 --> 00:02:23.583
two of these groups of 1/3.

00:02:23.583 --> 00:02:26.150
So that's one group of 1/3 of six,

00:02:26.150 --> 00:02:28.556
and that's another 1/3 of six.

00:02:28.556 --> 00:02:31.016
1/3 of six.

00:02:31.016 --> 00:02:32.537
So if you took both of these together,

00:02:32.537 --> 00:02:35.807
you're going to have 2/3 of six,

00:02:35.807 --> 00:02:39.834
and we see that that is one,
two, three, four triangles.

00:02:39.834 --> 00:02:42.310
So that is four triangles.

00:02:42.310 --> 00:02:43.558
Let's do another one,

00:02:43.558 --> 00:02:46.323
and this one's a little
bit more interesting.

00:02:46.323 --> 00:02:49.016
What is 5/8 of four?

00:02:49.016 --> 00:02:51.462
And we have drawn here four holes.

00:02:51.462 --> 00:02:53.323
This is a hole, this is
a hole, this is a hole,

00:02:53.323 --> 00:02:55.766
and that's a hole, and we
see that they're split.

00:02:55.766 --> 00:03:00.520
Each of the holes is split
in two, so this is 1/2.

00:03:00.520 --> 00:03:02.025
This is another 1/2.

00:03:02.025 --> 00:03:03.212
This is 1/2.

00:03:03.212 --> 00:03:04.857
That's another 1/2.

00:03:04.857 --> 00:03:06.134
That's 1/2.

00:03:06.134 --> 00:03:07.666
That's another 1/2.

00:03:07.666 --> 00:03:08.850
That's 1/2.

00:03:08.850 --> 00:03:10.221
That's another 1/2.

00:03:10.221 --> 00:03:13.307
Now, why do they do this?
Why did they give us a hint?

00:03:13.307 --> 00:03:16.099
Well, we see that each half is actually

00:03:16.099 --> 00:03:20.139
1/8 of the entire four holes.

00:03:20.139 --> 00:03:21.613
What am I talking about?

00:03:21.613 --> 00:03:23.697
Well, if I were to take
all of these four holes,

00:03:23.697 --> 00:03:27.760
and if I were to divide them
into eight equal sections,

00:03:27.760 --> 00:03:31.062
this is one of the eight equal
sections right over here.

00:03:31.062 --> 00:03:33.825
It just happens to be half of a whole.

00:03:33.825 --> 00:03:37.607
So if I'm talking about, so
one way to think about it is

00:03:37.607 --> 00:03:41.542
if I say 1/8 of four,

00:03:41.542 --> 00:03:43.809
that's going to be equal to 1/2.

00:03:43.809 --> 00:03:44.552
You see it here.

00:03:44.552 --> 00:03:47.139
We have four holes, and if
I were to divide that into

00:03:47.139 --> 00:03:50.528
eight equal sections, that
would mean taking each hole

00:03:50.528 --> 00:03:54.773
and dividing it into two,
so half is 1/8 of four,

00:03:54.773 --> 00:03:55.909
but that's not what they're asking us.

00:03:55.909 --> 00:03:58.301
They're asking, "What's 5/8 of four?"

00:03:58.301 --> 00:04:00.065
So this is 1/8.

00:04:00.065 --> 00:04:02.532
This is 2/8.

00:04:02.532 --> 00:04:03.675
This is 3/8.

00:04:03.675 --> 00:04:04.665
This is 4/8.

00:04:04.665 --> 00:04:06.325
This is 5/8.

00:04:06.325 --> 00:04:08.745
So what is that going to be all together?

00:04:08.745 --> 00:04:10.707
So I just wanna remind you 'cause

00:04:10.707 --> 00:04:11.775
it could be a little confusing.

00:04:11.775 --> 00:04:13.809
I'm saying eighths and
halfs and all of that.

00:04:13.809 --> 00:04:18.800
Remember, 1/8 of four is 1/2, so two.

00:04:18.800 --> 00:04:20.636
Let me shade this in, and we'll do this

00:04:20.636 --> 00:04:22.679
with a different color.

00:04:22.679 --> 00:04:24.623
So 1/8 of four,

00:04:24.623 --> 00:04:26.778
I wanna do a lighter color.

00:04:26.778 --> 00:04:30.193
So 1/8 of four is 1/2,

00:04:30.193 --> 00:04:32.778
2/8 is two halfs,

00:04:32.778 --> 00:04:35.699
3/8 is three halfs.

00:04:35.699 --> 00:04:38.790
4/8 is four halfs,

00:04:38.790 --> 00:04:43.179
and 5/8 is five halfs.

00:04:43.179 --> 00:04:45.458
So I've just shaded. Notice
I took the whole four.

00:04:45.458 --> 00:04:47.458
I divided it into eight equal sections,

00:04:47.458 --> 00:04:49.393
and I've shaded in five of them.

00:04:49.393 --> 00:04:53.022
One, two, three, four, five.

00:04:53.022 --> 00:04:55.309
Now, how many is that total?

00:04:55.309 --> 00:04:57.643
Well, each of those are a half.

00:04:57.643 --> 00:04:59.944
This is (mumbles).

00:04:59.944 --> 00:05:02.099
This is 1/2,

00:05:02.099 --> 00:05:04.723
this is another half, that's another half,

00:05:04.723 --> 00:05:06.995
that's another half,
and that's another half,

00:05:06.995 --> 00:05:10.950
so altogether I have five
halfs that I have shaded in.

00:05:10.950 --> 00:05:13.546
So that is five halfs.

00:05:13.546 --> 00:05:15.646
Let's do one more.

00:05:15.646 --> 00:05:18.722
What is 4/5 of ten?

00:05:18.722 --> 00:05:20.180
And, once again, I encourage
you to pause the video

00:05:20.180 --> 00:05:21.814
and try to think about it.

00:05:21.814 --> 00:05:23.857
Well, we see here on the number line,

00:05:23.857 --> 00:05:27.316
they've taken from zero to ten,
and they have split it into

00:05:27.316 --> 00:05:30.784
five equal sections, so each of this,

00:05:30.784 --> 00:05:33.664
so this right over here,

00:05:33.664 --> 00:05:36.462
this is 1/5 of 10.

00:05:36.462 --> 00:05:37.944
If you go from zero to 10,

00:05:37.944 --> 00:05:39.415
you'd write into five equal sections.

00:05:39.415 --> 00:05:41.285
This is one 1/5 of ten.

00:05:41.285 --> 00:05:42.802
Let me write that down.

00:05:42.802 --> 00:05:46.165
This is 1/5

00:05:46.165 --> 00:05:48.259
of 10 right over here.

00:05:48.259 --> 00:05:49.174
Now, what number is that?

00:05:49.174 --> 00:05:52.048
Well, we see that that number is two.

00:05:52.048 --> 00:05:54.002
Two is 1/5 of 10.

00:05:54.002 --> 00:05:57.836
1/5 of 10 is equal to two.

00:05:57.836 --> 00:05:59.561
Well, they're not just
asking us 1/5 of 10.

00:05:59.561 --> 00:06:01.867
They're asking us 4/5 of 10.

00:06:01.867 --> 00:06:03.736
So this is 1/5 of 10,

00:06:03.736 --> 00:06:05.551
2/5 if we go,

00:06:05.551 --> 00:06:07.064
actually, let me go this way.

00:06:07.064 --> 00:06:10.969
So let me do it with another color.

00:06:10.969 --> 00:06:13.623
So this is 1/5 of 10.

00:06:13.623 --> 00:06:15.806
Now we're gonna go 2/5 of 10,

00:06:15.806 --> 00:06:20.197
3/5 of 10, and we go all
the way to 4/5 of 10,

00:06:20.197 --> 00:06:22.695
which get us to this
number right over there.

00:06:22.695 --> 00:06:24.556
Now, what number is that?

00:06:24.556 --> 00:06:27.143
Well, we see this is zero,
one, two, three, four,

00:06:27.143 --> 00:06:30.923
five, six, seven, eight,
so it's the number eight.

00:06:30.923 --> 00:06:32.788
So what is 4/5 of 10?

00:06:32.788 --> 00:06:34.937
It is the number eight.


Multiplying fractions by whole numbers on the number line

WEBVTT
Kind: captions
Language: en

00:00:00.482 --> 00:00:02.442
- Whereas to move the orange dot

00:00:02.442 --> 00:00:07.039
to the number that equals five times 1/3.

00:00:07.039 --> 00:00:08.572
Alright, so one way to think about it,

00:00:08.572 --> 00:00:11.753
we just have to move 1/3 five times.

00:00:11.753 --> 00:00:14.098
So let's do it once,

00:00:14.098 --> 00:00:15.514
so that's going to be 1/3,

00:00:15.514 --> 00:00:17.534
you do it twice, you get to 2/3,

00:00:17.534 --> 00:00:20.274
you do it three times, you get to 3/3,

00:00:20.274 --> 00:00:22.851
four times, you get to 4/3,

00:00:22.851 --> 00:00:25.614
five times, you get to 5/3.

00:00:25.614 --> 00:00:30.614
Five times 1/3 is gonna be 5/3.

00:00:30.740 --> 00:00:32.534
Or you could say five 1/3,

00:00:32.534 --> 00:00:34.925
which is the same thing as 5/3,

00:00:34.925 --> 00:00:36.667
hopefully that makes some sense.

00:00:36.667 --> 00:00:39.499
Let's do some more examples here.

00:00:39.499 --> 00:00:41.194
So let's say we need to figure out,

00:00:41.194 --> 00:00:43.240
so let's see, it says move the orange dot

00:00:43.240 --> 00:00:47.231
to the number that equals two times 4/3.

00:00:47.231 --> 00:00:52.231
Alright, so one times 4/3
is just gonna get us to 4/3,

00:00:52.252 --> 00:00:54.499
and then if we have another 4/3,

00:00:54.499 --> 00:00:56.055
we're gonna add 4/3 to that,

00:00:56.055 --> 00:00:58.506
so we're gonna move
another 4/3 to the right.

00:00:58.506 --> 00:01:02.904
So 4/3 plus 4/3 would get us to 8/3.

00:01:02.904 --> 00:01:06.201
8/3, I'm having trouble moving this, 8/3.

00:01:06.201 --> 00:01:09.196
So one times 4/3 is just 4/3,

00:01:09.196 --> 00:01:12.679
and then two times 4/3 is 8/3.

00:01:12.679 --> 00:01:15.512
And notice that's the same
thing as two times four,

00:01:15.512 --> 00:01:19.018
which is eight, over three, 8/3.

00:01:19.018 --> 00:01:21.154
Let's do one more of these.

00:01:21.154 --> 00:01:23.081
So, move the orange dot to the number

00:01:23.081 --> 00:01:26.262
that equals three times 3/2.

00:01:26.262 --> 00:01:29.251
So this is gonna be 0/2, that's just zero,

00:01:29.251 --> 00:01:32.044
so you could do that as zero times 3/2.

00:01:32.044 --> 00:01:35.922
One times 3/2, well that
will just get us to 3/2.

00:01:35.922 --> 00:01:38.824
Two times 3/2, we'll add another 3/2,

00:01:38.824 --> 00:01:41.123
so that'll get us to 6/2.

00:01:41.123 --> 00:01:42.771
And then three times 3/2,

00:01:42.771 --> 00:01:44.327
we'll add another 3/2,

00:01:44.327 --> 00:01:48.498
that gets us to 9/2, and we're done.


Adding mixed numbers with regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.419 --> 00:00:04.500
- So we have here 2 2/3 plus 8 3/4

00:00:04.500 --> 00:00:06.073
and I encourage you to pause the video

00:00:06.073 --> 00:00:08.848
and see if you can compute what this is.

00:00:08.848 --> 00:00:10.246
So now let's work through this together

00:00:10.246 --> 00:00:12.503
and there's a bunch of ways
that you can tackle this.

00:00:12.503 --> 00:00:14.721
I'll do it in a couple of different ways.

00:00:14.721 --> 00:00:16.409
So one way that you might see

00:00:16.409 --> 00:00:18.887
is to just rewrite this expression,

00:00:18.887 --> 00:00:23.211
so just rewrite 2 2/3 plus,

00:00:23.859 --> 00:00:26.220
plus 8 3/4.

00:00:26.794 --> 00:00:30.033
Plus 8 3/4 and you might be saying

00:00:30.033 --> 00:00:32.064
why whould I rewrite it this way?

00:00:32.064 --> 00:00:34.014
Well, when you rewrite it
this way, it's very clear

00:00:34.014 --> 00:00:36.781
that these are the fraction
parts of the mixed numbers

00:00:36.781 --> 00:00:38.807
and these are the whole number
parts of the mixed numbers,

00:00:38.807 --> 00:00:40.630
and maybe we could add them separately,

00:00:40.630 --> 00:00:42.899
but before we even start
adding the fraction parts

00:00:42.899 --> 00:00:44.617
of the mixed numbers, we notice

00:00:44.617 --> 00:00:46.323
that we have different denominators here.

00:00:46.323 --> 00:00:48.151
We have a three and we have a four,

00:00:48.151 --> 00:00:50.821
so it would be nice to
find a common denominator

00:00:50.821 --> 00:00:52.719
and we've seen multiple times in the past,

00:00:52.719 --> 00:00:55.157
when you're adding fractions
with unlike denominators,

00:00:55.157 --> 00:00:56.892
a good common denominator would be

00:00:56.892 --> 00:00:58.266
their least common multiple,

00:00:58.266 --> 00:01:00.710
the least common multiple
of three and four.

00:01:00.710 --> 00:01:02.115
Well, what's that going to be?

00:01:02.115 --> 00:01:04.314
Well, let's look at the multiples of four.

00:01:04.314 --> 00:01:07.947
Four is not perfectly divisible by three.

00:01:07.947 --> 00:01:10.443
Eight is not perfectly divisible by three.

00:01:10.443 --> 00:01:13.628
12 is, in fact, 12 is four times three,

00:01:13.628 --> 00:01:16.291
so I can rewrite both
of these fraction parts

00:01:16.291 --> 00:01:18.433
with 12 as their denominator.

00:01:18.433 --> 00:01:20.757
So this is going to be equal to,

00:01:20.757 --> 00:01:22.425
let me just write it this way.

00:01:22.425 --> 00:01:26.662
So I can rewrite it as two

00:01:26.662 --> 00:01:29.686
and something over 12.

00:01:29.686 --> 00:01:31.264
Now what would be that something over 12?

00:01:31.264 --> 00:01:33.133
To go from three to 12 in the denominator,

00:01:33.133 --> 00:01:35.048
we multiply it by four,

00:01:35.048 --> 00:01:37.492
so we have to multiply the
numerator by four as well.

00:01:37.492 --> 00:01:40.278
So two times four is eight.

00:01:40.278 --> 00:01:42.605
2/3 is the same thing as 8/12,

00:01:42.605 --> 00:01:46.535
so 2 2/3 is the same thing as 2 8/12,

00:01:46.535 --> 00:01:48.316
and then we can do the
same thing down here.

00:01:48.316 --> 00:01:50.460
8 3/4 is gonna be the same thing as

00:01:50.460 --> 00:01:53.707
eight and something over 12.

00:01:53.707 --> 00:01:56.235
Well, to go from four to
12 in the denominator,

00:01:56.235 --> 00:01:57.703
you multiply it by three,

00:01:57.703 --> 00:02:00.019
so we have to multiply the
numerator by three as well.

00:02:00.019 --> 00:02:04.726
Three times three is
nine, and now, we can add.

00:02:05.259 --> 00:02:07.407
We can add, and what do we get?

00:02:07.407 --> 00:02:11.063
Well, we have 8/12 plus 9/12.

00:02:11.063 --> 00:02:14.294
That's going to be 17,

00:02:14.294 --> 00:02:16.737
17/12.

00:02:16.737 --> 00:02:18.582
Well, you might recognize, 17/12,

00:02:18.582 --> 00:02:22.204
this is greater than or equal
to one, so we can actually,

00:02:22.204 --> 00:02:24.781
this is an improper
fraction right over here,

00:02:24.781 --> 00:02:27.265
so we can actually rewrite this,

00:02:27.265 --> 00:02:29.081
we can rewrite this as a mixed number.

00:02:29.081 --> 00:02:30.207
Let me make it clear.

00:02:30.207 --> 00:02:34.334
17/12, so, do it right over here.

00:02:34.334 --> 00:02:37.551
17/12 is the same thing

00:02:37.551 --> 00:02:40.053
as 1 5/12.

00:02:40.053 --> 00:02:41.341
How did I figure that out?

00:02:41.341 --> 00:02:45.971
12 goes into 17 one time,
and then I still 5 left over.

00:02:45.971 --> 00:02:48.633
17/12 is the same thing as 1 5/12.

00:02:48.633 --> 00:02:51.564
So what I could do is, I
could write the 5/12 part

00:02:51.564 --> 00:02:54.802
in the fraction part of
our, or I guess, say,

00:02:54.802 --> 00:02:57.501
the fraction place on our numbers,

00:02:57.501 --> 00:02:59.293
and then I could regroup the one,

00:02:59.293 --> 00:03:01.158
put it in the whole number column,

00:03:01.158 --> 00:03:03.681
and now I can just add the one plus two

00:03:03.681 --> 00:03:07.159
is three, plus eight is 11.

00:03:07.159 --> 00:03:10.961
So we get 11 5/12,

00:03:10.961 --> 00:03:12.965
11 5/12.

00:03:12.965 --> 00:03:16.057
Now there is other ways that
we could have tackled this.

00:03:16.057 --> 00:03:18.808
What we could've done

00:03:18.808 --> 00:03:23.595
is gone from this place
right over here, the 2 8/12,

00:03:23.595 --> 00:03:26.637
and the 8 9/12, and we
could've converted these

00:03:26.637 --> 00:03:29.998
into improper fractions then
added those improper fractions,

00:03:29.998 --> 00:03:32.075
so we could've said, and
this is the same thing,

00:03:32.075 --> 00:03:33.990
let me rewrite it as, let's see,

00:03:33.990 --> 00:03:37.952
two is the same as 24/12 plus 8,

00:03:37.952 --> 00:03:39.775
is going to be 32/12,

00:03:39.775 --> 00:03:43.141
so I could rewrite this top mixed number

00:03:43.141 --> 00:03:46.650
as 32/12,

00:03:46.650 --> 00:03:48.399
clearly an improper fraction,

00:03:48.399 --> 00:03:51.574
and I could rewrite this bottom
mixed number as, let's see,

00:03:51.574 --> 00:03:53.867
eight is how many 12s?

00:03:53.867 --> 00:03:57.973
That's going to be
96/12 plus another 9/12,

00:03:57.973 --> 00:04:00.057
is 105/12.

00:04:00.057 --> 00:04:01.637
105/12.

00:04:01.637 --> 00:04:04.597
If what I just did looks
a little bit confusing,

00:04:04.597 --> 00:04:06.834
you should review the Khan Academy videos

00:04:06.834 --> 00:04:08.114
on converting mixed numbers

00:04:08.114 --> 00:04:10.741
to improper fractions or vice versa,

00:04:10.741 --> 00:04:13.648
but now I could think
about adding these two,

00:04:13.648 --> 00:04:14.973
and what, what would I get out?

00:04:14.973 --> 00:04:18.762
32/12 plus 105/12, which would be, what,

00:04:18.762 --> 00:04:22.577
137,

00:04:22.577 --> 00:04:25.015
137/12,

00:04:25.015 --> 00:04:29.495
and then if we wanted to
rewrite this as a mixed number,

00:04:29.495 --> 00:04:33.909
we could say, OK let's
see, 12 will go to 137,

00:04:33.909 --> 00:04:36.608
11 times, so it goes 11 times.

00:04:36.608 --> 00:04:39.214
11 times 12 is 132, so you
would have five left over,

00:04:39.214 --> 00:04:42.609
so it's 11 5/12.

00:04:42.609 --> 00:04:43.844
So you could've done it either way.

00:04:43.844 --> 00:04:44.957
I think this one,

00:04:44.957 --> 00:04:46.318
the way we did it the first time

00:04:46.318 --> 00:04:47.542
would've been a little bit easier

00:04:47.542 --> 00:04:48.811
because you didn't have to deal with

00:04:48.811 --> 00:04:52.891
these really big numbers
like 137 and 105 and 32,

00:04:52.891 --> 00:04:54.689
and you just dealt with more
straightforward numbers.

00:04:54.689 --> 00:04:58.664
Say, hey look, 8/12 plus 9/12 is 17/12,

00:04:58.664 --> 00:05:00.770
which is the the same thing as 1 5/12,

00:05:00.770 --> 00:05:03.736
and then you had 1 5/12
plus two plus eight,

00:05:03.736 --> 00:05:06.520
is gonna give you 11 5/12.


Subtracting mixed numbers with regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.260
- [Voiceover] So we have
the expression 17 4/9

00:00:03.260 --> 00:00:05.450
minus 12 and 2/3.

00:00:05.450 --> 00:00:06.830
And I encourage to pause the video

00:00:06.830 --> 00:00:09.905
and see if you can
figure out what this is.

00:00:09.905 --> 00:00:11.660
So now let's work through it.

00:00:11.660 --> 00:00:13.655
So what I'm going to do
is I'm going to rewrite

00:00:13.655 --> 00:00:15.095
these mixed numbers.

00:00:15.095 --> 00:00:19.675
So I'm going to write this as 17 and 4/9

00:00:20.550 --> 00:00:22.695
minus 12 and 2/3.

00:00:22.695 --> 00:00:24.960
I'm going to write the
12 right under the 17,

00:00:24.960 --> 00:00:27.810
and I'm going to write the
2/3 right under the 4/9.

00:00:27.810 --> 00:00:30.555
Let me make it very clear,
we are subtracting 12 and 2/3

00:00:30.555 --> 00:00:32.685
from 17 and 4/9.

00:00:32.685 --> 00:00:33.985
So the first thing that
we might want to do

00:00:33.985 --> 00:00:35.385
is we could look at the fraction parts

00:00:35.385 --> 00:00:37.950
and we might want to start
subtracting until we see,

00:00:37.950 --> 00:00:39.555
look, we have different denominators here.

00:00:39.555 --> 00:00:41.850
We have ninths and we have thirds.

00:00:41.850 --> 00:00:43.150
So the first thing we'd want to do

00:00:43.150 --> 00:00:44.950
is let's get to a common denominator,

00:00:44.950 --> 00:00:46.690
and a good common denominator
would be the least

00:00:46.690 --> 00:00:49.885
common multiple of nine and three.

00:00:49.885 --> 00:00:50.980
Well, what's that going to be?

00:00:50.980 --> 00:00:52.690
Well to think about that,

00:00:52.690 --> 00:00:55.315
I like to start with
a larger number, nine.

00:00:55.315 --> 00:00:57.595
And say, well, is that divisible by three?

00:00:57.595 --> 00:00:59.215
Well, yes, it is divisible by three.

00:00:59.215 --> 00:01:00.715
It is divisible by the other denominator,

00:01:00.715 --> 00:01:02.755
so this actually is the
least common multiple.

00:01:02.755 --> 00:01:04.975
If it wasn't, I would keep
taking higher and higher

00:01:04.975 --> 00:01:05.990
multiples of 9.

00:01:05.990 --> 00:01:08.730
I would go to 18 and
then I would go to 27,

00:01:08.730 --> 00:01:10.155
and I would keep going until I found one

00:01:10.155 --> 00:01:12.125
that's divisible by three.

00:01:12.125 --> 00:01:13.135
But I didn't have to do that

00:01:13.135 --> 00:01:15.090
because nine is divisible by three.

00:01:15.090 --> 00:01:17.415
So I can rewrite both
of these fraction parts

00:01:17.415 --> 00:01:19.830
in terms of ninths.

00:01:19.830 --> 00:01:22.110
Now the one on top already is
written in terms of ninths,

00:01:22.110 --> 00:01:23.640
so I can just rewrite that.

00:01:23.640 --> 00:01:27.040
17 and 4/9.

00:01:27.825 --> 00:01:30.555
And the one on the
bottom, I can write as 12

00:01:30.555 --> 00:01:33.510
and some number of ninths.

00:01:33.510 --> 00:01:36.585
So 2/3 is how many ninths?

00:01:36.585 --> 00:01:39.875
Well, to go from thirds to ninths

00:01:39.875 --> 00:01:44.150
I had to multiply by three,
had to multiply by three,

00:01:44.150 --> 00:01:46.955
so the numerator, I need to
multiply by three as well.

00:01:46.955 --> 00:01:49.775
Two times three is six.

00:01:49.775 --> 00:01:52.790
2/3 is the same thing as 6/9.

00:01:52.790 --> 00:01:56.090
And now I can try to subtract.

00:01:56.090 --> 00:01:57.695
But even here, when I try to subtract,

00:01:57.695 --> 00:01:59.630
I have a larger fraction down here

00:01:59.630 --> 00:02:01.325
that I'm trying to subtract
from a smaller one.

00:02:01.325 --> 00:02:04.325
I have 4/9 minus 6/9.

00:02:04.325 --> 00:02:05.495
So what can I do?

00:02:05.495 --> 00:02:08.330
Well, be answer is I can regroup.

00:02:08.330 --> 00:02:12.095
I can take a whole from the 17.

00:02:12.095 --> 00:02:13.790
Let me do that.

00:02:13.790 --> 00:02:16.325
So if I take a whole from the 17,

00:02:16.325 --> 00:02:19.520
that's going to become 16,

00:02:19.520 --> 00:02:21.710
and then that whole that I just took from,

00:02:21.710 --> 00:02:23.660
I guess you could say
the whole number place,

00:02:23.660 --> 00:02:25.325
I can add it to the fraction.

00:02:25.325 --> 00:02:28.785
Well, a whole is just going to be 9/9.

00:02:28.785 --> 00:02:30.750
So all I did was I regrouped here.

00:02:30.750 --> 00:02:33.150
I took 9/9 from 17.

00:02:33.150 --> 00:02:36.895
9/9 is one, so I took 9/9
from 17, I'm left with 16,

00:02:36.895 --> 00:02:38.690
and then I regrouped them and I added them

00:02:38.690 --> 00:02:41.450
to the fractions place, as
one way to think about it.

00:02:41.450 --> 00:02:43.760
Well, what's 4/9 plus 9/9?

00:02:43.760 --> 00:02:46.220
Well, that's going to be 13/9.

00:02:46.220 --> 00:02:49.590
So this right over here is 13/9.

00:02:49.590 --> 00:02:51.285
13/9.

00:02:51.285 --> 00:02:53.980
It's a very strange way to
write it, but 17 and 4/9

00:02:53.980 --> 00:02:58.690
is the same thing as 16
and 13/9, because notice,

00:02:58.690 --> 00:03:00.295
this is greater than one.

00:03:00.295 --> 00:03:02.855
This is the same thing as one and 4/9.

00:03:02.855 --> 00:03:06.310
One and 4/9 plus 16 is
going to be 17 and 4/9.

00:03:06.310 --> 00:03:07.930
Now why did I do all of this?

00:03:07.930 --> 00:03:12.425
Well 13/9 is larger than
6/9, so I can subtract.

00:03:12.425 --> 00:03:16.850
What's 13/9 minus 6/9 going to be?

00:03:16.850 --> 00:03:19.415
Well, 13 of something,
in this case, ninths,

00:03:19.415 --> 00:03:23.780
minus six of that same
something is going to be 7/9.

00:03:23.780 --> 00:03:27.015
Seven, let me write
that in a neutral color.

00:03:27.015 --> 00:03:30.695
So that's going to be 7/9.

00:03:30.695 --> 00:03:34.325
13/9 minus 6/9 is 7/9.

00:03:34.325 --> 00:03:35.955
And then I can look over
in the whole number place.

00:03:35.955 --> 00:03:37.785
All I have left a 16 here.

00:03:37.785 --> 00:03:41.295
16 minus 12 is four, and I'm done.

00:03:41.295 --> 00:03:44.395
17 and 4/9 minus 12 and 2/3

00:03:44.395 --> 00:03:49.395
is equal to four and 7/9.


Visually subtracting fractions with unlike denominators

WEBVTT
Kind: captions
Language: en

00:00:00.095 --> 00:00:04.160
- Let's see if we can figure
out with 3/4 minus 5/8 is.

00:00:04.160 --> 00:00:07.055
And we have 3/4 depicted right over here.

00:00:07.055 --> 00:00:11.980
You could view this entire bar

00:00:11.980 --> 00:00:15.325
as a whole, and we see that it is divided

00:00:15.325 --> 00:00:17.440
into four equal sections,

00:00:17.440 --> 00:00:18.990
and that three of them are shaded in.

00:00:18.990 --> 00:00:20.820
So those three that are shaded in,

00:00:20.820 --> 00:00:23.250
those represent 3/4 of the whole.

00:00:23.250 --> 00:00:24.840
So you see that right over there,

00:00:24.840 --> 00:00:27.030
and then this bar down
here, you could view this

00:00:27.030 --> 00:00:29.540
as another whole.

00:00:29.540 --> 00:00:31.195
This is another whole right over here

00:00:31.195 --> 00:00:33.830
and you could see this divided
into eight equal pieces,

00:00:33.830 --> 00:00:36.125
and five of them are shaded in.

00:00:36.125 --> 00:00:40.145
So that represents,
that represents the 5/8.

00:00:40.145 --> 00:00:43.685
So we want to have 3/4,
this green shaded area,

00:00:43.685 --> 00:00:46.215
and we want to take away the 5/8.

00:00:46.215 --> 00:00:47.190
So how could we do it?

00:00:47.190 --> 00:00:48.385
And even when you look at it visually,

00:00:48.385 --> 00:00:49.960
it might jump out at you.

00:00:49.960 --> 00:00:51.850
Whenever we add or subtract fractions,

00:00:51.850 --> 00:00:54.790
we like to think in terms of
having the same denominator.

00:00:54.790 --> 00:00:57.925
Are we going to deal in
fourths or eighths or 16s,

00:00:57.925 --> 00:00:59.215
or whatever else?

00:00:59.215 --> 00:01:01.585
So let's think about having
a common denominator.

00:01:01.585 --> 00:01:03.685
And a good common denominator

00:01:03.685 --> 00:01:06.595
is going to be a common
multiple of the two denominators

00:01:06.595 --> 00:01:09.955
right over here, and ideally
their least common multiple.

00:01:09.955 --> 00:01:11.830
And one way that I like to tackle that,

00:01:11.830 --> 00:01:13.360
there's many ways to do it,

00:01:13.360 --> 00:01:15.325
is look at the larger
of the two denominators,

00:01:15.325 --> 00:01:18.250
look at eight, and then keep
looking at increasing multiples

00:01:18.250 --> 00:01:20.795
of eight until you find one
that's also divisible by four,

00:01:20.795 --> 00:01:22.925
perfectly divisible by four.

00:01:22.925 --> 00:01:24.515
But with eight, you immediately say,

00:01:24.515 --> 00:01:26.450
"Well, eight is divisible by four,"

00:01:26.450 --> 00:01:28.720
and that's clearly
divisible by itself as well,

00:01:28.720 --> 00:01:31.180
so eight is actually the
least common multiple

00:01:31.180 --> 00:01:32.455
of four and eight.

00:01:32.455 --> 00:01:34.495
So you can rewrite both of these,

00:01:34.495 --> 00:01:37.260
both of these fractions
as something over eight.

00:01:37.260 --> 00:01:41.115
So the 3/4, you can write
it as something over eight,

00:01:41.115 --> 00:01:43.620
and then subtracting from that,

00:01:43.620 --> 00:01:46.095
the 5/8, if you want to write
that as something over eight,

00:01:46.095 --> 00:01:49.285
well, that's just going to be 5/8.

00:01:49.285 --> 00:01:51.850
And then you can figure
out your actual answer.

00:01:51.850 --> 00:01:54.695
So how can we rewrite 3/4
to something over eight?

00:01:54.695 --> 00:01:57.155
Well, there's a couple of
ways to think about doing it.

00:01:57.155 --> 00:01:58.060
One way, look.

00:01:58.060 --> 00:01:59.470
I had four in the denominator,

00:01:59.470 --> 00:02:02.090
now I'm going to have twice
as many equal sections.

00:02:02.090 --> 00:02:04.475
I'm multiplied by two,
so I'm going to have

00:02:04.475 --> 00:02:07.850
twice as many of the
sections actually shaded in.

00:02:07.850 --> 00:02:12.045
So times two, 3/4 is
the same thing as 6/8.

00:02:12.485 --> 00:02:14.525
And we can also see that visually.

00:02:14.525 --> 00:02:16.575
If we're going to have twice
as many equal sections,

00:02:16.575 --> 00:02:18.720
here we have everything in fourths,

00:02:18.720 --> 00:02:20.350
but I'm going to divide,
I'm going to turn this

00:02:20.350 --> 00:02:23.240
into twice as many equal
sections so I have eighths.

00:02:23.240 --> 00:02:24.845
So let's do that.

00:02:24.845 --> 00:02:26.550
So let me...

00:02:26.550 --> 00:02:29.085
So you have this right here.

00:02:29.085 --> 00:02:30.495
Let me divide that.

00:02:30.495 --> 00:02:32.430
Let me divide that.

00:02:32.430 --> 00:02:36.720
Let me divide that, and
then let me divide that,

00:02:36.720 --> 00:02:38.925
and now I went from fourths to eights.

00:02:38.925 --> 00:02:41.925
I have one, two, three, four, five, six,

00:02:41.925 --> 00:02:44.010
seven, eight equal sections,

00:02:44.010 --> 00:02:46.755
and we see that six of them are shaded in,

00:02:46.755 --> 00:02:49.425
that 3/4 is the same thing as 6/8.

00:02:49.425 --> 00:02:51.660
But regardless, now we can subtract.

00:02:51.660 --> 00:02:56.055
We have 6/8 and we want to
take away five of the eighths.

00:02:56.055 --> 00:02:59.250
So we have 6/8, and we want to take away

00:02:59.250 --> 00:03:03.025
one, two, three, four, five of them,

00:03:03.025 --> 00:03:04.985
and those five of them
correspond to these purple five

00:03:04.985 --> 00:03:06.290
right over here.

00:03:06.290 --> 00:03:08.665
We're taking away one,
two, three, four, five.

00:03:08.665 --> 00:03:09.555
We're taking these away.

00:03:09.555 --> 00:03:11.445
So if you're just looking at the green,

00:03:11.445 --> 00:03:14.020
we started with 6/8, we're taking away

00:03:14.020 --> 00:03:19.020
one, two, three, four, five of them,

00:03:19.195 --> 00:03:22.360
and you can see that
corresponds to the 5/8 down here

00:03:22.360 --> 00:03:23.830
and what are you left with?

00:03:23.830 --> 00:03:25.795
Well, you're just going to be left with,

00:03:25.795 --> 00:03:30.190
you're just going to be left
with this 1/8 right over there.

00:03:30.190 --> 00:03:31.850
So it's just going to be 1/8.

00:03:31.850 --> 00:03:34.350
And you could see that
numerically up here.

00:03:34.350 --> 00:03:37.185
If I have six of something,
in this case it's 6/8,

00:03:37.185 --> 00:03:38.895
and I want to subtract
five of that something,

00:03:38.895 --> 00:03:40.225
in this case 5/8,

00:03:40.225 --> 00:03:45.001
I'm going to be left with one
of that something, or 1/8.