Background and introduction to the United States Declaration of Independence

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Kind: captions
Language: en

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Male: I'm here with Walter Isaacson and

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what are we about to talk about?

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Walter: We're talking
about the Declaration

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of Independence, which
happens, as it says up top,

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on July 4, 1776, but
what we have to remember

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is that for more than a
year, since April 1775 there

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had been a lot of fighting going on.

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There was a revolution happening,

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but up until this point
the fighting was mainly

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against what they
considered to be the acts

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of Parliament and Parliament's
ministers and taxing

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the colonies.

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Finally, with this document they decide

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they're going to become ...

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The American colonies are going to become

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free and independent, a separate country,

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which means rebelling
against the King himself,

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George the III.

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Male: And this is George the
III in all of his regality.

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(laughing)

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Walter: Right. You know up until they met

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in Congress with the Continental Congress

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gathering themselves
together they pretended

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at least to respect George
the III, and they were

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blaming everything on
the British Parliament,

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but it was a pretty
difficult thing to decide you

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were going to overthrow the King himself.

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Male: And just to be clear,

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this Continental Congress it's easy now

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for us in hindsight.

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It seems like a very official thing,

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but this was really a rebel congress.

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It wasn't sanctioned in any way by

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kind of the formal government,

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by the government of Great Britain.

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Walter: Right, and as
you see it says the 13

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United States of America.

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This is the first time
they really start using

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the phrase United States of America.

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They weren't really a country yet.

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They were 13 different colonies,

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and not all of them wanted to come to

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this Continental Congress.

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Getting them all together
was quite difficult.

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They do so partly to
help George Washington's

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troops get funded because
they've started the

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skirmishes up in Lexington
and Concord and Massachusetts,

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but by 1776 George
Washington has a real army

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and they have to fund
him, and eventually they'd

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figure out well this
congress ought to decide are

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we really having a revolution?

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Are we trying to break
away from the King himself?

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And the answer here is yes.

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Male: Wow. And so this is the beginning

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of the Declaration of Independence

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and these three fellows
on the right-hand side

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look very familiar.

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Walter: They're on the
committee that the Congress

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appoints to draft the Declaration.

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Actually there were five
people on the committee,

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but these of course are
the three most important.

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Thomas Jefferson only
33 years old, by far the

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youngest person on the committee

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and he's chosen to write the first draft.

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Then Benjamin Franklin, who is a mentor

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of Jefferson's, a printer
from Philadelphia.

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Franklin had just been spending
the past two decades almost

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going back and forth to
England to try to prevent

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a revolution.

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Then John Adams, the
very passionate sage from

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the State of Massachusetts
who was the one who

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was most in favor of revolution.

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In fact, when Franklin comes back

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to Philadelphia in early
1776 after having tried

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to hold Britain and America
together most of these

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people didn't know whether
he would be on the side

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of revolution or not.

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In fact, his own son, William Franklin,

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is, at this time of this Declaration,

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the Royal Governor of New Jersey

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and is staying loyal to King George.

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Male: And just to, once
again emphasize the context,

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the Royal Governor of New Jersey.

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This wasn't like the
Governor of New Jersey

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we imagine now.

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He wasn't elected by the people.

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He was appointed.

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Walter: No, he was appointed by the King.

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He was the Royal Governor, and you know

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Franklin was proud of
his son, but they have

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this incredible split
starting in 1776 where

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William Franklin remains
loyal to the crown

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and loyal to King George the III who had

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made him Governor of New Jersey.

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Male: Now, one thing
that you had mentioned

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a few seconds ago that
I think is surprising

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is when you mentioned that
Jefferson was 33 years

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old when he wrote the
Declaration of Independence.

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How normal was that?

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You know back then it seemed like people

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did I guess mature faster, but he was

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still perceived as a fairly young person.

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Walter: Yes. They all loved to be thinking

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of themselves as young rebels too.

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In fact, Jefferson I think
is the second youngest

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person in the Continental Congress,

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but there was a third person
who lied about his age

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to pretend to be younger
and actually wasn't.

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Jefferson was a good wordsmith.

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He was from Virginia and
it was very important

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of this person from ...

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Franklin from Pennsylvania and Adams from

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Massachusetts to make
sure we got Virginia in

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because Virginia, there
was a chance it would

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remain loyal and ...

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Male: It was a large wealthy ...

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Walter: A large wealthy land-owning colony

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and so getting the
Virginians in, and there were

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very strong rebels from Virginia.

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The Lees of Virginia as
well as Jefferson were

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in favor of declaring independence

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so they decide they want
to make sure that Jefferson

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gets to write the first draft.

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Male: Interesting.

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And so what we see here,
this is the final text.

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This is the official
Declaration of Independence.

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Walter: Right.

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Male: In the future videos
we can talk about previous

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drafts.

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Walter: Right. They
went through five drafts

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to get to this draft and this is the

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one that they do after unaninously

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all 13 colonies, now
called the 13 United States

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in this document, declare
this to be the cause

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of the colonies.

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And what you can see
in the first paragraph

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is they have to explain why
are we writing this document.

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They say well if you're
going to have a revolution,

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if you're going to dissolve
the political bands

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which have connected you
with another state, then

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the laws of nature and
of nature's God entitle

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them a decent respect.

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This is the equal station
they get because of nature

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and nature's God.

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It's a pretty interesting phrase.

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Male: Let's read the whole thing.

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Walter: Yeah.

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Male: "When in the course of human events

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it becomes necessary for
one people to dissolve

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the political bands," this is
what you were talking about,

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"which have connected them with another

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and to assume among
the powers of the earth

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the separate and equal station
which the laws of nature

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and of nature's God entitle them."

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Walter: What they're saying right there

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is that the laws of
nature and the fact that

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nature's God created us
all equally means that

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one set of people don't
have to be subserviant to

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or occupied by or colonies
of another set of people.

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They want to be free and independent,

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and it's interesting
that they use the phrase,

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"Laws of nature and nature's
God," because this is

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sort of the beginning of
the enlightenment where

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we're supposed to understand that nature

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gives us our rights and reasons.

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John Locke, the great British philosopher,

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believed in the laws of nature, and these

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were deists.

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They kind of were religious
a bit, but they didn't

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subscribe to any
particular religious dogma,

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and so they just talk
about nature's God allowing

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us all to be free and equal.

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Male: Right.

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"A decent respect to
the opinions of mankind

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requires that they
should declare the causes

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which impel them to a separation."

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So that's really just saying ...

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Walter: What they're
saying is we care what the

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rest of you think.

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And by the way, it's directed
at one particular people,

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the French, because we're not going to win

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this revolution in the
United States unless

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the French help support us.

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The French, by the way,
were already at war

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with the British.

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There's been a long set of wars throughout

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the 18th century where France and Britian

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were fighting each other.

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So, this document is particularly aimed

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at saying to the French,
you've been fighting

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Britain for a long time so
we have a decent respect

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for all the opinions of
mankind, but yours in particular

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and we're going to tell
you why we're fighting this

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and of course in France at the
time this notion of liberty,

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equality, fraternity,
that's bubbling up as well.

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So, the document is to
try to persuade other

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nations please support us.

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We're explaining why we're
having this revolution.

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Male: And that's I think
important to remember

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for someone in 21st century America.

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It's obviously a major world power now,

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the major world power.

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But back then this was
like a little colony.

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It's kind of a sideshow.

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Walter: Right, right.

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And it's important for
us to remember now too

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that whenever we do something, whenever

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we get involved in the
world we should have

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a decent respect to the
opinions of mankind.

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Male: Right.

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Walter: That's how we
started as a nation saying

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when we do something
we're going to be open.

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We're going to be honest.

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We're going to explain to you why.


Comparing fractions with greater than and less than symbols

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Kind: captions
Language: en

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When you write a fraction,
there are actually

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words for the top number
and the bottom number.

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And the words are
a lot more fancy

00:00:10.160 --> 00:00:13.100
than the word "top number"
and "bottom number."

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What mathematicians
typically use

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is the word "numerator" for the
top number and "denominator"

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for the bottom number.

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And what I want to do now that
we know that the top number is

00:00:31.450 --> 00:00:33.900
the numerator of the fraction
and the bottom number is

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the denominator,
I want to compare

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pairs of fractions that have
either the same denominator

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or the same numerator.

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So let's look at
this first pair.

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I want to compare 4/7 to 3/7.

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And I have two wholes
right over here.

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They're the same hole, and I've
divided them into sevenths.

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I've divided them
into 7 equal chunks.

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And I want to see what's
larger, 4/7 or 3/7.

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So what I can do is,
I can fill in 4/7.

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Let me select 4 out of the 7.

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So, that's 1, 2, 3, 4.

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And the fact that trying to even
get to 4/7 I had to have 3/7

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first gives you good clue
that 4/7 is probably larger,

00:01:16.790 --> 00:01:18.030
or it is larger.

00:01:18.030 --> 00:01:20.790
But now let's color in 3/7,
just so we can compare.

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So 1, 2, 3/7.

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And so it's pretty clear
that on the left-hand side,

00:01:28.970 --> 00:01:33.250
we are shading in
more of the whole

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than on the right-hand side.

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So, 4/7 represents
a larger fraction,

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more of the whole than 3/7.

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And the way that we can state
that comparison mathematically

00:01:43.550 --> 00:01:45.610
is with the greater than symbol.

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We can write 4/7 is
greater than 3/7.

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Now, the greater than
and less than symbols

00:01:54.500 --> 00:01:56.010
can sometimes be confusing.

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This is greater than.

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This is less than.

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And the way that
I remember it is

00:02:06.730 --> 00:02:10.122
that the greater than symbol,
either symbol, the small pointy

00:02:10.122 --> 00:02:11.830
side is always on the
side of the smaller

00:02:11.830 --> 00:02:13.770
number, and the big
open side is always

00:02:13.770 --> 00:02:15.520
on the side of
the larger number.

00:02:15.520 --> 00:02:17.810
So here, big open
side is opening

00:02:17.810 --> 00:02:21.690
towards the 4/7, small pointy
side opening to the 3/7.

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4/7 is greater than 3/7.

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Now, what about 3/7 and 3/4?

00:02:27.600 --> 00:02:29.710
So, here I have
different denominators,

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but I have the same numerator.

00:02:31.781 --> 00:02:33.530
And so I encourage you
to pause this video

00:02:33.530 --> 00:02:36.060
and draw maybe little
boxes like this

00:02:36.060 --> 00:02:39.300
and try to judge for yourself
which of these is a larger

00:02:39.300 --> 00:02:43.330
fraction, represents
a larger number.

00:02:43.330 --> 00:02:45.110
Well, let's color them in.

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So, let's think about 3/7 first.

00:02:47.069 --> 00:02:48.610
And we actually
already drew it here,

00:02:48.610 --> 00:02:50.443
but I can do it really
fast right over here.

00:02:50.443 --> 00:02:52.040
So that is 3/7.

00:02:52.040 --> 00:02:56.610
I've colored in 3 of
the 7 equal groups.

00:02:56.610 --> 00:02:58.230
And what would 3/4 be?

00:02:58.230 --> 00:03:03.930
Well, that's 1/3, 2/4, and 3/4.

00:03:03.930 --> 00:03:07.590
And so it's pretty clear
that 3/4 represents a larger

00:03:07.590 --> 00:03:11.060
fraction of the whole,
that 3/4 is larger,

00:03:11.060 --> 00:03:13.420
or that 3/7 is smaller.

00:03:13.420 --> 00:03:18.120
So we could write that
3/7 is less than 3/4.

00:03:18.120 --> 00:03:20.420
So, notice, same
exact numerator.

00:03:20.420 --> 00:03:22.870
When I divided it-- because
this fraction symbol

00:03:22.870 --> 00:03:24.360
could also be
viewed as division.

00:03:24.360 --> 00:03:27.230
When I have it as
more equal groups,

00:03:27.230 --> 00:03:29.320
so it's a fraction of
more equal groups, so 3

00:03:29.320 --> 00:03:32.370
out of 7 versus 3 out of 4,
this is a smaller number,

00:03:32.370 --> 00:03:34.512
which which makes sense.

00:03:34.512 --> 00:03:35.720
Now, let's compare these two.

00:03:35.720 --> 00:03:39.360
We have the same denominator,
different numerators-- 3/4

00:03:39.360 --> 00:03:41.520
versus 2/4.

00:03:41.520 --> 00:03:43.610
Well, 3/4 we've
already looked at.

00:03:43.610 --> 00:03:46.650
We can just shade in 3 of these.

00:03:46.650 --> 00:03:48.810
So 3 of these fourths.

00:03:48.810 --> 00:03:51.035
So that's 3/4 right over there.

00:03:51.035 --> 00:03:52.660
And then 2/4, well,
we're going to only

00:03:52.660 --> 00:03:55.660
have 2 of the fourths, 1, 2.

00:03:55.660 --> 00:03:58.010
So 2/4 is clearly
the smaller number.

00:03:58.010 --> 00:04:00.040
3/4 is the greater number.

00:04:00.040 --> 00:04:03.710
So, once again, we could
write 3/4 is greater than 2/4.

00:04:03.710 --> 00:04:06.120
And then finally, I encourage
you to pause the video.

00:04:06.120 --> 00:04:10.475
Try to come up with whether
2/4 or 3/6 is a larger number.

00:04:10.475 --> 00:04:12.630
Well, let's color it in again.

00:04:12.630 --> 00:04:14.050
We've already seen 2/4.

00:04:14.050 --> 00:04:16.910
We just have to color
in 2 of our fourths.

00:04:16.910 --> 00:04:20.130
So let's just color in 2 of
our fourths right over there.

00:04:20.130 --> 00:04:23.420
And then 3/6, we've
split our whole

00:04:23.420 --> 00:04:27.370
into 6 equal sections--
1, 2, 3, 4, 5, 6.

00:04:27.370 --> 00:04:28.670
We need to color in 3 of them.

00:04:31.700 --> 00:04:35.750
And as you see, we're coloring
in the exact same amount

00:04:35.750 --> 00:04:36.900
of the whole.

00:04:36.900 --> 00:04:39.640
These two fractions
are equivalent.

00:04:39.640 --> 00:04:41.350
These are equivalent fractions.

00:04:41.350 --> 00:04:44.140
2/4 is equal to 3/6.

00:04:44.140 --> 00:04:46.230
And as you see
here, they're both

00:04:46.230 --> 00:04:48.670
filling in half of the whole.

00:04:48.670 --> 00:04:51.530
If we were to just draw
the whole and split

00:04:51.530 --> 00:04:54.060
it only-- let me do this
in a different color.

00:04:54.060 --> 00:04:57.210
If we have our whole, and
we were to split it only

00:04:57.210 --> 00:05:01.320
into two sections,
we are shading

00:05:01.320 --> 00:05:03.740
in exactly 1 out
of the 2 sections.

00:05:06.520 --> 00:05:10.870
So you could say that
2/4 is equal to 3/6,

00:05:10.870 --> 00:05:13.450
and they both equal 1/2.

00:05:13.450 --> 00:05:17.799
So 1/2 is equal to
2/4 is equal to 3/6.


Whole numbers as fractions

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:05.840
Let's say that this circle right
over here represents one whole.

00:00:05.840 --> 00:00:09.140
And we've divided this circle
into one, two, three, four,

00:00:09.140 --> 00:00:11.220
five equal sections.

00:00:11.220 --> 00:00:20.784
So each of these sections
represents 1/5 of the circle.

00:00:20.784 --> 00:00:21.950
And we've seen this already.

00:00:21.950 --> 00:00:26.680
1/5, 1/5, 1/5, and 1/5.

00:00:26.680 --> 00:00:29.161
And if we were to then
color in some of this-- So,

00:00:29.161 --> 00:00:31.410
let's say we were to color
in three of these sections.

00:00:31.410 --> 00:00:34.620
So, that's one of the
sections right over there,

00:00:34.620 --> 00:00:36.480
another one right over here.

00:00:36.480 --> 00:00:39.270
So, we've colored
in two of the fifths

00:00:39.270 --> 00:00:41.520
and then three of the fifths.

00:00:41.520 --> 00:00:43.800
Notice, you put those
three 1/5's together,

00:00:43.800 --> 00:00:45.450
how much have we now shaded in?

00:00:45.450 --> 00:00:49.440
We've shaded in
3/5 of the whole.

00:00:49.440 --> 00:00:56.200
So the fraction that's
actually shaded in now is 3/5.

00:00:56.200 --> 00:00:58.960
3/5 is what's shaded in.

00:00:58.960 --> 00:01:00.960
Now let's do something
in some ways a little bit

00:01:00.960 --> 00:01:04.040
simpler but also in some
ways kind of interesting.

00:01:04.040 --> 00:01:05.620
Let's start with
the whole again.

00:01:05.620 --> 00:01:08.400
So, once again,
this is one whole.

00:01:08.400 --> 00:01:10.110
Let me label it.

00:01:10.110 --> 00:01:10.825
One whole.

00:01:13.950 --> 00:01:16.170
And instead of dividing it
into five equal sections,

00:01:16.170 --> 00:01:19.250
I'm just going to divide
it into one equal section.

00:01:19.250 --> 00:01:25.810
So, if I were to shade
this in just like

00:01:25.810 --> 00:01:31.360
that-- so I'm shading in my one
whole, my one equal section,

00:01:31.360 --> 00:01:35.020
how many of the equal
sections are now shaded in?

00:01:35.020 --> 00:01:39.080
Well, just to remind ourselves,
there is one equal section,

00:01:39.080 --> 00:01:42.870
and I have shaded in exactly
one of those one equal section.

00:01:42.870 --> 00:01:44.640
I've shaded in the whole thing.

00:01:44.640 --> 00:01:47.310
Or I could say that 1/1--
which you'll never hear someone

00:01:47.310 --> 00:01:48.970
actually say-- is
shaded in, or I

00:01:48.970 --> 00:01:50.950
could say that the whole
thing is shaded in.

00:01:50.950 --> 00:01:53.970
So this is equal to one whole.

00:01:53.970 --> 00:01:54.750
So that's a whole.

00:01:54.750 --> 00:01:55.700
That's interesting.

00:01:55.700 --> 00:01:57.820
And I want you to keep
in mind, remember, look.

00:01:57.820 --> 00:02:00.740
We literally have one,
two, three fifths,

00:02:00.740 --> 00:02:03.480
and we literally call that 3/5.

00:02:03.480 --> 00:02:05.830
Now, this is one whole.

00:02:05.830 --> 00:02:10.350
Now, what happens if we were
to do this multiple times?

00:02:10.350 --> 00:02:11.780
Let me copy and paste that.

00:02:11.780 --> 00:02:14.590
So now I have another
one whole and then

00:02:14.590 --> 00:02:17.240
another one whole
right over here.

00:02:17.240 --> 00:02:21.490
So now, in total, how
many wholes do I have?

00:02:21.490 --> 00:02:22.650
Well, I have three.

00:02:22.650 --> 00:02:26.800
One, two, three wholes.

00:02:26.800 --> 00:02:28.540
And I've actually
shaded in three holes.

00:02:28.540 --> 00:02:30.872
So this one right over
here is equal to-- Let

00:02:30.872 --> 00:02:33.390
me make sure I label it right.

00:02:33.390 --> 00:02:36.400
This right over
here is equal to,

00:02:36.400 --> 00:02:38.590
if I were to take
the combination,

00:02:38.590 --> 00:02:40.890
this is equal to 3.

00:02:40.890 --> 00:02:42.997
3 holes.

00:02:42.997 --> 00:02:45.580
Or if I were to think of it in
terms of numbers, just a number

00:02:45.580 --> 00:02:48.880
line, literally, this would
represent the number 3.

00:02:48.880 --> 00:02:50.740
But what's another way
I could represent it?

00:02:50.740 --> 00:02:54.610
Notice when I took a 1/5,
another 1/5, and another 1/5,

00:02:54.610 --> 00:02:56.660
I could call that 3/5.

00:02:56.660 --> 00:03:01.260
So now if I take one 1/1,
another 1/1, and another 1/1,

00:03:01.260 --> 00:03:06.260
well, I should be able to
call this 3/1, or 3 firsts,

00:03:06.260 --> 00:03:07.920
or however you want to call it.

00:03:07.920 --> 00:03:11.020
So I could call this 3 firsts.

00:03:11.020 --> 00:03:12.200
So this is interesting.

00:03:12.200 --> 00:03:14.730
Now we're seeing where the
top number on a fraction

00:03:14.730 --> 00:03:16.540
is larger than the bottom one.

00:03:16.540 --> 00:03:18.910
But another way of thinking
about this fraction symbol

00:03:18.910 --> 00:03:20.260
is that it's division.

00:03:20.260 --> 00:03:26.160
So you could view this as 3
divided by 1 is equal to 3.

00:03:26.160 --> 00:03:29.050
Or you could say, well,
look, 1 over 1 is a whole,

00:03:29.050 --> 00:03:32.800
and I now have three of them,
so this is equal to 3 wholes.

00:03:32.800 --> 00:03:36.147
So 3/1 is the same
thing as the number 3.

00:03:36.147 --> 00:03:37.980
And let me make sure,
let me emphasize that.

00:03:37.980 --> 00:03:40.050
Let me draw this
on a number line.

00:03:40.050 --> 00:03:45.120
So once again, let me
go all the way to 3.

00:03:45.120 --> 00:03:51.150
So 0, 1, 2, and 3.

00:03:51.150 --> 00:03:56.150
So one whole gets us exactly
one jump on the number line.

00:03:56.150 --> 00:03:59.630
So this right over here,
that gets us to 1/1.

00:03:59.630 --> 00:04:04.340
We do another jump, now
we've gotten 2 firsts,

00:04:04.340 --> 00:04:05.289
I guess you could say.

00:04:05.289 --> 00:04:07.080
We've essentially taken
two of these jumps.

00:04:07.080 --> 00:04:08.450
Each jump is 1/1.

00:04:08.450 --> 00:04:11.990
Now we are at 2/1, which
is the same thing as 2.

00:04:11.990 --> 00:04:16.300
You take another jump,
and we essentially get to,

00:04:16.300 --> 00:04:21.861
or we do get to 3/1, which
is the exact same thing as 3.


Fraction basics

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.370
What we're going to
talk about in this video

00:00:02.370 --> 00:00:05.070
is the idea of a fraction.

00:00:05.070 --> 00:00:08.270
And we'll see there's many
ways to think about a fraction.

00:00:08.270 --> 00:00:11.490
But first, we'll think
about the most fundamental.

00:00:11.490 --> 00:00:14.540
So let's say that
I have this square.

00:00:14.540 --> 00:00:17.397
And we can consider
this a whole.

00:00:17.397 --> 00:00:18.480
So let me write that down.

00:00:18.480 --> 00:00:20.030
This is a whole.

00:00:20.030 --> 00:00:22.390
It is a complete square.

00:00:22.390 --> 00:00:27.320
Now, what I'm going to do is
divide this into 4 equal parts.

00:00:27.320 --> 00:00:32.490
So with one cut like that, I've
divided it into 2 equal parts,

00:00:32.490 --> 00:00:34.170
and then with another
cut like this,

00:00:34.170 --> 00:00:38.460
I could divide it
into 4 equal parts.

00:00:38.460 --> 00:00:40.872
So there are 4 equal parts.

00:00:40.872 --> 00:00:42.330
And now what I'm
going to do is I'm

00:00:42.330 --> 00:00:44.880
going to select one
of those equal parts,

00:00:44.880 --> 00:00:47.240
so let's say this
part right over here.

00:00:47.240 --> 00:00:49.410
I am going to select that.

00:00:49.410 --> 00:00:52.380
So the question is, what
fraction of the whole

00:00:52.380 --> 00:00:55.930
is the part that I
have shaded in red?

00:00:55.930 --> 00:01:00.220
Well, it is 1 out of the
4 equal parts, right?

00:01:00.220 --> 00:01:03.960
I've shaded in 1 out of
1, 2, 3, 4 equal parts.

00:01:03.960 --> 00:01:06.350
So we write this
as this fraction.

00:01:06.350 --> 00:01:10.670
This piece represents
1/4 of the whole.

00:01:10.670 --> 00:01:12.900
And there's two ways that
you can think about this.

00:01:12.900 --> 00:01:16.530
You could view this as
1 of the 4 equal parts,

00:01:16.530 --> 00:01:20.900
or you could view this
as a whole divided by 4

00:01:20.900 --> 00:01:24.919
would get you exactly this much.

00:01:24.919 --> 00:01:25.960
Now let's do another one.

00:01:25.960 --> 00:01:27.900
And this time, let's
think about how

00:01:27.900 --> 00:01:37.670
we could represent 1
over 8, so 1 over 8.

00:01:37.670 --> 00:01:40.780
Well, we could divide
this whole, in this case,

00:01:40.780 --> 00:01:43.230
the whole is this
rectangle-looking thing.

00:01:43.230 --> 00:01:45.920
We could divide the
whole into 8 equal parts.

00:01:45.920 --> 00:01:46.980
So let's do that.

00:01:46.980 --> 00:01:49.820
So here I've divided
into 2 equal parts.

00:01:49.820 --> 00:01:51.390
That looks pretty good.

00:01:51.390 --> 00:01:54.350
And now I can divide each
of those into 2 equal parts

00:01:54.350 --> 00:01:56.730
to get me 4 equal parts.

00:01:56.730 --> 00:01:58.740
And then if I were to
divide each of those

00:01:58.740 --> 00:02:06.180
into 2 equal parts, I
will have 8 equal parts.

00:02:06.180 --> 00:02:07.055
And it's not exact.

00:02:07.055 --> 00:02:08.430
Obviously, I've
drawn it by hand,

00:02:08.430 --> 00:02:10.020
but hopefully this
gets you a sense.

00:02:10.020 --> 00:02:13.650
So now I have 8 equal parts.

00:02:13.650 --> 00:02:16.550
And now I'm going to
select exactly one of them.

00:02:16.550 --> 00:02:18.470
And that'll represent 1/8.

00:02:18.470 --> 00:02:20.030
And I could select
any one of these,

00:02:20.030 --> 00:02:21.571
but I'll just do
this one to show you

00:02:21.571 --> 00:02:24.390
it does not have to be
necessarily the first one.

00:02:24.390 --> 00:02:26.650
So once again, this
square right over here

00:02:26.650 --> 00:02:32.777
that I'm shading in red
represents 1/8 of the whole.

00:02:32.777 --> 00:02:34.360
Now, let's look at
a few more examples

00:02:34.360 --> 00:02:36.880
where I've shaded
them in ahead of time.

00:02:36.880 --> 00:02:39.800
And what I want you to do
right now is pause the video,

00:02:39.800 --> 00:02:42.080
and either in your head
or a piece of paper

00:02:42.080 --> 00:02:46.390
write down if you consider this
purple thing, the whole, what

00:02:46.390 --> 00:02:48.430
fraction does this
red part represent?

00:02:48.430 --> 00:02:50.660
If you consider this
blue part the whole,

00:02:50.660 --> 00:02:53.260
what fraction does this
red part represent?

00:02:53.260 --> 00:02:55.540
If you view this yellow
triangle as a whole,

00:02:55.540 --> 00:02:58.110
what fraction does this
red part represent?

00:02:58.110 --> 00:03:01.790
And so I encourage you
to pause the video now.

00:03:01.790 --> 00:03:03.580
Well, let's look
at each of these.

00:03:03.580 --> 00:03:06.470
So in this case,
for this rectangle,

00:03:06.470 --> 00:03:11.070
we have 3 equal parts, and
we've shaded in one of them.

00:03:11.070 --> 00:03:14.700
So this red rectangle
right over here

00:03:14.700 --> 00:03:17.941
represents 1/3 of the whole.

00:03:17.941 --> 00:03:18.440
Now,

00:03:18.440 --> 00:03:21.140
Over here, in this kind
of pie-looking thing,

00:03:21.140 --> 00:03:26.055
this circle-looking thing, we
have 1, 2, 3, 4, 5 equal parts.

00:03:28.660 --> 00:03:32.510
And we have shaded in 1
of those 5 equal parts.

00:03:32.510 --> 00:03:36.760
So this little slice of the
pie, this represents 1/5.

00:03:36.760 --> 00:03:45.260
This right over here is
1/5 of the entire pie.

00:03:45.260 --> 00:03:46.790
Now, this one's interesting.

00:03:46.790 --> 00:03:50.440
You might be tempted to
say, well, I've got 5 parts,

00:03:50.440 --> 00:03:52.760
and then I've shaded in 1.

00:03:52.760 --> 00:03:54.760
That must represent 1/4.

00:03:54.760 --> 00:03:58.130
But remember, it needs
to be 4 equal parts.

00:03:58.130 --> 00:03:59.700
And it's pretty
clear looking at this

00:03:59.700 --> 00:04:01.790
that this part right
over here is not

00:04:01.790 --> 00:04:05.081
equal in size to this part right
over here or this part right

00:04:05.081 --> 00:04:05.580
over here.

00:04:05.580 --> 00:04:07.850
These are not 4 equal parts.

00:04:07.850 --> 00:04:11.230
So we cannot say that this
is 1/4 of the triangle.

00:04:11.230 --> 00:04:13.989
So you cannot say that.


Equivalent fractions visually and on number line

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.690
So what I want you to
do is pause this video

00:00:02.690 --> 00:00:06.480
and think about what fraction
the red part represents

00:00:06.480 --> 00:00:10.010
in each of these shapes.

00:00:10.010 --> 00:00:12.840
Or what fraction of the whole
does the red part represent?

00:00:12.840 --> 00:00:15.700
And I also want to plot
it out on a number line,

00:00:15.700 --> 00:00:20.260
to plot that fraction as
a number on a number line.

00:00:20.260 --> 00:00:21.720
So let's go through
each of these.

00:00:21.720 --> 00:00:23.550
So in this pie
right over here, we

00:00:23.550 --> 00:00:27.040
have 1, 2, 3, 4,
5 equal sections.

00:00:27.040 --> 00:00:30.700
And 1 of those 5 equal
sections is shaded in.

00:00:30.700 --> 00:00:35.700
So we could say that 1/5
of this pie is shaded in.

00:00:35.700 --> 00:00:40.994
Now over here, we have 1,
2, 3, 4, 5, 6, 7, 8, 9,

00:00:40.994 --> 00:00:45.570
10 equal sections, and
2 of them are shaded in.

00:00:45.570 --> 00:00:49.917
So we could say that
2/10 are shaded in.

00:00:49.917 --> 00:00:52.000
And then, finally, right
over here, once again, we

00:00:52.000 --> 00:00:59.010
have 1, 2, 3, 4, 5, 6, 7, 8,
9, 10 equal sections, and 2

00:00:59.010 --> 00:01:00.600
of them are shaded in red.

00:01:00.600 --> 00:01:04.340
So in this situation,
the red slices

00:01:04.340 --> 00:01:07.330
represent 2/10 of the whole.

00:01:07.330 --> 00:01:10.080
And if we were to try to
plot this on a number line,

00:01:10.080 --> 00:01:14.430
so right over here, we do a
quick one right over here.

00:01:14.430 --> 00:01:16.865
Let me do it like this.

00:01:16.865 --> 00:01:20.160
Let me make a big
number line here.

00:01:20.160 --> 00:01:24.234
And let's take the
section between 0 and 1,

00:01:24.234 --> 00:01:25.900
that's what we want
to focus on, and I'm

00:01:25.900 --> 00:01:28.310
going to divide it
into 5 equal sections.

00:01:28.310 --> 00:01:37.350
So 1, 2, 3, 4, 5 equal sections
and then that gets us to 1.

00:01:37.350 --> 00:01:42.160
So this right over here, 1/5,
that would be 1 out of the 5

00:01:42.160 --> 00:01:42.880
equal sections.

00:01:42.880 --> 00:01:44.810
So that would get
us right over there.

00:01:44.810 --> 00:01:49.270
So this would be 1/5.

00:01:49.270 --> 00:01:51.940
Now, what I want to do, let me
copy and paste this same number

00:01:51.940 --> 00:01:53.795
line since I've
already drawn it.

00:01:53.795 --> 00:01:56.135
So copy and paste it.

00:01:56.135 --> 00:01:59.320
So let me put it
right over here.

00:01:59.320 --> 00:02:03.810
But now, I'm going to divide
it into 10 equal sections.

00:02:03.810 --> 00:02:04.310
Let's see.

00:02:04.310 --> 00:02:07.570
Let me divide it into 10
equal sections on the top one.

00:02:07.570 --> 00:02:18.460
So 1, 2, 3, 4, 5,
6, 7, 8, 9, 10.

00:02:18.460 --> 00:02:20.550
So I've divided it
into 10 equal sections.

00:02:20.550 --> 00:02:23.470
And I want to figure
out where 2/10 goes.

00:02:23.470 --> 00:02:27.700
So I'm going to go 2 of those
equal sections, so 1, 2.

00:02:27.700 --> 00:02:29.950
So once again, I've got
to that exact same point.

00:02:29.950 --> 00:02:35.040
So this 1/5 I could
also represent as 2/10.

00:02:35.040 --> 00:02:39.110
So I could represent this
as 2/10, this point right

00:02:39.110 --> 00:02:39.872
over here.

00:02:39.872 --> 00:02:41.330
And you might be
saying, hey, wait,

00:02:41.330 --> 00:02:43.500
but that means that those
are the same number.

00:02:43.500 --> 00:02:46.420
They're the exact same
point on the number line.

00:02:46.420 --> 00:02:49.570
And if you said that, you
would be absolutely correct.

00:02:49.570 --> 00:02:54.120
1/5 is equal to 2/10.

00:02:54.120 --> 00:02:57.470
They represent the
exact same number.

00:02:57.470 --> 00:02:59.370
And it makes sense
even when you visually

00:02:59.370 --> 00:03:02.090
look at them as a
fraction of these pies.

00:03:02.090 --> 00:03:06.510
Here going from this slice to
this slice, if you were just

00:03:06.510 --> 00:03:11.770
divide all of these
slices into 2,

00:03:11.770 --> 00:03:19.020
you see that you have the
exact same fraction shaded

00:03:19.020 --> 00:03:20.719
in as this one right over here.

00:03:20.719 --> 00:03:21.760
They've become identical.

00:03:21.760 --> 00:03:23.090
I didn't shade in anything else.

00:03:23.090 --> 00:03:24.589
I haven't taken any
of the red away.

00:03:24.589 --> 00:03:25.720
I haven't any of the red.

00:03:25.720 --> 00:03:29.960
I just divided all of
those pieces of pie into 2.

00:03:29.960 --> 00:03:32.850
And so you see that the exact
same part of the whole pie

00:03:32.850 --> 00:03:34.040
is shaded in.

00:03:34.040 --> 00:03:35.830
And here it's not
quite as obvious,

00:03:35.830 --> 00:03:39.780
but if you imagine taking
this, dividing it into 2

00:03:39.780 --> 00:03:42.700
and then splitting them up
so that they look like this,

00:03:42.700 --> 00:03:49.314
you still have the same part
of the circle shaded in red.

00:03:49.314 --> 00:03:51.230
So it makes complete
sense that they represent

00:03:51.230 --> 00:03:52.688
the same number on
the number line.

00:03:52.688 --> 00:03:55.600
That this number right over
here, it's not only 1/5,

00:03:55.600 --> 00:03:58.620
it also is 2/10.

00:03:58.620 --> 00:04:01.405
1/5 is equal to 2/10.


More than one equal section

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:04.585
I have a square here divided
into one, two, three, four,

00:00:04.585 --> 00:00:08.530
five, six, seven, eight,
nine equal sections.

00:00:08.530 --> 00:00:11.110
And we've already seen that
if we were to shade in one

00:00:11.110 --> 00:00:13.790
of these sections, if we were
to select one of these sections,

00:00:13.790 --> 00:00:16.000
let's say the middle
one right over here,

00:00:16.000 --> 00:00:18.560
this is one out of the
nine equal sections.

00:00:18.560 --> 00:00:21.230
So if someone said, what
fraction of the whole

00:00:21.230 --> 00:00:24.180
does this purple
square represent?

00:00:24.180 --> 00:00:29.150
Well, you would say, well, that
represents 1/9 of the whole.

00:00:29.150 --> 00:00:32.040
This thing right over
here represents 1/9.

00:00:32.040 --> 00:00:34.760
Now what would happen if we
shaded in more than that?

00:00:34.760 --> 00:00:39.820
So let's say we shaded in this
one and this one, let me shade

00:00:39.820 --> 00:00:40.970
it in a little bit better.

00:00:40.970 --> 00:00:46.210
And this one and this
one right over here.

00:00:46.210 --> 00:00:49.550
Now what fraction of the
whole have we shaded in?

00:00:49.550 --> 00:00:52.640
Well, each of these, we've
already seen, each of these

00:00:52.640 --> 00:00:54.630
represent 1/9.

00:00:54.630 --> 00:00:56.890
So that's 1/9, that's 1/9.

00:00:56.890 --> 00:01:00.440
When I say 1/9, I
could also say a ninth.

00:01:00.440 --> 00:01:03.770
So this is 1/9 or a
ninth, so each of these

00:01:03.770 --> 00:01:05.480
represents a ninth.

00:01:05.480 --> 00:01:08.720
But how many of these
ninths do we have shaded in?

00:01:08.720 --> 00:01:11.470
Well we have one, two,
three, four shaded in.

00:01:11.470 --> 00:01:15.630
So now we have a total
of 4/9 shaded it.

00:01:15.630 --> 00:01:18.900
4 of the 9 equal
sections are shaded in.

00:01:18.900 --> 00:01:22.460
So 4/9 of the
whole is shaded in.

00:01:22.460 --> 00:01:25.550
Now let's make things a
little bit more interesting.

00:01:25.550 --> 00:01:26.280
Let's shade in.

00:01:26.280 --> 00:01:28.350
So here I have five
equal sections.

00:01:28.350 --> 00:01:29.790
Let me write this down.

00:01:29.790 --> 00:01:32.110
I have five equal sections.

00:01:32.110 --> 00:01:35.250
And let me shade
in five of them.

00:01:35.250 --> 00:01:41.200
So one, two, three, four, five.

00:01:41.200 --> 00:01:44.100
We already know that each
of these sections, each

00:01:44.100 --> 00:01:47.610
of these situated in
sections represent 1/5.

00:01:47.610 --> 00:01:51.980
So 1/5, another way of saying
that is a fifth, is 1/5.

00:01:51.980 --> 00:01:53.980
But now how much do
I have shaded in?

00:01:53.980 --> 00:01:57.580
Well I have five out of the
five equal sections shaded,

00:01:57.580 --> 00:02:00.190
or I have 5/5 shaded in.

00:02:00.190 --> 00:02:01.690
And you might be
saying, wait, wait,

00:02:01.690 --> 00:02:04.420
if I gave five out of the
five equal sections shaded in,

00:02:04.420 --> 00:02:08.350
if I have 5/5 shaded in, I've
got the whole thing shaded in.

00:02:08.350 --> 00:02:10.250
And you would be
absolutely right.

00:02:10.250 --> 00:02:14.230
5/5 is equal to the whole.

00:02:14.230 --> 00:02:16.654
Now what I want you to
do is pause this video

00:02:16.654 --> 00:02:18.070
and write down on
a piece of paper

00:02:18.070 --> 00:02:19.810
or at least think
in your head, what

00:02:19.810 --> 00:02:24.660
fraction of each of these
wholes is shaded in?

00:02:24.660 --> 00:02:26.320
So let's go to this first one.

00:02:26.320 --> 00:02:32.190
We have one, two, three, four,
five, six equal sections.

00:02:32.190 --> 00:02:36.280
And we see that one, two,
three, four are shaded in.

00:02:36.280 --> 00:02:39.590
So 4/6 of this
figure is shaded in.

00:02:39.590 --> 00:02:40.630
Let's go over here.

00:02:40.630 --> 00:02:49.350
We have one, two, three,
four, five equal sections.

00:02:49.350 --> 00:02:52.500
And one, two, three,
four are shaded in.

00:02:52.500 --> 00:02:57.600
So here, 4/5 of this
circle is shaded in.

00:02:57.600 --> 00:03:04.170
Now in this figure, I have two
equal sections and both of them

00:03:04.170 --> 00:03:06.770
are shaded in.

00:03:06.770 --> 00:03:10.690
And this we would say two halves
of this figure are shaded in.

00:03:10.690 --> 00:03:12.610
And once again, if two
halves are shaded in,

00:03:12.610 --> 00:03:16.050
that means everything
is shaded in,

00:03:16.050 --> 00:03:18.619
that this represents a whole.

00:03:18.619 --> 00:03:20.160
Now this one right
over here it might

00:03:20.160 --> 00:03:22.250
be tempting to say, I
have one, two, three,

00:03:22.250 --> 00:03:25.140
four sections and
one, two, three

00:03:25.140 --> 00:03:28.280
have been shaded in so
maybe the red represents

00:03:28.280 --> 00:03:29.940
3/4 of the figure.

00:03:29.940 --> 00:03:33.450
But remember, the sections
have to be equal sections.

00:03:33.450 --> 00:03:34.989
And this red section
is way bigger.

00:03:34.989 --> 00:03:37.280
It actually looks like it's
bigger than the other three

00:03:37.280 --> 00:03:38.320
combined.

00:03:38.320 --> 00:03:41.110
So you do not have four
equal sections here.

00:03:41.110 --> 00:03:43.110
So at least based
on how it's drawn,

00:03:43.110 --> 00:03:47.690
you can't say that 3/4
is actually filled in.


Fractions on a number line

WEBVTT
Kind: captions
Language: en

00:00:00.749 --> 00:00:02.540
We've already seen that
if we take a whole,

00:00:02.540 --> 00:00:06.360
and in this example, the whole
is this entire green circle.

00:00:06.360 --> 00:00:10.151
And if we were to split it into
5 equal sections-- 1, 2, 3, 4,

00:00:10.151 --> 00:00:10.650
5.

00:00:10.650 --> 00:00:13.220
So we've split it into
5 equal sections--

00:00:13.220 --> 00:00:16.160
and if we were to select 1
of those 5 equal sections.

00:00:16.160 --> 00:00:19.450
So let's say we select this
section right over here,

00:00:19.450 --> 00:00:24.120
that we have selected
1/5 of the whole, 1 out

00:00:24.120 --> 00:00:26.840
of the 5 equal sections.

00:00:26.840 --> 00:00:29.482
We could do the exact same
thing on a number line.

00:00:29.482 --> 00:00:31.940
Everything we've been doing so
far has to deal with shapes,

00:00:31.940 --> 00:00:34.890
but we could do the exact
same idea on a number line.

00:00:34.890 --> 00:00:37.460
So let me draw a
number line here.

00:00:37.460 --> 00:00:42.875
So let me draw it pretty big
so we get a sense of things.

00:00:42.875 --> 00:00:45.240
So it will go all
the way to there.

00:00:45.240 --> 00:00:54.144
And let's say that this is
0, this is 1, and this is 2.

00:00:54.144 --> 00:00:55.560
And of course, we
could keep going

00:00:55.560 --> 00:00:59.320
if we had more space to 3,
4, and on and on and on.

00:00:59.320 --> 00:01:01.480
And what I want to do,
instead of taking a circle

00:01:01.480 --> 00:01:03.700
and dividing it into
5 equal sections,

00:01:03.700 --> 00:01:07.550
I want to take the section of
our number line between 0 and 1

00:01:07.550 --> 00:01:09.969
and divide it into
5 equal sections.

00:01:09.969 --> 00:01:11.260
So let me see if I can do this.

00:01:11.260 --> 00:01:16.940
So 1, 2, 3, 4, 5.

00:01:16.940 --> 00:01:18.180
That looks pretty good.

00:01:18.180 --> 00:01:20.820
I'm drawing it as exact
as I can with my hand.

00:01:20.820 --> 00:01:23.540
But let's just assume
these are 5 equal sections.

00:01:23.540 --> 00:01:28.360
So what would you think would
be a good label for this number

00:01:28.360 --> 00:01:30.070
right over here?

00:01:30.070 --> 00:01:32.430
Well, it's the exact same idea.

00:01:32.430 --> 00:01:35.360
Between 0 and 1,
I've traveled 1 out

00:01:35.360 --> 00:01:37.752
of the 5 equal
sections towards 1.

00:01:37.752 --> 00:01:40.210
And actually, let me make it
a little bit neater than that.

00:01:40.210 --> 00:01:42.626
We could make the equal sections
look a little bit better.

00:01:42.626 --> 00:01:46.647
1, 2, 3, 4, 5.

00:01:46.647 --> 00:01:48.230
And what we're
thinking about is this.

00:01:48.230 --> 00:01:50.260
What should we call
this number here?

00:01:50.260 --> 00:01:52.980
This number is clearly
between 0 and 1.

00:01:52.980 --> 00:01:54.790
It's clearly closer to 0.

00:01:54.790 --> 00:01:59.810
And we've gone 1 out of the
5 equal sections towards 1.

00:01:59.810 --> 00:02:01.880
Well, it makes complete
sense that, look,

00:02:01.880 --> 00:02:05.250
we had 5 equal sections here.

00:02:05.250 --> 00:02:07.990
And we've traveled
1 of them towards 1.

00:02:07.990 --> 00:02:11.740
So we should call this
number right over here 1/5.

00:02:11.740 --> 00:02:15.360
So when we're talking
about a fraction, 1/5,

00:02:15.360 --> 00:02:18.390
it's not just talking about,
hey, what part of a pizza pie

00:02:18.390 --> 00:02:21.400
have I eaten or
something like that.

00:02:21.400 --> 00:02:23.430
This is actually a number.

00:02:23.430 --> 00:02:24.260
This is a number.

00:02:24.260 --> 00:02:27.895
And we can actually plot
it on the number line.

00:02:27.895 --> 00:02:30.020
Now you might say, OK,
well, that's fair about 1/5.

00:02:30.020 --> 00:02:31.645
But what about all
these other slashes?

00:02:31.645 --> 00:02:33.250
What numbers would we call that?

00:02:33.250 --> 00:02:36.940
Well, we can make
the exact same idea.

00:02:36.940 --> 00:02:39.810
If up here, instead of
shading in 1 out of the 5

00:02:39.810 --> 00:02:44.970
equal sections, if I shaded in
2 of the 5 equal sections, then

00:02:44.970 --> 00:02:46.620
I wouldn't say this
is 1/5 any more.

00:02:46.620 --> 00:02:49.260
I would say that this 2/5.

00:02:49.260 --> 00:02:53.860
And so if I go 2 of the
equal sections towards 1,

00:02:53.860 --> 00:02:59.670
then I should call this
number right over here 2/5.

00:02:59.670 --> 00:03:01.240
And I could keep going.

00:03:01.240 --> 00:03:05.796
This right over here
should be 3, 3/5.

00:03:05.796 --> 00:03:09.230
This right over here,
I've gone 1, 2, 3, 4

00:03:09.230 --> 00:03:11.680
out of the 5 sections towards 1.

00:03:11.680 --> 00:03:13.202
So I could call this 4/5.

00:03:16.112 --> 00:03:17.070
And I could keep going.

00:03:17.070 --> 00:03:18.528
I could call this
right over here--

00:03:18.528 --> 00:03:22.020
I've traveled 5 out of the
5 equal sections towards 5,

00:03:22.020 --> 00:03:25.110
so I could call this
right over here 5.

00:03:25.110 --> 00:03:26.850
Let me do it in that red color.

00:03:26.850 --> 00:03:31.360
I could call this
right over here 5/5.

00:03:31.360 --> 00:03:34.160
You might say, wait, but
5/5, we've gotten to 1.

00:03:34.160 --> 00:03:35.600
And that's exactly right.

00:03:35.600 --> 00:03:38.070
If I were to shade in
5 things over here--

00:03:38.070 --> 00:03:39.820
let me do that
little bit cleaner.

00:03:42.420 --> 00:03:44.310
That's not the
color I want to use.

00:03:44.310 --> 00:03:47.170
If I were to shade in
5 things over here,

00:03:47.170 --> 00:03:51.040
we've already seen that
shading in 5 things--

00:03:51.040 --> 00:03:52.770
let me make this a
little bit neater--

00:03:52.770 --> 00:03:57.110
if this is now 5
over 5 or 5/5, we've

00:03:57.110 --> 00:03:58.830
already seen that
this is a whole.

00:03:58.830 --> 00:04:03.130
And over here, if we've traveled
5/5 of the way towards 1,

00:04:03.130 --> 00:04:05.170
we've gotten to the whole 1.

00:04:05.170 --> 00:04:11.760
5/5 is the exact
same thing as 1.

00:04:11.760 --> 00:04:14.942
It is equal to a whole.


Generating equivalent fractions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.390
So we've got this fraction
written here, 2/3.

00:00:03.390 --> 00:00:05.850
What I want you to do
is pause this video

00:00:05.850 --> 00:00:08.400
and try to think of any
other fractions that

00:00:08.400 --> 00:00:10.930
are the same that are
equivalent to this fraction

00:00:10.930 --> 00:00:15.800
right over here that essentially
represent the same number.

00:00:15.800 --> 00:00:19.200
So to do that, let's
visualize what 2/3 is.

00:00:19.200 --> 00:00:21.570
So let me draw a whole here.

00:00:21.570 --> 00:00:23.090
Let me draw a
whole, and I'm going

00:00:23.090 --> 00:00:25.820
to divide it into
three equal sections.

00:00:25.820 --> 00:00:27.965
So that is my whole.

00:00:27.965 --> 00:00:29.670
I'm drawing three
equal sections I'm

00:00:29.670 --> 00:00:31.950
going to try to draw and
make them as equal as i--

00:00:31.950 --> 00:00:34.570
I can do a little
better job than that.

00:00:34.570 --> 00:00:39.580
So that's 1, 2, 3.

00:00:39.580 --> 00:00:41.130
Three equal sections.

00:00:41.130 --> 00:00:43.080
And so 2/3 would
represent two out

00:00:43.080 --> 00:00:44.840
of those three equal sections.

00:00:44.840 --> 00:00:46.630
So actually I can
spray paint this.

00:00:46.630 --> 00:00:51.140
So that's 1/3 right over
here, and then this is 2/3.

00:00:51.140 --> 00:00:55.070
So we have two of the
three equal sections.

00:00:55.070 --> 00:00:56.195
So that is 2/3.

00:00:56.195 --> 00:00:58.260
And now let me
copy and paste that

00:00:58.260 --> 00:01:02.600
so we can think about other
ways to represent this fraction.

00:01:02.600 --> 00:01:04.540
So copy.

00:01:04.540 --> 00:01:05.790
And then let me paste it.

00:01:05.790 --> 00:01:08.482
I'll do it once, and then
I'll do it another time.

00:01:08.482 --> 00:01:09.940
And I could do this
multiple times,

00:01:09.940 --> 00:01:12.270
but I'll do it two other
times right over here.

00:01:12.270 --> 00:01:14.190
So there's a couple
of things we could do.

00:01:14.190 --> 00:01:17.070
The first option is
we could take this

00:01:17.070 --> 00:01:19.700
and we could draw a
horizontal line that

00:01:19.700 --> 00:01:22.860
divides each of these three
sections into two sections.

00:01:22.860 --> 00:01:23.635
So let's do that.

00:01:26.750 --> 00:01:29.095
So now, how many equal
sections do I have?

00:01:29.095 --> 00:01:35.530
I have 1, 2, 3, 4,
5, 6 equals sections.

00:01:35.530 --> 00:01:37.720
And how many of those
equal sections are actually

00:01:37.720 --> 00:01:39.380
shaded in now.

00:01:39.380 --> 00:01:42.500
Well, we see it's 1, 2, 3, 4.

00:01:42.500 --> 00:01:43.000
4/6.

00:01:43.000 --> 00:01:47.870
So notice, 4/6 is the exact same
fraction of the whole as 2/3.

00:01:47.870 --> 00:01:49.340
These are equivalent fractions.

00:01:49.340 --> 00:01:53.814
We could say that
2/3 is equal to 4/6.

00:01:53.814 --> 00:01:55.480
Now, we could do
something very similar.

00:01:55.480 --> 00:01:59.111
Instead of dividing each
of these thirds into two,

00:01:59.111 --> 00:02:01.110
we could divide each of
these thirds into three.

00:02:01.110 --> 00:02:03.980
So I could draw three
horizontal lines here.

00:02:03.980 --> 00:02:09.139
So let's see 1, 2, 3.

00:02:09.139 --> 00:02:12.840
So now I have divided what
was in three equal sections,

00:02:12.840 --> 00:02:15.040
I now have three times
as many sections.

00:02:15.040 --> 00:02:20.640
I have 1, 2,, 3, 4, 5, 6,
7, 8, 9 equals sections.

00:02:20.640 --> 00:02:23.410
And then which of those
are actually shaded in?

00:02:23.410 --> 00:02:27.254
We have 1, 2, 3, 4, 5, 6.

00:02:27.254 --> 00:02:28.020
6.

00:02:28.020 --> 00:02:34.732
So 2/3, which is equal to
4/6, is also equal to 6/9.

00:02:34.732 --> 00:02:38.680
All three of these are
equivalent fractions, 2/3, 4/6,

00:02:38.680 --> 00:02:39.292
and 6/9.

00:02:39.292 --> 00:02:41.250
And if you were to put
them on the number line,

00:02:41.250 --> 00:02:42.420
the same thing would happen.

00:02:42.420 --> 00:02:43.631
So let's do that.

00:02:43.631 --> 00:02:44.880
Let's draw a number line here.

00:02:44.880 --> 00:02:46.070
Let's say that's 0.

00:02:46.070 --> 00:02:48.610
And I'm just going to
focus on between 0 and 1.

00:02:48.610 --> 00:02:52.780
And obviously we
can go beyond that.

00:02:52.780 --> 00:02:56.090
And let's divide it
first into thirds.

00:02:56.090 --> 00:02:59.940
So this is 1/3 and 2/3.

00:02:59.940 --> 00:03:05.550
So we already know this would
represent 1/3 and this is 2/3.

00:03:05.550 --> 00:03:09.050
We've gone two of the equal
spaces of the three on the way

00:03:09.050 --> 00:03:09.870
to 1.

00:03:09.870 --> 00:03:11.740
We've divided the
section between 0 and 1

00:03:11.740 --> 00:03:14.510
into three equal spaces.

00:03:14.510 --> 00:03:15.879
Now, what for 4/6 be?

00:03:15.879 --> 00:03:17.670
Well, now we would just
have to divide this

00:03:17.670 --> 00:03:19.300
into 6 equal spaces.

00:03:19.300 --> 00:03:24.190
So 1, 2, 4, 4, 5, 6.

00:03:24.190 --> 00:03:29.350
And 4/6, that would be 4 out of
the 6 spaces on the way to 1.

00:03:29.350 --> 00:03:31.620
So 1, 2, 3, 4.

00:03:31.620 --> 00:03:34.637
So this number is
also equal to 4/6.

00:03:34.637 --> 00:03:36.220
And you could do the
same thing if you

00:03:36.220 --> 00:03:37.705
want to think about ninths.

00:03:42.120 --> 00:03:51.450
So what we could do, we could
put 1, 2, 3, 4, 5, 6, 7, 8, 9.

00:03:51.450 --> 00:03:55.480
Now I've split this part of
our number line between 0 and 1

00:03:55.480 --> 00:03:57.290
into 9 equal spaces.

00:03:57.290 --> 00:03:59.230
Well, what would 6/9 be?

00:03:59.230 --> 00:04:03.330
Well, 1, 2, 3, 4, 5, 6.

00:04:03.330 --> 00:04:05.540
Once again, the exact same
point on the number line.

00:04:05.540 --> 00:04:06.706
It's an equivalent fraction.

00:04:06.706 --> 00:04:10.994
6/9 is equal to 2/3
is equal to 4/6.


Comparing fractions visually and on number line

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.460
We have four fractions
written over here.

00:00:03.460 --> 00:00:05.830
And what I want
you to do is think

00:00:05.830 --> 00:00:10.480
about which of these fractions
is the smallest value, which

00:00:10.480 --> 00:00:12.160
of these is the largest
value, and which

00:00:12.160 --> 00:00:14.001
of these might be equal.

00:00:14.001 --> 00:00:15.750
And there's two ways
that you could do it.

00:00:15.750 --> 00:00:17.910
You could try to plot
them on a number line,

00:00:17.910 --> 00:00:20.270
or you could try to
depict them visually.

00:00:20.270 --> 00:00:23.730
So pause the video now
and try it on your own.

00:00:23.730 --> 00:00:25.940
So let's plot these
on a number line.

00:00:25.940 --> 00:00:29.270
So let me give ourselves
another little number line

00:00:29.270 --> 00:00:30.230
right over here.

00:00:30.230 --> 00:00:32.850
Let me see, I can draw one.

00:00:32.850 --> 00:00:36.460
So let's say this is my number
line-- draw it as straight

00:00:36.460 --> 00:00:38.070
as I can.

00:00:38.070 --> 00:00:40.060
So that's my number line.

00:00:40.060 --> 00:00:41.490
That is 0.

00:00:41.490 --> 00:00:46.330
Let's say that this is
1, and then that is 2.

00:00:46.330 --> 00:00:51.150
So let's first try to plot
where 1/2 is on the number line.

00:00:51.150 --> 00:00:54.050
So we can split the
section between 0 and 1

00:00:54.050 --> 00:00:55.255
into 2 equal sections.

00:00:57.830 --> 00:01:00.410
And 1/2, would mean
that I would have

00:01:00.410 --> 00:01:04.099
gone one of those
2 equal sections.

00:01:04.099 --> 00:01:07.750
So if I just go that
far, I should be at 1/2.

00:01:07.750 --> 00:01:10.160
So this right over here is 1/2.

00:01:10.160 --> 00:01:11.990
Now let's think about 2/4.

00:01:11.990 --> 00:01:13.840
Well, to do 2/4,
we want to split

00:01:13.840 --> 00:01:16.980
the section between 0 and
1 into 4 equal sections.

00:01:16.980 --> 00:01:18.550
So let's do that.

00:01:18.550 --> 00:01:24.820
So 0-- so that's 1 equal
section, 2 equal sections,

00:01:24.820 --> 00:01:28.720
3 equal sections,
and 4 equal sections.

00:01:28.720 --> 00:01:32.800
So where we do end up if we go
2 out of those 4 equal sections?

00:01:32.800 --> 00:01:37.910
Well we would end up--
1, 2 equal sections,

00:01:37.910 --> 00:01:39.820
so we end up right over here.

00:01:39.820 --> 00:01:42.342
2/4, we end up at
the exact same place.

00:01:42.342 --> 00:01:44.550
So at least based on how
we've drawn the number line,

00:01:44.550 --> 00:01:47.272
it looks like 1/2
is equal to 2/4.

00:01:47.272 --> 00:01:48.230
Let me write that down.

00:01:48.230 --> 00:01:51.370
1/2 is equal to 2/4.

00:01:51.370 --> 00:01:53.430
Now what about 4/8?

00:01:53.430 --> 00:01:57.340
Well let's split the part
of our number line between 0

00:01:57.340 --> 00:01:59.177
and 1 to 8 equal sections.

00:01:59.177 --> 00:02:01.010
So if we take each of
those 4 equal sections

00:02:01.010 --> 00:02:12.120
and split them into 2-- So now
we have 1, 2, 3, 4, 5, 6, 7, 8.

00:02:12.120 --> 00:02:14.230
I can't draw it
perfectly, but I'm

00:02:14.230 --> 00:02:15.840
trying to make them
equal sections.

00:02:15.840 --> 00:02:17.550
Now it's 8 equal sections.

00:02:17.550 --> 00:02:19.920
And now we're going
to go 4 of them.

00:02:19.920 --> 00:02:24.340
Starting at 0, we're
going to go 1, 2, 3, 4.

00:02:24.340 --> 00:02:25.910
So we end up at the
same place again.

00:02:25.910 --> 00:02:28.750
This is also equal to 4/8.

00:02:28.750 --> 00:02:35.480
So 1/2 is equal to 2/4,
which is equal to 4/8.

00:02:35.480 --> 00:02:37.670
Now, what about 3/8?

00:02:37.670 --> 00:02:39.630
Well, we've already
split our number line

00:02:39.630 --> 00:02:42.780
into 8 equal sections--
let's go 3 of them.

00:02:42.780 --> 00:02:44.870
1, 2, 3.

00:02:44.870 --> 00:02:47.350
So 3/8 is right over here.

00:02:47.350 --> 00:02:52.860
3/8 is less than 1/2, it's less
than 2/4, it's less than 4/8.

00:02:52.860 --> 00:02:55.170
It is a smaller number.

00:02:55.170 --> 00:02:56.930
Let's see if that
also makes sense

00:02:56.930 --> 00:02:59.160
when we try to
visually depict or try

00:02:59.160 --> 00:03:01.090
to visually draw
these fractions.

00:03:01.090 --> 00:03:03.400
So I've done that
right over here.

00:03:03.400 --> 00:03:04.460
And this is one 1/2.

00:03:04.460 --> 00:03:08.040
So all of these rectangles
are the exact same size.

00:03:08.040 --> 00:03:09.540
And this purple
one over here I've

00:03:09.540 --> 00:03:12.690
split into 2 equal sections,
and we shaded in one of them.

00:03:12.690 --> 00:03:15.480
So we see that this is 1/2.

00:03:15.480 --> 00:03:18.680
Now here I shaded in 2 out
of the 4 equal sections,

00:03:18.680 --> 00:03:21.510
and you see that this
looks the exact same size

00:03:21.510 --> 00:03:23.850
as the 1/2 right over here.

00:03:23.850 --> 00:03:26.180
We started with rectangles
of the same size.

00:03:26.180 --> 00:03:29.569
If you shade in 1/2 or 2/4,
it looks exactly the same.

00:03:29.569 --> 00:03:32.110
And that makes sense, because
if you took this one right over

00:03:32.110 --> 00:03:34.530
here, if you took
this first one and you

00:03:34.530 --> 00:03:37.680
divided each of your 2 equal
sections into 2 equal sections,

00:03:37.680 --> 00:03:42.970
so you split it again, then you
see that this is equal to 2/4.

00:03:42.970 --> 00:03:43.950
Now what about 4/8?

00:03:43.950 --> 00:03:46.860
Here I've split it up where
I have this one, this one,

00:03:46.860 --> 00:03:48.780
this one, and this
one shaded in.

00:03:48.780 --> 00:03:50.340
But if you rearrange
them, you could

00:03:50.340 --> 00:03:53.080
see that you could get
exactly the same amount

00:03:53.080 --> 00:03:54.590
of the rectangle shaded in.

00:03:54.590 --> 00:03:56.220
And if you want to
see that, divide

00:03:56.220 --> 00:04:00.650
each of these 4 into 2-- so
divide those into 2 and those

00:04:00.650 --> 00:04:01.480
into 2.

00:04:01.480 --> 00:04:07.750
Notice I now have 1, 2, 3, 4 out
of the 8 equal squares shaded

00:04:07.750 --> 00:04:08.250
in.

00:04:08.250 --> 00:04:09.560
These two things are equal.

00:04:09.560 --> 00:04:10.960
So let me make it clear.

00:04:10.960 --> 00:04:13.920
This is an equal fraction, which
is an equal fraction of that

00:04:13.920 --> 00:04:15.760
which is equal fraction of that

00:04:15.760 --> 00:04:17.640
And here, this is 3 out of 8.

00:04:17.640 --> 00:04:22.250
You see that it's less than
this region right over here.

00:04:22.250 --> 00:04:24.210
This one we've
literally filled in half

00:04:24.210 --> 00:04:26.609
of the entire rectangle--
4 out of the 8.

00:04:26.609 --> 00:04:29.150
If we did 4 out of the 8 here,
we would have also had to fill

00:04:29.150 --> 00:04:31.025
in this one right over
here, which we didn't.

00:04:31.025 --> 00:04:32.920
We only filled in
3 out of the 8.

00:04:32.920 --> 00:04:35.760
So it makes sense
that 3/8 is a smaller

00:04:35.760 --> 00:04:39.360
part of our whole than
4/8, or 2/4, or 1/2.

00:04:39.360 --> 00:04:42.840
And likewise, it's
a smaller number.


Different ways to represent 1 as a fraction

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:04.100
So let's say that this
circle-looking thing represents

00:00:04.100 --> 00:00:04.800
a whole.

00:00:04.800 --> 00:00:06.550
We've already seen a
couple of situations.

00:00:06.550 --> 00:00:08.216
We said, hey, look,
we could divide this

00:00:08.216 --> 00:00:11.150
into two equal pieces.

00:00:11.150 --> 00:00:14.470
And if we shade in
one of them-- so one

00:00:14.470 --> 00:00:18.090
of these equal pieces--
that would be 1/2.

00:00:18.090 --> 00:00:21.107
And then if we shade in two
of them, this would be 2/2.

00:00:21.107 --> 00:00:22.940
So let me color that
in a little bit better.

00:00:22.940 --> 00:00:26.640
So this right over here, I've
divided into two equal pieces.

00:00:26.640 --> 00:00:29.800
And I'm shading in two
of these equal pieces.

00:00:29.800 --> 00:00:33.680
So what fraction of the
whole do I have shaded in?

00:00:33.680 --> 00:00:35.550
Well, we've seen
this multiple times.

00:00:35.550 --> 00:00:41.490
This is 2/2 of
the entire figure.

00:00:41.490 --> 00:00:43.540
And we see we've shaded
in the entire figure.

00:00:43.540 --> 00:00:49.025
So this is equal to 1 whole.

00:00:49.025 --> 00:00:49.900
And we could do that.

00:00:49.900 --> 00:00:52.360
We don't have to just split
it into two equal parts.

00:00:52.360 --> 00:00:56.930
We could split it into
three equal parts.

00:00:56.930 --> 00:01:00.410
So let's say we were to split
it into three equal parts--

00:01:00.410 --> 00:01:04.550
let me do that, my best attempt
to draw three equal parts.

00:01:04.550 --> 00:01:08.710
Three equal parts looks
like a Mercedes symbol.

00:01:08.710 --> 00:01:09.510
And that's my best.

00:01:09.510 --> 00:01:12.730
I could draw a little
bit better job of that.

00:01:12.730 --> 00:01:16.530
Let me be clear that I'm
trying to make them equal.

00:01:16.530 --> 00:01:18.620
So that's three equal parts.

00:01:18.620 --> 00:01:21.820
And then if we were to
actually shade in the three

00:01:21.820 --> 00:01:26.600
equal parts-- so that
would be one of the three,

00:01:26.600 --> 00:01:33.870
so that's 1/3, 2/3, and 3/3.

00:01:33.870 --> 00:01:43.160
So once again, 3/3
is equal to 1 whole.

00:01:43.160 --> 00:01:45.900
Now, what if we were to do
something, on some level,

00:01:45.900 --> 00:01:47.230
even simpler?

00:01:47.230 --> 00:01:49.850
What if we just take our
whole and we divide it

00:01:49.850 --> 00:01:51.652
into only one equal section?

00:01:51.652 --> 00:01:52.860
Well, I've already done that.

00:01:52.860 --> 00:01:55.660
This is one equal
section right over here.

00:01:55.660 --> 00:02:00.750
And then I were to select
that one equal section-- so

00:02:00.750 --> 00:02:02.550
let me color it in.

00:02:02.550 --> 00:02:06.390
So I have one section, and
I'm going to shade it in.

00:02:06.390 --> 00:02:13.140
So what fraction of the whole
do I now have shaded in?

00:02:13.140 --> 00:02:15.480
Well, I had one equal
section to begin with.

00:02:18.350 --> 00:02:21.530
And I shaded in that
one equal section.

00:02:21.530 --> 00:02:25.180
So I have 1/1 of this shaded in.

00:02:25.180 --> 00:02:29.100
And this is also
clearly an entire whole.

00:02:29.100 --> 00:02:33.020
So this is also
equal to a whole.

00:02:33.020 --> 00:02:34.770
And I think you
see a pattern here.

00:02:34.770 --> 00:02:40.210
2/2, 3/3, or 1/1-- these all
represent the exact same value.

00:02:40.210 --> 00:02:42.280
They all represent a whole.

00:02:42.280 --> 00:02:44.930
And you would also see this if
you were to draw a number line.

00:02:47.750 --> 00:02:49.280
So this is 0.

00:02:49.280 --> 00:02:50.290
This is 1.

00:02:50.290 --> 00:02:52.020
We could keep on going.

00:02:52.020 --> 00:02:54.230
Well, this is 1/1.

00:02:54.230 --> 00:02:56.550
If I were to say, well,
between 0 and 1, I just have 1,

00:02:56.550 --> 00:02:58.390
I divide it into one
equal chunk, well,

00:02:58.390 --> 00:03:00.490
that's just this whole
thing right over here.

00:03:00.490 --> 00:03:03.020
And if I were to move one
of those equal chunks,

00:03:03.020 --> 00:03:08.290
I would get to 1.

00:03:08.290 --> 00:03:15.600
If I divide it into two equal
chunks and if I make two

00:03:15.600 --> 00:03:19.950
jumps-- one, two jumps--
I still end up at 1.

00:03:19.950 --> 00:03:23.010
If I divide it into three
equal sections-- so let's

00:03:23.010 --> 00:03:27.850
say one, two, three
equal sections-- and I

00:03:27.850 --> 00:03:33.530
make three jumps-- one, two,
three-- I end up at 1 again.

00:03:33.530 --> 00:03:38.090
So 2/2, 3/3, 1/1,
or 1 over 1-- these

00:03:38.090 --> 00:03:42.920
all are different ways of
representing the number 1,

00:03:42.920 --> 00:03:44.732
or 1 whole.


Mental technique for subtraction without regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.630
What I want to do in this
video is show you a way

00:00:02.630 --> 00:00:06.540
of subtracting numbers that is
different than the regrouping

00:00:06.540 --> 00:00:07.040
technique.

00:00:07.040 --> 00:00:09.880
And this is closer to what
I actually do in my head.

00:00:09.880 --> 00:00:12.060
And this might not be
what you see in school,

00:00:12.060 --> 00:00:14.210
so be careful while
you're looking at this.

00:00:14.210 --> 00:00:16.129
Some people might
not fully approve

00:00:16.129 --> 00:00:17.610
of what I'm about to show you.

00:00:17.610 --> 00:00:20.240
But the idea, the reason
why I'm showing you this

00:00:20.240 --> 00:00:22.750
is to understand that there's
not just one way to do things.

00:00:22.750 --> 00:00:25.340
As long as you understand
the underlying principles,

00:00:25.340 --> 00:00:27.200
what these numbers
represent, and you

00:00:27.200 --> 00:00:29.645
do things that make
sense, you should be OK.

00:00:29.645 --> 00:00:31.215
And what's neat
about this technique

00:00:31.215 --> 00:00:32.590
is that we don't
have to regroup.

00:00:32.590 --> 00:00:34.631
We're just going to start
with the hundreds place

00:00:34.631 --> 00:00:35.750
and keep on moving.

00:00:35.750 --> 00:00:38.440
So, for example,
if I think of 301,

00:00:38.440 --> 00:00:39.990
I first want to subtract 100.

00:00:39.990 --> 00:00:45.000
So 300 minus 100 is
going to get me to 200.

00:00:45.000 --> 00:00:47.095
Now I need to subtract 60.

00:00:47.095 --> 00:00:51.760
And so I can think about what
is 20-- two, zero-- minus 6.

00:00:51.760 --> 00:00:54.910
So this is essentially
saying what is 200 minus 60?

00:00:54.910 --> 00:00:58.530
Well, 20 minus 6 is 14.

00:00:58.530 --> 00:01:01.480
So I just subtract, and
now I'm left with 14.

00:01:01.480 --> 00:01:06.350
And so now the problem has
resulted in 141 minus 9.

00:01:06.350 --> 00:01:09.012
So I really just have to
think about what-- well,

00:01:09.012 --> 00:01:11.580
or I could do what
141 minus 9 is.

00:01:11.580 --> 00:01:14.287
Well, 141, it's a little
bit more mental computation

00:01:14.287 --> 00:01:15.370
than you might be used to.

00:01:15.370 --> 00:01:21.070
But 141 minus 9 is
going to be 132.

00:01:21.070 --> 00:01:26.714
So we are left with 132.

00:01:26.714 --> 00:01:28.130
Let's do this one,
same technique.

00:01:28.130 --> 00:01:29.755
And I encourage you
to pause the video

00:01:29.755 --> 00:01:32.130
and try it yourself,
this same technique.

00:01:32.130 --> 00:01:35.630
Well, 9 minus 2, it's
really 900 minus 200.

00:01:35.630 --> 00:01:37.550
You're going to
be left with 700.

00:01:37.550 --> 00:01:43.550
Then 71 minus 8, let's
see, 11 minus 8 is 13.

00:01:43.550 --> 00:01:48.025
So you're going to
be left with 63.

00:01:48.025 --> 00:01:51.700
And now we have to
subtract 633 minus 6.

00:01:51.700 --> 00:01:55.270
13 minus 6 is 7.

00:01:55.270 --> 00:01:58.870
So it's going be 627.

00:01:58.870 --> 00:02:01.840
And once again, try to pause
the video and do it on your own.

00:02:01.840 --> 00:02:04.890
So I don't have any hundreds to
subtract, so I can immediately

00:02:04.890 --> 00:02:08.960
go to 72 minus 8, which
is really 720 minus 80.

00:02:08.960 --> 00:02:11.750
But let's just think
in terms of 72 minus 8.

00:02:11.750 --> 00:02:19.200
Well, 12 minus 8 is 4, so 72
minus 8 is going to be 64.

00:02:19.200 --> 00:02:22.900
So now this is the same
thing as 641 minus 8.

00:02:22.900 --> 00:02:28.250
Well, 41 minus 8
is going to be 33.

00:02:28.250 --> 00:02:30.660
I have to do a little bit
of mental computation here.

00:02:30.660 --> 00:02:34.470
So this will result in 633.


Subtracting: three digit numbers and basic regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:04.810
Let's try to subtract
659 from 971.

00:00:04.810 --> 00:00:07.856
And as soon as you start trying
to do it, you face a problem.

00:00:07.856 --> 00:00:09.230
You go to the ones
place, and you

00:00:09.230 --> 00:00:13.210
say, how am I going to
subtract a 9 from a 1?

00:00:13.210 --> 00:00:16.400
And the answer lies in
regrouping, taking value

00:00:16.400 --> 00:00:18.890
from one of the other
places here and giving it

00:00:18.890 --> 00:00:20.210
to the ones place.

00:00:20.210 --> 00:00:22.230
And to understand that
a little bit better,

00:00:22.230 --> 00:00:24.050
let me rewrite
these two numbers.

00:00:24.050 --> 00:00:25.520
Let me expand it out.

00:00:25.520 --> 00:00:30.080
So this 9 is in the hundreds
place, so it represents 900.

00:00:30.080 --> 00:00:35.030
The 7 is in the tens place,
so it represents 7 tens.

00:00:35.030 --> 00:00:39.110
And then, this 1 is in the ones
place, so it just represents 1.

00:00:39.110 --> 00:00:46.260
And then down here,
this 6 represents 600.

00:00:46.260 --> 00:00:50.070
This 5 represents 5 tens, or 50.

00:00:50.070 --> 00:00:51.750
And then, this
9-- well, it still

00:00:51.750 --> 00:00:54.386
just represents 9 ones, or 9.

00:00:54.386 --> 00:00:55.510
And we're subtracting this.

00:00:55.510 --> 00:00:58.840
We're subtracting
600 plus 50 plus 9.

00:00:58.840 --> 00:01:01.320
Or another way of thinking
about it, we're subtracting 600,

00:01:01.320 --> 00:01:05.680
we're subtracting 50,
we are subtracting 9.

00:01:05.680 --> 00:01:07.070
So let's work it out over here.

00:01:07.070 --> 00:01:08.560
So this is the
exact same problem,

00:01:08.560 --> 00:01:10.180
just written a little
bit differently.

00:01:10.180 --> 00:01:11.554
And we still have
the same issue.

00:01:11.554 --> 00:01:14.840
How do we subtract a larger
number from a smaller number?

00:01:14.840 --> 00:01:17.050
And the solution lies
in trying to take value

00:01:17.050 --> 00:01:18.360
from one of the other places.

00:01:18.360 --> 00:01:21.240
And the easiest place to go
is-- look, we've got 70 here.

00:01:21.240 --> 00:01:24.930
Why don't we take 10 from here,
and we'll be left with 60,

00:01:24.930 --> 00:01:27.480
and give that 10
to the ones place.

00:01:27.480 --> 00:01:30.250
So if you add 10 to
1, what do we have?

00:01:30.250 --> 00:01:34.390
Well, then we're
going to have 11.

00:01:34.390 --> 00:01:37.160
Notice, I have not changed
the value of the number.

00:01:37.160 --> 00:01:42.220
971 is the same thing
as 900 plus 60 plus 11.

00:01:42.220 --> 00:01:44.700
It's still 971.

00:01:44.700 --> 00:01:48.812
And now we can
actually subtract.

00:01:48.812 --> 00:01:51.545
11 minus 9 is 2.

00:01:51.545 --> 00:01:55.520
60 minus 50 is 10.

00:01:55.520 --> 00:02:01.160
And 900 minus 600 is 300.

00:02:01.160 --> 00:02:05.480
So this subtraction should
result in 300 plus 10

00:02:05.480 --> 00:02:07.990
plus 2, which is 312.

00:02:07.990 --> 00:02:10.389
Now, let's do the
exact same thing here,

00:02:10.389 --> 00:02:13.280
but we're going to do it
without expanding it out.

00:02:13.280 --> 00:02:16.240
So same issue-- how do
we subtract a 9 from a 1?

00:02:16.240 --> 00:02:18.080
Well, let's take a 10
from the tens place.

00:02:18.080 --> 00:02:19.242
We're going to regroup.

00:02:19.242 --> 00:02:21.200
So we're going to get
rid of one of these tens,

00:02:21.200 --> 00:02:24.300
so we're only going to have 6
tens left in the tens place.

00:02:24.300 --> 00:02:27.200
And we're going to give
that 10 to the ones place.

00:02:27.200 --> 00:02:30.140
So 10 plus 1 is 11.

00:02:30.140 --> 00:02:32.070
Now we are ready to subtract.

00:02:32.070 --> 00:02:34.480
11 minus 9 is 2.

00:02:34.480 --> 00:02:37.100
6 minus 5 is 1.

00:02:37.100 --> 00:02:40.350
9 minus 6 is 3.

00:02:40.350 --> 00:02:45.280
We get-- let me do
that same color-- 312.


Adding three digit numbers with regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.340
We have three different addition
problems right over here.

00:00:03.340 --> 00:00:06.250
And what I want you to do so
you get the hang of things

00:00:06.250 --> 00:00:08.740
is to pause the video
and try them on your own.

00:00:08.740 --> 00:00:11.180
But as you do them, I want
you to really keep in mind

00:00:11.180 --> 00:00:14.660
and think about what the
carrying actually means.

00:00:14.660 --> 00:00:16.520
So I assume you've
tried it on your own.

00:00:16.520 --> 00:00:20.780
Now I'll work through
them with you.

00:00:20.780 --> 00:00:22.420
So we have 9 plus 6.

00:00:22.420 --> 00:00:23.820
9 ones plus 6 ones.

00:00:23.820 --> 00:00:26.110
Well 9 plus 6 is 15.

00:00:26.110 --> 00:00:28.810
Well, we can write the
5 in the ones place,

00:00:28.810 --> 00:00:30.580
and then we can carry the one.

00:00:30.580 --> 00:00:31.650
But what did we just do?

00:00:31.650 --> 00:00:33.100
W.hat does this 1 represent?

00:00:33.100 --> 00:00:36.670
Well, we put it in the tens
place-- one 10 represents 10.

00:00:36.670 --> 00:00:41.280
So all we've said is that 9
plus 6 is equal to 10 plus 5.

00:00:41.280 --> 00:00:45.250
Is equal to one 10 plus
5, which is equal to 15.

00:00:45.250 --> 00:00:50.000
Now, in the tens place, we have
1 plus 0 plus 9 which is 10..

00:00:50.000 --> 00:00:56.350
So we can write 0 and carry
the 1 1 plus 0 plus 9 is 10.

00:00:56.350 --> 00:00:58.020
Now what does that really mean?

00:00:58.020 --> 00:01:03.240
Well this is 1 ten plus 0 tens
plus 9 tens, which is 10 tens.

00:01:03.240 --> 00:01:04.970
10 tens is 100.

00:01:04.970 --> 00:01:08.500
Or another way to think
about 100, it's 1 hundred,

00:01:08.500 --> 00:01:09.920
and 0 tens.

00:01:09.920 --> 00:01:12.030
So that's all that
carrying represents.

00:01:12.030 --> 00:01:14.870
So now we have a 1,
plus a 7, plus a 9.

00:01:14.870 --> 00:01:16.754
That is going to be 17.

00:01:16.754 --> 00:01:19.730
Now we have to remind ourselves,
this is in the hundreds place.

00:01:19.730 --> 00:01:23.390
This is actually 1 hundred
plus 7 hundred plus 9 hundred

00:01:23.390 --> 00:01:25.350
17 hundred.

00:01:25.350 --> 00:01:28.430
Or 1 thousand, 7 hundred.

00:01:28.430 --> 00:01:30.380
And of course we
have this 5 here.

00:01:30.380 --> 00:01:32.220
And we are done.

00:01:32.220 --> 00:01:34.690
Now let's try to
tackle this one.

00:01:34.690 --> 00:01:36.540
Now the reason why
I wrote it that way

00:01:36.540 --> 00:01:39.640
is to make sure that we
remind ourselves to align

00:01:39.640 --> 00:01:42.440
the proper places under
the appropriate places.

00:01:42.440 --> 00:01:49.388
So this we rewrite as-- let me
do that same green color-- we

00:01:49.388 --> 00:01:53.187
could rewrite this
as 373 plus-- we

00:01:53.187 --> 00:01:55.270
want to write the ones
place under the ones place,

00:01:55.270 --> 00:01:57.810
and the tens place
under the tens place,

00:01:57.810 --> 00:02:00.880
so that we're adding
the appropriate values.

00:02:00.880 --> 00:02:03.220
So 3 plus 8 is 11.

00:02:03.220 --> 00:02:06.090
The 1 in the ones place,
and the 1 in the tens place.

00:02:06.090 --> 00:02:08.090
10 plus 1 is 11.

00:02:08.090 --> 00:02:10.100
1 plus 7 is 8.

00:02:10.100 --> 00:02:11.890
8 plus 8 is 16.

00:02:11.890 --> 00:02:13.830
This is actually 16 tens.

00:02:13.830 --> 00:02:15.610
So let me just
write down the 16.

00:02:15.610 --> 00:02:16.800
What is 16 tens?

00:02:16.800 --> 00:02:17.860
Well, that's 160.

00:02:17.860 --> 00:02:21.290
So this 6 is a 60, and
then I have a hundred.

00:02:21.290 --> 00:02:22.940
1 plus 3 is 4.

00:02:22.940 --> 00:02:25.320
But these are actually
hundreds, so it's 4 hundreds.

00:02:25.320 --> 00:02:27.610
So we get 461.

00:02:27.610 --> 00:02:31.620
Now finally, 9 plus 3 is 12.

00:02:31.620 --> 00:02:33.770
2 ones, and 1 ten.

00:02:33.770 --> 00:02:37.120
12 is the same
thing as 10 plus 2.

00:02:37.120 --> 00:02:41.780
Now in the tens place,
1 plus 4 plus 9 is 14.

00:02:41.780 --> 00:02:44.500
So we write the 4,
and carry the 1.

00:02:44.500 --> 00:02:47.550
But remind ourselves this
is actually 10 plus 40

00:02:47.550 --> 00:02:50.340
plus 90, which is 140.

00:02:50.340 --> 00:02:53.500
Which is the same
thing as 40 plus 100.

00:02:53.500 --> 00:02:55.559
And then 1 plus 1 plus 2 is 4.

00:02:55.559 --> 00:02:56.850
But this is the hundreds place.

00:02:56.850 --> 00:02:59.940
So this is actually 1 hundred,
plus 1 hundred, plus 2 hundred,

00:02:59.940 --> 00:03:02.180
or 4 hundred.

00:03:02.180 --> 00:03:04.290
And we're done.


Subtracting: Regrouping from zero

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.390
Let's try to subtract
164 from 301,

00:00:03.390 --> 00:00:05.380
and I encourage you
to pause this video

00:00:05.380 --> 00:00:08.370
and try it on your own first.

00:00:08.370 --> 00:00:09.930
So let's go place
by place, and we

00:00:09.930 --> 00:00:13.450
can realize where we have to do
some borrowing or regrouping.

00:00:13.450 --> 00:00:15.440
So in the ones place,
we have an issue.

00:00:15.440 --> 00:00:17.080
4 is larger than 1.

00:00:17.080 --> 00:00:19.660
How do we subtract a larger
number from a smaller number?

00:00:19.660 --> 00:00:21.430
We also an issue
in the tens place.

00:00:21.430 --> 00:00:23.300
6 is larger than 0.

00:00:23.300 --> 00:00:25.840
How do we subtract 6 from 0?

00:00:25.840 --> 00:00:28.170
So the answer that might
be jumping into your head

00:00:28.170 --> 00:00:30.930
is oh, we've got to do some
borrowing or some regrouping.

00:00:30.930 --> 00:00:33.160
But then you might be
facing another problem.

00:00:33.160 --> 00:00:35.881
You'd say, OK, well, let's try
to borrow from the tens place

00:00:35.881 --> 00:00:36.380
here.

00:00:36.380 --> 00:00:37.160
So we have a 1.

00:00:37.160 --> 00:00:39.870
If we could borrow 10 from the
tens place, it could be 11.

00:00:39.870 --> 00:00:41.890
But there's nothing
here in the tens place.

00:00:41.890 --> 00:00:45.040
There's nothing to
borrow, so what do we do?

00:00:45.040 --> 00:00:47.000
So the way I would
tackle it is first

00:00:47.000 --> 00:00:49.070
borrow for the tens place.

00:00:49.070 --> 00:00:52.360
So we have nothing here,
so let's regroup 100

00:00:52.360 --> 00:00:54.220
from the hundreds place.

00:00:54.220 --> 00:00:56.100
So that's equivalent
to borrowing a 1

00:00:56.100 --> 00:00:57.230
from the hundreds place.

00:00:57.230 --> 00:00:58.940
So that's now a 2.

00:00:58.940 --> 00:01:02.940
And now in the tens
place, instead of a 0,

00:01:02.940 --> 00:01:06.420
we are going to have a 10.

00:01:06.420 --> 00:01:08.610
Now, let's make sure that
this still makes sense.

00:01:08.610 --> 00:01:11.480
This is 200 plus 10 tens.

00:01:11.480 --> 00:01:15.250
10 tens is 100, plus 1.

00:01:15.250 --> 00:01:20.100
200 plus 100 plus
1 is still 301.

00:01:20.100 --> 00:01:21.730
So this still makes sense.

00:01:21.730 --> 00:01:23.270
Now, the reason why
this is valuable

00:01:23.270 --> 00:01:26.430
is now we have something to
regroup from the tens place.

00:01:26.430 --> 00:01:31.670
If we take one of these tens,
so now we're left with 9 tens,

00:01:31.670 --> 00:01:35.980
and we give it to the ones
place, so you give 10 plus 1,

00:01:35.980 --> 00:01:39.860
you're going to be left with 11.

00:01:39.860 --> 00:01:41.900
And we can verify that
we still haven't changed

00:01:41.900 --> 00:01:44.180
the value of the actual number.

00:01:44.180 --> 00:01:51.160
200 plus 90 is 290,
plus 11 is still 301.

00:01:51.160 --> 00:01:53.890
And what was neat about
this is now up here,

00:01:53.890 --> 00:01:57.100
all of these numbers are larger
than the corresponding number

00:01:57.100 --> 00:01:58.560
in the same place.

00:01:58.560 --> 00:02:00.620
So we're ready to subtract.

00:02:00.620 --> 00:02:09.199
11 minus 4 is-- let's see, 10
minus 3 is 7, so 11 minus 4

00:02:09.199 --> 00:02:11.030
is 7.

00:02:11.030 --> 00:02:18.220
9 minus 6 is 3,
and 2 minus 1 is 1.

00:02:18.220 --> 00:02:20.890
So we are left with 137.


Subtracting: regrouping twice

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:05.310
So let's subtract 286 from 913.

00:00:05.310 --> 00:00:07.840
But first I'm going to do it
in a slightly different way.

00:00:07.840 --> 00:00:08.950
I've taken each
of these numbers,

00:00:08.950 --> 00:00:10.080
and I've expanded them out.

00:00:10.080 --> 00:00:12.960
This 9 in the hundreds
place represents 900.

00:00:12.960 --> 00:00:14.950
This 1 in the tens
place represents 10.

00:00:14.950 --> 00:00:17.580
This 3 in the ones
place represents 3.

00:00:17.580 --> 00:00:23.010
Likewise, 286 is the same
thing as 200 plus 80 plus 6.

00:00:23.010 --> 00:00:25.666
So let's try to subtract
going place by place.

00:00:25.666 --> 00:00:27.040
So if we start in
the ones place,

00:00:27.040 --> 00:00:28.760
we have a problem immediately.

00:00:28.760 --> 00:00:30.570
3 is less than 6.

00:00:30.570 --> 00:00:33.530
How do we subtract a larger
number from a smaller number?

00:00:33.530 --> 00:00:39.520
We also have a problem
in the tens place,

00:00:39.520 --> 00:00:40.730
80 is larger than 10.

00:00:40.730 --> 00:00:43.610
How do we subtract a larger
number from a smaller number?

00:00:43.610 --> 00:00:46.540
And you might guess the answer
here is regrouping, sometimes

00:00:46.540 --> 00:00:47.414
called borrowing.

00:00:47.414 --> 00:00:49.080
We're going to take
value from one place

00:00:49.080 --> 00:00:50.830
and give it to another.

00:00:50.830 --> 00:00:53.220
So let's say this
scenario right over here,

00:00:53.220 --> 00:00:55.870
where we have this
3, and we want

00:00:55.870 --> 00:00:58.650
to take some value from
one of the other places.

00:00:58.650 --> 00:01:03.300
Well, I could take 10
from the tens place,

00:01:03.300 --> 00:01:05.519
so then this is
going to become 0.

00:01:05.519 --> 00:01:10.246
And if I give that 10 to the
ones place, so 10 plus 3 is 13.

00:01:12.890 --> 00:01:14.700
Notice I haven't
changed the value.

00:01:14.700 --> 00:01:19.870
900 plus 0 plus 13 is still 913.

00:01:19.870 --> 00:01:22.480
Now, this solved the
problem for the ones place.

00:01:22.480 --> 00:01:24.812
I can now subtract 6 from 13.

00:01:24.812 --> 00:01:27.020
But it made the problem in
the tens place even worse.

00:01:27.020 --> 00:01:29.110
I now have to
subtract 80 from 0.

00:01:29.110 --> 00:01:30.580
What do I do?

00:01:30.580 --> 00:01:32.990
Well, luckily, I can go
to the hundreds place.

00:01:32.990 --> 00:01:39.150
I could take 100 from 900,
so then I'm left with 800.

00:01:39.150 --> 00:01:41.810
And I could you give
it to the tens place.

00:01:41.810 --> 00:01:43.970
So if I give it to
the tens place, then

00:01:43.970 --> 00:01:49.650
this is going to be 100.

00:01:49.650 --> 00:01:52.140
Notice this still
adds up to 913.

00:01:52.140 --> 00:01:56.730
800 plus 100 plus 13 is 913.

00:01:56.730 --> 00:01:58.000
Why is this valuable?

00:01:58.000 --> 00:02:01.367
Well, now in every column, I'm
subtracting a smaller number

00:02:01.367 --> 00:02:01.950
from a larger.

00:02:01.950 --> 00:02:04.200
You might say, wait, isn't
there a positive sign here?

00:02:04.200 --> 00:02:05.950
But we have this
negative out here.

00:02:05.950 --> 00:02:07.750
So we're subtracting 6 from 13.

00:02:07.750 --> 00:02:11.780
We're subtracting 80 from
100, subtracting 200 from 800.

00:02:11.780 --> 00:02:13.740
So let's do it.

00:02:13.740 --> 00:02:16.260
13 minus 6 is 7.

00:02:16.260 --> 00:02:21.210
100 minus 80 is 20.

00:02:21.210 --> 00:02:25.520
800 minus 200 is 600.

00:02:25.520 --> 00:02:31.170
So we're left with 600 plus
20 plus 7, which is 627.

00:02:31.170 --> 00:02:32.960
Now let's do the
exact same thing here,

00:02:32.960 --> 00:02:35.290
but we're not going to
expand out the numbers.

00:02:35.290 --> 00:02:37.590
So 6 is greater than
3, what do we do?

00:02:37.590 --> 00:02:40.460
Well, we can regroup
from the tens place.

00:02:40.460 --> 00:02:44.230
We can take 10 from here
so we're left with 0 tens

00:02:44.230 --> 00:02:47.560
and give that 1 ten
to the ones place.

00:02:47.560 --> 00:02:51.460
So you give 10 to
the 3, it becomes 13.

00:02:54.580 --> 00:02:56.560
But now we have a problem
in the tens place.

00:02:56.560 --> 00:02:58.390
How do we subtract 8 from 0?

00:02:58.390 --> 00:03:01.720
Well, we could take 100
from the hundreds place,

00:03:01.720 --> 00:03:06.740
so 900 becomes 800, and give
that 100 to the tens place.

00:03:06.740 --> 00:03:11.866
So you give the 100 to the tens
place, 100 plus 0 tens is 100.

00:03:11.866 --> 00:03:15.215
100 is the same
thing as 10 tens.

00:03:19.790 --> 00:03:22.500
And so now we are
ready to subtract.

00:03:22.500 --> 00:03:28.340
13 minus 6 is 7,
10 minus 8 is 2.

00:03:28.340 --> 00:03:32.510
Remember, this is really 10
tens minus 8 tens to get 2 tens.

00:03:36.050 --> 00:03:38.310
100 minus 80 to get 20.

00:03:38.310 --> 00:03:46.380
And then finally, we have 800
minus 200 to get 600-- 627.


Subtracting: three digit numbers and regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.240
I've written the same
subtraction problem twice.

00:00:03.240 --> 00:00:08.750
Here we see we're
subtracting 172 from 629.

00:00:08.750 --> 00:00:10.950
And all I did here is I
expanded out the numbers.

00:00:10.950 --> 00:00:17.400
I wrote 629 as 600 plus 20
plus 9, and I rewrote 172,

00:00:17.400 --> 00:00:19.000
the one is 100.

00:00:19.000 --> 00:00:20.190
So that's there.

00:00:20.190 --> 00:00:21.430
This is 7/10.

00:00:21.430 --> 00:00:23.360
It's in the tens
place, so it's 70.

00:00:23.360 --> 00:00:26.610
And then the 2 is 2 ones,
so it just represents 2.

00:00:26.610 --> 00:00:29.031
And we'll see why this
is useful in a second.

00:00:29.031 --> 00:00:30.780
So let's just start
subtracting, and we'll

00:00:30.780 --> 00:00:33.160
start with the ones place.

00:00:33.160 --> 00:00:35.090
So we have 9 minus 2.

00:00:35.090 --> 00:00:37.390
Well, that's clearly just 7.

00:00:37.390 --> 00:00:40.410
And over here we could
also say, well, 9 minus 2,

00:00:40.410 --> 00:00:42.090
we have the
subtraction out front.

00:00:42.090 --> 00:00:44.130
That is going to be 7.

00:00:44.130 --> 00:00:45.527
Pretty straightforward.

00:00:45.527 --> 00:00:47.110
But then something
interesting happens

00:00:47.110 --> 00:00:48.620
when we get to the tens place.

00:00:48.620 --> 00:00:51.770
We're going to try to
subtract 2 minus 7,

00:00:51.770 --> 00:00:54.660
or we're going to try
to subtract 7 from 2.

00:00:54.660 --> 00:00:56.790
And we haven't learned
yet how to do things

00:00:56.790 --> 00:00:59.110
like negative numbers, which
we'll learn in the future,

00:00:59.110 --> 00:01:00.340
so we have a problem.

00:01:00.340 --> 00:01:04.360
How do you subtract a larger
number from a smaller number?

00:01:04.360 --> 00:01:07.900
Well, luckily we have
something in our toolkit called

00:01:07.900 --> 00:01:10.490
regrouping, sometimes
called borrowing.

00:01:10.490 --> 00:01:11.840
And that's why this is valuable.

00:01:11.840 --> 00:01:14.180
When we're trying to
subtract a 7 from a 2,

00:01:14.180 --> 00:01:17.930
we're really trying to
subtract this 70 from this 20.

00:01:17.930 --> 00:01:20.220
Well, we can't subtract
the 70 from the 20,

00:01:20.220 --> 00:01:22.030
but we have other
value in the number.

00:01:22.030 --> 00:01:24.150
We have value in
the hundreds place.

00:01:24.150 --> 00:01:29.180
So why don't we take 100 from
the 600, so that becomes 500,

00:01:29.180 --> 00:01:32.230
and give that 100
to the tens place?

00:01:32.230 --> 00:01:37.570
If we give that 100 to the tens
place, what is 100 plus 20?

00:01:37.570 --> 00:01:43.370
Well, it's going to be 120.

00:01:43.370 --> 00:01:46.920
So all I did, I didn't
change the value of 629.

00:01:46.920 --> 00:01:48.890
I took 100 from
the hundreds place

00:01:48.890 --> 00:01:50.760
and I gave it to the tens place.

00:01:50.760 --> 00:01:56.020
Notice 500 plus 120
plus 9 is still 629.

00:01:56.020 --> 00:01:57.680
We haven't changed the value.

00:01:57.680 --> 00:02:00.130
So how would we do
that right over here?

00:02:00.130 --> 00:02:02.750
Well, if we take 100
from the hundreds place,

00:02:02.750 --> 00:02:06.870
this 600 becomes
a 5, 500, and we

00:02:06.870 --> 00:02:08.949
give that hundred
to the tens place,

00:02:08.949 --> 00:02:10.690
it's going to be 10 hundreds.

00:02:10.690 --> 00:02:13.390
So this will now become a 12.

00:02:13.390 --> 00:02:15.920
This will now become a 12.

00:02:15.920 --> 00:02:22.010
But notice, this 12 in the
tens place represents 12 tens,

00:02:22.010 --> 00:02:23.760
or 120.

00:02:23.760 --> 00:02:26.090
So this is just another
way of representing

00:02:26.090 --> 00:02:27.100
what we've done here.

00:02:27.100 --> 00:02:28.080
There's no magic here.

00:02:28.080 --> 00:02:30.760
This is often called borrowing,
where you say hey, I took a 1

00:02:30.760 --> 00:02:33.020
from the 6, and I
gave it to the 2.

00:02:33.020 --> 00:02:35.970
But wait, why did
this 2 become a 12?

00:02:35.970 --> 00:02:37.900
Why was I able to add 10?

00:02:37.900 --> 00:02:41.610
Well, you've added
10 tens, or 100.

00:02:41.610 --> 00:02:46.180
You took 100 from here, so
600 became 500, and then 20

00:02:46.180 --> 00:02:47.990
became 120.

00:02:47.990 --> 00:02:49.780
But now we're ready to subtract.

00:02:49.780 --> 00:02:55.030
12 tens minus 7 tens is 5 tens.

00:02:55.030 --> 00:03:00.030
Or you could say
120 minus 70 is 50.

00:03:00.030 --> 00:03:02.460
And then finally, you
have the hundreds place.

00:03:02.460 --> 00:03:07.880
5 minus 1 is equal to 4, but
that's really 500 minus 100

00:03:07.880 --> 00:03:08.880
is equal to 400.

00:03:08.880 --> 00:03:12.630
500 minus 100 is equal to 400.

00:03:12.630 --> 00:03:16.600
And so you get 457, which
is the same thing as 400

00:03:16.600 --> 00:03:19.710
plus 50 plus 7.