The Eighth Amendment

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- [Kim] Hi, this is Kim from Khan Academy.

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Today, I'm learning about
the Eighth Amendment

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to the US Constitution,

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which prohibits the government

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from imposing excessive fines and bail

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or inflicting cruel and unusual punishment

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on individuals accused
or convicted of a crime.

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But what counts as excessive
or cruel and unusual?

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To learn more, I sought out the help

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of two experts on the Eighth Amendment.

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John Stinneford is the assistant director

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of The Criminal Justice Center

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at the University of Florida Law School.

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John Bessler is an
associate professor of law

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at the University of Baltimore Law School.

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So, Professor Bessler,

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why were the Framers

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so keen to include the Eighth Amendment?

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Why did they wanna protect
these rights in particular?

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- [Bessler] Well, these
rights were actually enshrined

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in the English Bill of Rights of 1689.

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And so when the Americans

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got in the dispute with Great Britain,

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they decided they wanted
to have the same rights

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that Englishmen had.

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And so it was not too surprising

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that when George Mason actually wrote

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the Virginia Declaration
of Rights in 1776,

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that he looked to English law

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to see what rights the English had

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because he wanted exactly the same rights.

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And so these 16 words
in the Eighth Amendment

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have been subject to a lot of
controversy over the years.

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But I think one of the reasons

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that the Founders wanted this was that,

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like the English would have problems

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with the Monarch imposing
excessive bail, excessive fines,

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inflicting cruel and unusual punishments,

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the Founders also knew that was a risk

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that there was abuse from the government

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in the United States

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and so they also wanted those rights.

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Because originally in the Constitution,

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these rights were not protected against.

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And the Bill of Rights
was ratified in 1791

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which ensured that there'd be protections

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against these cruel
and unusual punishments

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and excessive bail and excessive fines.

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- [Stinneford] If you look
at the Eighth Amendment,

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there's three clauses.

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The excessive bail clause,

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the excessive fines clause,

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and the cruel and unusual
punishments clause.

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And these all have one thing in common

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which is that these are
all penalties essentially

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that the government inflicts on people

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usually as the result of
either being accused of crime

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or of being convicted of crime.

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So if you've been arrested,
you're waiting for trial,

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very often your only way to
get out of jail before trial

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is to make bail,

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and of course after you've been convicted

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the court might impose a fine on you

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or some other kind of punishment.

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So the Eighth Amendment is designed

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to prevent the government

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from doing things that are excessive.

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When the government punishes a person,

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that's the most coercive
thing the government does

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short of war.

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Other than shooting you in battle,

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picking you up and
throwing you in a jail cell

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is about as bad as it gets.

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And so the Framers wanted to make sure

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that we had a constitutional protection

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when it comes to criminal punishment.

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- [Kim] So, in a lot of
cases in the Bill of Rights,

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you see the Framers reacting
to some historical evil

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that they hoped to prevent.

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For example, the Third Amendment says

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you can't quarter soldiers
in private citizens homes

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because that had been such
an important tipping point

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in the American Revolution.

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Was there something that
the Framers had in mind

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as a particular historical evil
that they wanted to prevent?

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- [Bessler] Well, there
was some historical evils.

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The English Bill of Rights
went into place in 1689,

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and when that went into place

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it was actually a controversy in England

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around a person by the
name of Titus Oates.

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Titus Oates was somebody who
had made false accusations,

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Oates had committed perjury,

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resulted actually that allegation

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in the execution of 15 Catholics.

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And the plot that he had alleged

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was one to assassinate
the King of England.

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- [Stinneford] The question is
what to do with Oates, right?

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Because as a moral matter,
he's about as bad as it gets.

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In fact, in 2005, English
historians voted him

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the worst Briton of the 17th century

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and the third worst Briton
of the last thousand years

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or something like that.

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So a very bad guy.

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You can think of him as
sort of a serial killer.

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But the problem is that the
actual crime he committed

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was the crime of perjury.

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And even though his perjury

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resulted in the deaths
of many innocent people,

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nonetheless he could only
be convicted of perjury

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which at the time was a misdemeanor,

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which meant that he could not
be executed for his crime.

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So when it came time for his sentencing,

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the judge, Chief Justice Jeffreys,

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who was a famous hanging
judge from English history,

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says to Oates,

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"Well, Oates, we can't take your life,

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"we can't take your limb,

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"but we have something
special prepared for you."

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And it turns out that what
they had prepared for him was,

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number one, a huge fine.

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They fined him like 2,000 marks.

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They sentenced that he be
dragged across the city of London

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while being flogged.

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He was dragged from Aldgate to
Newgate while being flogged.

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And then two days later,

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just as the scabs were
starting to form on his wounds,

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he's dragged back across
the city of London

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from Newgate to Tyburn
again while being flogged.

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Many people think the hope was

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that he would die from the flogging

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but, like a cockroach in a
nuclear war, he survived.

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- [Bessler] This punishment

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was actually a very severe punishment.

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And after the English Bill
of Rights was promulgated,

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the Titus Oates punishment was challenged.

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And some of the members
of the House of Lords

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actually called the punishment barbarous,

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inhuman and unchristian

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and contrary to the
English Bill of Rights.

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So there was no precedent to warrant

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the punishments of whipping

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and committing to prison for life

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for the crime of perjury.

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This punishment was eventually remitted

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in the sense that Oates
was later released.

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- [Stinneford] Although the House of Lords

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refused to vote to suspend
the judgment against Oates

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'cause they hated his so much.

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They said so ill a man

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shouldn't get the benefit of any relief.

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But they all agreed

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that the punishment was cruel and unusual.

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And what's interesting is

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in the debate they say things like,

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this punishment is contrary
to law and ancient practice.

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It is without precedent

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and it'll be a bad
precedent for the future.

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So, in other words, it's cruel,

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and the way we know it's cruel

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is because it's so much harsher

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than has previously been inflicted

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for the crime of perjury.

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So the Oates case shows us

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that when the words cruel
and unusual were first used,

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they were used to describe punishments

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that are harsher than the
common law would permit,

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or harsher than longstanding
prior practice would permit.

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And this means, among other things,

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that the cruel and
unusual punishments clause

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is not limited to gruesome punishments

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like torture and the rack
and all that kind of thing.

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'Cause in fact the punishments
inflicted on Oates,

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although they were very harsh
for the crime of perjury,

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were not as harsh as
some other punishments

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that the common law permitted
for other crimes like treason.

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- [Kim] Wow.

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So that tells us a lot
about the English context

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of cruel and unusual punishment.

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Do we know what cruel and unusual meant

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to the Framers of the US Bill of Rights?

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- [Stinneford] Punishments
were cruel and unusual, again,

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if they're too harsh

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in light of longstanding prior practice

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for the crime for which they're inflicted.

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There's another problem with statutory law

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or with decisions of a judge or a king,

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or a president for that matter,

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that's also really relevant

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when we think about the
Eighth Amendment and that is

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sometimes the government
gets really mad at someone.

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They either think of a person
as an enemy of the state

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and they wanna inflict the
worst punishment they can

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on that person,

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or perhaps there's a panic about
a certain group in society.

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For example, in American society recently,

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there have been panics about drug crime

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or panics about sex offenses.

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And every time that happens

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the government tries to respond
with new forms of punishment

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that are much, much harsher
than what came before.

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So the insight behind
the Eighth Amendment is

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that when the government wants
to inflict a new punishment,

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you have to compare it against
longstanding prior practice,

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that is you have to compare
it against the common law.

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So the common law was called

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the law of custom and long usage.

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So if something comported
with the common law,

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it was usual.

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If it was contrary to the common law,

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it was unusual.

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And that's where we get the phrase

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cruel and unusual punishments.

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It's basically punishments that are cruel

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in light of or in comparison to

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longstanding prior practice.

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So the basic point of the Eighth Amendment

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from a historical point of view

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is to prohibit the government

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from innovating in a cruel manner,

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making up new cruel punishments

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in response to some actual
or perceived provocation

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by a criminal.

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- [Bessler] When the US
Bill of Rights was adopted

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many years later, many decades later,

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they had their own issues that
they were struggling with.

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And so the history shows
that the American Founders

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probably meant something
different than the English meant

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because it was done
over a 100 years later,

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but no one knows exactly what was meant

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when they adopted that wording.

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To give an example,

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when the Bill was debated in Congress,

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a representative from South Carolina

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said that he objected to the words

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nor cruel and unusual punishments

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because he said the import
of them was too indefinite.

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And there was another legislator,

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Mr. Livermore from New Hampshire,

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who said that the clause seemed to express

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"A great deal of humanity,

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"in which account I
have no objection to it.

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"But as it seems to have no meaning in it,

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"I do not think it necessary.

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"What is meant by the
terms excessive bail?"

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He asked.

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"Who are to be the judges?"

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He also asked.

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"What is understood by excessive fines?"

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It lies to the court to determine.

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And so that's really where we
are today in a lot of ways.

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The court is still deciding,

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in this case the US Supreme
Court is still deciding

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what the Eighth Amendment
language actually means today.

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- [Stinneford] The modern case law,

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specially starting in the 1970s,

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revolved a lot around the death penalty.

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And so the question was,

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is it still okay to execute people

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for various crimes short of murder?

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Since the 70s,

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the Court has continued to
do that in a number of areas.

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So it's said

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you can't execute the
mentally disabled anymore,

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you can't execute minors,

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you can't execute anyone
for a non-homicide offense.

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Although it's limited the death penalty

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in the name of current
standards of decency,

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it's really not clear how
the Court has set about

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to determine whether a punishment

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meets current standards of decency.

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When the Court is kind of on its own

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saying that a punishment

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violates current standards of decency,

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despite the fact

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that most democratically
elected legislators

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actually approve the practice,

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then it looks like the Court
is acting as a political body.

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It's led the court to ignore

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what I think is the
real danger of cruelty,

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which is that when there's a public panic

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and the legislature responds
by rationing up punishment

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to new and unprecedented
levels of punishment.

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And that's actually happened quite a lot

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in the last 40 years.

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Every time there's a panic,

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you predictably see the legislatures

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coming up with new punishments

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that are much, much harsher
than what came before.

00:11:33.590 --> 00:11:36.000
And so for example with sex offenders,

00:11:36.000 --> 00:11:37.290
there are now a bunch of states

00:11:37.290 --> 00:11:39.880
that actually impose chemical castration

00:11:39.880 --> 00:11:42.560
as a form of punishment for sex offenders.

00:11:42.560 --> 00:11:45.020
Now, castration as a form of punishment

00:11:45.020 --> 00:11:48.020
fell out of usage in the 13th century.

00:11:48.020 --> 00:11:50.520
We're literally getting
medieval on sex offenders.

00:11:51.380 --> 00:11:54.210
But the Court can't do
anything really to stop it,

00:11:54.210 --> 00:11:55.960
or at least hasn't,

00:11:55.960 --> 00:12:00.710
because these are very
popular forms of punishment.

00:12:00.710 --> 00:12:02.100
Everyone hates sex offenders.

00:12:02.100 --> 00:12:04.520
- [Bessler] The UN has actually decided

00:12:04.520 --> 00:12:08.280
that anything more than 15-days
use of solitary confinement

00:12:08.280 --> 00:12:10.160
should not be permitted.

00:12:10.160 --> 00:12:11.570
Justice Anthony Kennedy

00:12:11.570 --> 00:12:14.210
actually raised the issue
of solitary confinement

00:12:14.210 --> 00:12:15.610
in recent opinion.

00:12:15.610 --> 00:12:17.280
He authored for the Supreme Court

00:12:17.280 --> 00:12:19.060
and he actually raised
the issue on his own,

00:12:19.060 --> 00:12:21.010
an oral argument at one point,

00:12:21.010 --> 00:12:22.360
talking about how long

00:12:22.360 --> 00:12:24.650
people actually spend
in solitary confinement

00:12:24.650 --> 00:12:26.340
in American prisons,

00:12:26.340 --> 00:12:29.200
including on death row.

00:12:29.200 --> 00:12:32.510
You have cases actually
where people are spending

00:12:32.510 --> 00:12:35.840
not just years but sometimes
decades on death row

00:12:35.840 --> 00:12:37.720
in these kinds of conditions.

00:12:37.720 --> 00:12:41.040
Justice Breyer just
wrote a dissent in a case

00:12:41.040 --> 00:12:43.140
where the person had been on death row

00:12:43.140 --> 00:12:44.890
for more than 40 years,

00:12:44.890 --> 00:12:46.810
so literally four decades

00:12:46.810 --> 00:12:50.220
in these kinds of very harsh
conditions of confinement.

00:12:50.220 --> 00:12:53.100
Other countries have decided

00:12:53.100 --> 00:12:56.410
that that is not something
that they want to permit

00:12:56.410 --> 00:12:58.150
and they've actually set a rule

00:12:58.150 --> 00:12:59.560
that anybody that's on death row

00:12:59.560 --> 00:13:01.370
for a certain number of years for example

00:13:01.370 --> 00:13:03.800
would have their sentence
commuted to a life sentence

00:13:03.800 --> 00:13:05.630
because of the psychological aspect

00:13:05.630 --> 00:13:10.110
of waiting for one's own death.

00:13:10.110 --> 00:13:11.100
- [Kim] In the Bill of Rights,

00:13:11.100 --> 00:13:14.620
this is the last of four
amendments, actually,

00:13:14.620 --> 00:13:18.740
that are concerned with
protections for the accused.

00:13:18.740 --> 00:13:22.080
So, why do you think there is
so much in the Bill of Rights

00:13:22.080 --> 00:13:24.130
about the justice system?

00:13:24.130 --> 00:13:27.000
Were the Framers particularly interested

00:13:27.000 --> 00:13:29.760
in making sure that
the accused had rights?

00:13:29.760 --> 00:13:30.837
- [Stinneford] Yeah, they were.

00:13:30.837 --> 00:13:34.180
And in particular
Americans were very devoted

00:13:34.180 --> 00:13:38.210
to the idea of the common
law as a source of rights.

00:13:38.210 --> 00:13:39.620
In fact, that's why we had

00:13:39.620 --> 00:13:42.080
the American Revolution
in the first place,

00:13:42.080 --> 00:13:46.030
was that England was denying
to Americans common law rights

00:13:46.030 --> 00:13:48.480
like the right not to be taxed

00:13:48.480 --> 00:13:50.500
without representation in Parliament

00:13:50.500 --> 00:13:53.100
but also more specifically
to the criminal law

00:13:53.100 --> 00:13:55.980
they were denying them
the right to a jury trial

00:13:55.980 --> 00:13:57.460
in criminal cases.

00:13:57.460 --> 00:13:59.780
And so Americans wanted to make sure

00:13:59.780 --> 00:14:03.020
that when the US Constitution was adopted

00:14:03.020 --> 00:14:04.670
that those common law rights

00:14:04.670 --> 00:14:06.880
that had built up over time in England

00:14:07.750 --> 00:14:12.750
would be preserved in the new
American constitutional order.

00:14:13.100 --> 00:14:16.320
And many of those rights
had to do with criminal law,

00:14:16.320 --> 00:14:18.440
both criminal procedures

00:14:18.440 --> 00:14:20.630
and to some degree
subsequence of criminal law

00:14:20.630 --> 00:14:22.280
and of course criminal punishments.

00:14:22.280 --> 00:14:24.010
And, again, the reason gets back

00:14:24.010 --> 00:14:26.020
to what I said at the beginning

00:14:26.020 --> 00:14:28.590
which is that when the
government punishes someone,

00:14:28.590 --> 00:14:31.040
that's about the worse thing it can do.

00:14:31.040 --> 00:14:35.470
And because the early Americans
who framed the Constitution

00:14:35.470 --> 00:14:38.360
were very powerfully
concerned with liberty,

00:14:38.360 --> 00:14:39.193
they wanted to make sure

00:14:39.193 --> 00:14:42.170
the government would
preserve their liberty,

00:14:42.170 --> 00:14:43.810
protect their liberty,

00:14:43.810 --> 00:14:46.280
and not become tyrannical.

00:14:46.280 --> 00:14:48.740
And so one of the main ways

00:14:48.740 --> 00:14:50.950
that they wanted to
make sure this happened

00:14:50.950 --> 00:14:53.580
was by limiting the
power of the government

00:14:53.580 --> 00:14:55.660
to punish whoever it wanted to

00:14:55.660 --> 00:14:57.370
for any reason that it wanted to.

00:14:57.370 --> 00:15:00.900
And so we have really the
majority of the protections

00:15:00.900 --> 00:15:02.090
in the Bill of Rights

00:15:02.090 --> 00:15:05.440
have to do with the protections
for criminal defendants.

00:15:05.440 --> 00:15:08.600
- [Kim] What about excessive
bail and excessive fines?

00:15:08.600 --> 00:15:13.600
How can we define what
kind of financial penalty

00:15:14.410 --> 00:15:17.720
is proportionate to a crime?

00:15:17.720 --> 00:15:19.290
- [Bessler] The courts
have said, essentially,

00:15:19.290 --> 00:15:21.070
they've looked at dictionary definitions,

00:15:21.070 --> 00:15:24.550
that excessive means
more than it's necessary.

00:15:24.550 --> 00:15:25.800
One of the core principles

00:15:25.800 --> 00:15:29.660
actually go back to look at
Beccaria's work in the 1760s.

00:15:29.660 --> 00:15:32.537
He talked about this
idea of a scale of crimes

00:15:32.537 --> 00:15:34.330
and a scale of punishments,

00:15:34.330 --> 00:15:36.470
and he said that there
should be proportionality

00:15:36.470 --> 00:15:37.913
between the two.

00:15:37.913 --> 00:15:39.470
And so that proportionality principle

00:15:39.470 --> 00:15:43.290
is one that we're really
still wrestling with today.

00:15:43.290 --> 00:15:44.760
- [Stinneford] Now, the point of bail

00:15:44.760 --> 00:15:47.520
is not to punish someone

00:15:47.520 --> 00:15:51.840
but rather just to make sure
that they will appear at trial.

00:15:51.840 --> 00:15:55.130
And so the amount of money
you have to impose for bail

00:15:55.130 --> 00:15:57.590
doesn't depend so much on
what crime you committed

00:15:57.590 --> 00:16:00.790
but what your financial resources are.

00:16:00.790 --> 00:16:02.200
It depends partially on the crime too

00:16:02.200 --> 00:16:04.200
but largely on your financial resources.

00:16:04.200 --> 00:16:06.190
So the amount of money necessary

00:16:06.190 --> 00:16:09.490
to make sure that a poor
man appears at trial

00:16:09.490 --> 00:16:10.840
is probably gonna be much lower

00:16:10.840 --> 00:16:12.100
than the amount of money necessary

00:16:12.100 --> 00:16:14.740
to make sure that a rich
man appears at trial.

00:16:14.740 --> 00:16:16.850
And so it's a standard

00:16:16.850 --> 00:16:18.500
that depends partly on
the nature of the crime

00:16:18.500 --> 00:16:22.510
but also partially on the
nature of the offender.

00:16:22.510 --> 00:16:24.230
- [Kim] I imagine that what seemed

00:16:24.230 --> 00:16:28.880
like cruel and unusual
punishment in the 18th century

00:16:28.880 --> 00:16:32.910
might not be what we consider
cruel and unusual today.

00:16:32.910 --> 00:16:37.690
For example, we don't do
whipping as a punishment anymore.

00:16:37.690 --> 00:16:41.660
How has what counts as
cruel and unusual punishment

00:16:41.660 --> 00:16:43.130
changed over time?

00:16:43.130 --> 00:16:45.530
- [Bessler] The law really
changes gradually over time,

00:16:45.530 --> 00:16:46.453
as you know.

00:16:46.453 --> 00:16:48.650
And so when you look back at history,

00:16:48.650 --> 00:16:50.340
they actually had a large collection

00:16:50.340 --> 00:16:53.010
of pretty gruesome punishments

00:16:53.010 --> 00:16:56.470
back in the 18th century.

00:16:56.470 --> 00:16:58.400
We had non-lethal corporal punishments,

00:16:58.400 --> 00:17:01.020
they used things like branding people.

00:17:01.020 --> 00:17:04.440
They of course were
whipping slaves back then.

00:17:04.440 --> 00:17:06.100
Slavery was still around.

00:17:06.100 --> 00:17:08.860
This is well before the Civil War ended

00:17:08.860 --> 00:17:10.510
and the destitution of slavery.

00:17:10.510 --> 00:17:12.740
And we also had things like ear cropping.

00:17:12.740 --> 00:17:15.440
People would get their ears cut off.

00:17:15.440 --> 00:17:18.400
In the Crimes Act of 1790,

00:17:18.400 --> 00:17:19.700
which was passed just the year before

00:17:19.700 --> 00:17:22.403
the ratification of the Eighth Amendment,

00:17:23.320 --> 00:17:25.820
Congress actually
authorized public whipping,

00:17:25.820 --> 00:17:27.240
lashing of people,

00:17:27.240 --> 00:17:29.240
and it also authorized the pillory,

00:17:29.240 --> 00:17:32.140
the same punishment that had
been used against Titus Oates.

00:17:32.140 --> 00:17:36.560
So there was these non-lethal
corporal punishments

00:17:36.560 --> 00:17:37.430
and really the death penalty

00:17:37.430 --> 00:17:40.960
is the last vestige of bodily punishment

00:17:40.960 --> 00:17:44.720
that the Supreme Court has
read the Eighth Amendment

00:17:44.720 --> 00:17:47.570
to allow the use of capital punishment.

00:17:47.570 --> 00:17:49.210
That issue is still a very live one

00:17:49.210 --> 00:17:50.120
before the Supreme Court,

00:17:50.120 --> 00:17:54.510
renouncing challenges about
lethal injection protocols.

00:17:54.510 --> 00:17:58.720
We saw a challenge to a
protocol in Kentucky in 2008.

00:17:58.720 --> 00:18:03.400
We saw one to protocol
in Oklahoma in 2015.

00:18:03.400 --> 00:18:05.330
These of course are things
that the Founding Fathers

00:18:05.330 --> 00:18:06.760
would never have envisioned.

00:18:06.760 --> 00:18:08.020
- [Stinneford] Lethal injection

00:18:08.020 --> 00:18:11.030
typically involves a three-drug cocktail.

00:18:11.030 --> 00:18:13.770
There's a barbiturate which is
supposed to put you to sleep.

00:18:13.770 --> 00:18:16.520
There's a paralyzing agent
which paralyzes your body

00:18:16.520 --> 00:18:18.300
and also stops your lungs from moving.

00:18:18.300 --> 00:18:19.950
And there's a heart-stopping agent.

00:18:19.950 --> 00:18:21.840
Give the offender all three

00:18:21.840 --> 00:18:25.300
and they're supposed to
die quickly and painlessly.

00:18:25.300 --> 00:18:26.830
But the problem is,

00:18:26.830 --> 00:18:28.350
if the sedative doesn't put you

00:18:28.350 --> 00:18:32.180
completely, deeply unconscious,

00:18:32.180 --> 00:18:34.870
then the other two drugs

00:18:34.870 --> 00:18:38.310
are likely to make you suffer
quite a bit before you die.

00:18:38.310 --> 00:18:39.180
And so the question is,

00:18:39.180 --> 00:18:43.890
is this cruel and unusual or
is it not cruel and unusual?

00:18:43.890 --> 00:18:47.710
And to date the Supreme
Court has twice held

00:18:47.710 --> 00:18:50.630
that lethal injection
is not cruel and unusual

00:18:50.630 --> 00:18:52.120
and their main reason has been

00:18:52.120 --> 00:18:55.240
that the state's not trying
to torture you to death

00:18:55.240 --> 00:18:56.850
and so if maybe

00:18:56.850 --> 00:18:58.710
you're sometimes accidentally
tortured to death

00:18:58.710 --> 00:19:00.780
well, that's just too bad,

00:19:00.780 --> 00:19:02.854
but it's not a cruel
and unusual punishment.

00:19:02.854 --> 00:19:04.120
- [Bessler] So the (mumbles) has approved

00:19:04.120 --> 00:19:05.660
various methods of execution.

00:19:05.660 --> 00:19:06.780
At the same time, however,

00:19:06.780 --> 00:19:10.330
the Eighth Amendment has been
read to protect prisoners.

00:19:10.330 --> 00:19:11.513
So in general the Eighth Amendment

00:19:11.513 --> 00:19:13.370
is sort of a protective shield

00:19:13.370 --> 00:19:15.040
that prohibits prison guards

00:19:15.040 --> 00:19:17.810
from gratuitously beating up inmates.

00:19:17.810 --> 00:19:21.360
It requires prisons to provide

00:19:21.360 --> 00:19:23.730
some level of health care to prisoners

00:19:23.730 --> 00:19:25.440
because they cannot get it themselves,

00:19:25.440 --> 00:19:27.150
they are sort of wards of the state

00:19:27.150 --> 00:19:29.000
once they're put in prison.

00:19:29.000 --> 00:19:32.730
It requires inmates be fed and sheltered.

00:19:32.730 --> 00:19:33.730
So in a lot of ways

00:19:33.730 --> 00:19:36.560
the Eighth Amendment
is a protective shield

00:19:36.560 --> 00:19:37.740
protecting inmates,

00:19:37.740 --> 00:19:39.970
but then in the use of
the capital punishment

00:19:39.970 --> 00:19:41.651
it becomes what I like to call,

00:19:41.651 --> 00:19:45.150
it's kind of a Dr. Jekyll
and Mr. Hyde jurisprudence

00:19:45.150 --> 00:19:47.650
that the Eighth Amendment has right now.

00:19:47.650 --> 00:19:49.570
- [Kim] So, we've learned
that the Eighth Amendment

00:19:49.570 --> 00:19:52.090
seeks to limit the power of the government

00:19:52.090 --> 00:19:54.070
in meeting out punishment to people

00:19:54.070 --> 00:19:57.300
who have been accused
or convicted of a crime.

00:19:57.300 --> 00:19:58.660
Although it's hard to tell

00:19:58.660 --> 00:20:01.980
exactly what constitutes
excessive fines or bail,

00:20:01.980 --> 00:20:04.390
in general it's accepted
that those punishments

00:20:04.390 --> 00:20:07.240
should be proportional to
the crimes in question.

00:20:07.240 --> 00:20:09.150
Today, one of the biggest debates

00:20:09.150 --> 00:20:12.620
concerns whether or how
the Eighth Amendment

00:20:12.620 --> 00:20:14.860
may limit the death penalty.

00:20:14.860 --> 00:20:16.510
To learn more about the Eighth Amendment,

00:20:16.510 --> 00:20:18.620
visit the National Constitution Center's

00:20:18.620 --> 00:20:20.320
Interactive Constitution

00:20:20.320 --> 00:20:23.883
and Khan Academy's resources
on US government and politics.


Regrouping with decimals

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:01.810
- [Instructor] We are told

00:00:01.810 --> 00:00:04.150
fill in the table with whole numbers

00:00:04.150 --> 00:00:09.150
to make 10.74 in two different ways.

00:00:09.180 --> 00:00:11.860
So pause this video and see
if you can figure that out.

00:00:11.860 --> 00:00:12.930
So you really need to fill out

00:00:12.930 --> 00:00:14.900
what would you put in here

00:00:14.900 --> 00:00:17.923
for this to be a representation of 10.74?

00:00:19.150 --> 00:00:20.020
All right, now let's do it together

00:00:20.020 --> 00:00:22.690
and I'm gonna rewrite the
number a little bit larger

00:00:22.690 --> 00:00:27.150
so we can really inspect the place values.

00:00:27.150 --> 00:00:30.150
So one way to think about
it is we have zero ones

00:00:30.150 --> 00:00:32.390
and then we have one 10.

00:00:32.390 --> 00:00:33.890
But they don't express it that way.

00:00:33.890 --> 00:00:36.080
They don't say one 10
and zero ones up here.

00:00:36.080 --> 00:00:38.750
They just say 10 ones,
but that's reasonable.

00:00:38.750 --> 00:00:40.860
So when they're saying 10 ones here,

00:00:40.860 --> 00:00:43.920
they are referring to those 10 ones,

00:00:43.920 --> 00:00:45.870
or you could view that
as one 10 and zero ones.

00:00:45.870 --> 00:00:47.450
Either way, you have 10 ones,

00:00:47.450 --> 00:00:49.700
and then if you move
one decimal place over,

00:00:49.700 --> 00:00:52.160
this 7/10, well that makes sense.

00:00:52.160 --> 00:00:54.070
You have a seven in the tenths place

00:00:54.070 --> 00:00:57.640
and then this 4/100 makes sense.

00:00:57.640 --> 00:01:01.350
You have four in the hundredths place.

00:01:01.350 --> 00:01:02.900
But now let's see what
they're doing over here.

00:01:02.900 --> 00:01:06.560
So once again, they're
saying no 10s, but 10 ones.

00:01:06.560 --> 00:01:09.010
So that is actually the same.

00:01:09.010 --> 00:01:12.760
We could say one 10 and zero
ones, or we could say 10 ones.

00:01:12.760 --> 00:01:13.890
And now over here,

00:01:13.890 --> 00:01:16.720
they've reduced the number of tenths.

00:01:16.720 --> 00:01:17.960
You know, you're like,
hey, what's going on?

00:01:17.960 --> 00:01:21.130
I have, there looks
like there's 7/10 here.

00:01:21.130 --> 00:01:23.940
But one way to think about
it is they're regrouping

00:01:23.940 --> 00:01:25.970
from one place to another.

00:01:25.970 --> 00:01:27.710
And so what they've done is

00:01:27.710 --> 00:01:30.030
they've taken that extra 10th

00:01:30.030 --> 00:01:31.890
and they've put it someplace.

00:01:31.890 --> 00:01:33.380
And the only other place they could put it

00:01:33.380 --> 00:01:35.830
is in the hundredths place.

00:01:35.830 --> 00:01:40.830
So if I were to take a
tenth from the tenths place,

00:01:41.000 --> 00:01:44.070
a tenth is worth 10 hundredths.

00:01:44.070 --> 00:01:46.130
So one way to think about it is,

00:01:46.130 --> 00:01:49.130
you're taking a tenth from there,

00:01:49.130 --> 00:01:51.040
so now this is going to be,

00:01:51.040 --> 00:01:53.603
this is going to be 6/10,

00:01:54.470 --> 00:01:57.750
and if you, where you're
going to put that tenth,

00:01:57.750 --> 00:01:59.910
well you could put it
in the hundredths place,

00:01:59.910 --> 00:02:01.973
but a tenth is going to be 10/100.

00:02:02.980 --> 00:02:06.920
So if you add 10/100 to the
4/100 that are already there,

00:02:06.920 --> 00:02:11.200
well, that is going to give you 14/100,

00:02:11.200 --> 00:02:14.350
so we would put a 14 right over there.

00:02:14.350 --> 00:02:16.120
Let's do another example

00:02:16.120 --> 00:02:17.900
just to really make
sure we're understanding

00:02:17.900 --> 00:02:19.250
what's going on.

00:02:19.250 --> 00:02:22.220
So once again, they say fill
in the table with whole numbers

00:02:22.220 --> 00:02:25.930
to make 5.4 in three different ways.

00:02:25.930 --> 00:02:29.330
So pause this video and see
if you can have a go at it.

00:02:29.330 --> 00:02:30.260
All right.

00:02:30.260 --> 00:02:31.970
So I'm gonna write the number,

00:02:31.970 --> 00:02:33.653
so we have 5.4,

00:02:34.730 --> 00:02:38.550
and in this first row, this
is maybe the most standard way

00:02:38.550 --> 00:02:41.270
or traditional way of interpreting 5.4.

00:02:41.270 --> 00:02:43.160
In our ones place, you have a five

00:02:43.160 --> 00:02:44.550
and you see that, five ones,

00:02:44.550 --> 00:02:46.460
and in our tenths place,

00:02:46.460 --> 00:02:48.550
we have a four,

00:02:48.550 --> 00:02:50.810
and that's what they have
right over here, 4/10.

00:02:50.810 --> 00:02:53.470
But what are they doing
in this second row?

00:02:53.470 --> 00:02:56.551
Well, they're saying 24/10.

00:02:56.551 --> 00:03:00.920
24/10, so somehow, in
way to think about it,

00:03:00.920 --> 00:03:02.690
in the tenths place,

00:03:02.690 --> 00:03:06.140
they were able to add 20/10.

00:03:06.140 --> 00:03:11.140
So they went from 4/10 to 24/10.

00:03:11.530 --> 00:03:14.490
So if you're adding 20/10 here,

00:03:14.490 --> 00:03:18.187
they must have taken it
away from some other place.

00:03:18.187 --> 00:03:22.870
20/10 is the same thing as two ones,

00:03:22.870 --> 00:03:25.700
so they must have taken
two ones away from here.

00:03:25.700 --> 00:03:27.720
So they took two ones away from here,

00:03:27.720 --> 00:03:32.250
so this would be three ones and now 24/10.

00:03:32.250 --> 00:03:34.760
So we could put a three right over there.

00:03:34.760 --> 00:03:36.183
And you can verify that.

00:03:36.183 --> 00:03:39.940
24/10 is the same thing as two wholes

00:03:39.940 --> 00:03:43.823
and, or two ones and 4/10 or 2.4.

00:03:44.992 --> 00:03:49.473
2.4 plus three is going
to be equal to 5.4.

00:03:51.430 --> 00:03:52.460
All right.

00:03:52.460 --> 00:03:55.670
So here, we only have one one,

00:03:55.670 --> 00:03:58.770
and so what happened
to the other four ones?

00:03:58.770 --> 00:04:02.270
Well, they must have been
transferred to the tenths place.

00:04:02.270 --> 00:04:04.320
So let me rewrite the number.

00:04:04.320 --> 00:04:08.190
So if we were to take four
ones away from the ones place,

00:04:08.190 --> 00:04:09.910
so we're taking four away from this,

00:04:09.910 --> 00:04:11.580
we only have one left,

00:04:11.580 --> 00:04:14.030
and what would happen if I
transferred those four ones

00:04:14.030 --> 00:04:15.630
to the tenths place?

00:04:15.630 --> 00:04:18.440
Well, then there would be 40/10.

00:04:18.440 --> 00:04:21.160
Four ones are 40/10,

00:04:21.160 --> 00:04:25.860
so we would add 40 to 40/10 over here,

00:04:25.860 --> 00:04:27.490
so we already have 4/10,

00:04:27.490 --> 00:04:30.920
so this would become 44/10.

00:04:30.920 --> 00:04:32.580
44/10.

00:04:32.580 --> 00:04:33.503
And we're done.


Expressing decimals in multiple forms

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.040
- [Instructor] We're asked
which two of the following

00:00:03.040 --> 00:00:07.850
expressions have the same value as 8.76?

00:00:07.850 --> 00:00:10.130
Pause this video and see
if you can figure this out

00:00:10.130 --> 00:00:11.560
on your own.

00:00:11.560 --> 00:00:12.930
All right now let's do it together.

00:00:12.930 --> 00:00:15.390
Before I even look at these
choices, I'm just going

00:00:15.390 --> 00:00:18.000
to really understand what
this number represents

00:00:18.000 --> 00:00:19.440
and let me just rewrite it.

00:00:19.440 --> 00:00:24.440
So we have 8.76 so there's
a couple of ways we could

00:00:27.096 --> 00:00:28.600
think about it, we can look
at our place values, this

00:00:28.600 --> 00:00:31.760
right over here this is the
ones place, this right over here

00:00:31.760 --> 00:00:35.510
this is the tenths place,
and this right over here

00:00:35.510 --> 00:00:39.430
is the hundredths, hundredths place.

00:00:39.430 --> 00:00:43.830
And so we could view this as
eight ones and seven tenths

00:00:45.145 --> 00:00:47.645
and six hundredths or eight
ones plus seven tenths

00:00:48.663 --> 00:00:49.770
plus six hundredths.

00:00:49.770 --> 00:00:51.950
Well that's exactly what
they wrote right over here.

00:00:51.950 --> 00:00:56.020
Eight ones, seven tenths, and
six hundredths, so I would

00:00:56.020 --> 00:00:58.130
choose that one for sure.

00:00:58.130 --> 00:01:00.730
Now this second choice
looks like an expanded form,

00:01:00.730 --> 00:01:02.400
but before I even look at
it let's see how we would

00:01:02.400 --> 00:01:03.800
think about it over here.

00:01:03.800 --> 00:01:06.030
If we wanted to essentially
write the same idea

00:01:06.030 --> 00:01:09.000
but in expanded form, eight
ones is the same thing

00:01:09.000 --> 00:01:12.660
as eight times one and
actually let me color code it

00:01:12.660 --> 00:01:14.279
so you see where things are coming from.

00:01:14.279 --> 00:01:19.279
So eight ones that's the
same thing as eight times one

00:01:19.654 --> 00:01:24.654
and to that we would add
seven tenths, so that's plus

00:01:24.855 --> 00:01:29.855
seven times 1/10 so seven
tenths plus and I'll do this

00:01:32.980 --> 00:01:37.010
in this orange color, six
hundredths so that's plus

00:01:37.010 --> 00:01:41.880
six times a hundredth, six hundredths.

00:01:41.880 --> 00:01:44.590
So this would be this number
in expanded form is that

00:01:44.590 --> 00:01:46.624
what they put right over here,
yes it is indeed what they

00:01:46.624 --> 00:01:49.062
put right over here, so
I will circle that in.

00:01:49.062 --> 00:01:52.040
Now if you're doing this on
your own we know that we just

00:01:52.040 --> 00:01:54.610
picked two answers, but
let's see whether these other

00:01:54.610 --> 00:01:56.600
forms or let's see if we can
write this in these other

00:01:56.600 --> 00:02:00.540
forms and see how these might
not be the exact answer.

00:02:00.540 --> 00:02:03.530
So if we were to write
this out and take out each

00:02:03.530 --> 00:02:08.530
of the decimal parts, so the
eight ones you'd just write

00:02:08.960 --> 00:02:13.720
that as eight, the seven
tenths, seven tenths well that

00:02:13.720 --> 00:02:18.720
would be plus 0.7 this is
seven tenths right over here,

00:02:18.730 --> 00:02:21.870
this and this and this part
right over here are all

00:02:21.870 --> 00:02:24.470
equivalent and then last
but not least you have your

00:02:24.470 --> 00:02:28.820
six hundredths, so plus so
that's our ones, that's our

00:02:28.820 --> 00:02:30.960
tenths, and then we're in our
hundredths place and we're

00:02:30.960 --> 00:02:32.570
going to have six of them.

00:02:32.570 --> 00:02:37.570
So this would be equal to our
original value but that's not

00:02:38.150 --> 00:02:39.542
what they wrote over here,
they did write eight ones,

00:02:39.542 --> 00:02:44.542
they did write seven tenths,
but they did not write six

00:02:46.550 --> 00:02:49.840
hundredths, they wrote
six thousandths here.

00:02:49.840 --> 00:02:51.630
So we could rule that one out.

00:02:51.630 --> 00:02:54.679
And then if we were to write
it out in words we would say

00:02:54.679 --> 00:02:59.679
this is eight eight and
do and in a neutral color,

00:03:02.147 --> 00:03:06.850
now you might say and seven
tenths and six hundredths,

00:03:06.850 --> 00:03:09.493
or you could often what's
normally is you express it

00:03:09.493 --> 00:03:14.380
in the lowest place that
you have or the most precise

00:03:14.380 --> 00:03:18.790
place that you have so you
could do seven tenths as 70

00:03:18.790 --> 00:03:23.250
hundredths or you could view,
you could view this whole

00:03:24.425 --> 00:03:27.983
thing as 76 hundredths so
it could be eight and 70,

00:03:29.760 --> 00:03:34.760
70 six hundredths, hundredths.

00:03:37.190 --> 00:03:39.156
But what they wrote over
here they did write eight

00:03:39.156 --> 00:03:43.525
but instead of 76 hundredths,
they wrote eight and 67

00:03:43.525 --> 00:03:46.550
hundredths a little tricky
so we would rule that one

00:03:46.550 --> 00:03:47.423
out as well.


Visual understanding of regrouping decimals

WEBVTT
Kind: captions
Language: en

00:00:00.370 --> 00:00:01.960
- [Instructor] What we're
going to do in this video

00:00:01.960 --> 00:00:05.550
is explore place value involving decimals,

00:00:05.550 --> 00:00:07.120
and in particular, we're gonna think about

00:00:07.120 --> 00:00:11.070
how you can regroup value
from one place to another,

00:00:11.070 --> 00:00:13.210
which is gonna be very
useful later in your life

00:00:13.210 --> 00:00:16.830
when you start doing some
more arithmetic with decimals.

00:00:16.830 --> 00:00:20.250
So let's first think
about what this number is

00:00:20.250 --> 00:00:21.083
right over here.

00:00:21.083 --> 00:00:23.490
So each square represents a one

00:00:23.490 --> 00:00:27.650
or represents a whole,
so what number is this?

00:00:27.650 --> 00:00:31.510
Well, you could see that
we have three whole ones,

00:00:31.510 --> 00:00:33.250
so we could write three,

00:00:33.250 --> 00:00:36.590
and then we have right over here,

00:00:36.590 --> 00:00:40.910
a whole is divided into 10
equal bars, these vertical bars,

00:00:40.910 --> 00:00:42.930
and so each of these are 1/10

00:00:42.930 --> 00:00:46.570
and four of them are shaded
in, so it's three ones,

00:00:46.570 --> 00:00:51.570
4/10, and in this part right over here,

00:00:51.740 --> 00:00:54.590
we've divided a whole
into 100 equal sections.

00:00:54.590 --> 00:00:55.920
It's a 10 by 10 grid,

00:00:55.920 --> 00:00:58.640
and we can see that one, two, three, four,

00:00:58.640 --> 00:01:01.490
five, six, seven of them are shaded in.

00:01:01.490 --> 00:01:03.230
So this represents 7/100,

00:01:03.230 --> 00:01:05.840
so this is 3.47,

00:01:05.840 --> 00:01:08.660
or three and 4/10 and 7/100

00:01:08.660 --> 00:01:11.470
or three and 47/100.

00:01:11.470 --> 00:01:14.050
Now what we're going to do
is explore how are other ways

00:01:14.050 --> 00:01:18.010
you can put the value
into the different places?

00:01:18.010 --> 00:01:20.700
So let me set up a little table here.

00:01:20.700 --> 00:01:22.820
I don't think I'm gonna
need all of that space,

00:01:22.820 --> 00:01:25.230
but we have, in this first example,

00:01:25.230 --> 00:01:27.380
so we have our ones,

00:01:27.380 --> 00:01:29.470
we have our ones place,

00:01:29.470 --> 00:01:33.890
we have our tenths place,

00:01:33.890 --> 00:01:38.890
and we have our hundredths,
hundredths, hundredths place.

00:01:40.170 --> 00:01:41.400
I have to curve up a little bit.

00:01:41.400 --> 00:01:43.240
And so what we just did is we said, hey,

00:01:43.240 --> 00:01:44.470
this was pretty straightforward.

00:01:44.470 --> 00:01:47.880
This was three ones, three ones,

00:01:47.880 --> 00:01:52.880
4/10, and 7/100, 7/100,

00:01:53.160 --> 00:01:55.950
but are there other ways
that we could look at it?

00:01:55.950 --> 00:01:59.640
For example, is there a
way of re-imagining this,

00:01:59.640 --> 00:02:01.740
so instead of three ones,

00:02:01.740 --> 00:02:04.360
we have two ones,

00:02:04.360 --> 00:02:08.120
and we still have 7/100,

00:02:08.120 --> 00:02:09.640
so how many tenths would we have

00:02:09.640 --> 00:02:11.450
in order for it to be the same value?

00:02:11.450 --> 00:02:13.870
Pause this video and think about that.

00:02:13.870 --> 00:02:15.800
All right, we'll do it
together and to help us,

00:02:15.800 --> 00:02:19.090
I will put what we had here just now.

00:02:19.090 --> 00:02:21.210
Now what's different instead
of having three ones,

00:02:21.210 --> 00:02:22.280
we now have two ones,

00:02:22.280 --> 00:02:26.520
so we could say that these
are two ones right over here,

00:02:26.520 --> 00:02:27.830
our two ones.

00:02:27.830 --> 00:02:30.170
We have our 7/100 right over here.

00:02:30.170 --> 00:02:33.650
So essentially, we would
have to express all of this

00:02:33.650 --> 00:02:36.400
in terms of tenths.

00:02:36.400 --> 00:02:39.520
So how would we express
it all in terms of tenths?

00:02:39.520 --> 00:02:42.560
Well, this, what used to be a one,

00:02:42.560 --> 00:02:44.390
this is equivalent to 10/10.

00:02:44.390 --> 00:02:45.660
Let me make that very clear.

00:02:45.660 --> 00:02:47.940
I could, let me see if
I could shade this in

00:02:47.940 --> 00:02:50.870
with that green color, so there you go.

00:02:50.870 --> 00:02:51.960
I'm gonna shade with the green color

00:02:51.960 --> 00:02:54.360
and then I'm gonna draw
a bunch of lines here

00:02:54.360 --> 00:02:56.180
to make it very clear,

00:02:56.180 --> 00:02:59.230
so I'm just gonna hand draw it so one,

00:02:59.230 --> 00:03:03.340
two, three, four,

00:03:03.340 --> 00:03:08.020
five, six, seven,

00:03:08.020 --> 00:03:10.150
eight, nine, and 10.

00:03:10.150 --> 00:03:10.983
Did I do that right?

00:03:10.983 --> 00:03:15.400
One, two, three, four, five,
six, seven, eight, nine, 10.

00:03:15.400 --> 00:03:17.140
And I didn't draw it
as straight I need to.

00:03:17.140 --> 00:03:18.917
They really should be 10 equal sections,

00:03:18.917 --> 00:03:21.270
but it's hand drawn, so
I think you understand.

00:03:21.270 --> 00:03:22.103
So notice.

00:03:22.103 --> 00:03:23.930
I took the exact same value,

00:03:23.930 --> 00:03:24.910
but I have regrouped.

00:03:24.910 --> 00:03:28.470
I have regrouped this one
right over here into tenths.

00:03:28.470 --> 00:03:30.870
So how many tenths are here?

00:03:30.870 --> 00:03:34.000
Well, now I have 10 plus 4/10.

00:03:34.000 --> 00:03:38.220
So now I have shaded in 14/10.

00:03:39.360 --> 00:03:40.950
That was interesting.

00:03:40.950 --> 00:03:44.100
Let's do another example.

00:03:44.100 --> 00:03:45.910
So now let's imagine another scenario.

00:03:45.910 --> 00:03:49.670
Let's imagine a scenario where
we have three ones again,

00:03:49.670 --> 00:03:52.110
but this time, instead of having 7/100,

00:03:52.110 --> 00:03:54.560
we have 27/100.

00:03:54.560 --> 00:03:56.350
So in that circumstance,

00:03:56.350 --> 00:03:58.520
how many tenths would we have?

00:03:58.520 --> 00:04:01.520
Pause this video and see
if you can work it through.

00:04:01.520 --> 00:04:05.060
All right, well let's get
that same number again.

00:04:05.060 --> 00:04:07.530
And now let's think about
how we might have to regroup

00:04:07.530 --> 00:04:08.540
between the places.

00:04:08.540 --> 00:04:10.880
So we have our three ones,

00:04:10.880 --> 00:04:13.990
so that was just like the
first case right over there,

00:04:13.990 --> 00:04:16.200
so we have our three ones.

00:04:16.200 --> 00:04:18.270
But now we have 27/100.

00:04:18.270 --> 00:04:20.590
So in addition to these 7/100,

00:04:20.590 --> 00:04:24.150
we have to find another 20/100 someplace.

00:04:24.150 --> 00:04:26.750
Well, the most natural place
to go would be right over here

00:04:26.750 --> 00:04:30.150
and 20/100 is the same thing as 2/10.

00:04:30.150 --> 00:04:31.500
So what we want to do

00:04:31.500 --> 00:04:34.970
is we would convert these
two tenths into hundredths,

00:04:34.970 --> 00:04:37.703
so let me actually just
shade it in a little bit.

00:04:38.620 --> 00:04:43.620
So I'm going to convert this
right over here to hundredths.

00:04:43.940 --> 00:04:45.250
And so did I do that right?

00:04:45.250 --> 00:04:49.260
One, two, three, four, five,
six, seven, eight, nine, 10.

00:04:49.260 --> 00:04:50.800
And so there you have it.

00:04:50.800 --> 00:04:55.330
This plus this gives us our 27/100,

00:04:55.330 --> 00:04:58.010
and so how many tenths do we have left?

00:04:58.010 --> 00:05:02.010
We have 2/10, 2/10.

00:05:02.010 --> 00:05:03.170
So in this situation,

00:05:03.170 --> 00:05:06.190
we regrouped two from the tenths place

00:05:06.190 --> 00:05:08.200
and we expressed them as hundredths.

00:05:08.200 --> 00:05:11.250
So 2/10 became 20/100,

00:05:11.250 --> 00:05:13.950
added to the 7/100 that you already have

00:05:13.950 --> 00:05:16.800
and we now have 27/100.

00:05:16.800 --> 00:05:18.150
Hopefully that makes sense.


Ordering decimals

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:01.290
- [Instructor] What we're
gonna do in this video

00:00:01.290 --> 00:00:03.350
is do a few examples ordering numbers

00:00:03.350 --> 00:00:04.970
that involve decimals.

00:00:04.970 --> 00:00:07.280
So let's say that we have the numbers

00:00:07.280 --> 00:00:10.410
1.001, 0.113, and 1.101.

00:00:18.350 --> 00:00:20.460
What I would like you to do is order

00:00:20.460 --> 00:00:23.250
these numbers from least to greatest.

00:00:23.250 --> 00:00:24.430
Take out some paper and try to do it

00:00:24.430 --> 00:00:27.030
on your own before we do it together.

00:00:27.030 --> 00:00:28.590
All right, now let's do it together.

00:00:28.590 --> 00:00:31.290
And the way I would
tackle ordering numbers

00:00:31.290 --> 00:00:34.160
is I would go to the largest place value

00:00:34.160 --> 00:00:35.910
that the numbers have in common.

00:00:35.910 --> 00:00:37.460
And so, in this situation,

00:00:37.460 --> 00:00:41.290
we have a ones place value in all of them.

00:00:41.290 --> 00:00:44.250
And so we can see that this has one one,

00:00:44.250 --> 00:00:48.040
this has zero ones, and
this one has one one.

00:00:48.040 --> 00:00:50.070
So the thing that has the least ones

00:00:50.070 --> 00:00:52.850
is going to be the
smallest of the numbers.

00:00:52.850 --> 00:00:54.180
So this one over here is going to be

00:00:54.180 --> 00:00:55.300
the smallest of the numbers.

00:00:55.300 --> 00:00:56.490
So let me just write that over here.

00:00:56.490 --> 00:00:58.083
0.113.

00:01:00.220 --> 00:01:02.340
And now we have to figure
out which one is next

00:01:02.340 --> 00:01:06.380
between 1.001 and 1.101.

00:01:06.380 --> 00:01:08.470
Well then we just go to
the next place value.

00:01:08.470 --> 00:01:10.940
So we go to the tenths place

00:01:10.940 --> 00:01:12.130
and we right over here,

00:01:12.130 --> 00:01:13.740
they're equal on the ones place.

00:01:13.740 --> 00:01:15.810
So if you go to the tenths place,

00:01:15.810 --> 00:01:20.160
this one has zero tenths
while this one has one tenth.

00:01:20.160 --> 00:01:21.470
So the number on the right here

00:01:21.470 --> 00:01:23.210
is going to be larger.

00:01:23.210 --> 00:01:24.570
It has more tenths.

00:01:24.570 --> 00:01:27.030
Same number of ones,
but it has more tenths.

00:01:27.030 --> 00:01:28.510
It doesn't really matter what happens

00:01:28.510 --> 00:01:29.690
to the right of that.

00:01:29.690 --> 00:01:31.630
So the next smallest number,

00:01:31.630 --> 00:01:33.580
if we're ordering from least to greatest,

00:01:33.580 --> 00:01:35.573
is going to be 1.001.

00:01:37.420 --> 00:01:41.130
And then last, but not
least, would be this one.

00:01:41.130 --> 00:01:45.810
That is the largest, 1.101.

00:01:45.810 --> 00:01:47.330
Let's do another example.

00:01:47.330 --> 00:01:51.403
Let's say we had the numbers 0.424,

00:01:53.100 --> 00:01:57.737
0.343, and 0.443.

00:01:59.680 --> 00:02:01.170
Pause this video and try to order these

00:02:01.170 --> 00:02:03.420
from least to greatest on your own.

00:02:03.420 --> 00:02:05.950
And once again, the idea here is always

00:02:05.950 --> 00:02:08.890
start with the largest place
value and then compare,

00:02:08.890 --> 00:02:10.090
and then keep moving to the right

00:02:10.090 --> 00:02:12.120
if some things are equal.

00:02:12.120 --> 00:02:14.410
All right, now let's do this together.

00:02:14.410 --> 00:02:16.180
So they all have zero ones,

00:02:16.180 --> 00:02:17.300
so they're all equal there,

00:02:17.300 --> 00:02:18.870
so that's not gonna tell us much.

00:02:18.870 --> 00:02:21.990
So now let's go to the tenths place.

00:02:21.990 --> 00:02:24.290
So here I have four tenths.

00:02:24.290 --> 00:02:26.480
Here I have three tenths.

00:02:26.480 --> 00:02:28.560
Here I have four tenths.

00:02:28.560 --> 00:02:30.460
So I don't even have to
look at the hundredths

00:02:30.460 --> 00:02:32.200
or the thousandths place.

00:02:32.200 --> 00:02:35.010
This one has the least tenths.

00:02:35.010 --> 00:02:37.690
So I can put that as the least,

00:02:37.690 --> 00:02:41.860
or the smallest of the
three numbers, 0.343

00:02:41.860 --> 00:02:44.170
And now, so I've already used that one,

00:02:44.170 --> 00:02:45.760
and so I need to compare
these two numbers.

00:02:45.760 --> 00:02:47.050
They have the same number of ones,

00:02:47.050 --> 00:02:49.000
they have the same number of tenths,

00:02:49.000 --> 00:02:51.150
so then we move to the hundredths.

00:02:51.150 --> 00:02:53.640
So here I have two hundredths,

00:02:53.640 --> 00:02:56.120
here I have four hundredths.

00:02:56.120 --> 00:02:59.370
This one has less
hundredths than this one.

00:02:59.370 --> 00:03:01.300
So the one on the left is going to be

00:03:01.300 --> 00:03:02.680
the next smallest number.

00:03:02.680 --> 00:03:04.163
So then we have 0.424.

00:03:06.170 --> 00:03:09.650
And then, last but not least,
this one right over here,

00:03:09.650 --> 00:03:12.740
it has the same number of
ones as everything else.

00:03:12.740 --> 00:03:15.630
It has more tenths than this middle one,

00:03:15.630 --> 00:03:18.400
and the same number of
tenths as the left one.

00:03:18.400 --> 00:03:20.820
But then it has, but then it also has

00:03:20.820 --> 00:03:23.020
more hundredths than the
left one right over here.

00:03:23.020 --> 00:03:26.683
So this is the largest
of the numbers, 0.443.

00:03:28.830 --> 00:03:29.763
And we're done.


Comparing decimals in different representations

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.130
- [Instructor] So what we're
going to do in this video

00:00:02.130 --> 00:00:03.917
is build our muscles at comparing numbers

00:00:03.917 --> 00:00:07.810
that are represented in different ways.

00:00:07.810 --> 00:00:12.305
So, for example, over here
on the left we have 0.37,

00:00:12.305 --> 00:00:15.480
you could also view this
as thirty-seven hundredths,

00:00:15.480 --> 00:00:19.540
and on the right we have 307 thousandths.

00:00:19.540 --> 00:00:21.840
And so, what I want you
to do is pause this video,

00:00:21.840 --> 00:00:24.830
and figure out, are these
equal to each other?

00:00:24.830 --> 00:00:26.630
Or is one of the larger than the other?

00:00:26.630 --> 00:00:28.780
And if one of them is,
which one is larger,

00:00:29.663 --> 00:00:30.496
and which one is smaller?

00:00:30.496 --> 00:00:31.621
Pause this video and
try to figure that out.

00:00:31.621 --> 00:00:32.710
All right,

00:00:32.710 --> 00:00:33.980
now let's try to do this together,

00:00:33.980 --> 00:00:35.470
and the way that my brain works is,

00:00:35.470 --> 00:00:39.141
I try to put them into
a common representation.

00:00:39.141 --> 00:00:41.110
So, one way we could do it is

00:00:41.110 --> 00:00:45.760
we could try to re-write this
one on the right as a decimal.

00:00:45.760 --> 00:00:47.420
So let's do that.

00:00:47.420 --> 00:00:50.920
So, we could re-write this as,

00:00:50.920 --> 00:00:54.300
it's expressing a certain
number of thousandths,

00:00:54.300 --> 00:00:57.878
so let me just make some
blanks for our various placeS.

00:00:57.878 --> 00:00:58.711
So let's say that's the one's place,

00:00:58.711 --> 00:00:59.750
and that's our decimal,

00:00:59.750 --> 00:01:02.090
that's going to be our tenth's place,

00:01:02.090 --> 00:01:04.150
that's going to be our hundredth's place,

00:01:04.150 --> 00:01:07.380
and that's going to be
our thousandth's place.

00:01:07.380 --> 00:01:10.410
So, one way to view 307 thousandths

00:01:10.410 --> 00:01:15.410
is that we have 307 of this
place, right over there.

00:01:15.580 --> 00:01:18.030
So, we could just write the seven there,

00:01:18.030 --> 00:01:18.863
the zero there,

00:01:18.863 --> 00:01:19.696
and the three over there.

00:01:19.696 --> 00:01:22.980
This over here would be 307 thousandths,

00:01:22.980 --> 00:01:25.140
and so we would have no ones.

00:01:25.140 --> 00:01:26.847
And so when you look at it this way,

00:01:26.847 --> 00:01:29.790
it's a little bit easier to compare.

00:01:29.790 --> 00:01:31.049
You can say, "all right,

00:01:31.049 --> 00:01:33.646
we have the same number of ones,

00:01:33.646 --> 00:01:36.080
we have the same number of tenths,

00:01:36.080 --> 00:01:37.940
let me compare the like
one to the like ones."

00:01:37.940 --> 00:01:40.180
So, our tenths are equal,

00:01:40.180 --> 00:01:43.350
but what happens when we
get to the hundredths?

00:01:43.350 --> 00:01:45.850
Here, we have seven hundredths,

00:01:45.850 --> 00:01:48.705
and here we have zero hundredths.

00:01:48.705 --> 00:01:53.190
So, this number on the
left is going to be larger.

00:01:53.190 --> 00:01:58.190
So, 37 hundredths is greater
than 307 thousandths.

00:01:59.800 --> 00:02:01.530
Another that we could have done this

00:02:01.530 --> 00:02:04.180
is we could have
re-expressed this left number

00:02:04.180 --> 00:02:05.890
in terms of thousandths.

00:02:05.890 --> 00:02:09.200
We could've re-written it
as, instead of 37 hundredths,

00:02:09.200 --> 00:02:11.880
we could've just said
zero point three seven,

00:02:11.880 --> 00:02:13.600
and just put another zero on the right,

00:02:13.600 --> 00:02:17.400
and this is 370 thousandths.

00:02:17.400 --> 00:02:18.240
I'll write it out.

00:02:18.240 --> 00:02:22.290
370 thousandths.

00:02:22.290 --> 00:02:23.870
And when you look at it this way,

00:02:23.870 --> 00:02:26.520
once again it's clear
that 370 of something

00:02:26.520 --> 00:02:29.580
is more than 307 of that something.

00:02:29.580 --> 00:02:32.640
So, this quantity on the left is larger.

00:02:32.640 --> 00:02:33.770
Let's do another example,

00:02:33.770 --> 00:02:35.776
but I'll use different formats.

00:02:35.776 --> 00:02:40.130
So, let's say on the left,

00:02:40.130 --> 00:02:41.690
I'll use decimal format,

00:02:41.690 --> 00:02:44.646
I'll have zero point six, or six tenths,

00:02:44.646 --> 00:02:46.150
and then on the right,

00:02:46.150 --> 00:02:51.150
I'm going to have six
times one over a hundred.

00:02:51.290 --> 00:02:52.942
Pause this video and tell me,

00:02:52.942 --> 00:02:55.900
which of these quantities,
if either, are greater,

00:02:55.900 --> 00:02:57.573
or are they equal to each other?

00:02:59.130 --> 00:03:01.410
All right, so once again,
in order to tackle this,

00:03:01.410 --> 00:03:02.780
you really just have to think about

00:03:02.780 --> 00:03:04.570
what are different way to represent them?

00:03:04.570 --> 00:03:07.026
And really just get to
a common representation.

00:03:07.026 --> 00:03:12.026
And so, I could re-write six
tenths and six times one tenth.

00:03:13.150 --> 00:03:15.280
Six times one tenth.

00:03:15.280 --> 00:03:18.149
And this might be enough to
be able to compare the two,

00:03:18.149 --> 00:03:21.540
because six times one tenth,

00:03:21.540 --> 00:03:23.680
is that going to be
greater than, or less than,

00:03:23.680 --> 00:03:25.806
or equal to six times a hundredth?

00:03:25.806 --> 00:03:30.324
Well, a tenth is ten times
larger than a hundredth,

00:03:30.324 --> 00:03:33.840
so because this is ten
times larger than that,

00:03:33.840 --> 00:03:36.010
if you multiply is by six, well,

00:03:36.010 --> 00:03:37.620
this is going to be a larger quantity.

00:03:37.620 --> 00:03:41.147
So we could go and say, "Hey,
this is greater than that."

00:03:41.147 --> 00:03:43.970
Another way that you
might have realized that

00:03:43.970 --> 00:03:46.452
is if you were to express
this right quantity

00:03:46.452 --> 00:03:48.970
as a decimal like this.

00:03:48.970 --> 00:03:53.140
So, this is six times a
hundredth, or six hundredths.

00:03:53.140 --> 00:03:54.867
So, we could write that's our ones,

00:03:54.867 --> 00:03:56.460
that's our tenths,

00:03:56.460 --> 00:04:00.030
and then in our hundredths
place, you would have six.

00:04:00.030 --> 00:04:02.970
And if it isn't obvious
that this is less than that,

00:04:02.970 --> 00:04:04.843
you could add a zero here,

00:04:04.843 --> 00:04:08.165
and this, we would read as 60 hundredths,

00:04:08.165 --> 00:04:13.165
and 60 hundredths is for sure
larger than six hundredths.

00:04:14.150 --> 00:04:16.544
So, these are all very reasonable ways

00:04:16.544 --> 00:04:19.050
of re-representing these
numbers and putting them

00:04:19.050 --> 00:04:21.110
in the same format so we
can make the comparison

00:04:21.110 --> 00:04:23.910
and realize the one on the left,
actually in both scenarios,

00:04:23.910 --> 00:04:25.710
is larger than the one on the right.


Thousandths on the number line

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:03.430
- [Instructor] We're asked
what is the value of the point

00:00:03.430 --> 00:00:06.670
graphed on the number line
and this is the point right

00:00:06.670 --> 00:00:09.317
over here, so pause this video
and see if you can figure

00:00:09.317 --> 00:00:11.983
that out before we figure it out together.

00:00:13.090 --> 00:00:16.350
All right so let's try to
figure it out together now.

00:00:16.350 --> 00:00:21.250
So let's see this point is
between this is going to be

00:00:21.250 --> 00:00:25.150
this is the tenths place
this is the hundredths place

00:00:25.150 --> 00:00:29.350
so this over here is three
hundredths and this over here

00:00:29.350 --> 00:00:31.240
is four hundredths.

00:00:31.240 --> 00:00:36.050
So our point is between three
hundredths and four hundredths

00:00:36.050 --> 00:00:39.980
and the space between three
hundredths and four hundredths

00:00:39.980 --> 00:00:42.910
is divided into ten equal spaces.

00:00:42.910 --> 00:00:47.910
So if this whole thing is
a hundredth, so this whole

00:00:49.170 --> 00:00:52.610
thing is equal to 0.01 the
difference between that,

00:00:52.610 --> 00:00:56.790
that's a hundredth so then
each tenth of that is going

00:00:56.790 --> 00:00:58.268
to be a thousandth.

00:00:58.268 --> 00:01:01.030
So let me, I don't want to
make this too messy so this

00:01:01.030 --> 00:01:04.890
right over here will be
one tenth of a hundredth

00:01:04.890 --> 00:01:09.890
which is going to be a thousandth,
so it's going to be 0.001

00:01:10.330 --> 00:01:13.088
we have a one in the thousandths place.

00:01:13.088 --> 00:01:16.205
So one way to think about it,
this number is going to be

00:01:16.205 --> 00:01:21.205
0.03 so we have our three
hundredths but then let's see

00:01:23.530 --> 00:01:26.890
how many thousandths we have
to add to it to get to this

00:01:26.890 --> 00:01:28.210
value right over here.

00:01:28.210 --> 00:01:31.370
We have to add one
thousandth, two thousandths,

00:01:31.370 --> 00:01:34.921
three thousandths, four
thousandths, five thousandths,

00:01:34.921 --> 00:01:39.921
six thousandths, seven
thousandths, and eight thousandths.

00:01:40.540 --> 00:01:44.267
So we're starting at three
hundredths and then we are adding

00:01:44.267 --> 00:01:49.267
eight thousandths so you add
0.008 to get to this value.

00:01:51.310 --> 00:01:53.990
So if you were to add these
two together you're going

00:01:53.990 --> 00:01:57.120
to have three hundredths
and then the next place over

00:01:57.120 --> 00:02:00.390
is the thousandths place
and we have eight of them.

00:02:00.390 --> 00:02:05.390
So this point right over
here is 0.038 or you could do

00:02:05.610 --> 00:02:10.610
this as 38 thousandths
because three hundredths is 30

00:02:12.010 --> 00:02:15.040
thousandths, so any way you
want to view it, this is

00:02:15.040 --> 00:02:18.653
the value of the point graphed
on the number line 0.038.


Place value with decimals

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:02.190
- [Instructor] What we're
going to do in this video

00:00:02.190 --> 00:00:06.150
is refresh our understanding
of place value but we're

00:00:06.150 --> 00:00:07.960
going to dig a little bit
deeper and think about place

00:00:07.960 --> 00:00:10.860
value in the context of decimals.

00:00:10.860 --> 00:00:14.537
So just as a refresher
if I have the number 973,

00:00:17.080 --> 00:00:18.670
this should be review for you.

00:00:18.670 --> 00:00:22.631
We already know that this
rightmost space right over here,

00:00:22.631 --> 00:00:27.631
this is the ones place and if
we move one space to the left

00:00:27.970 --> 00:00:31.230
of that, this is the tens place.

00:00:31.230 --> 00:00:34.700
Notice we went from ones
to tens, tens are ten times

00:00:34.700 --> 00:00:35.840
as much as ones.

00:00:35.840 --> 00:00:39.020
And then we move one
space to the left of that,

00:00:39.020 --> 00:00:42.020
we go to the we multiply by tens again.

00:00:42.020 --> 00:00:46.840
We get to the hundreds space
and so this nine doesn't just

00:00:46.840 --> 00:00:51.800
represent nine, it represents
nine hundred or we could write

00:00:51.800 --> 00:00:53.580
that as 900.

00:00:53.580 --> 00:00:55.920
Similarly the seven doesn't
just represent seven,

00:00:55.920 --> 00:00:58.600
it represents seven tens or 70.

00:00:58.600 --> 00:01:01.850
This three represents three
ones, so it actually does

00:01:01.850 --> 00:01:03.670
represent three.

00:01:03.670 --> 00:01:06.110
But as I promised we're
now going to extend our

00:01:06.110 --> 00:01:09.570
understanding and what we
do is we put a decimal here

00:01:09.570 --> 00:01:12.210
which you've probably
seen before at the right

00:01:12.210 --> 00:01:14.822
and the reason why we even
need a decimal is to really

00:01:14.822 --> 00:01:17.710
tell us where our ones place is.

00:01:17.710 --> 00:01:20.790
We say okay if we go right
to the left of the decimal

00:01:20.790 --> 00:01:23.130
that's going to be our one
space because once we start

00:01:23.130 --> 00:01:25.920
introducing decimals we can
introduce as many spaces

00:01:25.920 --> 00:01:28.730
as we want to the right of the decimal.

00:01:28.730 --> 00:01:31.000
And so let's think about
those a little bit.

00:01:31.000 --> 00:01:34.410
If when we went from hundreds
to tens, notice we divided

00:01:34.410 --> 00:01:37.208
by ten, when we go from tens
to ones, notice you divide

00:01:37.208 --> 00:01:38.410
by ten.

00:01:38.410 --> 00:01:40.550
So what do you think this
place over here is going to be

00:01:40.550 --> 00:01:41.640
called?

00:01:41.640 --> 00:01:44.599
Well what happens if you
take one divided by ten?

00:01:44.599 --> 00:01:48.062
Well then you get a tenth
so as you might imagine

00:01:48.062 --> 00:01:50.400
this is the tenths place.

00:01:50.400 --> 00:01:52.610
And then if you were to
go one place to the right

00:01:52.610 --> 00:01:55.670
of that, what would this place be?

00:01:55.670 --> 00:02:00.109
Well it'd be tenths divided
by ten or 1/10 of a tenth,

00:02:00.109 --> 00:02:05.109
so this would be a
hundredth, hundredths place.

00:02:05.960 --> 00:02:08.120
And then if you were to
go one space to the right

00:02:08.120 --> 00:02:11.330
we could keep doing this
forever, but if we were to go

00:02:11.330 --> 00:02:14.023
one space to the right of
that, what would it be?

00:02:14.023 --> 00:02:19.023
Well a hundredth divided by
ten or 1/10 of a hundredth

00:02:19.570 --> 00:02:24.570
is a thousandth, thousandth space.

00:02:24.590 --> 00:02:27.320
And so for example if I
were to extend this number

00:02:27.320 --> 00:02:32.113
instead of if just being 973,
if I were to write 973.526,

00:02:36.250 --> 00:02:39.610
what do these numbers
these digits represent?

00:02:39.610 --> 00:02:44.060
This five doesn't just
represent five, it represents

00:02:45.110 --> 00:02:49.050
five tenths or another
way of writing five tenths

00:02:49.050 --> 00:02:51.960
you could write it like this
0.5 you just have a five

00:02:51.960 --> 00:02:53.080
in the tenths place.

00:02:53.080 --> 00:02:56.600
Or you could write it as five tenths.

00:02:56.600 --> 00:02:59.490
This two I think you
get where this is going,

00:02:59.490 --> 00:03:01.800
this doesn't just represent
two, it represents

00:03:01.800 --> 00:03:05.610
two hundredths I'm just going
to make it very explicit

00:03:05.610 --> 00:03:09.730
in this video, so it's
very clear two hundredths.

00:03:09.730 --> 00:03:12.270
Another way to write that
is you just write a two

00:03:12.270 --> 00:03:14.540
in the hundredths place.

00:03:14.540 --> 00:03:17.216
So we're going one two spaces
to the right of the decimal

00:03:17.216 --> 00:03:22.022
or you could write it as two
over 100, two hundredths.

00:03:22.022 --> 00:03:25.300
And so for kicks, pause
the video what are all

00:03:25.300 --> 00:03:27.474
the different ways of
representing this six?

00:03:27.474 --> 00:03:29.592
What does this six represent?

00:03:29.592 --> 00:03:34.592
Well this is six thousandths,
six thousandths, thousandths,

00:03:37.380 --> 00:03:38.880
there you go.

00:03:38.880 --> 00:03:40.980
I could also write that as
zero point, let's see it's

00:03:40.980 --> 00:03:43.910
the tenths place,
hundredths place, and then

00:03:43.910 --> 00:03:47.410
in the thousandths place I
have six or I could write

00:03:47.410 --> 00:03:52.410
this as six over 1000, six thousandths.

00:03:53.280 --> 00:03:57.470
So big picture place value we
can keep going to the right

00:03:57.470 --> 00:04:00.672
of the decimal and we can
start representing things

00:04:00.672 --> 00:04:03.463
that are I guess you
could say more precise.


Writing decimals and fractions from number lines

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:02.290
- [Instructor] We're
told express the point

00:00:02.290 --> 00:00:06.310
on the number line as both
a fraction and a decimal.

00:00:06.310 --> 00:00:08.923
So pause this video and have a go at that.

00:00:09.840 --> 00:00:11.420
All right, now let's do this together.

00:00:11.420 --> 00:00:14.650
So we can see that the point in question,

00:00:14.650 --> 00:00:18.300
it's at a higher value than
four and it's less than five.

00:00:18.300 --> 00:00:20.410
So greater than four, less than five.

00:00:20.410 --> 00:00:22.900
And the space between four and five

00:00:22.900 --> 00:00:26.730
is divided into one,
two, three, four, five,

00:00:26.730 --> 00:00:30.740
six, seven, eight,
nine, 10 equal sections.

00:00:30.740 --> 00:00:34.770
So each of these hash marks
represent an extra tenth.

00:00:34.770 --> 00:00:38.350
So this is four, then this
is four and one tenth,

00:00:38.350 --> 00:00:42.040
and now this is four right
over here and two tenths.

00:00:42.040 --> 00:00:43.840
So we could write this,
if we wanted to write it

00:00:43.840 --> 00:00:45.360
as a fraction or as a mixed number,

00:00:45.360 --> 00:00:48.010
this would be four and two tenths.

00:00:48.010 --> 00:00:50.090
And if we wanted to
write that as a decimal,

00:00:50.090 --> 00:00:53.590
that would be four, and
then in the tenths place,

00:00:53.590 --> 00:00:55.690
well we have two tenths.

00:00:55.690 --> 00:00:57.220
And that's it, we're done.

00:00:57.220 --> 00:00:58.473
Let's do another example.

00:00:59.800 --> 00:01:02.870
So here, we're once again asked

00:01:02.870 --> 00:01:04.580
to express the point on the number line

00:01:04.580 --> 00:01:06.400
as both a fraction and a decimal,

00:01:06.400 --> 00:01:07.840
but this one's a little bit different.

00:01:07.840 --> 00:01:09.230
Let's see how you can, if you can identify

00:01:09.230 --> 00:01:11.450
how it is different and
answer the question.

00:01:11.450 --> 00:01:14.050
So pause this video, and
once again have a go at it.

00:01:14.930 --> 00:01:18.440
All right, so here, our point,

00:01:18.440 --> 00:01:21.260
it's not between two whole numbers.

00:01:21.260 --> 00:01:24.420
It's actually between two tenths.

00:01:24.420 --> 00:01:26.400
We're between three and two tenths

00:01:26.400 --> 00:01:28.780
and three and three tenths.

00:01:28.780 --> 00:01:30.810
So this is between three and two tenths

00:01:30.810 --> 00:01:32.170
and three and three tenths.

00:01:32.170 --> 00:01:36.290
So each of these hash marks,
which are a tenth of a tenth.

00:01:36.290 --> 00:01:39.360
So they would actually be a hundredth.

00:01:39.360 --> 00:01:42.490
So one way to think about
it, you could view 3.2

00:01:42.490 --> 00:01:46.980
or three and two tenths as
three and 20 hundredths.

00:01:46.980 --> 00:01:48.820
And you could view three and three tenths

00:01:48.820 --> 00:01:51.540
as three and 30 hundredths.

00:01:51.540 --> 00:01:54.775
And so this is three and 20 hundredths.

00:01:54.775 --> 00:01:59.450
This is three and 21 hundredths,
three and 22 hundredths.

00:01:59.450 --> 00:02:04.450
So this point right over here
is three and 22 hundredths,

00:02:05.600 --> 00:02:08.130
22 hundredths, and of
course, you could also

00:02:08.130 --> 00:02:09.650
write that as a mixed number.

00:02:09.650 --> 00:02:14.650
That is three and 22 over 100.

00:02:16.590 --> 00:02:19.030
Now another way that you
could've approached it is

00:02:19.030 --> 00:02:24.030
hey, I'm starting at 3.2 or
three and two tenths, and

00:02:24.030 --> 00:02:25.873
so I'm starting here at 3.2.

00:02:26.960 --> 00:02:30.580
And then I'm going to add to
that, not just one hundredth,

00:02:30.580 --> 00:02:34.640
but two hundredths, so it would be three,

00:02:34.640 --> 00:02:38.350
two tenths, and then two hundredths.

00:02:38.350 --> 00:02:39.910
And there you have it, we've expressed it

00:02:39.910 --> 00:02:41.953
as both a fraction and a decimal.


Writing numbers in words and standard form

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:01.660
- What we're going to do in this video

00:00:01.660 --> 00:00:04.800
is get some practice writing
reasonably large numbers

00:00:04.800 --> 00:00:06.850
in different ways.

00:00:06.850 --> 00:00:09.920
So, for example, let's
say we had this number

00:00:09.920 --> 00:00:12.570
and I'm going to not say
it out loud on purpose.

00:00:12.570 --> 00:00:15.723
So, this number right over here.

00:00:16.600 --> 00:00:18.090
What I wanna do is think

00:00:18.090 --> 00:00:21.130
about how would I write this out in words?

00:00:21.130 --> 00:00:22.970
And I encourage you to pause this video

00:00:22.970 --> 00:00:24.720
and try to figure that out on your own

00:00:24.720 --> 00:00:25.610
before we do it together.

00:00:25.610 --> 00:00:28.110
And if you take out paper
and actually write this out,

00:00:28.110 --> 00:00:29.823
how would you write this number?

00:00:31.080 --> 00:00:33.150
Well now, we can do it together.

00:00:33.150 --> 00:00:35.220
Actually before I even
write it out in words,

00:00:35.220 --> 00:00:37.700
I'll write this out in a place value table

00:00:37.700 --> 00:00:39.350
'cause it might help
us, or it just helps us

00:00:39.350 --> 00:00:42.910
break down exactly what
this number represents.

00:00:42.910 --> 00:00:45.310
So, just as a bit of a refresher,

00:00:45.310 --> 00:00:47.960
if you're not familiar
with place value tables.

00:00:47.960 --> 00:00:50.660
You think of a column for
the different place values

00:00:50.660 --> 00:00:52.210
and here we're going from the ones place

00:00:52.210 --> 00:00:55.390
all the way to 10s, 1,000s, 10,000s,

00:00:55.390 --> 00:00:57.734
so we're going to have a 10,0000s

00:00:57.734 --> 00:00:59.456
place right over here.

00:00:59.456 --> 00:01:03.930
We're going to have a 1,000s
place right over here.

00:01:03.930 --> 00:01:07.110
This is just an S here, so
ten thousands, one thousands.

00:01:07.110 --> 00:01:08.760
Sometimes you'll see it written out.

00:01:08.760 --> 00:01:12.045
This is the 100s place.

00:01:12.045 --> 00:01:15.280
This is the 10s place.

00:01:15.280 --> 00:01:17.650
And this is the ones place.

00:01:17.650 --> 00:01:20.710
And so, how many 10,000s do we have?

00:01:20.710 --> 00:01:22.300
Actually, let's start at the ones place.

00:01:22.300 --> 00:01:23.580
How many ones do we have?

00:01:23.580 --> 00:01:24.413
We see it right here.

00:01:24.413 --> 00:01:26.720
We have five ones.

00:01:26.720 --> 00:01:28.220
How many 10s do we have?

00:01:28.220 --> 00:01:29.053
We see it right there,

00:01:29.053 --> 00:01:31.610
actually, we have zero 10s.

00:01:31.610 --> 00:01:34.280
How many 100s do we have?

00:01:34.280 --> 00:01:36.930
We see it right there, six 100s.

00:01:36.930 --> 00:01:38.610
How many 1,000s?

00:01:38.610 --> 00:01:39.837
We go to the 1,000s place

00:01:39.837 --> 00:01:42.160
and we see we have two 1,000s.

00:01:42.160 --> 00:01:45.460
And then, how many 10,000s we have?

00:01:45.460 --> 00:01:46.390
Well, we see right there,

00:01:46.390 --> 00:01:48.510
we have eight 10,000s.

00:01:48.510 --> 00:01:50.660
So now that we have this representation,

00:01:50.660 --> 00:01:51.493
let's write it out in words

00:01:51.493 --> 00:01:53.030
and you don't have to do this every time,

00:01:53.030 --> 00:01:54.380
but I am just trying to make sure we see

00:01:54.380 --> 00:01:57.200
every possible way of thinking about it.

00:01:57.200 --> 00:01:59.698
So, you could say eight 10s,

00:01:59.698 --> 00:02:04.140
two 1,000s, six 100s, and five ones,

00:02:04.140 --> 00:02:07.260
but that's not exactly how
people would say it out in words.

00:02:07.260 --> 00:02:09.180
What is typical is that people would say

00:02:09.180 --> 00:02:14.040
82 1,000s, 600 and five.

00:02:14.040 --> 00:02:15.790
And so I'm gonna write it that way.

00:02:15.790 --> 00:02:17.490
And when you write it out in words,

00:02:17.490 --> 00:02:19.900
the standard way of doing
it is actually put commas

00:02:19.900 --> 00:02:22.000
wherever you put the commas in,

00:02:22.000 --> 00:02:24.430
this is right over here
written in standard form.

00:02:24.430 --> 00:02:27.240
So what we would do is,
we would express this part

00:02:27.240 --> 00:02:30.310
right over here as eighty-two thousand,

00:02:30.310 --> 00:02:31.850
so let me write it out.

00:02:31.850 --> 00:02:33.537
Eighty two thousand.

00:02:42.330 --> 00:02:45.400
And then, put this comma there.

00:02:45.400 --> 00:02:47.260
Eighty two thousand, comma,

00:02:47.260 --> 00:02:50.940
and then we have 600
and then we have no 10s,

00:02:50.940 --> 00:02:52.430
so it's 605.

00:02:52.430 --> 00:02:53.800
If there was a one 10 here,

00:02:53.800 --> 00:02:55.100
then it would have been 615,

00:02:55.100 --> 00:02:56.820
but this is just 605.

00:02:56.820 --> 00:02:59.980
And so I will do it in a different color.

00:02:59.980 --> 00:03:01.730
Let's do it in this blue color.

00:03:01.730 --> 00:03:06.730
So eighty-two thousand,
and then six hundred five.

00:03:10.560 --> 00:03:11.810
And we're done!

00:03:11.810 --> 00:03:13.433
Let's do another example.

00:03:14.670 --> 00:03:16.770
So now I'm gonna write
out a number in words

00:03:16.770 --> 00:03:18.340
and I want you to write
it in standard form.

00:03:18.340 --> 00:03:22.983
So, the number is fifty
thousand, six hundred thirty-one.

00:03:35.310 --> 00:03:38.240
And I want you to write
this in standard form.

00:03:38.240 --> 00:03:40.120
So it's really just
write it out as a number

00:03:40.120 --> 00:03:41.810
and put commas in the appropriate places.

00:03:41.810 --> 00:03:44.073
Pause this video and
see if you can do that.

00:03:44.980 --> 00:03:47.190
All right, now let's do it together.

00:03:47.190 --> 00:03:51.120
So, this first part, fifty thousand,

00:03:51.120 --> 00:03:53.420
well, we could just write that as,

00:03:53.420 --> 00:03:55.150
actually, let's put the
place values out here,

00:03:55.150 --> 00:03:56.621
so there's a 10,000s place,

00:03:56.621 --> 00:04:00.843
this is a 1,000s place,
100s, 10s, and ones.

00:04:00.843 --> 00:04:02.493
So, fifty thousand, so 50 1,000s,

00:04:04.080 --> 00:04:05.350
you could write like this,

00:04:05.350 --> 00:04:07.430
you could say that's five 10,000s,

00:04:07.430 --> 00:04:09.849
or you could say this is 50 1,000s.

00:04:09.849 --> 00:04:13.830
Then they say six hundred thirty-one.

00:04:13.830 --> 00:04:18.410
So, six hundred, well that's
six 100s right over here

00:04:18.410 --> 00:04:23.220
and then thirty-one,
that is thirty and one,

00:04:23.220 --> 00:04:25.700
or three 10s and one one,

00:04:25.700 --> 00:04:28.930
three 10s and one one, or 31.

00:04:28.930 --> 00:04:31.240
And of course, you wanna put your comma

00:04:31.240 --> 00:04:33.840
in the same place, and you put your comma

00:04:33.840 --> 00:04:36.360
every three places, so
the first time you do it,

00:04:36.360 --> 00:04:37.860
if you're moving from the right,

00:04:37.860 --> 00:04:42.180
you put it between the
100s and the 1,000s place.

00:04:42.180 --> 00:04:43.013
So, there you have it.

00:04:43.013 --> 00:04:47.623
This is the same thing as 50,631.

00:04:49.080 --> 00:04:52.593
This is that number written in stand form.


Examples writing decimals and fractions greater than 1 shown on grids

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:02.470
- [Instructor] We're told each big square

00:00:02.470 --> 00:00:05.420
below represents one whole.

00:00:05.420 --> 00:00:07.560
Express the shaded area

00:00:07.560 --> 00:00:11.180
as both a mixed number and a decimal.

00:00:11.180 --> 00:00:13.160
So pause this video and
see if you can do that.

00:00:13.160 --> 00:00:15.190
What would this be as a mixed number,

00:00:15.190 --> 00:00:17.140
and then what would it be as a decimal?

00:00:18.140 --> 00:00:20.520
All right, now let's do it together.

00:00:20.520 --> 00:00:22.500
So let's first start
with the mixed number.

00:00:22.500 --> 00:00:24.980
So we see that we have one whole here,

00:00:24.980 --> 00:00:27.090
the whole thing is filled out,

00:00:27.090 --> 00:00:28.820
so this is going to be one whole,

00:00:28.820 --> 00:00:30.700
and then over here, we have part

00:00:30.700 --> 00:00:33.040
of this second whole filled out,

00:00:33.040 --> 00:00:35.600
and it looks like we're dividing

00:00:35.600 --> 00:00:38.190
this whole into 10 equal sections,

00:00:38.190 --> 00:00:41.370
and then two of those are filled out.

00:00:41.370 --> 00:00:44.150
So as a mixed number, we have one

00:00:44.150 --> 00:00:49.150
and then you have two
of the 10ths filled out.

00:00:49.160 --> 00:00:53.770
So this is gonna be 1
2/10, and we're done.

00:00:53.770 --> 00:00:55.520
You can see here, this
is split into 10ths,

00:00:55.520 --> 00:00:57.460
and we filled in two of them.

00:00:57.460 --> 00:00:59.770
Now, what about as a decimal?

00:00:59.770 --> 00:01:02.860
Well, we could just express
1 2/10 as a decimal.

00:01:02.860 --> 00:01:04.850
We could say, hey, that's going to be one,

00:01:04.850 --> 00:01:06.770
and then we get to the 10th place,

00:01:06.770 --> 00:01:08.470
and then how many 10ths do we have?

00:01:08.470 --> 00:01:09.960
We have two of them.

00:01:09.960 --> 00:01:11.260
So that's going to be 1.2.

00:01:12.340 --> 00:01:13.823
Let's do another example.

00:01:14.710 --> 00:01:16.830
So here, this is a
little bit more involved.

00:01:16.830 --> 00:01:18.830
They say, once again, each big square

00:01:18.830 --> 00:01:21.050
below represents one whole.

00:01:21.050 --> 00:01:22.290
And once again, they want us

00:01:22.290 --> 00:01:25.870
to express the shaded area as
both a fraction and a decimal.

00:01:25.870 --> 00:01:28.193
So pause this video and have a go at this.

00:01:29.030 --> 00:01:31.930
All right, so let's start
with the fraction again.

00:01:31.930 --> 00:01:35.350
So we have one whole, two wholes,

00:01:35.350 --> 00:01:38.170
and then partially shaded
in this third whole,

00:01:38.170 --> 00:01:39.760
so if I'm gonna express
this as a fraction,

00:01:39.760 --> 00:01:41.330
really it's going to be a mixed number,

00:01:41.330 --> 00:01:45.500
I would say that this over
here, this is two wholes.

00:01:45.500 --> 00:01:48.140
Now, this third whole is
only partially filled in,

00:01:48.140 --> 00:01:51.530
and we can see that it has
been divided into 100ths,

00:01:51.530 --> 00:01:53.540
you can see it's a 10 by 10 grid,

00:01:53.540 --> 00:01:57.190
so each of these squares
represent 100th of a whole,

00:01:57.190 --> 00:02:00.560
and how many of these
100ths are filled in?

00:02:00.560 --> 00:02:05.560
Well, let's see, you have
10, 20, 30, 40, 50, 60, 70,

00:02:12.440 --> 00:02:16.130
and then you have 71, two, three, four.

00:02:16.130 --> 00:02:21.130
We can see that 74 of
the 100ths are filled in.

00:02:21.620 --> 00:02:23.230
So as a mixed number, this

00:02:23.230 --> 00:02:28.103
whole thing would represent 2 74/100.

00:02:29.340 --> 00:02:31.280
Now, if we wanna write it as a decimal,

00:02:31.280 --> 00:02:32.760
we would have two wholes,

00:02:32.760 --> 00:02:34.780
and then we could go to the 10ths place.

00:02:34.780 --> 00:02:37.560
You could just write two
and 74 100ths like that,

00:02:37.560 --> 00:02:38.920
if you're pretty familiar with it.

00:02:38.920 --> 00:02:39.950
You could also think about it

00:02:39.950 --> 00:02:42.960
in terms of how many 10ths and
how many 100ths did you have.

00:02:42.960 --> 00:02:47.440
We have one, two, three,

00:02:47.440 --> 00:02:52.000
four, five, six, seven 10ths,

00:02:52.000 --> 00:02:54.420
and then we have four
more 100ths beyond that,

00:02:54.420 --> 00:02:58.770
so you could think of it as
74 100ths or 7/10 and 400ths,

00:02:58.770 --> 00:03:00.563
but either way, we are done.


Examples relating decimals and fractions in words

WEBVTT
Kind: captions
Language: en

00:00:00.250 --> 00:00:03.340
- [Instructor] We are told
to write seven hundredths

00:00:03.340 --> 00:00:06.260
as a fraction and a decimal.

00:00:06.260 --> 00:00:08.380
Why don't you get some
paper and a pencil out

00:00:08.380 --> 00:00:11.530
and see if you can do that
before we do it together.

00:00:11.530 --> 00:00:14.150
All right, so let's do
it first as a fraction.

00:00:14.150 --> 00:00:17.450
So what is going to be the
denominator of our fraction

00:00:17.450 --> 00:00:20.200
if they're saying seven hundredths?

00:00:20.200 --> 00:00:22.130
And the way I'm saying it
is a little bit of a hint.

00:00:22.130 --> 00:00:25.530
Seven hundredths, oh, I
think you got the picture.

00:00:25.530 --> 00:00:27.480
We're dealing with hundredths.

00:00:27.480 --> 00:00:30.600
So our denominator is going to be 100.

00:00:30.600 --> 00:00:34.140
And then how many hundredths do we have?

00:00:34.140 --> 00:00:35.580
Well we have seven of them.

00:00:35.580 --> 00:00:37.270
I'll do that in a different
color just to be clear.

00:00:37.270 --> 00:00:39.840
We have seven of those hundredths.

00:00:39.840 --> 00:00:42.520
So there you have it, seven hundredths.

00:00:42.520 --> 00:00:46.080
That's this expressed as a fraction.

00:00:46.080 --> 00:00:49.030
Now what about as a decimal?

00:00:49.030 --> 00:00:52.380
Well, we could think
about our decimal places.

00:00:52.380 --> 00:00:56.180
If let's say that this is the ones place,

00:00:56.180 --> 00:00:57.670
and I'm just putting a little blank here.

00:00:57.670 --> 00:00:58.520
So this is the ones place,

00:00:58.520 --> 00:01:00.400
and you have a decimal right over here.

00:01:00.400 --> 00:01:03.550
And then this would be the tenths place.

00:01:03.550 --> 00:01:07.880
And then this would be
the hundredths place.

00:01:07.880 --> 00:01:11.220
Well we have, we want to
represent seven hundredths.

00:01:11.220 --> 00:01:14.150
So let me be clear, this
right over here is ones.

00:01:14.150 --> 00:01:19.150
This is tenths, and this is hundredths.

00:01:19.300 --> 00:01:23.500
I like saying, it's unusually
fun to say that, hundredths.

00:01:23.500 --> 00:01:25.360
All right, ones, tenths, hundredths.

00:01:25.360 --> 00:01:26.870
So how many ones do we have here?

00:01:26.870 --> 00:01:29.410
Well we have no ones, we have zero ones.

00:01:29.410 --> 00:01:30.900
How many tenths do we have here?

00:01:30.900 --> 00:01:32.900
Well we have no tenths.

00:01:32.900 --> 00:01:35.350
How many hundredths do we have?

00:01:35.350 --> 00:01:37.700
Well we have seven hundredths.

00:01:37.700 --> 00:01:38.990
Okay now it's getting annoying.

00:01:38.990 --> 00:01:40.600
We have seven hundredths.

00:01:40.600 --> 00:01:41.890
So you can write it that way as well.

00:01:41.890 --> 00:01:43.620
And if I wanted to just write
it a little bit cleaner,

00:01:43.620 --> 00:01:48.620
I could just write no ones, no
tenths, and seven hundredths.

00:01:49.250 --> 00:01:52.090
I said it the last time
like a normal person.

00:01:52.090 --> 00:01:53.823
Let's do another example.

00:01:56.570 --> 00:02:01.570
Here, we're told select the
written form of each number.

00:02:01.570 --> 00:02:04.640
And so they, on the left right over here,

00:02:04.640 --> 00:02:06.250
you have different representations here.

00:02:06.250 --> 00:02:09.110
We have things written
as a decimal, a fraction,

00:02:09.110 --> 00:02:11.230
another decimal, and
then we want to say hey,

00:02:11.230 --> 00:02:13.800
which of these are represented in words

00:02:13.800 --> 00:02:15.780
or a combination of
numbers and words up here.

00:02:15.780 --> 00:02:19.070
So pause this video and
have a go at this as well.

00:02:19.070 --> 00:02:21.470
Okay, so this first
number right over here,

00:02:21.470 --> 00:02:24.310
we have no ones, and
then as we go one space

00:02:24.310 --> 00:02:27.200
to the right of the decimal,
this is the tenths place.

00:02:27.200 --> 00:02:29.830
And it's clear we have
four of those tenths.

00:02:29.830 --> 00:02:32.960
So this right over here is four tenths.

00:02:32.960 --> 00:02:36.270
So that is this choice
right here, so we would,

00:02:36.270 --> 00:02:38.440
I'll shade it in, if you're
doing this on Khan Academy,

00:02:38.440 --> 00:02:41.090
you would just click there
and it would fill in.

00:02:41.090 --> 00:02:42.143
So what about this one?

00:02:42.143 --> 00:02:44.030
Well this one, we would read,

00:02:44.030 --> 00:02:47.610
you have four out of ten, or four tenths.

00:02:47.610 --> 00:02:50.460
So this again would be four tenths.

00:02:50.460 --> 00:02:53.173
So we would shade that one in.

00:02:54.410 --> 00:02:56.180
Now what's going on over here?

00:02:56.180 --> 00:02:59.100
We have no ones, we have no tenths.

00:02:59.100 --> 00:03:00.790
But we have four hundredths.

00:03:00.790 --> 00:03:03.100
I said it again, it's too much fun.

00:03:03.100 --> 00:03:04.760
So we have four hundredths.

00:03:04.760 --> 00:03:06.380
So that's this column.

00:03:06.380 --> 00:03:10.243
So we would fill that
one in, and we're done.