Spanish colonization

WEBVTT
Kind: captions
Language: en

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- [Instructor] Imagine that one day

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you are standing in your backyard

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when all of a sudden you
saw an alien ship land

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and the alien ship had
incredible technology

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and you saw aliens walking out of the ship

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bearing strange animals,

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maybe scary looking weapons

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and speaking a very strange language.

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What would you do?

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Would you try to be kind to the aliens,

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hope that maybe you could befriend them?

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Would you fear them

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and perhaps immediately try
to make war against them?

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Would you hope that perhaps
War of the Worlds style

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that they would die of the common cold

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or would you fear that
maybe they had some kind

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of common cold that you might die of?

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These are the choices that
were faced by Native Americans

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when they encountered the
Spanish at the end of the 1400s.

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Columbus arrived in the New World in 1492

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and when he got back in 1493

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the secret was out that
there were great riches

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to be had in the New World,

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so much so that as early as 1494

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Spain and Portugal were trying to decide

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how they would divide the riches

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of the Old World and the New between them,

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so in the years after
Columbus' first voyage

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many Spanish conquistadors or conquerors

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began exploring throughout the Americas

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and it's not necessary for you to memorize

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any of these names

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but I want you to get a sense

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that in the 50 years or so after Columbus,

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European explorers began
checking out everything

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in the Caribbean, North
America and South America

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and their motivations, like Columbus'

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were the three Gs of colonization,

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gold, that is to get rich,

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glory, that is to bring
glory to one's self

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for one's nation

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and a little bit of God,

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that is to bring Catholicism

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to native peoples living in the Americas.

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Now, as you can see from
the many individuals here,

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Spanish colonization was
a very complex process

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taking place in many different regions

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but in this video I wanna focus in

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on just a few aspects,

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the conquest of Mexico by Cortes,

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a society that came out of this blend

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of Spanish and Native American culture

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and a little bit about the resistance

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to colonization that
we'll see in New Mexico

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with the Pueblo Revolt.

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Now, like Columbus, Spanish
explorers originally

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were looking for a passage
to Asia through the Americas

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but quickly learned that
there was quite a lot

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of riches to be found in
the Americas themselves

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and one place that came to the attention

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of the conquistadors was Tenochtitlan

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which was the capital of the Aztec Empire.

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Now, the Aztecs were not
well loved in Mexico.

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They ruled over a vast territory

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with many smaller tribes

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they required to give them tribute

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and even human captives for sacrifice,

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so in 1519, Hernan Cortes,

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a Spanish conquistador landed

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with a group of about 600 men in Vera Cruz

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and with the help of some translators

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he worked his way across Mexico

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learning of the general
dislike of the Aztec Empire

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so that when he finally came upon the city

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of Tenochtitlan he had about
20,000 Native Americans

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who were ready to make war

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on this city along with him.

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Now, it's hard to
imagine what Tenochtitlan

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would have looked like to the Spanish.

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It's estimated that it had about 200,000

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to 300,000 inhabitants

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which made it one of the
larger cities in the world.

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There was nothing quite
so large as this city

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in all of Europe.

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It sat in the middle of a
lake with hanging gardens

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and an aqueduct and
had incredible pyramids

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that were many stories tall

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and at first, Moctezuma the Second

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who was emperor of the Aztec Empire

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treated the approaching
Spaniards with great kindness

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and generosity showering them with gifts.

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After all, the Spanish had things

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that Moctezuma had never seen before

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like horses and gigantic war dogs

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which they used to rip apart their enemies

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and cannons which even though
they only had a few of them

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and they didn't work very well

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were very frightening

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when they were fired,

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much like I think a ray gun
would be frightening to us now.

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Now, it didn't take very
long for the relationship

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between the Spanish and
the Aztecs to crumble

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and aided by their many
native American allies

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and also by the spread of
deadly disease like smallpox

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which decimated the Native
American population,

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by 1521 Tenochtitlan had fallen,

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in fact, was in ruins.

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Moctezuma had died

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and the Spanish began building
on the ruins of Tenochtitlan,

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Mexico City but much to the horror

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of the many Native American tribes

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that had allied with the Spanish,

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the Spanish would turn out

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to be much crueler imperial masters

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than the Aztecs had been

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and the smallpox that
had ravaged the Aztecs,

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ravaged the rest of the Native
American population as well

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as they lacked the immunity

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to European disease.

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Although much of the conquest

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of the Spanish Empire in the New World

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was done by conquistadors, adventurers,

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the Spanish Crown was
eager to place some control

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over this new territory

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and one way that they did was
through the Encomiendas System

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and the Encomiendas
System was a labor system

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that in a way was kind of a combination

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of feudalism from Europe and slavery.

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So, the idea was that the Spanish Crown

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would grant land holders
called encomenderos

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the right to the labor
of Native Americans,

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perhaps a village or two

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and anything that those
Native Americans produced

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through their slave labor,

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so gold if they mined it
or agricultural products

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if they were working on a plantation

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and in theory what the Native
Americans would get for this

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would be Christianization

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which to the Catholic Spanish Crown

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was an important goal to convert

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all of the world's people to Catholicism

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and also the protection

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of these Spanish feudal
lords or encomenderos.

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In practice the Encomiendas System

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was really just another
way of saying slavery

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and between the harsh
treatment of the Spanish lords

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and disease, the native
population of this region

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went from about 20 million

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when the Spanish arrived
to only about two million

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by 1600, so that not
very long after conquest,

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the Spanish began to
bring enslaved Africans

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to labor in the New World

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as the Native American
population had shrunk

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to a fraction of its former size.

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So, for the native people of Mexico

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the arrival of the Spanish
was about the worst outcome

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of an alien invasion
that you could imagine.

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Now, native people did resist
the Spanish in many ways.

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Some ways were more subtle

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like outwardly adopting Christianity

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while maintaining their
ancestral beliefs inwardly.

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The combination of Native American beliefs

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and Christianity together
is called syncretism

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or the blending of two
religious traditions

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but sometimes the Spanish
pushed native people too far

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as in the case of the
Pueblo Revolt in 1680

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when after a few generations

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of being forced to shed all
of their religious beliefs

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in favor of Christianity

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or face severe punishment,
the pueblo people rose up

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against the Spanish led
by a man named Pope,

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so that sometimes the
Pueblo Revolt is also called

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Pope's Rebellion and they
killed Spanish priests,

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burned churches, replaced them with kivas,

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their own place of worship

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and drove the Spanish out

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so that in the next 50 years

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it took the Spanish to reestablish control

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of this region, the Spanish took

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a much more accommodating
approach to pueblo society.

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The last aspect of
Spanish colonial society

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that I wanna point out is
the racial caste system

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that developed in the New World.

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Because the conquistadors
were on dangerous adventures,

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very few Spanish women came
with them in the New World

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and so, Spanish men had relationships

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with both native women and African women

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and Native Americans and
Africans had relationships

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such that there was
really an unprecedented

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mixing of peoples and
cultures in the New World

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and to account for this
incredibly diverse society

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the Spanish developed a caste system

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that very carefully ranked individuals

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by how much Spanish blood they had

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so that people with pure Spanish blood,

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criollos were at the top of the hierarchy,

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and people who had both
Native American heritage

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and European heritage were
called mestizo at the time

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whereas people with European
heritage and African heritage

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were called mulatto at the time

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and so, as you moved up or down this scale

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you had more legal rights

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than the groups below you

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and this is what is
known as a casta painting

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which very carefully categorized

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where every person fell
on this hierarchy of race,

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so we see in the Spanish caste system

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the beginnings of assigning legal status

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to individuals based on their race

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but I think it's also important

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to note here that the casta system

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made a place for many
different types of people

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in society and that
will be important later

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when we contrast it with
how English settlers

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treated Native Americans.

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The Spanish sought to Christianize

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and incorporate and enslave native people,

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whereas the English sought
to completely eradicate them

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from the landscape.


Native American societies before contact

WEBVTT
Kind: captions
Language: en

00:00:00.157 --> 00:00:02.290
- [Instructor] Often when we think about

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the beginning of American history,

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we think 1776, with the
Declaration of Independence,

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or maybe 1492, when Columbus
arrived in the Americas.

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But the history of America really begins

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about 15,000 years ago,

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when people first arrived in the Americas.

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In this video, I want to
provide a very brief overview

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of native societies before contact

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to give you an idea of just how diverse

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and complex these societies were,

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as native groups adapted to

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and interacted with their environments.

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Now there's recently
been a scholarly debate

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about how people first arrived.

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We know that maybe 12,000 years ago,

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during an ice age, the
sea level was lower,

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and so a spit of land in between

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the Americas and Asia was exposed,

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over which people may have traveled.

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But recent archeological evidence suggests

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that people were perhaps
already in the Americas

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at the time of this ice age,

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so it's possible

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that they may have come earlier in boats.

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Now however it was that they arrived,

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they spread north and south

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and east throughout the Americas

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so that by the time that Europeans
arrived in the late 1400s

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there were perhaps 50

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million people.

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That's kind of a mid-range
number for the estimates,

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that historians have made,

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living in the Americas.

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And of those, four to six million

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were living in North America.

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So how did these societies develop?

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Well, a really big moment
was around 5000 BCE,

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when people in Mexico domesticated corn,

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maize, as it's also known.

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And domesticating maize meant that people

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who had originally been
hunters, gatherers,

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following herds of animals

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could partake in settled agriculture.

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So they could develop
villages, complex societies.

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This isn't to say that they
stopped hunting or gathering,

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but they began staying in one place.

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So let's zoom in a little
bit and take a look

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at some of the major
societies in these regions.

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Native American societies developed

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around their natural environments,

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using the resources that
were available to them.

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For example, the Southwest,
Plains and Great Basin

00:02:47.951 --> 00:02:50.200
were quite dry.

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A lot of desert.

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And so societies in these regions

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adapted to the dry
climate in several ways.

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For example, Native American groups

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that lived on the Great Plains

00:03:01.450 --> 00:03:05.041
continued their hunting
and gathering way of life.

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Hunting bison,

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and following the herds of animals

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in teepees,

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which were dwellings that were easy

00:03:16.172 --> 00:03:18.257
to set up and then take down.

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People in the Southwest, like
the Ancestral Puebloan people,

00:03:24.703 --> 00:03:26.671
dealt with this dry environment

00:03:26.671 --> 00:03:31.500
by creating very complex
irrigation projects

00:03:31.550 --> 00:03:35.038
so that they could water their maize crops

00:03:35.038 --> 00:03:37.548
using what little moisture there was.

00:03:37.548 --> 00:03:41.619
The Puebloans lived in
large cave complexes

00:03:41.619 --> 00:03:45.047
as agriculture allowed them
to grow their population.

00:03:45.047 --> 00:03:51.600
In the Northwest,
fishing in the Pacific Ocean

00:03:51.628 --> 00:03:55.686
gave Native Americans a
plentiful source of food,

00:03:55.686 --> 00:04:03.093
while farming allowed the Mississippian peoples

00:04:03.093 --> 00:04:06.988
to develop large
settlements, like Cahokia,

00:04:06.988 --> 00:04:10.355
near modern day St.
Louis, which, at its peak,

00:04:10.355 --> 00:04:16.200
may have had as many as 25,000-40,000 residents.

00:04:16.220 --> 00:04:19.898
The Mississippians and other
East Coast native peoples

00:04:19.898 --> 00:04:25.100
relied a lot on what's known
as three-sister farming,

00:04:25.162 --> 00:04:36.300
in which people would plant
corn, beans and squash together,

00:04:36.351 --> 00:04:41.193
which was mutually beneficial
to all three plants,

00:04:41.200 --> 00:04:47.500
as the corn served as a trellis for the beans,

00:04:47.600 --> 00:04:55.640
and the squash protected the root system of the corn.

00:04:55.740 --> 00:04:58.957
All three together create
a very nutritious diet,

00:04:58.957 --> 00:05:02.013
which allowed for a relatively
high population density

00:05:02.013 --> 00:05:03.270
on the East Coast.

00:05:03.270 --> 00:05:06.347
So by the time that
Europeans began to arrive

00:05:06.347 --> 00:05:09.424
in the late 1400s and 1500s,

00:05:09.424 --> 00:05:14.285
native societies had been
evolving for over 14,000 years.

00:05:14.285 --> 00:05:18.019
But the introduction of
European people, pathogens,

00:05:18.019 --> 00:05:20.259
plants, and animals would introduce

00:05:20.259 --> 00:05:24.077
an unprecedented amount
of change in the Americas.


Command and market economies

WEBVTT
Kind: captions
Language: en

00:00:00.288 --> 00:00:01.466
- [Instructor] In this
video we're gonna talk about

00:00:01.466 --> 00:00:04.085
different ways of structuring an economy.

00:00:04.085 --> 00:00:05.979
In particular, who owns what

00:00:05.979 --> 00:00:08.377
and how does an economy
decide what to produce

00:00:08.377 --> 00:00:12.195
and who gets the output
of that production.

00:00:12.195 --> 00:00:16.362
So on one side, you have what's
known as a command economy

00:00:18.116 --> 00:00:20.860
and good examples of command economies

00:00:20.860 --> 00:00:22.762
are the communist states,

00:00:22.762 --> 00:00:24.845
especially during the 20th century,

00:00:24.845 --> 00:00:28.809
The USSR, the Soviet Union
being the best example of that.

00:00:28.809 --> 00:00:32.013
In a command economy,
the government controls

00:00:32.013 --> 00:00:34.890
what's often known as
the factors of production

00:00:34.890 --> 00:00:36.815
and sometimes, in an extreme case,

00:00:36.815 --> 00:00:39.390
there might not even be private ownership.

00:00:39.390 --> 00:00:42.741
So lets say that this right
over here is the government.

00:00:42.741 --> 00:00:43.949
I'll do this with some type

00:00:43.949 --> 00:00:46.509
of a building with pillars in it.

00:00:46.509 --> 00:00:48.787
So this is the government.

00:00:48.787 --> 00:00:50.286
And in a command economy,

00:00:50.286 --> 00:00:53.945
the government will often own
the factories and the farms,

00:00:53.945 --> 00:00:55.796
so the factors of production.

00:00:55.796 --> 00:00:58.964
So let me draw a little factory here.

00:00:58.964 --> 00:01:00.967
So with a little smoke stack...

00:01:00.967 --> 00:01:02.820
And so the government
will tell this factory,

00:01:02.820 --> 00:01:07.658
"This year, you have to
produce exactly 10,000 cars,"

00:01:07.658 --> 00:01:09.657
"No more and no less."

00:01:09.657 --> 00:01:10.793
and so that factory say,

00:01:10.793 --> 00:01:13.206
"Okay, you're our boss,
you're the government."

00:01:13.206 --> 00:01:16.350
"We will produce exactly 10,000 cars."

00:01:16.350 --> 00:01:19.326
Similarly, the government might tell some,

00:01:19.326 --> 00:01:22.057
let's say that this right
over here is a farm,

00:01:22.057 --> 00:01:26.017
and this is my best drawing
of a field really fast,

00:01:26.017 --> 00:01:28.417
and so it needs to say
"Hey, you need to produce",

00:01:28.417 --> 00:01:31.305
I don't know, "10,000,000 apples,"

00:01:31.305 --> 00:01:32.138
I don't even know

00:01:32.138 --> 00:01:34.398
if that's a reasonable
amount of apples to produce

00:01:34.398 --> 00:01:36.731
but you need to produce 10,000,000 apples.

00:01:36.731 --> 00:01:39.814
And so the farmers who essentially
work for the government,

00:01:39.814 --> 00:01:41.296
'cause the government would own that farm,

00:01:41.296 --> 00:01:43.320
they would produce those apples.

00:01:43.320 --> 00:01:45.124
And also in a command
economy, there's like

00:01:45.124 --> 00:01:47.904
well who gets these cars
and who gets these apples?

00:01:47.904 --> 00:01:50.281
Well, in an extreme command economy,

00:01:50.281 --> 00:01:53.291
they will be directly
allocated by the government.

00:01:53.291 --> 00:01:54.232
The government will say,

00:01:54.232 --> 00:01:57.900
"Alright, you get a car, you
get a car, you get a car."

00:01:57.900 --> 00:01:59.129
and it might not be based,

00:01:59.129 --> 00:02:01.801
and in fact it will not likely be based

00:02:01.801 --> 00:02:04.458
on any type who's willing
to pay for the car

00:02:04.458 --> 00:02:06.067
or who could pay the most for the car.

00:02:06.067 --> 00:02:08.861
Similarly, for the apples.

00:02:08.861 --> 00:02:10.616
So they are all gonna be allocated.

00:02:10.616 --> 00:02:12.347
So there will have to be
a lot of planning done

00:02:12.347 --> 00:02:14.712
on the part of the government to say,

00:02:14.712 --> 00:02:16.931
"Okay, how much should
we produce of one thing"

00:02:16.931 --> 00:02:19.540
"versus another thing
and how do we decide"

00:02:19.540 --> 00:02:22.344
"who gets the different things?".

00:02:22.344 --> 00:02:25.099
Now the other side of the spectrum,

00:02:25.099 --> 00:02:28.230
you have what's known as a market economy.

00:02:28.230 --> 00:02:30.194
And most economies in the world,

00:02:30.194 --> 00:02:32.253
especially the United
States, are much closer

00:02:32.253 --> 00:02:35.801
to being a market economy
than being a command economy.

00:02:35.801 --> 00:02:38.007
Rather than having the government owning

00:02:38.007 --> 00:02:40.995
the factors of production,
deciding who produces what

00:02:40.995 --> 00:02:42.969
and how much and who gets those things,

00:02:42.969 --> 00:02:46.846
in a market economy, it's
all based on the factory

00:02:46.846 --> 00:02:50.416
right over here, independent
of the government

00:02:50.416 --> 00:02:52.673
for the most part unless
you're in certain fields

00:02:52.673 --> 00:02:55.112
that might have strong
influence from the government,

00:02:55.112 --> 00:02:56.804
let's say you're a military contractor.

00:02:56.804 --> 00:02:58.159
But if you're not in one of those fields,

00:02:58.159 --> 00:02:59.963
if you're fairly independent
of the government,

00:02:59.963 --> 00:03:00.900
what they would say is,

00:03:00.900 --> 00:03:02.339
"Well, what does the market need?"

00:03:02.339 --> 00:03:06.506
they might say, "Hey look,
the market needs 5,000 cars"

00:03:08.049 --> 00:03:10.455
"That look like this
but then it also needs"

00:03:10.455 --> 00:03:14.038
"another 2,000 trucks
that look like that."

00:03:16.268 --> 00:03:17.462
and how are they basing it?

00:03:17.462 --> 00:03:18.842
Well, they might have looked in the past

00:03:18.842 --> 00:03:20.398
or how well is this car selling,

00:03:20.398 --> 00:03:21.460
how well is the truck selling.

00:03:21.460 --> 00:03:23.069
They might look at competitors,

00:03:23.069 --> 00:03:24.727
in fact in a market economy,

00:03:24.727 --> 00:03:27.408
you'll often have more competition.

00:03:27.408 --> 00:03:28.503
The command economies,

00:03:28.503 --> 00:03:30.693
you might have one
really ginormous factory

00:03:30.693 --> 00:03:31.965
that's owned by the government or a few

00:03:31.965 --> 00:03:34.344
while here you might
have competing factories

00:03:34.344 --> 00:03:35.472
where they're always

00:03:35.472 --> 00:03:38.164
"Hey make a pretty good
car, we're gonna make one"

00:03:38.164 --> 00:03:41.445
"that's just as good
and it comes in yellow."

00:03:41.445 --> 00:03:43.907
So once again, you have
this notion of competition

00:03:43.907 --> 00:03:46.930
for people to produce
things out in the market

00:03:46.930 --> 00:03:48.832
and they're also making
their best judgment

00:03:48.832 --> 00:03:51.771
of what does the market actually need.

00:03:51.771 --> 00:03:54.321
Similarly, you could have
your farmers here and you say,

00:03:54.321 --> 00:03:55.255
"Hey, you know what?"

00:03:55.255 --> 00:03:56.323
"I've been growing a lot of apples"

00:03:56.323 --> 00:03:57.874
"but that doesn't seem
to be in demand anymore"

00:03:57.874 --> 00:04:01.701
"so I'm going to grow more,
I don't know, peaches"

00:04:01.701 --> 00:04:03.379
"'cause it looks like the sale of peaches"

00:04:03.379 --> 00:04:06.228
"are in fashion this year."

00:04:06.228 --> 00:04:07.785
and similarly, who gets these

00:04:07.785 --> 00:04:10.470
is not dictated by the government,

00:04:10.470 --> 00:04:12.513
it's determined by the market.

00:04:12.513 --> 00:04:16.118
And so let's say this is
me, let's say this is you,

00:04:16.118 --> 00:04:18.795
let's say this is someone else.

00:04:18.795 --> 00:04:22.005
If I might not be interested
in getting a truck

00:04:22.005 --> 00:04:24.565
but I really need to get
a new car for my family

00:04:24.565 --> 00:04:27.401
so I can drive them around
and take vacations and things,

00:04:27.401 --> 00:04:29.856
and so let's say you wanna get a car too

00:04:29.856 --> 00:04:32.378
but the market will dictate who gets it.

00:04:32.378 --> 00:04:36.188
So for example, the factory
might set a price for this car,

00:04:36.188 --> 00:04:39.771
they might say it is
10,000 dollars per car

00:04:40.711 --> 00:04:43.361
and maybe I'm willing to
pay it, maybe I'm not.

00:04:43.361 --> 00:04:45.090
Maybe I'm just like "Well I
really like that blue car"

00:04:45.090 --> 00:04:46.739
"but I'm not willing
to pay 10,000 dollars"

00:04:46.739 --> 00:04:49.987
"and this yellow car is
available for 9,000 dollars"

00:04:49.987 --> 00:04:52.859
"so I'm going to get that."

00:04:52.859 --> 00:04:55.501
and the prices themselves
won't necessarily be fixed.

00:04:55.501 --> 00:04:56.951
If they're not selling
enough of these cars

00:04:56.951 --> 00:04:59.121
at 10,000 dollars, they
might lower the price

00:04:59.121 --> 00:05:02.161
to compete with the yellow
car at 9,000 dollars.

00:05:02.161 --> 00:05:04.194
Similarly, if there's a ton of people

00:05:04.194 --> 00:05:06.077
who are willing to pay
the price of this truck,

00:05:06.077 --> 00:05:08.039
they might raise the price on the truck

00:05:08.039 --> 00:05:09.371
or produce more of it.

00:05:09.371 --> 00:05:11.007
In this severe competition,

00:05:11.007 --> 00:05:14.535
if one firm doesn't do this
well on a regular basis,

00:05:14.535 --> 00:05:15.368
they might go out of business.

00:05:15.368 --> 00:05:17.880
So there's a strong motivation

00:05:17.880 --> 00:05:20.700
to be able to meet the needs of the market

00:05:20.700 --> 00:05:23.338
and then the prices adjust accordingly.

00:05:23.338 --> 00:05:24.171
And so you might say,

00:05:24.171 --> 00:05:26.738
"well what are the
positives and negatives"

00:05:26.738 --> 00:05:28.270
"that people generally associate"

00:05:28.270 --> 00:05:31.666
"with command economies
or market economies?".

00:05:31.666 --> 00:05:33.597
Well, the motivation that's often given

00:05:33.597 --> 00:05:37.896
for a command economy is
some notion of fairness

00:05:37.896 --> 00:05:40.570
or some notion of equality

00:05:40.570 --> 00:05:42.897
that when you have a market economy,

00:05:42.897 --> 00:05:44.859
as a byproduct of this mechanism

00:05:44.859 --> 00:05:46.855
where different people are competing,

00:05:46.855 --> 00:05:49.065
some people are innovating more or less,

00:05:49.065 --> 00:05:52.124
some people might be
working harder or less,

00:05:52.124 --> 00:05:54.112
some people might just
get luckier or unluckier,

00:05:54.112 --> 00:05:58.195
you naturally will have
some level of inequality.

00:06:00.187 --> 00:06:03.842
And some of the original
ideas behind command economies

00:06:03.842 --> 00:06:07.404
like communism where hey we
don't like this inequality,

00:06:07.404 --> 00:06:08.855
we wanna see more fairness.

00:06:08.855 --> 00:06:11.916
And so we wanna see everyone
get exactly the same car.

00:06:11.916 --> 00:06:15.661
We wanna see everyone get exactly
the same number of apples.

00:06:15.661 --> 00:06:18.258
We don't wanna see all
of these competing firms,

00:06:18.258 --> 00:06:21.309
we want one really big, efficient factory

00:06:21.309 --> 00:06:24.029
to just churn out these cars.

00:06:24.029 --> 00:06:26.222
Now it turns out in reality,

00:06:26.222 --> 00:06:27.918
that's a little bit utopian.

00:06:27.918 --> 00:06:31.546
Even at the peak of some of
these communist economies,

00:06:31.546 --> 00:06:34.717
there were people who had
better access to certain things.

00:06:34.717 --> 00:06:37.249
People who might have carried
favor with the leadership,

00:06:37.249 --> 00:06:40.615
had more power, who had better
apartments, better cars,

00:06:40.615 --> 00:06:43.713
had more access to resources
and people who didn't.

00:06:43.713 --> 00:06:45.335
In a market economy,

00:06:45.335 --> 00:06:48.105
yes, there will be some inequality,

00:06:48.105 --> 00:06:50.286
but the best thing going for it,

00:06:50.286 --> 00:06:52.438
and the reason why most
economies in the world,

00:06:52.438 --> 00:06:54.794
even ones that are nominally communist

00:06:54.794 --> 00:06:56.200
like the Chinese economy,

00:06:56.200 --> 00:06:58.604
have transitioned to a market economy

00:06:58.604 --> 00:07:00.896
because a market economy
is also associated

00:07:00.896 --> 00:07:03.146
with things like innovation

00:07:05.616 --> 00:07:08.283
and strong incentives for people

00:07:09.449 --> 00:07:12.199
to innovate or work or do things.

00:07:13.269 --> 00:07:16.107
Think about the situation
in a command economy.

00:07:16.107 --> 00:07:18.585
Let's say that you're a
manager at this factory here.

00:07:18.585 --> 00:07:21.536
The government told you to
produce exactly 10,000 cars

00:07:21.536 --> 00:07:23.476
and exactly the type of model,

00:07:23.476 --> 00:07:24.309
maybe you have an idea

00:07:24.309 --> 00:07:27.024
for how to make this car more efficient,

00:07:27.024 --> 00:07:29.745
but no one is really
gonna reward you for that.

00:07:29.745 --> 00:07:31.512
If you're not gonna be
able to get paid more

00:07:31.512 --> 00:07:33.064
so that you can buy more things,

00:07:33.064 --> 00:07:35.825
you might not be as
interested in doing it.

00:07:35.825 --> 00:07:36.933
While in this situation,

00:07:36.933 --> 00:07:38.989
you have extreme competition

00:07:38.989 --> 00:07:41.699
where as soon as this
factory or this company

00:07:41.699 --> 00:07:42.914
is outselling this one,

00:07:42.914 --> 00:07:45.531
everyone here is gonna think
"Wow, we need to innovate."

00:07:45.531 --> 00:07:47.110
"We need to really work harder"

00:07:47.110 --> 00:07:49.983
"so that we can take more market share"

00:07:49.983 --> 00:07:52.027
"from the other company."

00:07:52.027 --> 00:07:53.512
Similarly, at the individual level,

00:07:53.512 --> 00:07:54.690
different people might say

00:07:54.690 --> 00:07:56.489
"Hey, I can only afford
the yellow car now"

00:07:56.489 --> 00:07:59.164
"but gee, I really wish
I had the blue car"

00:07:59.164 --> 00:08:01.021
"so maybe I'm gonna work
a little bit harder"

00:08:01.021 --> 00:08:02.097
"or I'm gonna try to innovate"

00:08:02.097 --> 00:08:03.837
"or I'm gonna try to start a business."

00:08:03.837 --> 00:08:06.900
most of the world has
transitioned to something

00:08:06.900 --> 00:08:10.475
much closer to a market
economy because overall,

00:08:10.475 --> 00:08:12.695
even though you have this inequality,

00:08:12.695 --> 00:08:16.373
it has made more
productivity, more innovation,

00:08:16.373 --> 00:08:20.088
more goods and services
available to more people.

00:08:20.088 --> 00:08:22.671
Now the reality is most
economies, as I mentioned,

00:08:22.671 --> 00:08:24.340
they're close to a market economy,

00:08:24.340 --> 00:08:26.550
but aren't a perfect market economy.

00:08:26.550 --> 00:08:29.552
You still have the
government very involved

00:08:29.552 --> 00:08:30.949
in certain industries.

00:08:30.949 --> 00:08:32.001
In many economies,

00:08:32.001 --> 00:08:35.682
the government might be
in control of healthcare.

00:08:35.682 --> 00:08:37.305
The government in almost every country

00:08:37.305 --> 00:08:41.205
has significant influence
in things like the military.

00:08:41.205 --> 00:08:43.285
The government, in say the United States

00:08:43.285 --> 00:08:45.337
which is considered a fairly capitalist,

00:08:45.337 --> 00:08:47.164
a fairly strong market economy,

00:08:47.164 --> 00:08:50.286
the government still does
offer social safety nets

00:08:50.286 --> 00:08:52.407
when the inequality gets too extreme.

00:08:52.407 --> 00:08:54.382
If someone isn't able to feed themselves,

00:08:54.382 --> 00:08:55.626
they might get food stamps.

00:08:55.626 --> 00:08:57.964
If someone isn't able to get
healthcare because of poverty,

00:08:57.964 --> 00:08:59.419
they might get Medicaid.

00:08:59.419 --> 00:09:01.686
And so there's actually a spectrum

00:09:01.686 --> 00:09:05.709
between a command economy
and a market economy.

00:09:05.709 --> 00:09:07.757
With most economies
actually falling in this

00:09:07.757 --> 00:09:10.916
inbetween state which
is sometimes referred to

00:09:10.916 --> 00:09:12.499
as a mixed economy.

00:09:13.866 --> 00:09:15.461
For example, as I already mentioned,

00:09:15.461 --> 00:09:17.192
the United States which is considered

00:09:17.192 --> 00:09:19.533
a very capitalist country
with a market economy,

00:09:19.533 --> 00:09:22.846
it still does have some public ownership.

00:09:22.846 --> 00:09:26.130
The government still does
control certain aspects

00:09:26.130 --> 00:09:27.798
of the economy, it still represents

00:09:27.798 --> 00:09:30.602
a fairly large chunk of the economy.

00:09:30.602 --> 00:09:33.272
And so a pure command economy government

00:09:33.272 --> 00:09:34.722
would control everything,

00:09:34.722 --> 00:09:37.063
pure market economy, you
would have very little

00:09:37.063 --> 00:09:38.612
that's controlled by the government,

00:09:38.612 --> 00:09:40.549
but the reality in most of the world,

00:09:40.549 --> 00:09:43.559
things fall in this
mixed economy spectrum.


Adding and subtracting on number line 2

WEBVTT
Kind: captions
Language: en

00:00:00.388 --> 00:00:05.507
- Which number line shows 361 + 544?

00:00:05.507 --> 00:00:09.149
Let's see, in all of them we're starting at 361,

00:00:09.149 --> 00:00:11.660
so now let's add 544.

00:00:11.660 --> 00:00:15.804
This one starts with adding 400, and then 50, and then 4,

00:00:15.804 --> 00:00:20.181
it's adding 454, not 544.

00:00:20.181 --> 00:00:25.186
Now this one adds 500, then 40, and then 4,

00:00:25.186 --> 00:00:28.852
so this is adding 544,

00:00:28.852 --> 00:00:33.132
to 361, we see that it is 905.

00:00:33.132 --> 00:00:36.048
This makes sense, you add 500, you get 861,

00:00:36.048 --> 00:00:39.639
you add 40, you get to 901, and then add the 4,

00:00:39.639 --> 00:00:42.586
you get 905, so definitely go with that choice.

00:00:42.586 --> 00:00:45.442
This one over here, instead of add 500, you add 50,

00:00:45.442 --> 00:00:46.545
which doesn't make sense

00:00:46.545 --> 00:00:50.864
because we're clearly adding 5 hundreds, 4 tens, or 40,

00:00:50.864 --> 00:00:54.919
and then 4 ones, let's do some more examples.

00:00:54.919 --> 00:00:56.606
Which expression can be solved

00:00:56.606 --> 00:00:58.340
with the following number line?

00:00:58.340 --> 00:01:02.844
Let's see, we're starting at 718, and then we're adding 200,

00:01:02.844 --> 00:01:05.642
and then we're adding 40 to get to 958.

00:01:05.642 --> 00:01:10.042
This is really 718 + 240,

00:01:10.042 --> 00:01:13.095
which is this choice right over there.

00:01:13.095 --> 00:01:16.775
This is kind of, strangely fun.

00:01:16.775 --> 00:01:21.059
Which number line shows 585 - 368?

00:01:21.059 --> 00:01:24.344
Let's see, these second two number lines,

00:01:24.344 --> 00:01:28.454
both start at 585, we're starting our computation at 585,

00:01:28.454 --> 00:01:31.379
this one's starting at 558, so it swapped the numbers,

00:01:31.379 --> 00:01:32.482
so this one doesn't make sense,

00:01:32.482 --> 00:01:34.316
but let's look at these two choices.

00:01:34.316 --> 00:01:38.379
We want to subtract 3 hundreds, 6 tens, and 8 ones.

00:01:38.379 --> 00:01:42.060
In both of these, we subtract 3 hundreds,

00:01:42.060 --> 00:01:45.662
and then we want to subtract 6 tens,

00:01:45.662 --> 00:01:48.932
6 tens, and then 8 ones, that's this one over here.

00:01:48.932 --> 00:01:52.612
This one swaps the... instead of having 6 tens and 8 ones,

00:01:52.612 --> 00:01:55.201
it has 8 tens and 6 ones, so it swapped...

00:01:55.201 --> 00:01:58.660
this is subtracting 386, which isn't the original problem,

00:01:58.660 --> 00:02:01.156
so definitely like this third choice.

00:02:01.156 --> 00:02:04.059
Let's do one more example.

00:02:04.059 --> 00:02:05.637
Which expression has been solved

00:02:05.637 --> 00:02:07.135
with the following number line?

00:02:07.135 --> 00:02:11.116
We're starting at 935, and then we're going to subtract 400,

00:02:11.116 --> 00:02:13.089
and then we're going to subtract 20.

00:02:13.089 --> 00:02:16.817
This is 935 - 420.

00:02:16.817 --> 00:02:21.318
935 - 420,

00:02:21.318 --> 00:02:24.080
that choice right over there.


Addition using groups of 10 and 100

WEBVTT
Kind: captions
Language: en

00:00:00.248 --> 00:00:01.602
- [Voiceover] So, let's do some practice problems

00:00:01.602 --> 00:00:03.982
on Khan Academy exercises that make us

00:00:03.982 --> 00:00:05.782
rewrite an addition problem so that we

00:00:05.782 --> 00:00:07.459
can get them to rounder numbers.

00:00:07.459 --> 00:00:12.377
Numbers that might be multiples of 10, or multiples of 100.

00:00:12.377 --> 00:00:15.993
So, let's see here, I have 63 plus 427,

00:00:15.993 --> 00:00:18.628
and that seems kind of hairy, you know,

00:00:18.628 --> 00:00:20.555
it feels like I wanna write those things down,

00:00:20.555 --> 00:00:22.866
but maybe I could take away from one of them,

00:00:22.866 --> 00:00:24.433
and give it to the other one so that they both

00:00:24.433 --> 00:00:26.348
become round numbers, and it's clear

00:00:26.348 --> 00:00:29.970
that 63, if I were to give three away I would get to 60,

00:00:29.970 --> 00:00:32.733
and if I were to give those three to 427,

00:00:32.733 --> 00:00:35.914
it would get to 430, and 60 plus 430

00:00:35.914 --> 00:00:37.551
is a much simpler problem.

00:00:37.551 --> 00:00:39.292
So, let's think about what this question is saying.

00:00:39.292 --> 00:00:41.928
Well, it's just taking a step by step through that process.

00:00:41.928 --> 00:00:45.886
It's just saying 63 plus 427 is going to be equal

00:00:45.886 --> 00:00:49.810
to 60 plus what plus 427?

00:00:49.810 --> 00:00:51.586
So, the 60 plus what, this is gonna

00:00:51.586 --> 00:00:53.838
to be the same thing as 63, cause we have

00:00:53.838 --> 00:00:58.749
427 in both places, so 63 is 60 plus three.

00:00:58.749 --> 00:01:00.119
Well, that makes sense.

00:01:00.119 --> 00:01:02.267
On this next step, they just change the order.

00:01:02.267 --> 00:01:05.366
60 plus three, and then adding 427

00:01:05.366 --> 00:01:07.366
is the same thing as doing 60 plus,

00:01:07.366 --> 00:01:11.131
and then adding the three to the 427 first.

00:01:11.131 --> 00:01:12.374
So, we're just taking this three

00:01:12.374 --> 00:01:15.752
and moving it from the 63 to the 427.

00:01:15.752 --> 00:01:20.233
Well, three plus 427, that's just 430.

00:01:20.233 --> 00:01:22.450
And now this addition problem became a lot simpler.

00:01:22.450 --> 00:01:25.109
60 plus 430, we can do that in our head.

00:01:25.109 --> 00:01:27.291
We're just adding six 10's to this, right over here,

00:01:27.291 --> 00:01:31.470
so it's gonna be 490, and we're done.

00:01:31.470 --> 00:01:34.233
Let's do a couple more examples of this.

00:01:34.233 --> 00:01:38.192
So, over here, we want to, let's see,

00:01:38.192 --> 00:01:40.189
we want to add these two numbers,

00:01:40.189 --> 00:01:42.567
and let's maybe see if we can 'em

00:01:42.567 --> 00:01:44.585
a little bit a rounder number.

00:01:44.585 --> 00:01:46.129
So, over here, we're breaking up

00:01:46.129 --> 00:01:50.064
the 275 into 270 and something.

00:01:50.064 --> 00:01:52.885
Well, that's gonna be 270 plus five.

00:01:52.885 --> 00:01:54.243
Notice, the rest is still the same,

00:01:54.243 --> 00:01:57.552
plus 595, plus 595.

00:01:57.552 --> 00:01:59.433
Now, why are we doing that?

00:01:59.433 --> 00:02:03.124
Well, if we take five from 275 and give it to 595,

00:02:03.124 --> 00:02:04.088
which is what we're doing here,

00:02:04.088 --> 00:02:05.829
the 595 can get to 600.

00:02:05.829 --> 00:02:08.023
It's gonna make the calculation easier.

00:02:08.023 --> 00:02:09.486
So, once again, here we're adding

00:02:09.486 --> 00:02:13.271
the 270 and the five first, and then we're adding the 595,

00:02:13.271 --> 00:02:15.221
but we could change the order in which we do it.

00:02:15.221 --> 00:02:18.088
We could add the five to the 595 first,

00:02:18.088 --> 00:02:19.470
and then add the 270.

00:02:19.470 --> 00:02:21.826
So, this is the same thing as 270 plus,

00:02:21.826 --> 00:02:24.311
five plus 595 is 600, that's the whole reason

00:02:24.311 --> 00:02:27.608
why we took five from the 275 is so

00:02:27.608 --> 00:02:30.080
that we can add it to the 595 and get 600,

00:02:30.080 --> 00:02:31.949
and now we can do this in our head.

00:02:31.949 --> 00:02:34.805
270 plus six 100's, well we're just gonna increase

00:02:34.805 --> 00:02:39.201
our 100's by six, so it's gonna be 870.

00:02:39.201 --> 00:02:42.603
Let's do a couple of more examples here.

00:02:42.603 --> 00:02:43.799
Fill in the blank.

00:02:43.799 --> 00:02:48.802
51 plus 83 is the same thing as blank plus 84.

00:02:48.802 --> 00:02:51.960
Well, they increased 83 by one to 84,

00:02:51.960 --> 00:02:55.059
so we gotta decrease 51 by one.

00:02:55.059 --> 00:02:58.519
So, this is gonna be the same thing as 50 plus 84.

00:02:58.519 --> 00:03:00.469
Now, why would anyone care about this?

00:03:00.469 --> 00:03:01.688
Why would they do that?

00:03:01.688 --> 00:03:03.696
Well, I find this easier to calculate,

00:03:03.696 --> 00:03:04.927
because now I just have to say

00:03:04.927 --> 00:03:08.688
eight 10's plus four ones, plus another five 10's.

00:03:08.688 --> 00:03:12.763
Well, that's gonna be 13 10's and four ones, or 134.

00:03:12.763 --> 00:03:14.922
So, I find this a little bit easier,

00:03:14.922 --> 00:03:17.325
but the important thing is if we add to one number,

00:03:17.325 --> 00:03:19.369
we gotta take away the same amount from the other number

00:03:19.369 --> 00:03:21.330
in order to not change the value

00:03:21.330 --> 00:03:25.312
of this addition problem, of this expression.

00:03:25.312 --> 00:03:27.700
Let's do one more.

00:03:27.700 --> 00:03:32.693
138 plus 710 is the same as blank plus 700.

00:03:32.693 --> 00:03:35.559
So, we had 710, and it's now 700.

00:03:35.559 --> 00:03:37.916
So, we took 10 away from that number,

00:03:37.916 --> 00:03:39.738
so we have to give it to the other number.

00:03:39.738 --> 00:03:42.745
So, 138, we have to add 10 to that,

00:03:42.745 --> 00:03:45.380
so that's gonna be 148.

00:03:45.380 --> 00:03:46.785
Now, why was that useful?

00:03:46.785 --> 00:03:49.455
Well, 148 plus 700, you can do that in your head.

00:03:49.455 --> 00:03:53.042
That's gonna be 848, and that's easier to compute

00:03:53.042 --> 00:03:55.765
than what we just had over here.


Solving 3-digit addition in your head

WEBVTT
Kind: captions
Language: en

00:00:00.261 --> 00:00:01.294
- [Voiceover] What I want to do in this video

00:00:01.294 --> 00:00:04.893
is go over some techniques for doing mental addition.

00:00:04.893 --> 00:00:08.318
Now, if I saw something like 355 plus 480,

00:00:08.318 --> 00:00:10.153
if you have some paper around, you could

00:00:10.153 --> 00:00:13.612
write these numbers down and do your traditional addition,

00:00:13.612 --> 00:00:15.551
but you might want to be able to do these in your head,

00:00:15.551 --> 00:00:17.386
so how do you do it?

00:00:17.386 --> 00:00:18.953
Well, the way to think about it is,

00:00:18.953 --> 00:00:21.855
can we add or subtract from these numbers

00:00:21.855 --> 00:00:23.574
so that one of these numbers might become

00:00:23.574 --> 00:00:26.023
a little bit simpler?

00:00:26.023 --> 00:00:27.626
So for example, I have 480 here.

00:00:27.626 --> 00:00:29.727
What would I have to do to 480

00:00:29.727 --> 00:00:31.120
to, say, get it to 500?

00:00:31.120 --> 00:00:32.560
It's almost 500.

00:00:32.560 --> 00:00:36.020
Well, I could add 20 to 480, but I can't just add 20.

00:00:36.020 --> 00:00:38.051
I would would also have to take away 20,

00:00:38.051 --> 00:00:43.020
so what happens if I take away 20 from 355 and add it

00:00:43.020 --> 00:00:44.251
to 480?

00:00:44.251 --> 00:00:45.807
So what am I talking about?

00:00:45.807 --> 00:00:47.455
So this is going to be the same thing as,

00:00:47.455 --> 00:00:50.416
so let me take away 20 from this.

00:00:50.416 --> 00:00:55.013
So that's going to be 335

00:00:55.013 --> 00:00:57.092
and let me add 20 to this,

00:00:57.092 --> 00:01:00.157
so 480 plus 20 is 500,

00:01:00.157 --> 00:01:03.094
and the whole reason why I took away 20 and I added 20

00:01:03.094 --> 00:01:04.719
is I said, well what do I have to add to 480

00:01:04.719 --> 00:01:06.217
to get to 500?

00:01:06.217 --> 00:01:07.227
And I could have done that in my head.

00:01:07.227 --> 00:01:08.493
Okay, I need to add 20.

00:01:08.493 --> 00:01:11.384
355 minus 20 is 335.

00:01:11.384 --> 00:01:14.623
But now this problem is much, much simpler to compute.

00:01:14.623 --> 00:01:17.386
335 plus 500,

00:01:17.386 --> 00:01:20.254
well, it's going to be three hundreds plus five hundreds,

00:01:20.254 --> 00:01:22.785
it's going to be equal to

00:01:22.785 --> 00:01:27.487
800 and then we have 35.

00:01:27.487 --> 00:01:29.588
835.

00:01:29.588 --> 00:01:32.014
Now, to make it a little bit clearer what we did,

00:01:32.014 --> 00:01:34.186
remember, we wanted to take 20 from here

00:01:34.186 --> 00:01:35.718
and put it over here,

00:01:35.718 --> 00:01:38.690
so we could break up 355,

00:01:38.690 --> 00:01:41.221
we could say that it's going to be equal to,

00:01:41.221 --> 00:01:42.382
it's going to be equal to,

00:01:42.382 --> 00:01:46.411
let's break it up into 335

00:01:46.411 --> 00:01:48.118
and 20, and remember,

00:01:48.118 --> 00:01:49.789
the whole reason why I picked 20 is because

00:01:49.789 --> 00:01:52.251
I'm going to want to add that to 480,

00:01:52.251 --> 00:01:54.561
but I'm just doing it step-by-step here,

00:01:54.561 --> 00:01:59.124
so plus 480,

00:01:59.124 --> 00:02:02.827
and now I can just change the order with which I add,

00:02:02.827 --> 00:02:05.660
so I could say, this is going to be equal to

00:02:05.660 --> 00:02:09.224
335

00:02:09.224 --> 00:02:10.954
plus 20

00:02:10.954 --> 00:02:15.051
plus 480,

00:02:15.051 --> 00:02:17.126
and instead of adding the 335 and 20 first,

00:02:17.126 --> 00:02:19.506
I could add the 20 and 480 first,

00:02:19.506 --> 00:02:24.057
so I could add these two characters first,

00:02:24.057 --> 00:02:25.984
and so then I'm going to be left with,

00:02:25.984 --> 00:02:28.921
this is going to be equal to 335

00:02:28.921 --> 00:02:31.661
plus, what's 20 plus 480?

00:02:31.661 --> 00:02:32.486
Well, that was the whole point.

00:02:32.486 --> 00:02:34.587
I picked 20 so that I can get to 500.

00:02:34.587 --> 00:02:38.407
20 plus 480 is going to be 500

00:02:38.407 --> 00:02:40.555
and then, now, you can add them.

00:02:40.555 --> 00:02:43.887
This is going to be 835.

00:02:43.887 --> 00:02:47.288
So this is just a longer way of saying what I did here.

00:02:47.288 --> 00:02:49.518
I took 20 from 355,

00:02:49.518 --> 00:02:52.316
so I can make the 480 into a 500.

00:02:52.316 --> 00:02:54.858
Let's do a couple more examples of this,

00:02:54.858 --> 00:02:56.658
and remember, the key is just thinking about

00:02:56.658 --> 00:02:59.990
how could I add or take away from one of the numbers

00:02:59.990 --> 00:03:01.395
to make them simpler?

00:03:01.395 --> 00:03:04.123
So over here, I have 18 plus 704.

00:03:04.123 --> 00:03:05.992
So there's a couple ways we could tackle this.

00:03:05.992 --> 00:03:08.987
One way, we could try to get the 704 to be equal

00:03:08.987 --> 00:03:10.984
to 700.

00:03:10.984 --> 00:03:14.990
So, we could say that this is the same thing as

00:03:14.990 --> 00:03:17.625
18 plus,

00:03:17.625 --> 00:03:21.515
I could write 700 plus 4,

00:03:21.515 --> 00:03:23.453
or I could write it as 4

00:03:23.453 --> 00:03:26.553
plus 700, like that,

00:03:26.553 --> 00:03:30.489
and then I could put the parentheses around this first,

00:03:30.489 --> 00:03:33.206
and then I could just switch the order in which I add,

00:03:33.206 --> 00:03:36.480
so this is the same thing as 18

00:03:36.480 --> 00:03:40.155
plus 4

00:03:40.155 --> 00:03:42.855
plus 700,

00:03:42.855 --> 00:03:45.995
and now I could add the 18 and the 4 first.

00:03:45.995 --> 00:03:47.852
Now what's 18 plus 4?

00:03:47.852 --> 00:03:52.392
It's 22, and then I have plus 700,

00:03:52.392 --> 00:03:56.595
and now this is pretty easy to compute,

00:03:56.595 --> 00:03:58.487
and all of these are going to be equal to each other,

00:03:58.487 --> 00:03:59.892
so let me just write it like that.

00:03:59.892 --> 00:04:01.959
22 plus 700, I could do that in my head.

00:04:01.959 --> 00:04:04.455
That's going to be 722.

00:04:04.455 --> 00:04:05.360
Now, I know what you're thinking.

00:04:05.360 --> 00:04:07.253
Hey, this is supposed to be a technique for doing something

00:04:07.253 --> 00:04:07.972
in my head?

00:04:07.972 --> 00:04:10.260
I can't write all this stuff down in my head.

00:04:10.260 --> 00:04:11.792
The reason why I'm writing this down this way

00:04:11.792 --> 00:04:14.161
is that you see what's really going on.

00:04:14.161 --> 00:04:15.658
But you could do it a lot quicker.

00:04:15.658 --> 00:04:18.421
You could just say, hey, let me take 4 from the,

00:04:18.421 --> 00:04:21.394
I'm closing my eyes right now as I'm talking,

00:04:21.394 --> 00:04:23.483
just so I can pretend like I'm doing it in my head.

00:04:23.483 --> 00:04:26.920
Okay, let me add 4 to 18 and take away 4 from 700,

00:04:26.920 --> 00:04:29.892
well that's going to be the same thing if I add 4 to 18,

00:04:29.892 --> 00:04:31.761
this is going to be 22,

00:04:31.761 --> 00:04:34.060
and if I take away 4 from 704,

00:04:34.060 --> 00:04:36.951
that's going to be plus 700.

00:04:36.951 --> 00:04:40.620
22 plus 700 is 722.

00:04:40.620 --> 00:04:43.058
Now, there's another way that you could tackle it.

00:04:43.058 --> 00:04:44.927
You could say, well, let me actually,

00:04:44.927 --> 00:04:47.354
instead of taking away 4, let me add some number,

00:04:47.354 --> 00:04:48.654
some value here, so I get to,

00:04:48.654 --> 00:04:50.361
I don't know, say 710.

00:04:50.361 --> 00:04:52.288
That seems a little less ideal for me,

00:04:52.288 --> 00:04:53.228
but you could do it.

00:04:53.228 --> 00:04:55.620
Well, let's see, if I added 6 to 704,

00:04:55.620 --> 00:04:57.222
I'll get to 710.

00:04:57.222 --> 00:04:59.184
So how about I break off 6 from the 18

00:04:59.184 --> 00:05:01.251
and add it to the 704?

00:05:01.251 --> 00:05:02.423
So I could write it like this.

00:05:02.423 --> 00:05:06.951
I could say that 18 is 12 plus 6

00:05:06.951 --> 00:05:11.405
and then plus 704

00:05:11.405 --> 00:05:16.060
which is the same thing as 12 plus 6

00:05:16.060 --> 00:05:20.727
plus 704,

00:05:20.727 --> 00:05:22.817
and then I could add these first,

00:05:22.817 --> 00:05:27.925
and I could have 12 plus 710

00:05:27.925 --> 00:05:30.317
and I might be able to say,

00:05:30.317 --> 00:05:32.360
Okay, that's going to be 722,

00:05:32.360 --> 00:05:33.951
but this method right over here doesn't seem

00:05:33.951 --> 00:05:36.818
anywhere near as intuitive as doing this,

00:05:36.818 --> 00:05:41.184
as trying to get the 704 to 700

00:05:41.184 --> 00:05:43.854
by taking 4 from here and adding it

00:05:43.854 --> 00:05:45.456
over here.

00:05:45.456 --> 00:05:48.986
Let's do one more example of this.

00:05:48.986 --> 00:05:52.353
So, 275 plus 595.

00:05:52.353 --> 00:05:53.815
Well, my gut reaction is,

00:05:53.815 --> 00:05:57.461
can we get this 595 to be equal to 600?

00:05:57.461 --> 00:05:58.320
So this is going to be equal to,

00:05:58.320 --> 00:06:02.918
so if I add 5 to 595, I get 600.

00:06:02.918 --> 00:06:04.694
Let me do that in the same color.

00:06:04.694 --> 00:06:07.724
So I'm going to add 5 to it to get 600.

00:06:07.724 --> 00:06:10.522
I have trouble switching colors.

00:06:10.522 --> 00:06:12.589
So, 600,

00:06:12.589 --> 00:06:13.610
but I can't just add 5,

00:06:13.610 --> 00:06:15.352
I also have to take 5 away from here,

00:06:15.352 --> 00:06:19.659
so taking 5 from there, that would be 270,

00:06:19.659 --> 00:06:22.620
so this becomes 270 plus 600,

00:06:22.620 --> 00:06:24.559
and this is easier for me to compute in my head.

00:06:24.559 --> 00:06:27.647
This would be two hundreds plus six hundreds

00:06:27.647 --> 00:06:30.120
is equal to eight hundreds.

00:06:30.120 --> 00:06:32.186
I have seven 10's plus zero 10's,

00:06:32.186 --> 00:06:33.823
so 800, and then I have no one's.

00:06:33.823 --> 00:06:35.588
870.

00:06:35.588 --> 00:06:37.492
Now, another way that you could have tried to tackle it

00:06:37.492 --> 00:06:41.056
is like, let's try to get this 275 to,

00:06:41.056 --> 00:06:42.752
let's try to get it to 300.

00:06:42.752 --> 00:06:46.490
So, we could say this is the same thing as 300.

00:06:46.490 --> 00:06:50.321
We could say this is the same thing as 300.

00:06:50.321 --> 00:06:53.084
And to get there, I just added 25,

00:06:53.084 --> 00:06:55.023
but if I add 25 from this number, I'm going to have to

00:06:55.023 --> 00:06:57.519
take 25 from this number.

00:06:57.519 --> 00:07:01.049
So I could subtract 25 from here,

00:07:01.049 --> 00:07:05.554
and then I would get 300 plus 95 minus 25

00:07:05.554 --> 00:07:06.622
is going to be 70,

00:07:06.622 --> 00:07:08.723
so it's going to be 570.

00:07:08.723 --> 00:07:12.218
570, and this is also easy to compute.

00:07:12.218 --> 00:07:15.794
300 plus 570 is 870.

00:07:15.794 --> 00:07:17.326
And there's other ways that you can tackle this

00:07:17.326 --> 00:07:19.439
including the traditional way of writing it down on paper

00:07:19.439 --> 00:07:20.472
and whatnot.

00:07:20.472 --> 00:07:22.794
But hopefully this gives you a little bit of a toolkit

00:07:22.794 --> 00:07:26.376
for being able to solve these things maybe in your head.


Breaking apart 3-digit addition problems

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:04.682
- Mike isn't sure how to add 189 + 608,

00:00:04.682 --> 00:00:06.982
help Mike by choosing an addition problem

00:00:06.982 --> 00:00:10.650
that is the same as 189 + 608.

00:00:10.650 --> 00:00:12.287
Now let's look at these choices.

00:00:12.287 --> 00:00:15.722
Let's just start with this first choice,

00:00:15.722 --> 00:00:19.089
actually all of these choices start with having 1 hundred,

00:00:19.089 --> 00:00:22.340
they all have 1 hundred, so where do we see 1 hundred here?

00:00:22.340 --> 00:00:26.125
Well in 189, we have a 1 in the hundreds place,

00:00:26.125 --> 00:00:30.966
so this right over here, that is one 1 hundred,

00:00:30.966 --> 00:00:35.284
100, and then all of the choices actually have 6 hundreds,

00:00:35.284 --> 00:00:38.894
6 hundreds, 6 hundreds, where are they getting that from?

00:00:38.894 --> 00:00:42.054
Well in 608, the 6 is in the hundreds,

00:00:42.054 --> 00:00:45.385
so that 6, represents 6 hundreds,

00:00:45.385 --> 00:00:48.422
6 hundreds, so that's where they got the 1 hundred,

00:00:48.422 --> 00:00:51.943
and the 6 hundreds from, and then all of them have 8 tens,

00:00:51.943 --> 00:00:53.870
so they're actually all looking pretty similar,

00:00:53.870 --> 00:00:56.564
up to that point, they all have 8 tens, 8 tens, 8 tens,

00:00:56.564 --> 00:00:57.992
where do they get that from?

00:00:57.992 --> 00:01:01.486
Well in 189, this 8 is in the tens place,

00:01:01.486 --> 00:01:04.702
so it represents 8 tens,

00:01:04.702 --> 00:01:08.103
8 tens, and then what else do they have?

00:01:08.103 --> 00:01:09.427
Well this is where they all start

00:01:09.427 --> 00:01:12.810
to be a little bit different, so let's go one by one.

00:01:12.810 --> 00:01:17.251
So this next one then has 9 ones,

00:01:17.251 --> 00:01:19.886
9 ones, where are they getting that from?

00:01:19.886 --> 00:01:23.822
Well in 189, the 9 is in the ones place,

00:01:23.822 --> 00:01:26.968
so it's reasonable to write 9 ones,

00:01:26.968 --> 00:01:30.509
and then finally, they have 8 ones.

00:01:30.509 --> 00:01:32.228
8 ones, where do they get that from?

00:01:32.228 --> 00:01:35.547
Well in 608, the 8 is in the ones place,

00:01:35.547 --> 00:01:39.784
so it's 8 ones, so this first choice is looking quite good.

00:01:39.784 --> 00:01:40.783
Now you might be saying,

00:01:40.783 --> 00:01:43.063
'Okay, we took into account, all of these digits,

00:01:43.063 --> 00:01:45.898
except for this 0 over here, how come we didn't,

00:01:45.898 --> 00:01:48.745
how come there's a 0 here in the tens place?'

00:01:48.745 --> 00:01:50.848
Well that would just be 0 tens,

00:01:50.848 --> 00:01:53.158
0 tens, which is just zero,

00:01:53.158 --> 00:01:55.133
so it's not going to change the value of this,

00:01:55.133 --> 00:01:57.465
so this first choice is indeed the same,

00:01:57.465 --> 00:02:00.762
as the original addition problem.

00:02:00.762 --> 00:02:04.001
If we want to see where the other ones break down,

00:02:04.001 --> 00:02:08.057
this one has 8 tens written twice, we only have one,

00:02:08.057 --> 00:02:10.722
we have an 8 tens here, but this 8 is in ones,

00:02:10.722 --> 00:02:13.683
so this should say 'ones', right over here,

00:02:13.683 --> 00:02:17.038
and this choice, we have the 8 ones,

00:02:17.038 --> 00:02:19.360
but this 9 right over here, this is ones,

00:02:19.360 --> 00:02:21.577
this isn't tens, so this should be 'ones'.

00:02:21.577 --> 00:02:23.899
So that's why we wouldn't pick either of those.

00:02:23.899 --> 00:02:27.877
Let's do another one of these, or related type of problem.

00:02:27.877 --> 00:02:29.969
Which addition problem is the same as

00:02:29.969 --> 00:02:33.939
525 + 379?

00:02:33.939 --> 00:02:35.322
Let's just break it down.

00:02:35.322 --> 00:02:38.898
So we have a 5 in the hundreds place,

00:02:38.898 --> 00:02:41.556
so that would be 500,

00:02:41.556 --> 00:02:44.481
and then we have a 3 in the hundreds place,

00:02:44.481 --> 00:02:48.528
so + 300,

00:02:48.528 --> 00:02:51.738
then we have a 2 in the tens place,

00:02:51.738 --> 00:02:53.641
so it's two 10, or 20,

00:02:53.641 --> 00:02:56.543
and then we have a 7 in the tens place,

00:02:56.543 --> 00:03:01.408
so that's + 70, and then last but not least,

00:03:01.408 --> 00:03:05.660
we have a 5 in the ones place, so that's just going to be 5,

00:03:05.660 --> 00:03:07.839
and then we have a 9 in the ones place,

00:03:07.839 --> 00:03:10.299
so that's just going to be equal to 9.

00:03:10.299 --> 00:03:11.438
So which of these choices

00:03:11.438 --> 00:03:14.259
is the same of what I just wrote over here?

00:03:14.259 --> 00:03:19.053
500 + 300, so this first choice has no 500 or 300,

00:03:19.053 --> 00:03:20.191
so we can rule that out.

00:03:20.191 --> 00:03:24.092
500 + 300 + 20 + 70 + 5 +9,

00:03:24.092 --> 00:03:26.437
that's exactly what I wrote down.

00:03:26.437 --> 00:03:30.221
This last choice breaks down, they wrote 90 instead of 9,

00:03:30.221 --> 00:03:33.797
and then they wrote 5, and then they wrote 7, instead of 70.

00:03:33.797 --> 00:03:37.860
That 7 is in the tens place, it's not in the ones place,

00:03:37.860 --> 00:03:41.077
we ruled that one out as well.


Subtracting 3-digit numbers (no regrouping)

WEBVTT
Kind: captions
Language: en

00:00:00.110 --> 00:00:03.369
We have the number 357.

00:00:03.520 --> 00:00:05.806
So the three is in the hundreds place.

00:00:05.892 --> 00:00:07.726
So that represents three hundreds:

00:00:07.726 --> 00:00:10.478
one hundred, two hundred, three hundreds.

00:00:10.478 --> 00:00:14.553
Three hundreds right over here,

00:00:14.553 --> 00:00:16.050
that's what this three represents,

00:00:16.050 --> 00:00:18.012
'cause it's in the hundreds place.

00:00:18.012 --> 00:00:22.227
Let me write that. This is the hundreds place.

00:00:22.227 --> 00:00:25.826
Then we have a five in the tens place.

00:00:26.736 --> 00:00:31.294
So that represents one, two, three, four, five tens.

00:00:31.294 --> 00:00:33.396
Five tens.

00:00:33.396 --> 00:00:36.176
And then last but not least we have,

00:00:36.176 --> 00:00:38.922
in the ones place, we have seven.

00:00:38.922 --> 00:00:40.756
So that represents seven ones:

00:00:40.756 --> 00:00:45.372
one, two, three, four, five, six, seven. Seven ones.

00:00:45.726 --> 00:00:49.092
Now from that, we want to subtract 251;

00:00:49.092 --> 00:00:51.054
or, we want to subtract two hundreds,

00:00:51.054 --> 00:00:54.015
five tens, and one one.

00:00:54.015 --> 00:00:55.667
So pause this video and think about

00:00:55.667 --> 00:00:57.797
what that is going to be.

00:00:58.267 --> 00:01:00.508
So let's work through this together.

00:01:00.728 --> 00:01:02.542
Let's first focus on the ones.

00:01:02.542 --> 00:01:04.533
We're starting off with seven ones

00:01:04.533 --> 00:01:06.937
in three hundreds, five tens, and seven ones.

00:01:06.937 --> 00:01:09.183
357. Seven ones.

00:01:09.183 --> 00:01:12.472
And we are going to take away

00:01:12.472 --> 00:01:14.225
one one right over here.

00:01:14.225 --> 00:01:15.670
Let me do this in a different color.

00:01:15.670 --> 00:01:19.007
So we're gonna take away one one.

00:01:19.007 --> 00:01:20.650
So let's take away one one.

00:01:20.650 --> 00:01:25.439
And then we're gonna be left with six ones.

00:01:25.689 --> 00:01:27.582
Now let's go to the tens place.

00:01:27.582 --> 00:01:30.882
We start with five tens,

00:01:30.882 --> 00:01:33.575
but we're going to take away,

00:01:33.575 --> 00:01:35.736
we're going to subtract, five tens.

00:01:35.736 --> 00:01:38.424
So let's subtract one ten, two tens,

00:01:38.424 --> 00:01:40.877
three tens, four tens, five tens.

00:01:40.877 --> 00:01:42.605
We've taken all of the tens away.

00:01:42.605 --> 00:01:47.533
So now, we have zero tens.

00:01:47.683 --> 00:01:52.777
And then finally, we started with three hundreds

00:01:53.427 --> 00:01:56.050
but we want to take away two of them.

00:01:56.170 --> 00:01:58.213
We want to take away two of the hundreds.

00:01:58.213 --> 00:01:59.534
So let's do that; let's take away

00:01:59.534 --> 00:02:01.738
one hundred and two hundred.

00:02:01.738 --> 00:02:02.930
And we're just going to be left

00:02:02.930 --> 00:02:04.400
with this hundred right over here.

00:02:04.400 --> 00:02:06.046
We're going to be left with one hundred.

00:02:06.046 --> 00:02:08.820
Three hundreds minus two hundreds

00:02:08.820 --> 00:02:10.638
is one hundred.

00:02:10.638 --> 00:02:12.842
So hopefully this gives you an appreciation

00:02:12.842 --> 00:02:14.982
of what's happening in place value.

00:02:14.982 --> 00:02:17.860
You could just say, "Oh, I'll line up the numbers,

00:02:17.860 --> 00:02:19.821
"and I'll look at the place values, and then

00:02:19.821 --> 00:02:21.841
"over here I have a seven minus one is a six,

00:02:21.841 --> 00:02:23.431
"a five minus five is a zero,

00:02:23.431 --> 00:02:25.601
a three minus two is a one,"

00:02:25.601 --> 00:02:27.873
but I really want you to think about what's going on.

00:02:27.873 --> 00:02:30.598
This is seven ones minus one one,

00:02:30.598 --> 00:02:33.268
which gives us six ones;

00:02:33.268 --> 00:02:35.858
this is five tens minus five tens, so you're

00:02:35.858 --> 00:02:39.420
not gonna have any tens left over, so it's zero tens;

00:02:39.420 --> 00:02:41.972
and three hundreds, three groups of a hundred,

00:02:41.972 --> 00:02:45.273
minus two groups of a hundred is gonna leave us

00:02:45.273 --> 00:02:48.064
only one group of a hundred.

00:02:48.304 --> 00:02:50.802
So we're gonna be left with 106:

00:02:50.802 --> 00:02:56.077
one hundred, zero tens, and six ones.


Adding 3-digit numbers (no regrouping)

WEBVTT
Kind: captions
Language: en

00:00:00.412 --> 00:00:01.769
- [Voiceover] So I have two numbers here

00:00:01.769 --> 00:00:02.966
that I wanna add together.

00:00:02.966 --> 00:00:05.983
The first number is 327,

00:00:05.983 --> 00:00:08.667
and that means three hundreds,

00:00:08.667 --> 00:00:11.589
I have a three in the hundreds place.

00:00:11.589 --> 00:00:13.180
You see them right over here.

00:00:13.180 --> 00:00:15.629
You see the three hundreds,

00:00:15.629 --> 00:00:20.283
each of these big squares have a hundred

00:00:20.283 --> 00:00:22.072
little squares in 'em, so three of them

00:00:22.072 --> 00:00:23.930
is going to be three hundreds.

00:00:23.930 --> 00:00:26.704
And then I have two tens.

00:00:26.704 --> 00:00:28.780
Two tens, the two is in the tens place,

00:00:28.780 --> 00:00:31.521
so this is the two tens right over here.

00:00:31.521 --> 00:00:33.390
And then I have seven ones.

00:00:33.390 --> 00:00:36.385
The seven is in the ones place.

00:00:36.385 --> 00:00:39.089
Seven is in

00:00:39.089 --> 00:00:41.645
the ones place, and you see the seven ones.

00:00:41.645 --> 00:00:43.745
Seven ones, right over there.

00:00:43.745 --> 00:00:45.650
Now, this is three hundreds, two tens,

00:00:45.650 --> 00:00:48.506
and seven ones, or 327.

00:00:48.506 --> 00:00:52.455
Now, to that, I wanna add 251,

00:00:52.455 --> 00:00:53.941
or another way of thinking about it as

00:00:53.941 --> 00:00:56.692
I wanna add two hundreds, because the two

00:00:56.692 --> 00:01:00.462
is in the hundreds place, two hundreds.

00:01:00.872 --> 00:01:02.185
Two hundreds.

00:01:02.185 --> 00:01:04.528
One hundred, two hundred.

00:01:04.948 --> 00:01:07.310
50 is the same thing as five tens,

00:01:07.310 --> 00:01:09.549
we see a five in the tens place.

00:01:09.549 --> 00:01:12.904
Five, let me do that same color.

00:01:12.904 --> 00:01:16.354
Five tens, right over here.

00:01:16.354 --> 00:01:17.950
One ten, two tens, three tens,

00:01:17.950 --> 00:01:19.672
four tens, five tens.

00:01:19.672 --> 00:01:22.164
And then finally, I have a one in the ones place,

00:01:22.164 --> 00:01:24.182
so that just means one one.

00:01:24.182 --> 00:01:26.433
You see that right over there.

00:01:26.433 --> 00:01:29.257
This is one.

00:01:29.257 --> 00:01:30.777
One one.

00:01:30.777 --> 00:01:32.733
Now, pause the video,

00:01:32.733 --> 00:01:36.185
and see if you could add these two numbers together.

00:01:36.185 --> 00:01:37.962
All right, now let's do it together.

00:01:37.962 --> 00:01:41.666
So we have seven ones, and to that we're going

00:01:41.666 --> 00:01:46.177
to add one one, so how many ones is that going to be?

00:01:46.177 --> 00:01:48.929
Well, seven ones plus one one,

00:01:48.929 --> 00:01:50.765
is going to be eight ones.

00:01:50.765 --> 00:01:53.270
You see here, seven, and then you have an eighth one.

00:01:53.270 --> 00:01:56.763
So eight ones, seven plus one is eight.

00:01:56.763 --> 00:01:58.435
Now let's look at the tens.

00:01:58.435 --> 00:02:00.753
I had two tens, and I'm gonna add

00:02:00.753 --> 00:02:03.849
five more tens, you put 'em all together,

00:02:03.849 --> 00:02:07.252
you're going to have seven tens.

00:02:07.252 --> 00:02:09.133
Two tens plus five tens, you're going

00:02:09.133 --> 00:02:12.091
to have seven tens.

00:02:12.091 --> 00:02:13.669
And then, finally,

00:02:13.669 --> 00:02:16.642
if you have three hundreds and you add two more hundreds,

00:02:16.642 --> 00:02:18.709
how many hundreds are you going to have?

00:02:18.709 --> 00:02:20.671
Well, you're going to have five hundreds.

00:02:20.671 --> 00:02:23.989
You see these five groups of a hundred right over here.

00:02:23.989 --> 00:02:26.882
So you're gonna put a five in the hundreds place.

00:02:26.882 --> 00:02:30.132
So 327 plus 251

00:02:30.132 --> 00:02:32.502
is 578.

00:02:32.502 --> 00:02:35.720
Five hundreds, seven tens,

00:02:35.720 --> 00:02:38.522
and eight ones.


Missing numbers in addition and subtraction

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:02.390
- Let's say someone walks up to you on the street

00:00:02.390 --> 00:00:03.150
and says, "Quick!

00:00:03.150 --> 00:00:05.520
"73 plus blank is equal to 57."

00:00:05.520 --> 00:00:07.170
What would blank be?

00:00:07.390 --> 00:00:09.200
Well, there's a couple of ways to think about it.

00:00:09.390 --> 00:00:10.840
Blank is essentially what you have to

00:00:10.840 --> 00:00:12.620
add to 57 to get to 73.

00:00:12.620 --> 00:00:15.270
It's the difference between 57 and 73.

00:00:15.820 --> 00:00:17.620
And so you could think about it as,

00:00:17.620 --> 00:00:18.890
"Well, let me start with 57..."

00:00:18.890 --> 00:00:20.940
And this is actually how I would do it in my head.

00:00:21.320 --> 00:00:22.780
I would start with 57,

00:00:22.780 --> 00:00:24.450
and I would add numbers that get me

00:00:24.450 --> 00:00:27.170
to nice round numbers until I get to 73.

00:00:27.450 --> 00:00:29.820
So I'd start with 57, and I would say, "Let's see.

00:00:29.820 --> 00:00:32.590
I could get to 60 pretty easily by adding three."

00:00:32.590 --> 00:00:35.150
So that's going to get me to 60.

00:00:35.150 --> 00:00:38.990
And from 60 to 73, well, I can imagine I'm gonna add 13.

00:00:38.990 --> 00:00:41.250
That's easy for my brain to compute.

00:00:41.250 --> 00:00:43.790
Or I could say I'm gonna add 10 to get to 70.

00:00:43.790 --> 00:00:44.820
And then I could add another three.

00:00:44.820 --> 00:00:46.720
But my brain can say 60 plus 13.

00:00:46.720 --> 00:00:48.450
Yeah, that's gonna be 73.

00:00:48.450 --> 00:00:51.140
So now this whole thing is 73.

00:00:51.520 --> 00:00:54.700
So what did I add to 57 to actually bridge the gap?

00:00:55.020 --> 00:00:59.170
I added three plus 13.

00:00:59.320 --> 00:01:01.060
So I added 16.

00:01:01.350 --> 00:01:05.660
So 16 plus 57 is equal to 73.

00:01:05.660 --> 00:01:08.190
Now there's other ways that you could try to tackle it.

00:01:08.190 --> 00:01:09.030
You could say, "Look!

00:01:09.030 --> 00:01:12.620
This blank is the difference between 73 and 57."

00:01:12.620 --> 00:01:17.460
So you could write this as 73 minus 57

00:01:20.270 --> 00:01:22.260
is equal to blank.

00:01:24.590 --> 00:01:26.000
And this would get you the same value.

00:01:26.000 --> 00:01:28.090
It would get you 16.

00:01:28.090 --> 00:01:29.900
And there's multiple ways to compute this.

00:01:29.900 --> 00:01:31.290
But this is the way that I would

00:01:31.290 --> 00:01:33.970
actually try to tackle this in my brain.

00:01:34.190 --> 00:01:35.590
Let's do a few more examples.

00:01:35.590 --> 00:01:38.200
This is strangely, strangely fun.

00:01:38.950 --> 00:01:40.650
So let's say that someone,

00:01:40.650 --> 00:01:42.790
another person walks up to you on the street

00:01:42.790 --> 00:01:45.020
and says, "Quick!

00:01:45.020 --> 00:01:48.340
"94 minus blank

00:01:51.220 --> 00:01:53.870
"is equal to 57."

00:02:01.750 --> 00:02:03.990
So this is saying, "What do I have to subtract from 94

00:02:03.990 --> 00:02:05.660
"to get to 57?"

00:02:06.060 --> 00:02:07.420
Well, I could do it in the similar way,

00:02:07.420 --> 00:02:08.620
I'm subtracting this time,

00:02:08.620 --> 00:02:10.350
where I could keep subtracting numbers

00:02:10.350 --> 00:02:12.620
that make the math easy in my head until I get to 57.

00:02:12.620 --> 00:02:15.040
And this is actually how I would do it in my head.

00:02:15.450 --> 00:02:17.220
I would start at 94.

00:02:17.220 --> 00:02:21.650
And then I would subtract four to get to 90.

00:02:21.650 --> 00:02:23.590
So that's 90 right over there.

00:02:23.590 --> 00:02:24.520
And let's see.

00:02:24.520 --> 00:02:29.400
Then I could subtract 20 to get to 60.

00:02:30.020 --> 00:02:32.720
I don't want to get lower than 57.

00:02:33.090 --> 00:02:35.520
So I'm going to subtract 20, 90 minus 20.

00:02:35.520 --> 00:02:37.900
Oh, sorry, I could subtract 30 to get to 60.

00:02:38.290 --> 00:02:40.290
So I could subtract 30 now.

00:02:40.290 --> 00:02:44.970
So 90 minus 30, this is going to be equal to 60.

00:02:45.250 --> 00:02:47.790
And then to go from 60 to 57, pretty straightforward.

00:02:47.790 --> 00:02:49.770
I just have to subtract another three.

00:02:49.820 --> 00:02:53.740
60 minus three, this is going to be 57.

00:02:54.290 --> 00:02:56.190
So how much did I subtract?

00:02:56.190 --> 00:02:59.200
I subtracted four, 30, and three.

00:02:59.580 --> 00:03:00.420
So let's see.

00:03:00.420 --> 00:03:02.950
I subtracted, this is 34, 37.

00:03:02.950 --> 00:03:08.100
I subtracted 37, just like that.

00:03:08.450 --> 00:03:09.870
Now another way you could have done it,

00:03:09.870 --> 00:03:10.850
you could have said, "Look!

00:03:10.850 --> 00:03:13.750
"37 is the difference between 94 and 57."

00:03:13.750 --> 00:03:15.390
So you could have thought about it.

00:03:15.390 --> 00:03:19.600
Well, if I start at 57, what do I have to add

00:03:21.890 --> 00:03:24.940
to get to 94?

00:03:26.750 --> 00:03:28.020
And you could have tackled it exactly

00:03:28.020 --> 00:03:30.210
the way that we did the last problem,

00:03:30.210 --> 00:03:32.090
where you could, here I'm gonna add three.

00:03:34.170 --> 00:03:37.940
If I add three, I get to 60; then if I add 30, I get to 90.

00:03:37.940 --> 00:03:39.750
And then if I add another four, I get to 94.

00:03:39.750 --> 00:03:41.880
So I added 37.

00:03:43.850 --> 00:03:45.610
Now there are clearly other ways to do this,

00:03:45.610 --> 00:03:48.450
especially if you have paper around.

00:03:48.450 --> 00:03:50.750
But this is actually how my brain would tackle it

00:03:50.750 --> 00:03:52.850
if someone walked up to me in the street

00:03:52.850 --> 00:03:55.370
and asked me to solve these problems.

00:03:55.520 --> 00:03:56.910
Now let's do one more.

00:03:56.910 --> 00:03:58.550
Let's do one more of these.

00:03:58.550 --> 00:04:00.620
Let's see where I have a clear space.

00:04:00.620 --> 00:04:03.720
So let's say that someone asks,

00:04:03.720 --> 00:04:07.600
"You have 36 is equal to blank

00:04:09.490 --> 00:04:11.800
"minus 41."

00:04:15.810 --> 00:04:18.070
So the way I think about this is,

00:04:18.070 --> 00:04:19.390
they're saying that 36 is the difference

00:04:19.390 --> 00:04:20.990
between blank and 41.

00:04:20.990 --> 00:04:24.010
Or another way to think about it is 36 plus 41

00:04:31.020 --> 00:04:33.170
is going to be equal to blank.

00:04:36.520 --> 00:04:38.600
You could draw this on a number line.

00:04:38.600 --> 00:04:40.390
If these two statements being the same

00:04:40.390 --> 00:04:42.150
doesn't make full sense,

00:04:42.150 --> 00:04:43.940
you do a number line right over here.

00:04:44.440 --> 00:04:45.620
And then they're saying, "OK,

00:04:45.620 --> 00:04:48.020
"we're gonna start at some mystery number.

00:04:48.020 --> 00:04:50.390
"We're starting at some mystery number.

00:04:50.390 --> 00:04:51.650
"That's our blank.

00:04:51.650 --> 00:04:54.050
"We're gonna subtract 41.

00:04:54.050 --> 00:04:55.970
"We're subtracting 41,

00:04:59.150 --> 00:05:01.140
"and we're getting to 36."

00:05:05.150 --> 00:05:07.020
So this is the exact same thing as saying,

00:05:07.020 --> 00:05:10.940
"If we start at 36, and if we were to add 41,

00:05:14.750 --> 00:05:17.620
"we get back to our blank right over here."

00:05:17.620 --> 00:05:20.190
So what's 36 plus 41?

00:05:22.820 --> 00:05:24.000
Well, let's add our ones first.

00:05:24.000 --> 00:05:27.000
So we have six ones plus one one.

00:05:27.420 --> 00:05:29.620
That is seven ones.

00:05:29.620 --> 00:05:34.140
And then we have three tens plus four tens,

00:05:34.690 --> 00:05:37.610
which is seven tens, 77.

00:05:37.610 --> 00:05:40.810
So this is 77.

00:05:40.810 --> 00:05:42.350
And we're done.


Addition and subtraction with number lines

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.430
- [Voiceover] Which number line shows 47 plus 22?

00:00:04.430 --> 00:00:06.309
Let's see, in this first number line we're

00:00:06.309 --> 00:00:11.184
starting at 47, then we add 20 to get to 67.

00:00:11.184 --> 00:00:14.350
And then we add two to get to 69.

00:00:14.350 --> 00:00:16.194
So if you add 20, and then you're adding two,

00:00:16.194 --> 00:00:17.706
you're adding 22.

00:00:17.706 --> 00:00:19.477
So this one seems right.

00:00:19.477 --> 00:00:22.026
So let's see why these choices don't seem right.

00:00:22.026 --> 00:00:24.443
So this one, we add two to get to 49

00:00:24.443 --> 00:00:26.950
and then two more to get to 51.

00:00:26.950 --> 00:00:28.560
Well this isn't adding 22.

00:00:28.560 --> 00:00:31.389
This adding two plus two, this is adding four.

00:00:31.389 --> 00:00:34.287
To add 22, you have to add 20 and then two.

00:00:34.287 --> 00:00:36.342
And this choice right over here,

00:00:36.342 --> 00:00:39.163
we are adding 20 and then we're adding two,

00:00:39.163 --> 00:00:43.170
but we're not starting at 47, we're starting at 74.

00:00:43.170 --> 00:00:46.303
So this is 74 plus 22, so I definitely feel good

00:00:46.303 --> 00:00:48.806
about that choice right over there.

00:00:48.806 --> 00:00:52.047
Let's do a few more examples.

00:00:52.047 --> 00:00:56.190
Which number line shows 93 minus 76?

00:00:56.190 --> 00:00:59.877
So let's see, in all of these choices, we're starting at 93.

00:00:59.877 --> 00:01:04.296
Here, in this first choice we subtract seven and then six.

00:01:04.296 --> 00:01:07.696
But it's subtracting 76, you want to subtract 70.

00:01:07.696 --> 00:01:09.576
And then you want to subtract six, not just seven.

00:01:09.576 --> 00:01:11.804
So I don't like this first choice.

00:01:11.804 --> 00:01:16.253
Here we subtract 60, and then we subtract seven.

00:01:16.253 --> 00:01:19.073
But that's subtracting 67, not 76.

00:01:19.073 --> 00:01:21.279
We gotta subtract seven tens.

00:01:21.279 --> 00:01:23.664
Seven tens, and then six ones.

00:01:23.664 --> 00:01:27.110
So here we're subtracting seven tens, we're subtracting 70.

00:01:27.110 --> 00:01:29.794
And then we are subtracting six ones.

00:01:29.794 --> 00:01:33.493
So that's definitely going to be the right choice.

00:01:33.493 --> 00:01:35.580
So let's see, maybe

00:01:35.580 --> 00:01:38.540
maybe one more just for fun.

00:01:38.540 --> 00:01:41.600
Which number line shows 12 plus 87?

00:01:41.600 --> 00:01:43.824
So all of these start off at 12,

00:01:43.824 --> 00:01:46.297
and we're gonna want to add 80 and then add 7.

00:01:46.297 --> 00:01:48.570
Eight tens and seven ones.

00:01:48.570 --> 00:01:50.798
And let's see, this one adds 80

00:01:50.798 --> 00:01:52.604
and then we add seven, so yep.

00:01:52.604 --> 00:01:56.077
Feel good about that one right over there.


Strategies for adding 2-digit numbers

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.450
- So let's do a bunch of examples

00:00:01.450 --> 00:00:03.169
from the Khan Academy Exercises

00:00:03.169 --> 00:00:05.191
to get comfortable with different ways

00:00:05.191 --> 00:00:06.705
of adding numbers.

00:00:06.705 --> 00:00:08.509
So this says, select any strategy

00:00:08.509 --> 00:00:10.786
that can be used to add 78 plus 9.

00:00:10.786 --> 00:00:13.164
Select all that apply.

00:00:13.164 --> 00:00:16.444
So this first choice is 77 plus 10.

00:00:16.444 --> 00:00:18.284
Well, does that make sense?

00:00:18.284 --> 00:00:21.104
So instead of 78 they wrote a number one less

00:00:21.104 --> 00:00:22.962
which is 77, and instead of nine

00:00:22.962 --> 00:00:25.644
they wrote a number which is one more, that's 10.

00:00:25.644 --> 00:00:26.765
So that makes sense.

00:00:26.765 --> 00:00:30.844
If you take away one from 78 but add that one to nine,

00:00:30.844 --> 00:00:32.675
and you're going to get the same sum.

00:00:32.675 --> 00:00:35.443
It's still going to be the same thing as 78 plus nine.

00:00:35.443 --> 00:00:39.429
So this is legitimate, this is okay right over here.

00:00:39.429 --> 00:00:41.142
And this actually might be a good thing to do

00:00:41.142 --> 00:00:43.200
because this is easier to compute

00:00:43.200 --> 00:00:47.806
because you now are adding 77 plus a nice, round 10 here.

00:00:47.806 --> 00:00:49.919
So you're just going to increase your 10s by one,

00:00:49.919 --> 00:00:51.415
so this is going to be 87.

00:00:51.415 --> 00:00:54.423
A little bit easier to compute than this, right over here.

00:00:54.423 --> 00:00:57.143
Now what do they do here? Add seven and eight.

00:00:57.143 --> 00:01:00.854
So why would I add seven 10s plus eight ones?

00:01:00.854 --> 00:01:04.744
They're treating the seven like as if it's seven ones,

00:01:04.744 --> 00:01:06.430
so this doesn't make sense.

00:01:06.430 --> 00:01:09.474
So I would only choose this first choice.

00:01:09.474 --> 00:01:13.267
Let's do a few more examples.

00:01:13.267 --> 00:01:16.782
Select any strategy that can be used to add 35 plus 15.

00:01:16.782 --> 00:01:18.442
Select all that apply.

00:01:18.442 --> 00:01:22.993
So here they say add 35 plus five then add 10,

00:01:22.993 --> 00:01:26.406
so they've broken up the fifteen into five and 10

00:01:26.406 --> 00:01:28.705
which makes sense, 15 is five plus 10.

00:01:28.705 --> 00:01:31.823
And this is a decent strategy because this gets us to 40,

00:01:31.823 --> 00:01:32.938
which we can do in our head,

00:01:32.938 --> 00:01:35.381
and then we add 10, that gets us to 50.

00:01:35.381 --> 00:01:37.754
So that one definitely works out.

00:01:37.754 --> 00:01:40.202
Now what are they doing over here? 30 plus 20.

00:01:40.202 --> 00:01:42.624
So instead of writing 35 they wrote 30,

00:01:42.624 --> 00:01:44.357
so this is five less,

00:01:44.357 --> 00:01:47.526
and then they increased the second number 15 by five.

00:01:47.526 --> 00:01:49.780
So you take away five from 35, you get 30.

00:01:49.780 --> 00:01:52.120
You add that five to 15, you get 20.

00:01:52.120 --> 00:01:54.932
Yeah, that makes sense too and this is also a good strategy

00:01:54.932 --> 00:01:57.774
because 30 plus 20, that's three 10s plus two 10s,

00:01:57.774 --> 00:01:59.957
that's going to be five 10s, or 50.

00:01:59.957 --> 00:02:01.674
Very easy to compute, and we got that

00:02:01.674 --> 00:02:06.203
by taking a five from the 35 and giving it to the 15.

00:02:06.203 --> 00:02:07.723
So that makes a lot of sense.

00:02:07.723 --> 00:02:10.672
Let's do one more.

00:02:10.672 --> 00:02:14.965
Select any strategy that can be used to add 41 to 52.

00:02:14.965 --> 00:02:16.395
So, select all that apply.

00:02:16.395 --> 00:02:21.299
Add 40 plus 52 then add one.

00:02:21.299 --> 00:02:23.495
So let's see, 41 plus 52,

00:02:23.495 --> 00:02:26.018
yeah so 40,

00:02:26.018 --> 00:02:28.638
we've taken the one out of the 41,

00:02:28.638 --> 00:02:30.404
so we're going to have to deal with it later.

00:02:30.404 --> 00:02:32.202
52 hasn't changed, so yeah,

00:02:32.202 --> 00:02:34.041
41 plus 52 is the same thing

00:02:34.041 --> 00:02:36.777
as 40 plus 52, plus one.

00:02:36.777 --> 00:02:38.496
We're just adding this one later,

00:02:38.496 --> 00:02:39.936
and the reason why that might be easier

00:02:39.936 --> 00:02:42.176
is you could just think in terms of the 10's place first,

00:02:42.176 --> 00:02:46.030
40 plus 52 is 92, then you add one, is 93.

00:02:46.030 --> 00:02:50.017
Now let's think about this one. 43 plus 50.

00:02:50.017 --> 00:02:53.750
See, they added two to 41 and they took away two from 50.

00:02:53.750 --> 00:02:55.096
Yeah, that makes sense.

00:02:55.096 --> 00:02:58.951
You added two here and you took away two here,

00:02:58.951 --> 00:03:00.320
so that's going to give us the same value.

00:03:00.320 --> 00:03:03.691
It's still going to get us to 93

00:03:03.691 --> 00:03:05.371
and this is useful because it makes the,

00:03:05.371 --> 00:03:08.331
it turns the 52 into a nice, round number of 50

00:03:08.331 --> 00:03:11.419
by taking two away from it and giving it to the 41,

00:03:11.419 --> 00:03:14.107
turning it into 43.


Subtracting a 1-digit number with regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.329 --> 00:00:04.024
- So we have the number 35, which is 3 tens,

00:00:04.024 --> 00:00:06.459
because we have a 3 in the tens place,

00:00:06.459 --> 00:00:09.013
so we have 3 tens,

00:00:09.013 --> 00:00:11.597
and we have a 5 in the ones place.

00:00:11.597 --> 00:00:14.005
So, this is the ones place,

00:00:14.005 --> 00:00:16.814
I'll do the ones place in that same purple color.

00:00:16.814 --> 00:00:18.370
This is the ones place,

00:00:18.370 --> 00:00:21.272
and we see it represented here as 5,

00:00:21.272 --> 00:00:22.828
as 5 ones.

00:00:22.828 --> 00:00:24.961
So 35 is the same thing as

00:00:24.961 --> 00:00:27.651
3 tens and 5 ones.

00:00:27.651 --> 00:00:30.119
3 tens and 5 ones.

00:00:30.119 --> 00:00:33.393
Now, what we wanna do with this video is subtract 8 from 35,

00:00:33.393 --> 00:00:34.883
and I encourage you to pause the video and

00:00:34.883 --> 00:00:37.795
try to figure out what 35 minus 8 is.

00:00:38.385 --> 00:00:40.126
Alright, so let's think about it.

00:00:40.126 --> 00:00:44.375
We have 5 ones, and we wanna take away 8 ones from there.

00:00:44.375 --> 00:00:47.255
Well, we don't know how to take 8 ones if we only have 5.

00:00:47.255 --> 00:00:49.414
We could take away 5, but then we're all out of ones,

00:00:49.414 --> 00:00:51.527
so how do we take away 8?

00:00:51.527 --> 00:00:53.617
Well, this is where regrouping is valuable,

00:00:53.617 --> 00:00:56.426
because we don't only have 5 squares.

00:00:56.426 --> 00:00:58.864
We actually have 35 squares.

00:00:58.864 --> 00:01:00.675
We have 3 groups of ten squares,

00:01:00.675 --> 00:01:02.765
and then we have the 5 ones.

00:01:02.765 --> 00:01:05.574
So what if we were to take one of these groups of ten,

00:01:05.574 --> 00:01:07.385
right now it's in the tens place,

00:01:07.385 --> 00:01:09.916
and we were to regroup it into the ones place?

00:01:09.916 --> 00:01:12.610
So let's say we were to take this group right over here,

00:01:12.610 --> 00:01:15.257
if we were to take this group of 10,

00:01:15.257 --> 00:01:18.136
and instead of expressing it in the tens place,

00:01:18.136 --> 00:01:20.040
let's put it in the ones place.

00:01:20.040 --> 00:01:21.479
So let's move it over here,

00:01:21.479 --> 00:01:23.012
and in the ones place, it would be,

00:01:23.012 --> 00:01:25.961
so 1 ten is the same thing as 10 ones,

00:01:25.961 --> 00:01:30.744
1, 2, 3, 4, 5, 6,

00:01:30.744 --> 00:01:35.016
7, 8, 9, and 10.

00:01:35.016 --> 00:01:36.920
So what just happened there?

00:01:36.920 --> 00:01:38.801
I took away one of the tens,

00:01:38.801 --> 00:01:41.680
so now I only have 2 tens,

00:01:41.680 --> 00:01:44.745
and I took those 10 and put them in the ones place.

00:01:44.745 --> 00:01:46.718
So I have the original 5 ones,

00:01:46.718 --> 00:01:48.692
and then I have the new 10

00:01:48.692 --> 00:01:50.457
that I just took from the tens place.

00:01:50.457 --> 00:01:52.686
So 5 plus 10 is going to be 15,

00:01:52.686 --> 00:01:57.167
so I now have 15 ones.

00:01:57.167 --> 00:01:59.837
And now I can take away 8.

00:01:59.837 --> 00:02:02.647
So, if I take away 8 now,

00:02:02.647 --> 00:02:05.038
I would take away 1, 2, 3, 4,

00:02:05.038 --> 00:02:08.010
5, 6, 7, 8, just to be clear.

00:02:08.010 --> 00:02:10.773
I'm taking away 8.

00:02:10.773 --> 00:02:12.538
And so what am I going to be left with?

00:02:12.538 --> 00:02:15.580
Well, I'm gonna be left with 2 tens,

00:02:15.580 --> 00:02:18.296
I'm gonna be left with 2 tens,

00:02:18.296 --> 00:02:19.984
and how many ones?

00:02:19.984 --> 00:02:24.589
I have 1, 2, 3, 4, 5, 6, 7 ones.

00:02:24.589 --> 00:02:27.609
7 ones right over here.

00:02:27.609 --> 00:02:29.246
Now how do we do it if I didn't wanna

00:02:29.246 --> 00:02:31.243
draw all of these things?

00:02:31.243 --> 00:02:33.021
Well, you would've said: Alright, I have

00:02:33.021 --> 00:02:35.831
3 tens and 5 ones, this is where we started,

00:02:35.831 --> 00:02:37.953
but I took one of those tens,

00:02:37.953 --> 00:02:41.442
I took one of those tens away so I only have 2 tens now,

00:02:41.442 --> 00:02:45.151
and I took that group of 10 and put it in the ones place,

00:02:45.151 --> 00:02:46.660
so it's 10 ones.

00:02:46.660 --> 00:02:49.145
So the 5 ones, plus another 10 ones,

00:02:49.145 --> 00:02:51.420
give us 15 ones.

00:02:51.420 --> 00:02:52.953
15 ones.

00:02:52.953 --> 00:02:57.132
And so you can say I have 15 ones minus 8 ones is 7 ones,

00:02:57.132 --> 00:03:00.383
which we see over here, 1, 2, 3, 4, 5, 6, 7,

00:03:00.383 --> 00:03:03.308
so after doing the subtraction, we have 7 ones,

00:03:03.308 --> 00:03:06.396
and then I'll have 2 tens left in the tens place.

00:03:06.396 --> 00:03:10.079
So 2 tens, 7 ones. 27.


Subtracting two-digit numbers without regrouping (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.191 --> 00:00:02.223
- I would like you to pause the video

00:00:02.223 --> 00:00:06.286
and think about what 64 minus 31 is.

00:00:06.286 --> 00:00:09.456
Alright, now let's think about this together.

00:00:09.456 --> 00:00:13.090
So what does 64 actually mean?

00:00:13.090 --> 00:00:15.818
Well we can use place value to think about that.

00:00:15.818 --> 00:00:18.280
The six is in the tens place

00:00:18.280 --> 00:00:21.828
and the four is in the ones place.

00:00:21.828 --> 00:00:25.826
So the six over here, that means six tens.

00:00:25.826 --> 00:00:27.927
It means six groups of ten.

00:00:27.927 --> 00:00:31.596
And so, we can visualize that right over here.

00:00:31.596 --> 00:00:35.393
Six tens, one, two, three, four, five, six

00:00:35.393 --> 00:00:37.157
groups of ten.

00:00:37.157 --> 00:00:38.260
Or another way to think about it,

00:00:38.260 --> 00:00:41.650
six tens, these are 60 boxes right over here

00:00:41.650 --> 00:00:43.125
in groups of ten.

00:00:43.125 --> 00:00:44.994
And then we have four ones.

00:00:44.994 --> 00:00:49.893
Well this is four ones right over here.

00:00:49.893 --> 00:00:51.484
Four ones.

00:00:51.484 --> 00:00:54.352
So this is 64.

00:00:54.352 --> 00:00:55.907
Six tens, four ones.

00:00:55.907 --> 00:00:58.357
Altogether I have 64 boxes.

00:00:58.357 --> 00:01:00.284
Sixty of these blue ones

00:01:00.284 --> 00:01:01.887
and I have four ones.

00:01:01.887 --> 00:01:03.884
Four of these individual ones.

00:01:03.884 --> 00:01:06.519
Now I want to take away 31.

00:01:06.519 --> 00:01:08.226
Well what is 31?

00:01:08.226 --> 00:01:11.384
Thirty-one is three tens and one one.

00:01:11.384 --> 00:01:14.646
So let's first take away the one one.

00:01:14.646 --> 00:01:15.761
We're going to take this away,

00:01:15.761 --> 00:01:16.550
we're subtracting.

00:01:16.550 --> 00:01:17.850
We're subtracting three tens

00:01:17.850 --> 00:01:20.079
and we're subtracting one one.

00:01:20.079 --> 00:01:22.448
So let's take away that one one.

00:01:22.448 --> 00:01:24.921
And then how many ones are we left with?

00:01:24.921 --> 00:01:27.545
Well now we're left with three ones.

00:01:27.545 --> 00:01:30.354
So four ones minus one one

00:01:30.354 --> 00:01:32.583
is three ones.

00:01:32.583 --> 00:01:37.483
Now let's think about the tens.

00:01:37.483 --> 00:01:40.116
So I had six tens.

00:01:40.116 --> 00:01:42.243
I'm going to take away three of them.

00:01:42.243 --> 00:01:43.950
I'm going to take away three of the tens.

00:01:43.950 --> 00:01:44.658
So let's do that.

00:01:44.658 --> 00:01:47.176
Let's take away, that's one ten,

00:01:47.176 --> 00:01:49.923
two tens and then three tens.

00:01:49.923 --> 00:01:52.024
And so what am I going to have left over?

00:01:52.024 --> 00:01:53.452
Well I'm going to have left over,

00:01:53.452 --> 00:01:55.054
just three tens.

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Just these three, these three right over here.

00:01:58.653 --> 00:02:03.193
So six tens minus three tens is three tens.

00:02:03.193 --> 00:02:07.686
And so 64 minus 31 is 33.

00:02:07.686 --> 00:02:09.787
Three tens and three ones.

00:02:09.787 --> 00:02:12.086
Three tens, this is a group of ten,

00:02:12.086 --> 00:02:13.827
another group of ten, another group of ten.

00:02:13.827 --> 00:02:16.219
Three tens and then three ones.

00:02:16.219 --> 00:02:19.267
One, two, three.