Second Law of Thermodynamics and entropy

WEBVTT
Kind: captions
Language: en

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- [Voiceover] The Second
Law of Thermodynamics,

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one statement of it is

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that the entropy of the
universe only increases.

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And, I put an exclamation mark here,

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because it seems like a
very profound statement.

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And, on a lot of levels, it is.

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And, just to get us into
the right frame of mind,

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I have this image here
from the Hubble telescope

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of the night sky.

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And, each of these dots,

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these are not stars.

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These are galaxies.

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That's a galaxy.

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That's a galaxy there.

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That's a galaxy.

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So, hopefully this gets
you into little bit more

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of a cosmological scale.

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But, let's think about what
this is actually telling us.

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The entropy of the
universe only increases.

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So, entropy, we can define
that as the disorder

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of a system.

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And, we're really talking about the number

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of states that a system could take on.

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And then, we're saying the universe.

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But, we could also say the entropy

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of a closed system only increases.

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A system that is fully contained,

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that's not interacting
with its surroundings,

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because the universe is
the ultimate closed system.

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There's nothing for it to,
outside of it to interact

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with thermodynamically.

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And, I'll do a quick review
of open and closed systems,

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just so we make sure we understand that.

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So, if I had a campfire,

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so I have some logs and I had my,

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the flame going right over here.

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So, that's the campfire.

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If I were to just look
at the logs and the fire,

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that's going to be an open system.

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Because, it's clearly
interacting thermodynamically

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with its surroundings.

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It's releasing heat.

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It's warming up the air
molecules around it.

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It's releasing light
out into the universe.

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There could be interactions from the rest

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of the universe into the system.

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So, it isn't isolated from
the rest of everything else.

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But, a closed system, it is isolated.

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And, there are, it's very hard

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to create a true closed
system in our everyday life.

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But, we can approximate it.

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And, the one that you've
probably experienced

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in the not too distant
past is an ice cooler.

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And, an ice cooler,
we're at least attempting

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to thermodynamically isolate,

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isolate the inside of the cooler

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from the outside, from
the rest of the universe.

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So, this is, and the way we
do it is we have some type

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of an insulating material.

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Maybe some styrofoam.

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And, we could put,

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you know, we'd use it to maybe store ice.

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But, it's not a perfect closed system,

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because eventually, the heat from the rest

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of the universe will warm
up the walls of the cooler.

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And eventually, that heat will warm up,

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will be transferred to the ice,

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and it will warm it up.

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And, it will melt it.

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So, it's not a perfect closed system,

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but it's a good approximation,

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because we're at least attempting

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to isolate it thermodynamically

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from the rest of the universe.

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And, I can even make
a little cover of this

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to show that we really
wanted to isolate it.

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And, in research labs,

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you'll see things that are
much better approximations

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of closed systems.

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But, even those at some level
are, they're going to interact

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with the rest of the universe.

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The ultimate closed system,

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so this is a closed system,

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is really the universe.

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Nothing to interact with
outside of it thermodynamically.

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So, let's think a little
bit about this definition.

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The entropy of the
universe only increases.

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Why does this make intuitive sets?

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Well, the best example I can think

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of is just straight up diffusion.

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So, if I were to have, let's
say I have a container.

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So, I have a container,

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and I'll make it a,

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I'm gonna make it a closed container.

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We'll say this is some type

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of theoretical ideal closed system here.

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Now, let's say I had some ideal gas.

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So, I had some ideal gas
molecules right over here.

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They have some average temperature,

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but that means they all each
have their own individual,

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their own individual kinetic energy.

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They're all bouncing
around in different ways.

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What's going to happen over time?

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Well, over time, the
ones on the left here,

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they're gonna bounce off this wall.

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And then, they're eventually gonna go

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in this direction.

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And so, over time, you're
gonna have a situation

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where the system is going to
look something more like this.

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So, the system is going
to look more like this,

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where instance, let's see,
this is six particles.

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These six particles are gonna diffuse

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throughout the container.

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So, they're gonna diffuse
throughout the container.

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They're going to take up more
of the space of the container.

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Now, what just happened in that process?

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Well, when you knew that
the particles were confined

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to this little section of the container,

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there were fewer possible states.

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You had lower entropy
than when you are here,

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when you know that it's
filled up the container.

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There's more possible locations,

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more possible orientations for it.

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And so, you are going to have more states.

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You have higher entropy,

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higher, higher,

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higher, entropy.

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And, in general, these processes
where you have the entropy

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increasing, we call these
irreversible processes,

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irreversible, irreversible processes.

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And, why is it irreversible?

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Well, there's some probability

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that these molecules
might just gather back

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into this corner of it.

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But, it's very, very low probability.

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And, this is when we're dealing

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with six molecules.

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But, in real systems, we'd
be dealing with much larger

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than six molecules.

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We'll be dealing with millions of millions

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of millions of millions of molecules.

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So, things with, between 20
and 30 zero's of molecules.

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And there,

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it's very unlikely that
they just all bump together

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in the right way to start
taking a smaller volume,

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when they could actually
fill the container.

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And so, that's why you don't see,

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that's why you don't see
smoke just naturally turn

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into to some type of shaped particle,

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or take up less space,

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as opposed to filling its container.

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So, this is irreversible,

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because you went from,
you went from fewer number

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of potential states,

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as a smaller volume,

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to a higher number of potential states.

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And, the universe is
constantly doing this.

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That's why the entropy of the
universe is only increasing.

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Now, there's some processes

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that it feels like the entropy
isn't increasing that much.

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So, if you were to take one
billiard ball right over here,

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and you were to roll
it, you were to roll it

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into another billiard
ball right over here,

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and transfer the momentum to that one,

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it feels like that could
go the other way around.

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Like, that other billiard
ball could hit this one

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and go backwards.

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And, at a macro level,

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it feels like this is
a reversible process,

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and people will tend to
call this reversible.

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But, if you really were to
go on a microscopic level,

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and it looks like the entropy
isn't increasing that much,

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but if you were to look at
it on a microscopic level,

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and just to be clear, the entropy,

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you know, when this ball is moving

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and this is stationery,

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going to a state where this is moving

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and this is stationery,

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it doesn't look like the
entropy is increasing that much.

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And so, that's why they tend
to call this irreversible,

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because you tend to observe things

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where maybe this one,
it could go backwards.

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This could hit this one

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and then, this one could go,

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you can kind of run the film in rewind.

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But, even there, if you were
to look at a microscopic level,

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you would see that some
heat is being generated,

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and that some molecules in
the ball are getting excited

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as they collide, and as they
have friction with the air,

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and as they, roll on the ground over here.

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And, you're never going
to get those molecules

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to go back into the state
that they were before,

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that you actually do have
the entropy increasing

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in the system.

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So, even when in our every day lives,

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in thermodynamics,

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people talk about reversible processes.

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They're only approximately reversible,

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and that the entropy's only
increasing a little bit.

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It's not like there's
zero increase in entropy.

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Irreversible reactions,
these are the ones.

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Diffusion is a very obvious one,

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where it's very clear
that you have an increase

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in entropy, and it feels like it's a very,

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very low probability, or
almost zero probability

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of this thing ever going
back to where it was.

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And, you won't observe it,

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because we were talking
about that many molecules.

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Something with 20 or
30 zero's of molecules.

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The odds of all of them
just doing the right thing,

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you could wait around a very long time

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and never actually observe that happening.

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And so, hopefully this makes sense

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that the disorder in this way,

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the number of states only increases

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as you have more and more interactions.

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And, a lot of that is coming from heat.

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Everything you're doing right now,

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when I'm making this video,

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my body is generating heat.

00:08:01.623 --> 00:08:03.480
That heat is dissipating
into the universe.

00:08:03.480 --> 00:08:05.500
That is adding to the number of states

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that the universe can actually take on.

00:08:07.938 --> 00:08:09.064
As I move my hands up,

00:08:09.064 --> 00:08:13.058
my little digital pencil that
I'm using is causing friction.

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That's releasing heat into the universe.

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My computer is running and releasing heat

00:08:16.622 --> 00:08:17.481
into the universe.

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You watching this, releasing
heat into the universe.

00:08:19.338 --> 00:08:21.962
The electrons traveling on
the wire to your computer,

00:08:21.962 --> 00:08:23.355
releasing heat into the universe.

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And, all of that is increasing
the number of states.

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So, if you're thinking on a
molecular level of everything.


Counting 1 exercise

WEBVTT
Kind: captions
Language: en

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Brandon had to do
problems 45 through 75

00:00:04.820 --> 00:00:06.230
for homework tonight.

00:00:06.230 --> 00:00:07.960
If Brandon did all
of the problems

00:00:07.960 --> 00:00:12.010
he was assigned, how
many problems did he do?

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So there's a
temptation here to just

00:00:13.510 --> 00:00:16.280
say, well, what is 75 minus 45?

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So you might say, hey, 75
minus 45 is equal to 30.

00:00:20.540 --> 00:00:23.190
Maybe he did
exactly 30 problems.

00:00:23.190 --> 00:00:25.602
But we have to be very,
very, very careful here.

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And to understand why we
have to be careful here-- so

00:00:27.810 --> 00:00:30.562
let me put a question
mark right over here.

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To realize why we have
to be careful here

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is imagine-- let's do
some smaller numbers

00:00:34.130 --> 00:00:36.580
so that we can really
count things out.

00:00:36.580 --> 00:00:38.880
So let's say instead
of 45 through 75,

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let's say he had to do
problems 3 through 6.

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If we use this technique
that we just did up here,

00:00:49.610 --> 00:00:52.900
we would say, well, 6
minus 3 is equal to 3.

00:00:52.900 --> 00:00:54.707
Maybe he had to do 3 problems.

00:00:54.707 --> 00:00:56.040
But let's actually count it out.

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So there's problem 1, problem
2, problem 3, problem 4,

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problem 5, problem 6.

00:01:01.840 --> 00:01:05.830
He's going to have to do problem
3 through 6, including problem

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3.

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So he's going to do problem 3,
problem 4, problem 5, problem

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6.

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When I subtract 6
minus 3, I'm really

00:01:13.550 --> 00:01:15.870
only caring about
problems 4, 5, and 6.

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I'm looking at how much do I
have to go to go from 3 to 6.

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I'm not including the actual 3.

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So really, when he's doing
problems 3 through 6,

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he's not just doing 6 minus 3.

00:01:26.560 --> 00:01:31.000
He's doing 6 minus 3 plus 1.

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He's actually doing 4 problems.

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And you could do this
with multiple scenarios

00:01:36.830 --> 00:01:38.490
to realize when
you just subtract,

00:01:38.490 --> 00:01:42.250
you're not counting
the very first problem.

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You're just counting all the
incremental ones after that.

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So to answer this question, if
he's doing problems 45 to 75,

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it's not enough
just to subtract.

00:01:50.520 --> 00:01:53.290
That won't count the
first problem, the 45th.

00:01:53.290 --> 00:01:56.020
You actually have to add 1.

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So in this case, Brandon
does 31 problems.

00:02:02.370 --> 00:02:05.210
Let's do another
counting problem.

00:02:05.210 --> 00:02:07.290
A baker has a whole
loaf of bread.

00:02:07.290 --> 00:02:11.850
How many cuts must he make to
have exactly 7 even slices?

00:02:11.850 --> 00:02:14.320
Well, you could work
this one out directly,

00:02:14.320 --> 00:02:15.752
trying to cut a slice of bread.

00:02:15.752 --> 00:02:17.210
But let's think
about even if there

00:02:17.210 --> 00:02:20.310
were 70 even slices,
a much larger number.

00:02:20.310 --> 00:02:23.850
Well, we could do a
little loaf of bread.

00:02:23.850 --> 00:02:25.590
So that's our loaf of bread.

00:02:25.590 --> 00:02:30.950
If you want 2 slices, how
many cuts do you make?

00:02:30.950 --> 00:02:32.720
Well, you only make 1 cut.

00:02:32.720 --> 00:02:35.490
2 slices, you only make 1 cut.

00:02:35.490 --> 00:02:37.900
What if you wanted 3 slices?

00:02:37.900 --> 00:02:41.830
So if you wanted 3 slices, how
many cuts do you need to make?

00:02:41.830 --> 00:02:45.380
Well, you make 1 cut,
and you make 2 cuts.

00:02:45.380 --> 00:02:49.070
So you make 2 cuts, and I
think you see a pattern here.

00:02:49.070 --> 00:02:54.450
If we want 4 slices, how
many cuts do we make?

00:02:54.450 --> 00:03:00.172
Well, 1, 2, 3 cuts,
so it's 3 cuts.

00:03:00.172 --> 00:03:02.630
And you could go all the way
to 7, but you see the pattern.

00:03:02.630 --> 00:03:05.600
However many slices
you want, you only

00:03:05.600 --> 00:03:09.410
need to do that many
slices minus 1 cuts.

00:03:09.410 --> 00:03:14.770
So for 7 even slices, you
only have to do 6 cuts,

00:03:14.770 --> 00:03:15.500
and you see why.

00:03:15.500 --> 00:03:19.100
You don't have to cut
out that very last piece.

00:03:19.100 --> 00:03:21.540
It just falls off the edge.

00:03:21.540 --> 00:03:25.980
Let's do one more of these.

00:03:25.980 --> 00:03:30.510
Ishaan is building a straight
fence with posts 1 meter apart.

00:03:30.510 --> 00:03:33.440
If the fence is 18 meters
long, how many fence posts

00:03:33.440 --> 00:03:34.940
does he need?

00:03:34.940 --> 00:03:37.130
Well, we use the same idea.

00:03:37.130 --> 00:03:38.680
Let's do it with a
smaller situation.

00:03:38.680 --> 00:03:43.150
Let's imagine a
2-meter-long fence.

00:03:43.150 --> 00:03:45.950
For a 2-meter-long fence, he's
going to have 1 fence post.

00:03:45.950 --> 00:03:49.360
Then the fence is
going to go a meter.

00:03:49.360 --> 00:03:51.500
Then he's going to need
another fence post.

00:03:51.500 --> 00:03:53.240
Then the fence is
going to go a meter,

00:03:53.240 --> 00:03:55.220
and then he needs
another fence post.

00:03:55.220 --> 00:03:56.970
So if you compare it
to the cutting bread,

00:03:56.970 --> 00:03:58.900
you have your cut,
but you also have

00:03:58.900 --> 00:04:00.620
to do the ends right over here.

00:04:00.620 --> 00:04:05.260
So for a 2-meter fence,
he needs 3 posts.

00:04:05.260 --> 00:04:06.930
What about for a 3-meter fence?

00:04:09.397 --> 00:04:11.480
Well, you're going to have
to have the first post.

00:04:11.480 --> 00:04:13.410
You go a meter of fencing.

00:04:13.410 --> 00:04:16.660
Then you have the second
post, a meter of fencing.

00:04:16.660 --> 00:04:18.990
You have the third post,
but you're not done yet.

00:04:18.990 --> 00:04:20.615
Then you have another
meter of fencing.

00:04:20.615 --> 00:04:22.970
And then you need to have
the end post right over here.

00:04:22.970 --> 00:04:26.220
So you see you need 4 posts.

00:04:26.220 --> 00:04:29.540
You essentially need 1 post at
the beginning of each section,

00:04:29.540 --> 00:04:31.680
and then you need
1 to finish up.

00:04:31.680 --> 00:04:33.630
So what you see here
is however meters,

00:04:33.630 --> 00:04:36.390
you're going to need
that many plus 1 posts.

00:04:36.390 --> 00:04:38.460
So here, if you're
going 18 meters,

00:04:38.460 --> 00:04:42.600
you're going to
need 19 fence posts.


Place value relationships example

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:04.400
I've written out two numbers
here, and they both contain 4.

00:00:04.400 --> 00:00:06.570
They each have 4
as a digit, and I

00:00:06.570 --> 00:00:10.115
want to think about what
the values of these 4's are

00:00:10.115 --> 00:00:11.529
and to actually compare them.

00:00:11.529 --> 00:00:13.070
So I encourage you
to pause the video

00:00:13.070 --> 00:00:16.420
and try that out on your own.

00:00:16.420 --> 00:00:18.640
So let's look at
the two 4's here.

00:00:18.640 --> 00:00:23.490
So in 110,413, I have
a 4 right over here.

00:00:23.490 --> 00:00:25.700
And what does that
actually represent?

00:00:25.700 --> 00:00:27.230
Well, this is the ones place.

00:00:27.230 --> 00:00:29.010
You have three 1's.

00:00:29.010 --> 00:00:31.510
You have one 10's.

00:00:31.510 --> 00:00:33.060
And the 4 is in
the hundreds place.

00:00:33.060 --> 00:00:35.550
You have four 100's.

00:00:35.550 --> 00:00:41.930
So this 4 literally represents
4 times 100, or 400.

00:00:41.930 --> 00:00:43.330
Now, what about this 4?

00:00:43.330 --> 00:00:46.660
Well, once again,
you have zero 1's.

00:00:46.660 --> 00:00:49.970
You have five 10's.

00:00:49.970 --> 00:00:53.720
You have three 100's.

00:00:53.720 --> 00:01:00.550
And you have four 1,000's.

00:01:00.550 --> 00:01:08.940
So this literally represents
4 times 1,000, which is 4,000.

00:01:08.940 --> 00:01:11.440
So we can now compare the two.

00:01:11.440 --> 00:01:22.960
The 4 in 110,413 is 1/10 the
value of the 4 in 54,350.

00:01:22.960 --> 00:01:24.790
And it makes complete sense.

00:01:24.790 --> 00:01:28.490
This 4 is one place to
the right of this 4.

00:01:28.490 --> 00:01:29.790
This 4 is in the thousands.

00:01:29.790 --> 00:01:32.690
The magenta-circled
4 is in the hundreds.

00:01:32.690 --> 00:01:37.680
So every time we move a
place value to the right,

00:01:37.680 --> 00:01:42.700
each increment
represents 1/10 as much,

00:01:42.700 --> 00:01:44.510
assuming we have the same digit.

00:01:44.510 --> 00:01:47.131
So a 4 in thousands is 4,000.

00:01:47.131 --> 00:01:49.700
A 4 in hundreds would
be 1/10 as much.

00:01:49.700 --> 00:01:53.740
A 4 in the tens would
be 1/10 as much as that.

00:01:53.740 --> 00:01:56.140
So we could write a sentence.

00:01:56.140 --> 00:01:57.730
Let me write it this way.

00:01:57.730 --> 00:02:09.690
The value of the
4 in this number

00:02:09.690 --> 00:02:26.014
is 1/10 the value of
the 4 in this number.

00:02:26.014 --> 00:02:27.680
I know it's a little
bit weird sentence.

00:02:27.680 --> 00:02:33.790
The value of the 4 in 110,413
is 1/10 the value of the 4

00:02:33.790 --> 00:02:35.460
in 54,350.


Understanding place value 1 exercise

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:08.550
The 4 in the number
5,634 is blank times

00:00:08.550 --> 00:00:13.912
blank than the 4 in
the number 12,749.

00:00:13.912 --> 00:00:15.620
So let's think about
what they're saying.

00:00:15.620 --> 00:00:19.250
So the 4 in the number 5,634,
that's literally in the ones

00:00:19.250 --> 00:00:19.750
place.

00:00:19.750 --> 00:00:22.250
It literally just represents 4.

00:00:22.250 --> 00:00:27.900
Now, the 4 in the number 12,749,
that 4 is in the tens place.

00:00:27.900 --> 00:00:30.220
It represents 40.

00:00:30.220 --> 00:00:40.290
So this 4, it's a 10 times
smaller value than this 4.

00:00:40.290 --> 00:00:41.490
Or this 4, I should say.

00:00:41.490 --> 00:00:43.920
This 4 right over
here represents 4,

00:00:43.920 --> 00:00:46.850
while this represents 40.

00:00:46.850 --> 00:00:56.030
So it is 10 times smaller
than the 4 in 12,749.

00:00:56.030 --> 00:01:01.010
4 by itself is 10
times smaller than 40.

00:01:01.010 --> 00:01:02.536
Make sure I got
the right answer.

00:01:02.536 --> 00:01:03.410
Let's do another one.

00:01:08.410 --> 00:01:16.000
In the number 3,779,264,
how many times less

00:01:16.000 --> 00:01:20.740
is the value of the second 7
than the value of the first 7?

00:01:20.740 --> 00:01:24.320
How many times less is
the value of the second 7

00:01:24.320 --> 00:01:25.570
than the value of the first 7?

00:01:25.570 --> 00:01:29.650
So the second 7 right over here,
that's in the ten thousands

00:01:29.650 --> 00:01:30.150
place.

00:01:30.150 --> 00:01:36.190
It literally represents 70,000,
7 ten thousands, or 70,000,

00:01:36.190 --> 00:01:41.100
while this represents 700
thousands, or 700,000.

00:01:41.100 --> 00:01:48.520
So the second 7 is 1/10
the value of the first 7.

00:01:48.520 --> 00:01:50.790
Or another way of thinking,
it's 10 times less.

00:01:50.790 --> 00:01:54.460
This is 70,000, and
this is 700,000.

00:01:54.460 --> 00:01:59.190
So the value of the
second 7 is 10 times less

00:01:59.190 --> 00:02:02.190
than the value of the first 7.

00:02:02.190 --> 00:02:03.210
Let's do one more.

00:02:03.210 --> 00:02:04.430
This is fun.

00:02:04.430 --> 00:02:06.610
Fill in the following
blanks to complete

00:02:06.610 --> 00:02:12.310
the relationships
between 25,430 and 2,543.

00:02:12.310 --> 00:02:22.380
All right, so 25,430 is 10
times larger than 2,543.

00:02:22.380 --> 00:02:24.640
Literally, you take this,
you multiply it by 10,

00:02:24.640 --> 00:02:27.540
you're going to get 25,430.

00:02:27.540 --> 00:02:32.210
The digits in
25,430 are one place

00:02:32.210 --> 00:02:38.140
to the blank of the digits
in the number 2,543.

00:02:38.140 --> 00:02:39.300
Well, let's think about it.

00:02:39.300 --> 00:02:41.450
Here you have a 2 in
the thousands place.

00:02:41.450 --> 00:02:44.490
Here you have a 2 in
the ten thousands place.

00:02:44.490 --> 00:02:47.250
Here you have a 5 in
the hundreds place.

00:02:47.250 --> 00:02:49.420
Here you have a 5 in
the thousands place.

00:02:49.420 --> 00:02:50.650
And we could keep going.

00:02:50.650 --> 00:02:54.370
But what we see is a
corresponding digit.

00:02:54.370 --> 00:02:59.290
In 25,430, they're one place
to the left of the digits

00:02:59.290 --> 00:03:00.950
right over here.

00:03:00.950 --> 00:03:03.545
Now, finally, we're
going to take 25,430

00:03:03.545 --> 00:03:07.260
and divide it by 2,543.

00:03:07.260 --> 00:03:10.740
Well, we already know that the
first number is 10 times larger

00:03:10.740 --> 00:03:12.960
than this number
right over here.

00:03:12.960 --> 00:03:18.100
So literally, if you divide the
smaller number into the larger

00:03:18.100 --> 00:03:20.330
one, you're going to get 10.

00:03:20.330 --> 00:03:22.860
This right over here is
10 times larger than this.

00:03:22.860 --> 00:03:27.620
So that was the first part of
the question, and we are done.


Comparing place values

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:05.160
We have the number 43,249.

00:00:05.160 --> 00:00:08.670
Now, what I want you to think
about is what these two 4's

00:00:08.670 --> 00:00:11.480
actually represent and how
much more value is represented

00:00:11.480 --> 00:00:15.325
by this first 4, this 4 on the
left, than this 4 on the right.

00:00:15.325 --> 00:00:16.950
And I encourage you
to pause this video

00:00:16.950 --> 00:00:18.710
and think about that.

00:00:18.710 --> 00:00:21.480
So let's just think about what
each of these digits represent.

00:00:21.480 --> 00:00:23.270
So the 9 is in the ones place.

00:00:23.270 --> 00:00:26.420
So it literally
represents 9 ones.

00:00:26.420 --> 00:00:30.610
This 4 on the right, I should
say, is in the tens place

00:00:30.610 --> 00:00:36.610
so it literally represents 4
tens, or 4 times 10, or 40.

00:00:36.610 --> 00:00:40.500
This 2 is in the hundreds
place, so it literally

00:00:40.500 --> 00:00:43.780
represents 2 hundreds, or 200.

00:00:43.780 --> 00:00:45.950
This 3 is in the
thousands place,

00:00:45.950 --> 00:00:48.350
so it represents 3 thousands.

00:00:48.350 --> 00:00:53.480
And then the 4 on the left is
in the ten thousands place.

00:00:53.480 --> 00:00:56.690
So it literally
represents 4 times 10,000,

00:00:56.690 --> 00:01:02.510
or 4 ten thousands, or 40,000.

00:01:02.510 --> 00:01:05.000
So let's actually
compare the value

00:01:05.000 --> 00:01:07.660
that we're getting
here versus here.

00:01:07.660 --> 00:01:10.800
So what's the difference
between 40,000 and 40?

00:01:10.800 --> 00:01:14.220
Well, 40,000 has four zeroes
while 40 only has one.

00:01:14.220 --> 00:01:16.900
So if you wanted to
go from 40 to 40,000,

00:01:16.900 --> 00:01:19.002
you would have to add
three more zeroes.

00:01:19.002 --> 00:01:20.460
And we already know
how to do that.

00:01:20.460 --> 00:01:26.060
You could add three more
zeroes by multiplying by 1,000.

00:01:26.060 --> 00:01:29.240
So 40,000 is equal
to 1,000 times 40.

00:01:29.240 --> 00:01:31.610
Or we could say the
4 on the left here,

00:01:31.610 --> 00:01:40.160
this blue circled 4 represents
1,000 times the value

00:01:40.160 --> 00:01:43.170
as the yellow circled 4.

00:01:43.170 --> 00:01:45.570
Now, another way of
thinking about it

00:01:45.570 --> 00:01:48.790
is every time you move
place values to the left,

00:01:48.790 --> 00:01:51.750
as you see here-- this is
tens, hundreds, thousands, ten

00:01:51.750 --> 00:01:53.890
thousands-- you're
increasing what those place

00:01:53.890 --> 00:01:57.230
values represent
by a factor of 10.

00:01:57.230 --> 00:02:01.950
So if you're going from
this 4 to this 4, times 10

00:02:01.950 --> 00:02:07.979
times 10 times 10, you
multiply by 10 the place value.

00:02:07.979 --> 00:02:10.270
And you see that with the
place values right over here,

00:02:10.270 --> 00:02:13.470
the place values increase
by a factor of 10 each time.

00:02:13.470 --> 00:02:16.040
So if you're going from
this place to this place,

00:02:16.040 --> 00:02:18.070
and you have the exact
same digit there,

00:02:18.070 --> 00:02:20.610
multiplying by 10 three
times is the same thing

00:02:20.610 --> 00:02:22.730
as multiplying by 1,000.

00:02:22.730 --> 00:02:25.670
So whatever this represents,
multiply it by 1,000,

00:02:25.670 --> 00:02:28.478
and you're going to get
what this represents.


Representing numbers

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.580
We are asked what
number's represented

00:00:03.580 --> 00:00:05.260
by the blocks shown.

00:00:05.260 --> 00:00:06.930
So we could sit and
count all of these,

00:00:06.930 --> 00:00:09.730
but let's think about this a
little bit more constructively.

00:00:09.730 --> 00:00:12.754
So how many blocks do we have
in one of these squares here?

00:00:12.754 --> 00:00:13.420
Well, let's see.

00:00:13.420 --> 00:00:18.320
It's a 1, 2, 3, 4,
5, 6, 7, 8, 9, 10

00:00:18.320 --> 00:00:24.300
by 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, so this is a 10

00:00:24.300 --> 00:00:25.250
by 10 square.

00:00:25.250 --> 00:00:26.890
It's going to have 100 blocks.

00:00:26.890 --> 00:00:28.070
So this is 100.

00:00:28.070 --> 00:00:31.640
This is another 100,
so we have two 100's.

00:00:31.640 --> 00:00:35.540
So 200 blocks from these two.

00:00:35.540 --> 00:00:37.650
And then let's count
these right over here.

00:00:37.650 --> 00:00:45.500
We have 1, 2, 3, 4, 5, 6, 7,
8, 9, 10 in each of these rows.

00:00:45.500 --> 00:00:47.750
And we have four of these rows.

00:00:47.750 --> 00:00:50.310
So 4 times 10, that's 40.

00:00:50.310 --> 00:00:55.820
So we have 200 and
then 40-- 240 blocks.

00:00:55.820 --> 00:00:57.250
Let's check our answer.

00:00:57.250 --> 00:00:58.580
Now, you might be saying,
what's the whole point

00:00:58.580 --> 00:00:59.430
of this exercise?

00:00:59.430 --> 00:01:01.800
Isn't this just an exercise
in counting blocks?

00:01:01.800 --> 00:01:04.620
But you might notice something
interesting happening here.

00:01:04.620 --> 00:01:06.280
When I wrote this
2 here, it's making

00:01:06.280 --> 00:01:09.090
you appreciate that that 2
doesn't just represent 2.

00:01:09.090 --> 00:01:12.080
It's in the hundreds place,
so that 2 is actually

00:01:12.080 --> 00:01:13.760
representing 200.

00:01:13.760 --> 00:01:16.250
It's representing both of
these squares of blocks

00:01:16.250 --> 00:01:17.310
right over here.

00:01:17.310 --> 00:01:19.570
This 4 doesn't just represent 4.

00:01:19.570 --> 00:01:22.770
It's in the tens place, so
it actually represents 40.

00:01:22.770 --> 00:01:25.450
It's representing these
four groups of 10.

00:01:25.450 --> 00:01:28.080
This is these two groups of 100.

00:01:28.080 --> 00:01:31.250
This is these four groups of 10.

00:01:31.250 --> 00:01:33.250
Let's do another one.

00:01:33.250 --> 00:01:34.250
Oh, this is interesting.

00:01:34.250 --> 00:01:35.600
So let's do the hundreds first.

00:01:35.600 --> 00:01:37.940
So we have one group of
100 right over there,

00:01:37.940 --> 00:01:41.000
so it's going to be 100.

00:01:41.000 --> 00:01:45.890
And then let's see, we have
five groups of 10, so 150.

00:01:45.890 --> 00:01:49.900
And then we just have 4 blocks,
so we just have that 4 there.

00:01:49.900 --> 00:01:51.740
So let's check, make
sure we got it right.

00:01:51.740 --> 00:01:54.460
And you see that this 4--
that's in the ones place.

00:01:54.460 --> 00:01:57.190
So it literally
represents 4 blocks.

00:01:57.190 --> 00:01:59.730
This 5, that's in
the tens place.

00:01:59.730 --> 00:02:02.650
So it's literally representing
five groups of 10-- 1,

00:02:02.650 --> 00:02:04.810
2, 3, 4, 5.

00:02:04.810 --> 00:02:06.870
And this 1 is in
the hundreds place.

00:02:06.870 --> 00:02:10.020
It literally
represents one group

00:02:10.020 --> 00:02:13.640
of 100, this big
square right over here.


Orders of magnitude exercise example 2

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:02.720
Let's do a few more
examples from the orders

00:00:02.720 --> 00:00:04.940
of magnitude exercise.

00:00:04.940 --> 00:00:07.150
Earth is approximately
1 times 10

00:00:07.150 --> 00:00:10.080
to the seventh
meters in diameter.

00:00:10.080 --> 00:00:12.760
Which of the following
could be Earth's diameter?

00:00:12.760 --> 00:00:14.490
So this is just
an approximation.

00:00:14.490 --> 00:00:15.700
It's an estimate.

00:00:15.700 --> 00:00:17.700
And they're saying,
which of these,

00:00:17.700 --> 00:00:19.960
if I wanted to estimate
it, would be close

00:00:19.960 --> 00:00:22.640
or would be 1 times
10 to the seventh?

00:00:22.640 --> 00:00:27.300
And the key here is to realize
that 1 times 10 to the seventh

00:00:27.300 --> 00:00:30.050
is the same thing as one
followed by seven zeroes.

00:00:30.050 --> 00:00:33.542
One, two, three, four,
five, six, seven.

00:00:33.542 --> 00:00:36.000
Let me put some commas here so
we make it a little bit more

00:00:36.000 --> 00:00:36.680
readable.

00:00:36.680 --> 00:00:39.212
Or another way of
talking about it

00:00:39.212 --> 00:00:40.920
is that it is, 1 times
10 to the seventh,

00:00:40.920 --> 00:00:43.880
is the same thing as 10 million.

00:00:43.880 --> 00:00:47.160
So which of these, if we were
to really roughly estimate,

00:00:47.160 --> 00:00:49.710
we would go to 10 million.

00:00:49.710 --> 00:00:56.090
Well, this right over here is
1.271 million, or 1,271,543.

00:00:56.090 --> 00:00:57.907
If I were to really
roughly estimate it,

00:00:57.907 --> 00:00:59.490
I might go to one
million, but I'm not

00:00:59.490 --> 00:01:01.030
going to go to 10 million.

00:01:01.030 --> 00:01:03.040
So I'd rule that out.

00:01:03.040 --> 00:01:06.510
This is 12,715,430.

00:01:06.510 --> 00:01:09.580
If I were to roughly
estimate this, well, yeah.

00:01:09.580 --> 00:01:10.830
I would go to 10 million.

00:01:10.830 --> 00:01:14.380
10 million is if I wanted really
just one digit to represent it,

00:01:14.380 --> 00:01:16.550
if I were write this
in scientific notation.

00:01:16.550 --> 00:01:22.412
This right over here is 1.271543
times 10 to the seventh.

00:01:22.412 --> 00:01:23.370
Let me write that down.

00:01:23.370 --> 00:01:27.479
12,715,430.

00:01:27.479 --> 00:01:29.020
If I were to write
this in scientific

00:01:29.020 --> 00:01:36.642
notation as 1.271543
times 10 to the seventh.

00:01:36.642 --> 00:01:38.100
And when you write
it this way, you

00:01:38.100 --> 00:01:40.683
say, hey, well, yeah, if I was
to really estimate this and get

00:01:40.683 --> 00:01:43.160
pretty rough with it, and
I just rounded this down,

00:01:43.160 --> 00:01:45.210
I would make this 1
times 10 to the seventh.

00:01:45.210 --> 00:01:48.092
So this really does look
like our best choice.

00:01:48.092 --> 00:01:49.050
Now let me just verify.

00:01:49.050 --> 00:01:50.360
Well, this right over here,
if I were to write it,

00:01:50.360 --> 00:01:52.680
I would go to 100 million,
or 1 times 10 to the eighth.

00:01:52.680 --> 00:01:53.890
That's way too big.

00:01:53.890 --> 00:01:55.600
And this, if I were to write
it, I would go to a billion,

00:01:55.600 --> 00:01:56.730
or 1 times 10 to the ninth.

00:01:56.730 --> 00:01:58.270
So that's also too big.

00:01:58.270 --> 00:02:01.790
So once again, this feels
like the best answer.

00:02:01.790 --> 00:02:03.650
Now let's try a couple more.

00:02:03.650 --> 00:02:06.980
So here we're asked,
how many times larger

00:02:06.980 --> 00:02:11.160
is 7 times 10 to the fifth
than 1 times 10 to the fourth?

00:02:11.160 --> 00:02:13.250
Well, we could just divide
to think about that.

00:02:13.250 --> 00:02:19.640
So 7 times 10 to the fifth
divided by 1 times 10

00:02:19.640 --> 00:02:20.650
to the fourth.

00:02:20.650 --> 00:02:25.790
Well, this is the same
thing as 7 over 1 times 10

00:02:25.790 --> 00:02:29.370
to the fifth over 10 to
the fourth, which is just

00:02:29.370 --> 00:02:32.330
going to be equal to--
well, 7 divided by 1 is 7.

00:02:32.330 --> 00:02:38.480
And 10 to the fifth, that's
multiplying five 10's.

00:02:38.480 --> 00:02:40.317
And then you're
dividing by four 10's.

00:02:40.317 --> 00:02:41.900
You're going to have
one 10 left over.

00:02:41.900 --> 00:02:44.770
Or, if you remember your
exponent properties,

00:02:44.770 --> 00:02:46.300
this would be the
same thing as 10

00:02:46.300 --> 00:02:50.110
to the 5 minus 4 power,
or 10 to the first power.

00:02:50.110 --> 00:02:54.230
So this right over here,
all of this business,

00:02:54.230 --> 00:02:57.437
is going to simplify
to 10 to the first,

00:02:57.437 --> 00:02:59.020
or I could actually
write it this way.

00:02:59.020 --> 00:03:01.590
This would be the same thing
as 10 to the 5 minus 4,

00:03:01.590 --> 00:03:05.470
which is equal to 10 to the
1, which is just equal to 10.

00:03:05.470 --> 00:03:10.750
So this is 7 times 10,
which is equal to 70.

00:03:10.750 --> 00:03:17.050
So 7 times 10 to the fifth is
70 times larger than 1 times

00:03:17.050 --> 00:03:18.950
10 to the fourth.

00:03:18.950 --> 00:03:20.320
Let's do one more.

00:03:20.320 --> 00:03:23.620
So here, they're asking
us 3 times 10 to the ninth

00:03:23.620 --> 00:03:27.080
is 30,000 times larger
than what number?

00:03:27.080 --> 00:03:29.010
So once again, we can divide.

00:03:29.010 --> 00:03:34.470
So we have 3 times
10 to the ninth

00:03:34.470 --> 00:03:37.260
is 30,000 times larger
than what number?

00:03:37.260 --> 00:03:40.990
So let's just divide by
30,000 and see what we get.

00:03:40.990 --> 00:03:42.990
And here we've written
something in kind

00:03:42.990 --> 00:03:44.805
of an exponential
notation, or we

00:03:44.805 --> 00:03:46.830
should say scientific
notation actually.

00:03:46.830 --> 00:03:48.457
And here, we just
wrote the number out.

00:03:48.457 --> 00:03:50.290
So one way we could do
it is we could either

00:03:50.290 --> 00:03:52.320
write this number
out and then divide,

00:03:52.320 --> 00:03:54.280
or we could write this
in scientific notation.

00:03:54.280 --> 00:03:55.450
So let's do it either way.

00:03:55.450 --> 00:04:01.460
So if we were to expand
the top number out,

00:04:01.460 --> 00:04:04.660
we could write that as 3
followed by nine zeros.

00:04:04.660 --> 00:04:08.610
One, two, three, four, five,
six, seven, eight, nine.

00:04:08.610 --> 00:04:11.070
Let me put some commas
here to make it readable.

00:04:11.070 --> 00:04:16.570
And then we're dividing that
by 3 followed by four zeros.

00:04:16.570 --> 00:04:19.160
One, two, three, four.

00:04:19.160 --> 00:04:21.470
And then we could
cancel out the zeros.

00:04:21.470 --> 00:04:24.710
We could say, OK, let's divide
the top and the bottom by 10.

00:04:24.710 --> 00:04:29.950
Let's divide it by another 10,
by another 10, by another 10.

00:04:29.950 --> 00:04:33.470
And then, let's see, we've
done all the dividing by 10.

00:04:33.470 --> 00:04:35.840
And now let's divide the
top and the bottom by 3.

00:04:35.840 --> 00:04:37.340
So this would become a 1.

00:04:37.340 --> 00:04:38.770
This would become a 1.

00:04:38.770 --> 00:04:40.650
So on the bottom, we're
just left with a 1.

00:04:40.650 --> 00:04:44.830
And we'd have 1 followed by one,
two, three, four, five zeros.

00:04:44.830 --> 00:04:48.840
So this would be 1 followed
by one, two, three, four,

00:04:48.840 --> 00:04:52.260
five zeros, or 100,000.

00:04:52.260 --> 00:04:55.901
Now let's write it, both of
these, in scientific notation.

00:04:55.901 --> 00:04:57.400
So 3 times 10 to
the ninth, I'm just

00:04:57.400 --> 00:05:00.870
going to rewrite that as
3 times 10 to the ninth.

00:05:00.870 --> 00:05:03.430
And we're dividing
that by 30,000,

00:05:03.430 --> 00:05:07.390
which is the exact same thing
as 3 times 10 to the-- we

00:05:07.390 --> 00:05:09.840
have one, two, three,
four zeros here.

00:05:09.840 --> 00:05:11.290
3 times 10 to the fourth.

00:05:11.290 --> 00:05:12.950
Or I guess I really
should say, we

00:05:12.950 --> 00:05:15.500
have four places
after the three.

00:05:15.500 --> 00:05:17.670
So one, two, three, four.

00:05:17.670 --> 00:05:19.810
So 3 times 10 to the fourth.

00:05:19.810 --> 00:05:22.300
And so we could
divide the 3 by the 3,

00:05:22.300 --> 00:05:24.410
and then that will simplify.

00:05:24.410 --> 00:05:27.290
So 3 divided by 3 is just 1.

00:05:27.290 --> 00:05:30.390
And then 10 to the ninth
divided by 10 to the fourth,

00:05:30.390 --> 00:05:33.440
well that's going to be 10 to
the 9 minus 4, 10 to the fifth.

00:05:33.440 --> 00:05:36.690
So it's going to be 1 times
10 to the fifth, which,

00:05:36.690 --> 00:05:39.260
once again, is 1
followed by five zeros,

00:05:39.260 --> 00:05:42.840
or the exact same
thing as 100,000.

00:05:42.840 --> 00:05:46.620
So it's 30,000 times
larger than 100,000.


Orders of magnitude exercise example 1

WEBVTT
Kind: captions
Language: en

00:00:00.950 --> 00:00:06.010
So we're told that Mount
Everest is 29,029 feet tall.

00:00:06.010 --> 00:00:09.700
Estimate Mount Everest's
height by rewriting it

00:00:09.700 --> 00:00:14.970
as a number in the form x times
10 to the y-th power feet,

00:00:14.970 --> 00:00:18.670
where x and y are
single-digit numbers.

00:00:18.670 --> 00:00:21.140
So let's think about
how we would do this.

00:00:21.140 --> 00:00:24.080
So x and y have to be
single-digit numbers.

00:00:24.080 --> 00:00:26.437
So if we were to write it
in scientific notation,

00:00:26.437 --> 00:00:28.145
we could write something
like, let's see,

00:00:28.145 --> 00:00:38.727
it would be 2.9029 times 10.

00:00:38.727 --> 00:00:40.310
And actually, I'm
going to be careful.

00:00:40.310 --> 00:00:43.440
Times I should write like that.

00:00:43.440 --> 00:00:44.880
I'll use this little caret sign.

00:00:44.880 --> 00:00:48.070
I'm pressing shift and
6 to get the caret sign.

00:00:48.070 --> 00:00:51.640
Times 10 to the-- and
I moved the decimal.

00:00:51.640 --> 00:00:55.340
Let's see, in order to
go from 2 to 20,000,

00:00:55.340 --> 00:00:59.079
I have to add 1, 2, 3, 4 zeroes,
so times 10 to the fourth.

00:00:59.079 --> 00:01:00.870
So that's what I would
do if they were just

00:01:00.870 --> 00:01:02.750
asking us to write it
in scientific notation.

00:01:02.750 --> 00:01:03.805
But they're not
asking us to write it

00:01:03.805 --> 00:01:05.160
in scientific notation.

00:01:05.160 --> 00:01:08.136
They're saying estimate
Mount Everest's height

00:01:08.136 --> 00:01:09.510
by writing it in
this form, where

00:01:09.510 --> 00:01:12.620
x and y are
single-digit numbers.

00:01:12.620 --> 00:01:14.190
So as I wrote it
right over here,

00:01:14.190 --> 00:01:17.150
2.0929 is not a
single-digit number.

00:01:17.150 --> 00:01:18.670
4 is a single-digit number.

00:01:18.670 --> 00:01:21.160
So at least I got
the y part right.

00:01:21.160 --> 00:01:23.910
But I need to write this
part, which, in this form,

00:01:23.910 --> 00:01:25.010
this would be the x.

00:01:25.010 --> 00:01:26.926
I need to write that as
a single-digit number.

00:01:26.926 --> 00:01:28.840
And the key is that they
want me to estimate.

00:01:28.840 --> 00:01:33.520
So if I were to estimate
2.9029, I would write that as 3.

00:01:33.520 --> 00:01:36.330
So I'm going to go with
3 times 10 to the fourth.

00:01:36.330 --> 00:01:39.450
And if I were to expand out
3 times 10 to the fourth,

00:01:39.450 --> 00:01:40.800
it would be 30,000 feet.

00:01:40.800 --> 00:01:42.380
So my estimate
seems pretty good.

00:01:42.380 --> 00:01:43.610
So let's check.

00:01:43.610 --> 00:01:45.500
There we go.


Exponents and powers of zero patterns

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:06.210
We are asked what is 10 to
the fifth power equivalent to.

00:00:06.210 --> 00:00:11.190
Well, 10 to the fifth power is
the same thing as taking a 1

00:00:11.190 --> 00:00:13.860
and multiplying it
by 10 five times.

00:00:13.860 --> 00:00:15.180
So let's do that.

00:00:15.180 --> 00:00:19.600
So that's three, four, and five.

00:00:19.600 --> 00:00:22.420
So it's 1 times 10 times 10
times 10 times 10 times 10.

00:00:22.420 --> 00:00:25.360
Notice one, two,
three, four, five.

00:00:25.360 --> 00:00:26.730
And what's this going to be?

00:00:26.730 --> 00:00:28.670
Well, 1 times 10 is 10.

00:00:28.670 --> 00:00:31.081
10 times 10 is 100.

00:00:31.081 --> 00:00:33.800
100 times 10 is 1,000.

00:00:33.800 --> 00:00:37.070
1,000 times 10 is 10,000.

00:00:37.070 --> 00:00:39.450
10,000 times 10 is 100,000.

00:00:39.450 --> 00:00:45.060
So this is going to be 100,000.

00:00:45.060 --> 00:00:48.370
Now, you might have noticed,
every time we multiply it

00:00:48.370 --> 00:00:51.630
by 10, we're adding
another 0 to the product.

00:00:51.630 --> 00:00:54.930
So if we're multiplying
by 10 five times,

00:00:54.930 --> 00:00:58.920
we're going to add five
0's to the product.

00:00:58.920 --> 00:01:07.760
So this is literally going to
be 1 followed by five 0's, one,

00:01:07.760 --> 00:01:09.360
two, three, four, five.

00:01:09.360 --> 00:01:12.900
So 10 to the fifth is the
same thing as 100,000.

00:01:12.900 --> 00:01:16.300
Let's do another one that
is on a similar topic.

00:01:16.300 --> 00:01:20.460
How many 0's does the product
67 times 10 to the fifth have?

00:01:20.460 --> 00:01:22.850
Well, there's a bunch of
ways of thinking about that.

00:01:22.850 --> 00:01:29.430
67 times 10 to the fifth
is the exact same thing,

00:01:29.430 --> 00:01:33.805
this is equivalent to-- you
could view this as 67 times.

00:01:33.805 --> 00:01:36.130
And we could use
the same exact logic

00:01:36.130 --> 00:01:37.900
that we just saw in
the previous problem.

00:01:37.900 --> 00:01:45.620
So 67 times 1 times 10 times
10 times 10 times 10 times 10.

00:01:45.620 --> 00:01:47.680
And we already figured
out exactly what this is.

00:01:47.680 --> 00:01:51.050
If you have 1 multiplied
by 10 five times,

00:01:51.050 --> 00:01:54.720
this right over here is going
to be 1 followed by five 0's.

00:01:54.720 --> 00:01:57.710
One, two, three, four, five.

00:01:57.710 --> 00:02:00.309
Or it's going to be 100,000.

00:02:00.309 --> 00:02:02.100
So there's a couple
ways to think about it.

00:02:02.100 --> 00:02:04.391
If you're looking at just
this product right over here,

00:02:04.391 --> 00:02:07.126
you could say, well, 67 times
1 is just going to be 67.

00:02:07.126 --> 00:02:08.750
And then every time
you multiply by 10,

00:02:08.750 --> 00:02:10.000
you're going to add another 0.

00:02:10.000 --> 00:02:12.860
So you could say, well,
this is just going to be 67.

00:02:12.860 --> 00:02:15.800
And we're going to be
multiplying by 10 five times,

00:02:15.800 --> 00:02:17.440
so we're going to
add five 0's here.

00:02:17.440 --> 00:02:20.440
One, two, three, four, five.

00:02:20.440 --> 00:02:24.487
And so this would
come to 6,700,000.

00:02:24.487 --> 00:02:26.070
Another way you could
think about it--

00:02:26.070 --> 00:02:28.830
this is the same thing
as 67 times 100,000,

00:02:28.830 --> 00:02:32.660
so it's going to be 67, and
then we have five 0's here.

00:02:32.660 --> 00:02:35.300
So once again, one,
two, three, four, five.

00:02:35.300 --> 00:02:37.840
You get the exact same value.

00:02:37.840 --> 00:02:39.670
Let's do another one.

00:02:39.670 --> 00:02:44.400
How many 0's does the
quotient 5,700,000 divided

00:02:44.400 --> 00:02:48.020
by 10 to the third power have?

00:02:48.020 --> 00:02:50.550
Well, 10 to the third
power we already know.

00:02:50.550 --> 00:02:53.130
10 to the third power-- that's
the same thing as 1 times

00:02:53.130 --> 00:03:00.140
10 times 10 times 10, which
is the same thing as 1,000.

00:03:00.140 --> 00:03:02.894
And so if we're dividing by
that, that's the same thing.

00:03:02.894 --> 00:03:05.310
So another way of writing this
expression right over here.

00:03:05.310 --> 00:03:12.740
So 5,700,000 divided
by 10 to the third

00:03:12.740 --> 00:03:16.720
is the same thing as--
I could write this way.

00:03:16.720 --> 00:03:24.421
5,700,000 divided by
10 times 10 times 10.

00:03:24.421 --> 00:03:26.420
We could put a 1 out here,
but this won't really

00:03:26.420 --> 00:03:32.925
change the value, which is
equivalent to 5,700,000 divided

00:03:32.925 --> 00:03:35.634
by 1,000.

00:03:35.634 --> 00:03:37.050
And either way you
think about it,

00:03:37.050 --> 00:03:40.350
every time you divide by 10,
you're going to eliminate one

00:03:40.350 --> 00:03:41.850
of these 0's.

00:03:41.850 --> 00:03:44.130
So if you divide
by 10 three times--

00:03:44.130 --> 00:03:46.897
so if you divide by 10 once,
you're going to eliminate a 0.

00:03:46.897 --> 00:03:49.230
Divide by 10 again, you're
going to eliminate another 0.

00:03:49.230 --> 00:03:51.780
Divide by 10 again, you're
going to eliminate another 0.

00:03:51.780 --> 00:03:55.350
You're going to be
left with 5,700.

00:03:55.350 --> 00:03:57.210
Another way of thinking
about it is, well,

00:03:57.210 --> 00:04:00.440
if I'm dividing by something
that has three 0's, I'm going

00:04:00.440 --> 00:04:01.530
to eliminate three 0's.

00:04:01.530 --> 00:04:08.122
So if I eliminate three
0's, I am left with 5,700.

00:04:08.122 --> 00:04:09.580
Now, just to be
clear, I could only

00:04:09.580 --> 00:04:11.170
do this because this was 1,000.

00:04:11.170 --> 00:04:13.090
If this was like
3,000 or something,

00:04:13.090 --> 00:04:16.680
I could think of this
as 3 times 1,000.

00:04:16.680 --> 00:04:19.860
And then I could only cancel
out the 1,000 with these 0's.

00:04:19.860 --> 00:04:22.600
But the 3 I would then
have to divide separately.

00:04:22.600 --> 00:04:25.570
But only because I'm straight
up dividing by 1,000.

00:04:25.570 --> 00:04:27.890
I'm dividing by a power of 10.

00:04:27.890 --> 00:04:29.500
I have three 0's here.

00:04:29.500 --> 00:04:32.120
I can cancel it out
with three 0's there.

00:04:32.120 --> 00:04:33.980
Let's do another one.

00:04:33.980 --> 00:04:35.890
So it asks you to do a few.

00:04:35.890 --> 00:04:37.470
And these are the
type of questions

00:04:37.470 --> 00:04:39.350
that you might see
in the exercise.

00:04:39.350 --> 00:04:42.780
When 72.1 is multiplied
by 10 to the third,

00:04:42.780 --> 00:04:46.410
the decimal point moves
blank places to the blank.

00:04:46.410 --> 00:04:49.280
And in the exercise, you
might see a dropdown here.

00:04:49.280 --> 00:04:52.480
So remember, if you're
multiplying by--

00:04:52.480 --> 00:04:54.837
and this is essentially
10 to the third,

00:04:54.837 --> 00:04:56.420
so you're multiplying
by 1,000, you're

00:04:56.420 --> 00:04:57.830
going to get a larger number.

00:04:57.830 --> 00:05:00.550
You're not going to
get a smaller number.

00:05:00.550 --> 00:05:02.770
And so every time
you multiply by 10,

00:05:02.770 --> 00:05:05.070
your decimal point is
going to move to the right

00:05:05.070 --> 00:05:06.490
because you're getting larger.

00:05:06.490 --> 00:05:08.340
So we're going to move
the decimal point.

00:05:08.340 --> 00:05:12.877
So 72.1, if we
multiplied it by 10 once,

00:05:12.877 --> 00:05:14.835
then we would move the
decimal point over once.

00:05:14.835 --> 00:05:17.820
And we'd get 721,
which makes sense.

00:05:17.820 --> 00:05:20.170
72 times 10 is 720.

00:05:20.170 --> 00:05:22.810
72.1 times 10 should be 721.

00:05:22.810 --> 00:05:25.094
But if we want to multiply
it by 10 three times,

00:05:25.094 --> 00:05:27.010
we're going to move the
decimal place over not

00:05:27.010 --> 00:05:30.222
just once, but twice,
and three times.

00:05:30.222 --> 00:05:31.180
And you say wait, wait.

00:05:31.180 --> 00:05:31.870
How could I move it over?

00:05:31.870 --> 00:05:32.745
There's nothing here.

00:05:32.745 --> 00:05:33.997
Well, you throw a 0 in there.

00:05:33.997 --> 00:05:35.580
And so you're going
to get, it's going

00:05:35.580 --> 00:05:39.757
to be equal to-- so it's
going to be equal to 72,100.

00:05:39.757 --> 00:05:41.090
Now, they're not asking us that.

00:05:41.090 --> 00:05:42.850
They're just asking
how we would do it.

00:05:42.850 --> 00:05:45.860
And we saw the decimal
point moves three places

00:05:45.860 --> 00:05:47.220
to the right.

00:05:47.220 --> 00:05:48.720
And the big key
here is you're going

00:05:48.720 --> 00:05:51.600
to get a larger number when
you multiply by 10 three times,

00:05:51.600 --> 00:05:54.090
or multiply by 1,000.

00:05:54.090 --> 00:05:54.995
Let's do this one.

00:05:54.995 --> 00:05:57.470
When 56 is divided
by 10 to the third,

00:05:57.470 --> 00:06:00.860
the decimal point moves
blank places to the blank.

00:06:00.860 --> 00:06:02.590
Well, dividing by
10 to the third

00:06:02.590 --> 00:06:05.659
is the same thing as
dividing by 10 three times.

00:06:05.659 --> 00:06:07.200
And every time you
divide by 10, it's

00:06:07.200 --> 00:06:08.574
going to become
a smaller number.

00:06:08.574 --> 00:06:11.520
So 56, and you might say,
where is the decimal point?

00:06:11.520 --> 00:06:13.940
Well, there's implicitly a
decimal point right over there.

00:06:13.940 --> 00:06:16.520
You divide by 10 once, you're
going to get a smaller number.

00:06:16.520 --> 00:06:18.610
56 is going to become
5 and something,

00:06:18.610 --> 00:06:21.310
so it's going to
become literally 5.6.

00:06:21.310 --> 00:06:24.820
Divide by 10 again, it's
going to become 0.56.

00:06:24.820 --> 00:06:26.630
Divide by 10 again,
you're like, wait.

00:06:26.630 --> 00:06:29.762
If I keep moving the decimal
place again to the left,

00:06:29.762 --> 00:06:31.220
what am I moving
it to the left of?

00:06:31.220 --> 00:06:33.300
Well, you could throw
a 0 right over here.

00:06:33.300 --> 00:06:38.890
And so if you move the decimal
point three places to the left,

00:06:38.890 --> 00:06:41.990
you are left with 0.056.

00:06:41.990 --> 00:06:45.030
And you might want to put a
0 out here just for clarity.

00:06:45.030 --> 00:06:50.140
But what we did here is we moved
the decimal point three places

00:06:50.140 --> 00:06:52.750
to the left in this situation.

00:06:52.750 --> 00:06:55.360
We got a smaller value.


Constructing numerical expressions

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:06.400
Alan found 4 marbles
to add to his 5 marbles

00:00:06.400 --> 00:00:08.400
currently in his pocket.

00:00:08.400 --> 00:00:10.760
He then had a competition
with his friends

00:00:10.760 --> 00:00:13.230
and tripled his marbles.

00:00:13.230 --> 00:00:16.340
Write a numerical expression
to model the situation

00:00:16.340 --> 00:00:18.730
without performing
any operations.

00:00:18.730 --> 00:00:20.620
So let's think about
what's going on.

00:00:20.620 --> 00:00:23.120
So he already had 5
marbles in his pocket.

00:00:26.470 --> 00:00:29.380
And then he found 4 more
marbles to add to that.

00:00:29.380 --> 00:00:33.200
So we can add the 4
marbles to the 5 marbles.

00:00:33.200 --> 00:00:35.900
So 4 marbles plus the 5 marbles.

00:00:35.900 --> 00:00:37.900
So that's what happens
after the first sentence.

00:00:37.900 --> 00:00:40.490
He found 4 marbles to
add to his 5 marbles

00:00:40.490 --> 00:00:42.070
currently in his pocket.

00:00:42.070 --> 00:00:43.970
He then had a competition
with his friends

00:00:43.970 --> 00:00:45.890
and tripled his marbles.

00:00:45.890 --> 00:00:48.500
So this is how many marbles
he had before tripling.

00:00:48.500 --> 00:00:52.740
And now he's
tripling his marbles.

00:00:52.740 --> 00:00:56.980
So we want to multiply 3 times
the total number of marbles

00:00:56.980 --> 00:01:00.690
he has now-- times 4 plus 5.

00:01:00.690 --> 00:01:03.410
So this right over here is
the numerical expression

00:01:03.410 --> 00:01:05.950
that models the situation
without performing

00:01:05.950 --> 00:01:06.840
any operations.

00:01:06.840 --> 00:01:10.550
We, of course, could then
actually calculate this.

00:01:10.550 --> 00:01:13.590
He has 9 marbles
before tripling.

00:01:13.590 --> 00:01:16.041
And then you multiply
it by 3 and he has 27.

00:01:16.041 --> 00:01:17.540
But this is what
they're asking for.

00:01:17.540 --> 00:01:21.395
They want us to write
this expression.


Number patterns: interpreting and graphing relationships

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.660
Below are ordered pairs
that represent the first six

00:00:03.660 --> 00:00:06.290
terms of two given patterns.

00:00:06.290 --> 00:00:10.310
The first value in each pair
is a term from pattern A.

00:00:10.310 --> 00:00:13.750
And the second value is a term
from pattern B. In the answer

00:00:13.750 --> 00:00:15.820
box, there are different
statements about the two

00:00:15.820 --> 00:00:16.530
patterns.

00:00:16.530 --> 00:00:19.281
Choose all correct statements.

00:00:19.281 --> 00:00:21.030
So let's think about
what's going on here.

00:00:21.030 --> 00:00:25.470
They said the first
term is pattern A.

00:00:25.470 --> 00:00:28.750
So the first term in each of
these coordinates is pattern A,

00:00:28.750 --> 00:00:30.530
or in each pair is
pattern A. So pattern A

00:00:30.530 --> 00:00:37.390
goes from 1, to 2, to
4, to 8, to 16, to 32.

00:00:37.390 --> 00:00:38.970
So it looks like
pattern A, to go

00:00:38.970 --> 00:00:41.050
from the first term
to the second term,

00:00:41.050 --> 00:00:42.595
we multiplied by 2.

00:00:45.979 --> 00:00:48.020
And then to go from the
second to the third term,

00:00:48.020 --> 00:00:49.350
we also multiplied by 2.

00:00:49.350 --> 00:00:54.284
And we just keep
multiplying by 2.

00:00:54.284 --> 00:00:55.450
And we just keep doing that.

00:00:55.450 --> 00:00:56.230
8 times 2 is 16.

00:00:56.230 --> 00:00:59.140
16 times 2 is 32.

00:00:59.140 --> 00:01:03.580
Now let's think about what's
going on with pattern B.

00:01:03.580 --> 00:01:08.960
So pattern B is the second
number in each of these pairs.

00:01:08.960 --> 00:01:09.986
And it's just always 3.

00:01:09.986 --> 00:01:12.110
So there's a couple of ways
you can think about it.

00:01:12.110 --> 00:01:13.930
You could just say,
pattern B's always 3.

00:01:13.930 --> 00:01:15.710
You could say pattern
B starts at 3,

00:01:15.710 --> 00:01:18.910
and we're just
adding 0 every time.

00:01:18.910 --> 00:01:22.790
Or you could say that
pattern B starts at 3,

00:01:22.790 --> 00:01:25.850
and we are multiplying
by 1 every time.

00:01:25.850 --> 00:01:28.730
Either of those
would give you just 3

00:01:28.730 --> 00:01:32.630
showing up over and over again.

00:01:32.630 --> 00:01:35.020
So now that we've
looked at these pairs,

00:01:35.020 --> 00:01:38.280
we show the corresponding terms
for pattern A and pattern B,

00:01:38.280 --> 00:01:41.470
let's look at the choices here
and see which of these apply.

00:01:41.470 --> 00:01:43.710
In pattern A, you can get
from any term to the next

00:01:43.710 --> 00:01:45.962
by multiplying by
a constant number.

00:01:45.962 --> 00:01:46.920
Well, that looks right.

00:01:46.920 --> 00:01:48.870
We go from the first
term to the second term

00:01:48.870 --> 00:01:49.900
by multiplying by 2.

00:01:49.900 --> 00:01:51.840
Then we multiply by 2 again
to get to the third term.

00:01:51.840 --> 00:01:53.090
Then we keep multiplying by 2.

00:01:53.090 --> 00:01:55.210
So that constant number
that we're multiplying by

00:01:55.210 --> 00:01:56.710
to get to the next term is 2.

00:01:56.710 --> 00:01:58.600
So this looks right.

00:01:58.600 --> 00:02:01.610
The next pair should
be 52 comma 3.

00:02:01.610 --> 00:02:03.030
So let's think about this.

00:02:03.030 --> 00:02:06.565
If we keep doubling
for pattern A-- so this

00:02:06.565 --> 00:02:07.785
is going to be times 2.

00:02:07.785 --> 00:02:11.090
32 times 2 is 64.

00:02:11.090 --> 00:02:14.860
And then if we'd say that this
is 1 times the previous term,

00:02:14.860 --> 00:02:16.580
we're just going
to get a 3 again.

00:02:16.580 --> 00:02:19.640
So it should be 64 comma
3 should be the next one.

00:02:19.640 --> 00:02:21.760
They say the next pair
should be 52 comma 3.

00:02:21.760 --> 00:02:23.860
So that's not right.

00:02:23.860 --> 00:02:26.130
If we graph the
pairs, the points

00:02:26.130 --> 00:02:28.436
will be on the same line.

00:02:28.436 --> 00:02:30.060
So let's think about
that a little bit.

00:02:30.060 --> 00:02:32.250
Let's think about that.

00:02:32.250 --> 00:02:34.870
So this is my vertical axis.

00:02:34.870 --> 00:02:37.760
This is my horizontal axis.

00:02:37.760 --> 00:02:41.800
On the horizontal axis,
I will graph pattern A.

00:02:41.800 --> 00:02:47.430
And on my vertical axis, I will
graph pattern B. And let's see.

00:02:47.430 --> 00:02:52.160
Pattern A goes all
the way up to 32.

00:02:52.160 --> 00:02:54.090
So I'm going to
try my best here.

00:02:54.090 --> 00:02:56.770
So let's say that this is 32.

00:02:56.770 --> 00:02:59.856
Then half of that
is going to be 16.

00:02:59.856 --> 00:03:02.242
Half of that is going to be 8.

00:03:02.242 --> 00:03:05.000
Half of that is going to be 4.

00:03:05.000 --> 00:03:07.540
Half of that is going to be 2.

00:03:07.540 --> 00:03:10.870
And half of that
is going to be 1.

00:03:10.870 --> 00:03:14.840
So these are all the points on
pattern A. But for any of them,

00:03:14.840 --> 00:03:18.130
the corresponding term
on pattern B is 3.

00:03:18.130 --> 00:03:24.970
So we have, when pattern A
is 1, pattern B is 3-- 1,3.

00:03:24.970 --> 00:03:28.460
When pattern A is
2, pattern B is 3.

00:03:28.460 --> 00:03:31.400
When pattern A is
4, pattern B is 3.

00:03:31.400 --> 00:03:34.500
When pattern A is
8, pattern B is 3.

00:03:34.500 --> 00:03:38.670
When pattern A is 16, pattern--
this is like a tongue-- when

00:03:38.670 --> 00:03:42.350
pattern A is 16, pattern B is 3.

00:03:42.350 --> 00:03:45.910
When pattern A is
32, pattern B is 3.

00:03:45.910 --> 00:03:49.850
And you see, they
all sit on a line.

00:03:49.850 --> 00:03:52.590
They all sit on this horizontal
line, or at least the way

00:03:52.590 --> 00:03:54.140
that we've drawn it.

00:03:54.140 --> 00:03:57.070
They all sit on this
line that you probably

00:03:57.070 --> 00:03:58.580
can't see in yellow.

00:03:58.580 --> 00:04:00.250
So let me do it
in this red color.

00:04:00.250 --> 00:04:03.910
They all sit on this
line right over here.

00:04:03.910 --> 00:04:05.126
So this looks right.

00:04:05.126 --> 00:04:06.500
If we graph the
pairs, the points

00:04:06.500 --> 00:04:07.670
will be on the same line.

00:04:07.670 --> 00:04:09.110
So I'll go with that one.

00:04:09.110 --> 00:04:11.560
In pattern B, you can get
from any term to the next

00:04:11.560 --> 00:04:14.034
by multiplying by
a constant number.

00:04:14.034 --> 00:04:16.200
Well, yeah, even though
every term is the same term,

00:04:16.200 --> 00:04:19.550
but you can get from a 3 to a
3 by always multiplying by 1.

00:04:19.550 --> 00:04:21.089
1 is a constant number.

00:04:21.089 --> 00:04:23.110
So we're just multiplying
every term by 1.

00:04:23.110 --> 00:04:25.139
So that also seems to be right.

00:04:25.139 --> 00:04:27.180
So all of these are right,
except the second one.

00:04:27.180 --> 00:04:29.550
The next pair isn't 52 comma 3.

00:04:29.550 --> 00:04:32.810
It's going to be 64 comma 3.


Number patterns: interpreting relationships

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.210
The following graph represents
the first five terms

00:00:03.210 --> 00:00:04.499
of two given patterns.

00:00:04.499 --> 00:00:06.540
In the answer box, there
are different statements

00:00:06.540 --> 00:00:07.498
about the two patterns.

00:00:07.498 --> 00:00:09.320
Choose all correct statements.

00:00:09.320 --> 00:00:11.700
So here, for each point,
this point right over here,

00:00:11.700 --> 00:00:13.750
this represents its
horizontal coordinate

00:00:13.750 --> 00:00:16.086
is the first term of
pattern A, which is 4.

00:00:16.086 --> 00:00:17.960
And its vertical coordinate
is the first term

00:00:17.960 --> 00:00:19.149
in pattern B, which is 1.

00:00:19.149 --> 00:00:21.440
And then we could do that
for the other points as well.

00:00:21.440 --> 00:00:23.990
So actually, let's figure
out what the values are.

00:00:23.990 --> 00:00:29.160
So we have pattern A and
then we have pattern B.

00:00:29.160 --> 00:00:32.680
So the first term
for pattern A is 4.

00:00:32.680 --> 00:00:38.590
And when pattern A is 4, the
first term for pattern B is 1.

00:00:38.590 --> 00:00:42.330
The second term
for pattern A is 7.

00:00:42.330 --> 00:00:46.680
And when pattern A is
7, pattern B is also 7.

00:00:46.680 --> 00:00:53.080
Third term, pattern A is
10, and pattern B is 13.

00:00:53.080 --> 00:00:59.590
And then the fourth
term, pattern A is 13,

00:00:59.590 --> 00:01:03.130
and pattern B is 19.

00:01:03.130 --> 00:01:08.910
And then finally, fifth
term, pattern A is 16,

00:01:08.910 --> 00:01:14.534
and pattern B is 25.

00:01:14.534 --> 00:01:15.950
Now, before even
looking at these,

00:01:15.950 --> 00:01:18.700
let's see what we can think
about these patterns here.

00:01:18.700 --> 00:01:21.070
So it looks like
pattern A starts at 4,

00:01:21.070 --> 00:01:24.320
and it increases
by 3 every time.

00:01:24.320 --> 00:01:29.490
To go from one term to the
next, you just have to add 3.

00:01:29.490 --> 00:01:32.160
Now, what about for pattern B?

00:01:32.160 --> 00:01:35.170
Well pattern B starts at
1, and every term here it

00:01:35.170 --> 00:01:36.550
looks like you're adding 6.

00:01:36.550 --> 00:01:39.050
So when pattern A
increases by 3 and we're

00:01:39.050 --> 00:01:42.155
moving in the horizontal
direction based on the fact

00:01:42.155 --> 00:01:45.270
that pattern A is represented
on the horizontal axis,

00:01:45.270 --> 00:01:47.490
we're going to move up
6 in the vertical axis,

00:01:47.490 --> 00:01:48.460
and we see that here.

00:01:52.000 --> 00:01:55.230
Pattern A increases by 3
from one term to the next.

00:01:55.230 --> 00:01:57.220
And when that increased
by 3, pattern B

00:01:57.220 --> 00:01:59.820
increased by 6 from
one term to the next.

00:02:03.740 --> 00:02:09.680
And we see that it
keeps doing that.

00:02:09.680 --> 00:02:12.160
Now, let's think about
what we have over here

00:02:12.160 --> 00:02:16.100
to see which of these statements
actually apply to this.

00:02:16.100 --> 00:02:18.900
For every term in pattern
A, multiply the term by 2

00:02:18.900 --> 00:02:21.640
and then subtract 7 to get the
corresponding term from pattern

00:02:21.640 --> 00:02:24.690
B. So let's see
if that holds up.

00:02:24.690 --> 00:02:26.780
So according to this,
if this was true,

00:02:26.780 --> 00:02:28.910
I should be able to take
this, multiply it by 2

00:02:28.910 --> 00:02:30.990
and subtract 7 and get that.

00:02:30.990 --> 00:02:31.580
So let's see.

00:02:31.580 --> 00:02:37.380
Is 1 equal to 2 times 8 minus 7?

00:02:37.380 --> 00:02:39.490
Sorry, 2 times 4 minus 7.

00:02:39.490 --> 00:02:43.090
So 2 times this number,
2 times 4 minus 7.

00:02:43.090 --> 00:02:45.610
Well, 8 minus 7 is equal to 1.

00:02:45.610 --> 00:02:51.660
Is this right over here equal
to 2 times this 7 minus 7?

00:02:51.660 --> 00:02:53.510
Well, yeah, it's equal to 7.

00:02:53.510 --> 00:02:59.594
Is 13 equal to 2
times 10 minus 7?

00:02:59.594 --> 00:03:02.130
Well, yeah, 20 minus 7 is 13.

00:03:02.130 --> 00:03:06.870
Is 19 equal to 2
times 13 minus 7?

00:03:06.870 --> 00:03:09.210
26 minus 7 is 19.

00:03:09.210 --> 00:03:13.400
Is 25 equal to 2
times 16 minus 7?

00:03:13.400 --> 00:03:17.060
Well, 32 minus 7 is 25.

00:03:17.060 --> 00:03:20.360
So this first
statement checks out.

00:03:20.360 --> 00:03:24.880
For the corresponding term,
the value of pattern B

00:03:24.880 --> 00:03:29.475
is two times the value
of pattern A minus 7.

00:03:29.475 --> 00:03:30.850
Now let's look at
the second one.

00:03:30.850 --> 00:03:32.370
The terms of
pattern B are always

00:03:32.370 --> 00:03:34.470
greater than or equal to
their corresponding terms

00:03:34.470 --> 00:03:37.630
from pattern A. Well,
no, that's not right.

00:03:37.630 --> 00:03:39.130
It's true for a
couple of scenarios.

00:03:39.130 --> 00:03:41.380
Here for the third,
fourth, and fifth term,

00:03:41.380 --> 00:03:45.670
or actually for the second,
third, fourth, and fifth terms,

00:03:45.670 --> 00:03:47.977
pattern B is equal to or
greater than pattern A.

00:03:47.977 --> 00:03:49.560
But for the first
term, it's not true.

00:03:49.560 --> 00:03:53.520
Pattern A is greater,
so this is not right.

00:03:53.520 --> 00:03:55.190
To get from each
point to the next,

00:03:55.190 --> 00:03:58.080
you need to move 3 units to
the right and 6 units up.

00:03:58.080 --> 00:03:59.830
Well, that's exactly
what we talked about.

00:03:59.830 --> 00:04:01.550
From one term to
the next, pattern A,

00:04:01.550 --> 00:04:04.710
along our horizontal
axis, we increased by 3,

00:04:04.710 --> 00:04:08.620
while pattern B, which is
plotted on our vertical axis,

00:04:08.620 --> 00:04:09.620
by 6.

00:04:09.620 --> 00:04:12.580
So you move 3 to
the right and 6 up.

00:04:12.580 --> 00:04:14.630
So that is right.

00:04:14.630 --> 00:04:16.874
The second terms of
both patterns are 7.

00:04:16.874 --> 00:04:18.540
Well, yeah, we see
that right over here.

00:04:18.540 --> 00:04:19.950
The second terms are 7.

00:04:19.950 --> 00:04:24.770
We have 7 here, and
we have 7 there.

00:04:24.770 --> 00:04:26.090
And so that is right as well.

00:04:26.090 --> 00:04:28.730
So the only one that doesn't
apply is this second one.

00:04:28.730 --> 00:04:30.970
This is not right.


Number patterns: Seeing relationships

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:03.900
The following table contains
the first five terms

00:00:03.900 --> 00:00:07.410
of the given Pattern
A. Generate Pattern

00:00:07.410 --> 00:00:09.190
B according to this rule.

00:00:09.190 --> 00:00:11.390
For every term of Pattern
A-- so they give us

00:00:11.390 --> 00:00:15.290
the terms of Pattern A
here-- multiply the term by 3

00:00:15.290 --> 00:00:18.090
and add 1 to get the
corresponding term of Pattern

00:00:18.090 --> 00:00:22.120
B. Then graph the pairs
of corresponding terms.

00:00:22.120 --> 00:00:26.660
So for every term in Pattern A,
we want to multiply by 3 and 1.

00:00:26.660 --> 00:00:29.170
So if we multiply
0 by 3, we get 0.

00:00:29.170 --> 00:00:31.325
And you add 1, you get 1.

00:00:31.325 --> 00:00:34.710
If you multiply 1
by 3, you get 3.

00:00:34.710 --> 00:00:37.530
And then you add 1, you get 4.

00:00:37.530 --> 00:00:42.462
2 times 3 is 6, plus 1 is 7.

00:00:42.462 --> 00:00:47.510
3 times 3 is 9, plus 1 is 10.

00:00:47.510 --> 00:00:49.930
Remember, we're just
multiplying by 3 and adding 1.

00:00:49.930 --> 00:00:54.040
4 times 3 is 12, plus 1 is 13.

00:00:54.040 --> 00:00:56.380
So those are the corresponding
terms for Pattern B.

00:00:56.380 --> 00:00:57.920
And then they ask
us to graph them.

00:00:57.920 --> 00:01:01.510
So let's try to
graph these points.

00:01:01.510 --> 00:01:05.400
So when Pattern A is
0, Pattern B is 1.

00:01:05.400 --> 00:01:10.110
When Pattern A is 0-- so
this is Pattern A equaling 0.

00:01:10.110 --> 00:01:13.800
That's our horizontal axis, the
value of Pattern A-- Pattern B

00:01:13.800 --> 00:01:16.050
is the value of
our vertical axis.

00:01:16.050 --> 00:01:19.630
Pattern B is 1.

00:01:19.630 --> 00:01:21.700
When Pattern A is
1, Pattern B is 4.

00:01:24.420 --> 00:01:27.020
So when Pattern A is
1, Pattern B is 4.

00:01:27.020 --> 00:01:29.420
Pattern B is on
the vertical axis.

00:01:29.420 --> 00:01:31.845
When Pattern A is
2, Pattern B is 7.

00:01:37.040 --> 00:01:45.264
When Pattern A is 3,
Pattern B is 10, so 3

00:01:45.264 --> 00:01:46.430
in the horizontal direction.

00:01:46.430 --> 00:01:47.990
That's our Pattern A value.

00:01:47.990 --> 00:01:50.550
And our Pattern B value is 10.

00:01:50.550 --> 00:01:56.325
And then, finally, when Pattern
A is 4, Pattern B is 13.

00:01:58.795 --> 00:02:00.420
Now, let's just look
at these patterns.

00:02:00.420 --> 00:02:04.750
We see Pattern A is
increasing by 1 each time,

00:02:04.750 --> 00:02:08.479
while Pattern B is
increasing by it's-- well,

00:02:08.479 --> 00:02:11.400
Pattern A starts at
0 and increases by 1,

00:02:11.400 --> 00:02:16.400
while Pattern B starts
at 1 and increases by 3,

00:02:16.400 --> 00:02:17.550
which makes complete sense.

00:02:17.550 --> 00:02:19.925
It makes sense that it starts
at 1, because all of these,

00:02:19.925 --> 00:02:21.760
you multiply by 3 and add 1.

00:02:21.760 --> 00:02:23.162
So you start at 1.

00:02:23.162 --> 00:02:25.120
And then, the fact that
we're multiplying by 3,

00:02:25.120 --> 00:02:28.430
that's what's leading to the
distance between these points

00:02:28.430 --> 00:02:30.410
being 3.

00:02:30.410 --> 00:02:33.830
So let's check our answer to
make sure we got this right,

00:02:33.830 --> 00:02:35.640
and we did.


Math patterns example 2

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.900
I want to make little townhouse
shapes with toothpicks.

00:00:03.900 --> 00:00:06.370
So this would be
my first townhouse.

00:00:06.370 --> 00:00:10.900
I've used 3 toothpicks
so far-- 4, 5, and 6.

00:00:10.900 --> 00:00:13.280
So that is my first townhouse.

00:00:13.280 --> 00:00:16.280
Now, let me make a little table
here keeping track of things.

00:00:16.280 --> 00:00:17.990
So I'll do that in white.

00:00:17.990 --> 00:00:21.700
So here's my table to
keep track of things.

00:00:21.700 --> 00:00:29.210
So this is the number of houses,
and then this is the toothpicks

00:00:29.210 --> 00:00:31.250
that I'm using to
make that house.

00:00:31.250 --> 00:00:36.350
So this first house here,
took me 6 toothpicks--

00:00:36.350 --> 00:00:38.850
1, 2, 3, 4, 5, 6.

00:00:38.850 --> 00:00:41.319
Now let's make our second house.

00:00:41.319 --> 00:00:42.860
And these are going
to be townhouses.

00:00:42.860 --> 00:00:44.460
They're going to
share common walls.

00:00:44.460 --> 00:00:50.050
So I'm going to
add 1, 2, 3, 4, 5

00:00:50.050 --> 00:00:51.700
toothpicks for my second house.

00:00:51.700 --> 00:00:54.310
Now, why did I only
have to add 5 and not 6?

00:00:54.310 --> 00:00:55.810
Well, they shared
a common wall here

00:00:55.810 --> 00:00:57.726
so I didn't have to add
another toothpick here

00:00:57.726 --> 00:00:59.880
for this left-hand side wall.

00:00:59.880 --> 00:01:01.860
So starting with
the first house,

00:01:01.860 --> 00:01:04.430
I really just had
to add 5 toothpicks.

00:01:04.430 --> 00:01:08.970
I had to add 5 toothpicks to get
to now 11 total toothpicks if I

00:01:08.970 --> 00:01:10.550
want two houses.

00:01:10.550 --> 00:01:12.040
I think you see the trend here.

00:01:12.040 --> 00:01:15.400
What about 3 of these?

00:01:15.400 --> 00:01:17.690
Well, this is going
to be another 5-- 1,

00:01:17.690 --> 00:01:21.350
2, 3, 4, 5 toothpicks.

00:01:21.350 --> 00:01:25.550
So we're going to add
5 again and get to 16.

00:01:25.550 --> 00:01:27.820
Let's do 4 just
for good measure.

00:01:27.820 --> 00:01:31.070
So the fourth one, we're
going to add another 5-- 1,

00:01:31.070 --> 00:01:34.130
2, 3, 4, 5.

00:01:34.130 --> 00:01:41.240
So the fourth one, we're going
to add another 5 gets us to 21.

00:01:41.240 --> 00:01:44.760
Now, I want to think about,
can we, using this pattern,

00:01:44.760 --> 00:01:47.240
figure out how many
toothpicks it would take

00:01:47.240 --> 00:01:49.860
for us to, say, make
50 of these townhouses

00:01:49.860 --> 00:01:53.734
or even 500 of these townhouses,
or even 5,000 of them?

00:01:53.734 --> 00:01:55.400
Now we just have to
look at this pattern

00:01:55.400 --> 00:01:58.430
here and see can we come up
with an equation for each

00:01:58.430 --> 00:02:00.720
of these actual values?

00:02:00.720 --> 00:02:03.240
So, for example,
we see a pattern

00:02:03.240 --> 00:02:05.220
that-- well, we
already recognize

00:02:05.220 --> 00:02:06.900
that we started
with 6, and we're

00:02:06.900 --> 00:02:11.620
adding 5 every time
we add a house.

00:02:11.620 --> 00:02:14.080
So when you add the second
house, you add 5 once.

00:02:14.080 --> 00:02:18.120
The third house, you start
with 6, and you add 5 twice.

00:02:18.120 --> 00:02:22.430
The fourth house, you start with
6 and you add 5 three times.

00:02:22.430 --> 00:02:24.390
So let's actually
write that down.

00:02:24.390 --> 00:02:30.030
So 21 is equal to-- you start
with 6, you start with this 6

00:02:30.030 --> 00:02:38.710
here, and then you add 5
three times, plus 5 times 3.

00:02:38.710 --> 00:02:44.110
When you had the 3 houses,
once again, you started with 6

00:02:44.110 --> 00:02:47.820
and you added 5 two times.

00:02:47.820 --> 00:02:49.950
Let me do that same color.

00:02:49.950 --> 00:02:53.090
And you added 5 two times.

00:02:53.090 --> 00:02:57.070
Plus 5 times 2.

00:02:57.070 --> 00:03:00.250
When you had 2 houses,
you started with 6 again.

00:03:00.250 --> 00:03:10.260
This is equal to 6 and you
added 5 once, so plus 5 times 1.

00:03:10.260 --> 00:03:13.840
And then when you had
1 house-- and it'll

00:03:13.840 --> 00:03:15.870
fit the same pattern--
you started with 6,

00:03:15.870 --> 00:03:18.120
and how many times
did you add 5?

00:03:18.120 --> 00:03:19.160
Well, you didn't add 5.

00:03:19.160 --> 00:03:24.160
You could say that you
added 5 zero times.

00:03:24.160 --> 00:03:26.390
So you might see a
little pattern here.

00:03:26.390 --> 00:03:29.950
However many houses you needed,
you take one less than that

00:03:29.950 --> 00:03:33.020
and multiply it by
5, add that to 6,

00:03:33.020 --> 00:03:34.940
and you get the
number of toothpicks.

00:03:34.940 --> 00:03:36.800
And actually, let
me rewrite this.

00:03:36.800 --> 00:03:41.730
So I could rewrite this as
6 plus 5 times 4 minus 1.

00:03:41.730 --> 00:03:47.760
I could write this as 6
plus 5 times 3 minus 1.

00:03:47.760 --> 00:03:52.740
You could write this as
6 plus 5 times 2 minus 1.

00:03:52.740 --> 00:03:57.600
You could rewrite this as
6 plus 5 times 1 minus 1.

00:03:57.600 --> 00:04:01.140
And maybe that makes a
pattern a little bit clearer.

00:04:01.140 --> 00:04:03.550
This 4 is right over here.

00:04:03.550 --> 00:04:07.000
This 3 is right over here.

00:04:07.000 --> 00:04:11.240
This 2 is right over here.

00:04:11.240 --> 00:04:14.790
And then this 1 is
right over here.

00:04:14.790 --> 00:04:16.700
So now, I think we
are ready to think

00:04:16.700 --> 00:04:21.350
about what would happen if
we wanted to make 50 houses.

00:04:21.350 --> 00:04:24.890
So let's try to do that.

00:04:24.890 --> 00:04:26.300
Let me do that in orange.

00:04:26.300 --> 00:04:29.110
This right over here
is our 50th house.

00:04:29.110 --> 00:04:32.100
So this is the shared
left wall it has.

00:04:32.100 --> 00:04:34.220
This is the 50th
house right over here.

00:04:34.220 --> 00:04:37.020
So how many total
toothpicks for 50 houses?

00:04:37.020 --> 00:04:39.190
So if we have 50
houses, well, we

00:04:39.190 --> 00:04:41.470
can use the pattern
that we came up with.

00:04:41.470 --> 00:04:46.800
It's going to be equal
to, starting with our 6,

00:04:46.800 --> 00:04:48.630
the first house requires 6.

00:04:48.630 --> 00:04:53.120
And then we're going to add
5 for each incremental house,

00:04:53.120 --> 00:04:56.230
so plus 5 for each
incremental house.

00:04:56.230 --> 00:04:58.560
And how many incremental
houses are there going to be?

00:04:58.560 --> 00:05:02.680
Well, there are going to be
50 minus 1 incremental houses.

00:05:02.680 --> 00:05:03.710
Why minus 1?

00:05:03.710 --> 00:05:05.890
Well, you already built
one of them with the 6.

00:05:05.890 --> 00:05:07.650
Then for every extra
one-- so there's

00:05:07.650 --> 00:05:09.440
going to be 49 extra
houses-- you're

00:05:09.440 --> 00:05:11.850
going to add 5
toothpicks apiece.

00:05:11.850 --> 00:05:19.570
So this is going to be
equal to 6 plus 5 times 49.

00:05:19.570 --> 00:05:22.020
And that is 245.

00:05:22.020 --> 00:05:30.532
So 6 plus 245 is
equal to 251 sticks.

00:05:30.532 --> 00:05:32.240
And what's really neat
about this pattern

00:05:32.240 --> 00:05:34.490
we just came up with is you
could use it to figure out

00:05:34.490 --> 00:05:37.440
how many sticks you
would need for a million

00:05:37.440 --> 00:05:40.276
of these little
toothpick townhouses.


Math patterns example 1

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:05.440
So let's say I have tables where
I can fit one person at either

00:00:05.440 --> 00:00:07.390
of the short ends of the table.

00:00:07.390 --> 00:00:08.892
So I could fit one person there.

00:00:08.892 --> 00:00:10.100
I could fit one person there.

00:00:10.100 --> 00:00:13.350
You could view this as we're
looking from above the table

00:00:13.350 --> 00:00:13.850
here.

00:00:13.850 --> 00:00:15.410
So we could put one
person at either

00:00:15.410 --> 00:00:17.050
of the short ends of the table.

00:00:17.050 --> 00:00:18.980
And then on these longer
ends right over here,

00:00:18.980 --> 00:00:21.260
we can fit two people.

00:00:21.260 --> 00:00:23.810
We can fit two people
at the longer end.

00:00:23.810 --> 00:00:27.830
So when you have one table,
you could fit one, two, three,

00:00:27.830 --> 00:00:29.810
four, five, six people.

00:00:29.810 --> 00:00:31.704
You could fit six people.

00:00:31.704 --> 00:00:33.120
Now let's think
about what happens

00:00:33.120 --> 00:00:37.490
as we add tables end to end
to this table right over here.

00:00:37.490 --> 00:00:39.720
So let's imagine now two tables.

00:00:39.720 --> 00:00:42.980
So here we have
one table, and it's

00:00:42.980 --> 00:00:45.855
going to touch ends with
this table right over here.

00:00:45.855 --> 00:00:47.730
And because it touches
ends right over here--

00:00:47.730 --> 00:00:50.180
we're making it one
big continuous table--

00:00:50.180 --> 00:00:52.210
you can't fit
someone here anymore.

00:00:52.210 --> 00:00:54.190
So now how many
people can we fit?

00:00:54.190 --> 00:00:54.690
Let's see.

00:00:54.690 --> 00:00:59.259
We can fit one, two,
three, four, five.

00:00:59.259 --> 00:01:01.050
And then on this table,
which is identical,

00:01:01.050 --> 00:01:04.805
you could fit six,
seven, eight, nine.

00:01:04.805 --> 00:01:06.180
And then you could
put one person

00:01:06.180 --> 00:01:08.390
at the end right over here.

00:01:08.390 --> 00:01:11.730
So when you have two
tables end to end,

00:01:11.730 --> 00:01:14.840
you can fit a
total of 10 people.

00:01:14.840 --> 00:01:18.240
Let's keep going and see if we
can think of a pattern here.

00:01:18.240 --> 00:01:21.630
So let's put three
tables here-- so

00:01:21.630 --> 00:01:27.740
one table, two tables,
and three tables.

00:01:27.740 --> 00:01:30.460
So just as before, we
could put one at each end.

00:01:30.460 --> 00:01:32.600
So that's two people.

00:01:32.600 --> 00:01:40.140
Then we have 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13,

00:01:40.140 --> 00:01:43.750
14 folks-- 14 folks.

00:01:43.750 --> 00:01:46.089
So what does it look
like is happening here.

00:01:46.089 --> 00:01:47.630
Well if you just
look at the numbers,

00:01:47.630 --> 00:01:50.240
we went from 6 to 10 to 14.

00:01:50.240 --> 00:01:55.400
It looks like we're adding four
people every time we actually

00:01:55.400 --> 00:01:56.440
add a table.

00:01:56.440 --> 00:01:58.841
Now does that
actually makes sense?

00:01:58.841 --> 00:02:00.590
So let's think about
this first situation.

00:02:00.590 --> 00:02:02.090
Let's imagine these
are real people,

00:02:02.090 --> 00:02:04.320
and I'll make this person
in blue right over here.

00:02:04.320 --> 00:02:06.780
If you were to bring
over this new table,

00:02:06.780 --> 00:02:12.000
if you bring over table
two-- so this is table one--

00:02:12.000 --> 00:02:13.882
this blue person has to move.

00:02:13.882 --> 00:02:15.090
And so where could they move?

00:02:15.090 --> 00:02:16.465
Let's say that
they always insist

00:02:16.465 --> 00:02:17.950
on sitting at the
end of a table.

00:02:17.950 --> 00:02:20.800
So the blue person moves to
the new end of the table.

00:02:23.650 --> 00:02:25.290
They move right over here.

00:02:25.290 --> 00:02:29.480
So how many new people could
move to this combined table

00:02:29.480 --> 00:02:31.650
now that you brought
this second table in?

00:02:31.650 --> 00:02:37.310
Well the new people I will
do in this purple color.

00:02:37.310 --> 00:02:38.840
The new people are
that person-- let

00:02:38.840 --> 00:02:41.510
me do it in a more unique
color-- this person,

00:02:41.510 --> 00:02:45.820
this person, this
person, and this person.

00:02:45.820 --> 00:02:48.580
So you were able to add four
new people with the new table.

00:02:48.580 --> 00:02:51.250
One way to think about
it is a new table

00:02:51.250 --> 00:02:53.580
is going to have
one usable end here.

00:02:53.580 --> 00:02:55.830
That usable end is going to
be taken by the person who

00:02:55.830 --> 00:02:59.380
was already at the usable end
of when you had less tables.

00:02:59.380 --> 00:03:03.530
And so the real addition
is the two sides here.

00:03:03.530 --> 00:03:06.630
So you're adding four people
every time you add a table.

00:03:06.630 --> 00:03:08.890
So it makes complete sense.

00:03:08.890 --> 00:03:10.760
So based on this, you
could think about,

00:03:10.760 --> 00:03:15.080
without even having to draw
these diagrams, how many people

00:03:15.080 --> 00:03:17.650
you would be able to fit
if you had four or five

00:03:17.650 --> 00:03:19.670
or six or however many tables.

00:03:19.670 --> 00:03:24.590
So you could imagine,
if you have four tables,

00:03:24.590 --> 00:03:26.060
we just have to
add four, and you

00:03:26.060 --> 00:03:28.700
should be able to sit 18 people.

00:03:28.700 --> 00:03:32.420
If you have five
tables, you should

00:03:32.420 --> 00:03:37.960
be able to fit 22 people
and on and on and on.


Inequality word problem: one variable

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.960
The Karman line is
the point at which

00:00:02.960 --> 00:00:06.620
the Earth's atmosphere ends
and outer space begins.

00:00:06.620 --> 00:00:11.050
The Karman line is at 62
miles above sea level.

00:00:11.050 --> 00:00:14.760
And any distance from sea level
at the Karman line or above

00:00:14.760 --> 00:00:16.740
is considered outer space.

00:00:16.740 --> 00:00:20.470
Set c equals 62 and use
x to represent distance

00:00:20.470 --> 00:00:22.050
from sea level.

00:00:22.050 --> 00:00:25.640
Write an inequality
in terms of x and c

00:00:25.640 --> 00:00:28.460
that represents the
distances from sea level that

00:00:28.460 --> 00:00:31.210
are inside Earth's atmosphere.

00:00:31.210 --> 00:00:36.320
So we want to write an
inequality in terms of x and c.

00:00:36.320 --> 00:00:39.770
And x is the distance
above sea level,

00:00:39.770 --> 00:00:41.540
and c is the Karman line.

00:00:41.540 --> 00:00:43.060
And we want to
think about things--

00:00:43.060 --> 00:00:44.601
we want to think
about distances that

00:00:44.601 --> 00:00:47.170
are inside Earth's atmosphere.

00:00:47.170 --> 00:00:50.950
So that's going to be anything
that's below the Karman line.

00:00:50.950 --> 00:00:55.340
And so, let's see, I would
write x is less than c,

00:00:55.340 --> 00:00:57.530
because they're
telling us c is 62.

00:00:57.530 --> 00:00:58.820
Now, I want to make sure.

00:00:58.820 --> 00:01:02.170
So they say the Karman line is
the point at which the Earth's

00:01:02.170 --> 00:01:05.480
atmosphere ends and
outer space begins.

00:01:05.480 --> 00:01:07.620
So what if we're
right at 62 miles?

00:01:07.620 --> 00:01:09.920
The Karman line is 62
miles above sea level,

00:01:09.920 --> 00:01:14.530
and any distance from sea level
at the Karman line or above

00:01:14.530 --> 00:01:15.950
is considered outer space.

00:01:15.950 --> 00:01:16.730
OK.

00:01:16.730 --> 00:01:20.950
So if I'm at c or above,
I'm in outer space.

00:01:20.950 --> 00:01:22.700
So only the things
that are not, I'm sure,

00:01:22.700 --> 00:01:24.640
they're strictly below c.

00:01:24.640 --> 00:01:26.780
So I'm going to do a less
than, not a less than

00:01:26.780 --> 00:01:30.341
or equal sign right over here.

00:01:30.341 --> 00:01:31.340
All right, got it right.

00:01:31.340 --> 00:01:33.400
Let's do another one.

00:01:33.400 --> 00:01:36.220
A black hole is a region of
space where the pull of gravity

00:01:36.220 --> 00:01:38.500
is so strong that
nothing can get back out

00:01:38.500 --> 00:01:40.560
once it has fallen inside.

00:01:40.560 --> 00:01:43.030
Let's use E to represent
the largest distance

00:01:43.030 --> 00:01:47.910
from the center of the black
hole that pulls objects inside.

00:01:47.910 --> 00:01:49.960
If x represents the
distance of an object

00:01:49.960 --> 00:01:51.600
from the center
of the black hole,

00:01:51.600 --> 00:01:54.380
write an inequality
for safe distances

00:01:54.380 --> 00:01:58.720
from the center of the black
hole in terms of x and E.

00:01:58.720 --> 00:02:01.030
So x is the distance
from the center.

00:02:01.030 --> 00:02:03.010
e represents the
largest distance

00:02:03.010 --> 00:02:06.120
from the center of the black
hole that pulls objects inside.

00:02:06.120 --> 00:02:08.180
So if you're at
E away or closer,

00:02:08.180 --> 00:02:09.669
you're going to
be pulled inside.

00:02:09.669 --> 00:02:12.040
But we want to think
about safe distances.

00:02:12.040 --> 00:02:16.690
So if you're further away
than E, you should be cool.

00:02:16.690 --> 00:02:19.660
And it's strictly greater
than because if you're at E,

00:02:19.660 --> 00:02:21.660
that's the largest distance
that you would still

00:02:21.660 --> 00:02:24.300
be pulled inside.

00:02:24.300 --> 00:02:26.070
Very good.