Price discrimination for a monopoly

WEBVTT
Kind: captions
Language: en

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- [Instructor] Let's say
that you own the only hotel

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that is in a city,

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and for a wide variety of reasons,

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maybe all of the city council
members are your friends

00:00:09.970 --> 00:00:13.520
or whatever else, no one else
can build a hotel in the city.

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So there are insurmountable
barriers to entry.

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So in that situation, you
would have a monopoly.

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You are the only player in the market,

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and there are very, very
high barriers to entry.

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Now, this is a typical cost structure

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and demand curve for a monopoly.

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We've already talked
about your marginal cost.

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It might dip down a little initially,

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but then it might go up,

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and we could debate
whether that would be true

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for a hotel or not,

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but this is a typical model we see.

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And then while your marginal cost

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is below average total cost,

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average total cost trends down,

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and then hits a minimum point

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where marginal cost intersects it,

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and then it starts to trend up

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as marginal cost is higher than that.

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And the demand curve for
a monopoly looks familiar.

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When the prices are high,

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if the prices on the hotel
rooms per night are high,

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very few people will demand them,

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and if the prices are low, a
lot of folks would demand them.

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Now something that we've
talked about in a lot of detail

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in other videos is how
the marginal revenue curve

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is different than the
demand curve for a monopoly.

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And that's because, if you
were to charge a price of,

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let's say, $500 per room,

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you might be able to get one
room rented out for the night

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but no other rooms.

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And if you wanted to get
two rooms rented out,

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well, you would have to charge
$400, not just for that room.

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So now we'd get a little bit
further down this demand curve.

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When you charge $400 maybe
for that second room,

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because someone's
willingness to pay is $400.

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Well you might have to also
charge $400 for that first room.

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So in many monopoly industries,

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whatever you charge to one consumer,

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you have to charge to other consumers.

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Now, I know what some of you are thinking,

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hey, that doesn't always
happen in a hotel,

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and that's why I picked this example.

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Because we're going to
look at the situation

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where one, you do have to
charge the same to everyone,

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and then we'll look at another situation

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known as price discrimination,

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where you don't have to
charge the same to everyone.

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But let's just go with the model

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where you do have to change
the same to everyone.

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So when you go from one room
at 500 to two rooms at 400,

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your marginal revenue
isn't the incremental 400,

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because this 500 is now 400 as well.

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So you go from 500 to 800,

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so your marginal revenue
is an incremental 300,

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as you go from 500 total

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to 400 plus 400 or 800 total.

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And that's why, and we go
into significant detail

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in other videos on this,

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and we do it with tables of numbers,

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and I encourage you to do that.

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That is why your marginal
revenue curve for a monopoly

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has twice the slope, the negative slope,

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than your demand curve would have.

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So your marginal revenue curve

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would look something like this.

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And we've already talked
about it in multiple videos,

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for any firm, it's rational
to produce the quantity

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where marginal cost is
equal to a marginal revenue.

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So this monopoly would
produce this quantity,

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and the price they would get,

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well, that quantity, we go
to look at the demand curve.

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The price would be right over there.

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So this monopoly firm would
be able to get that price,

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and we can think about what
its economic profit would be.

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On every room in this case,
it charges that price.

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And its average total cost

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is this blue line right over here.

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So its average total cost are there,

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so the difference is how much
economic profit per room,

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and then you multiply that
times the total number of rooms,

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and so this area is the
firms economic profit.

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Now, there is still some
consumer surplus here.

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This is benefit that consumers are getting

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above and beyond what
they're paying for it.

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So the consumer surplus in this situation

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would be all of this.

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So that first person who's willing

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to pay maybe $500 per room is now able

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to get this market price
that everyone is able to get,

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which is maybe $300 per room,

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and so this benefit for that one unit,

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it goes to the consumer.

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And we've also seen that there
is dead weight loss here.

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Your allocatively efficient
when marginal cost

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is equal to the demand curve,

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and so, we study that in other videos.

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This right over here is
our dead weight loss.

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But now let's imagine the other scenario.

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Let's say that we are a hotel,

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where we try to capture as
much of someone's willingness

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to pay as possible.

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And I'll give a little bit
of a idealistic scenario

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that doesn't really
exist in the real world,

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but just to look at an extreme case.

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Let's say that you were
able to get a computer

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that can read people's minds,

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and every time they call
for a quote on a room,

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you know exactly what their
willingness to pay is.

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So if I call, the computer says,

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hey, Sal's willingness to
pay for that room is $375.

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So you quote me, all right $375,

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and I say, okay, sure.

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And then when that first person

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who has a high willingness to pay calls,

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it says, okay, why don't
we quote them $500?

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And so we quote them $500,
and so they get that room.

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And so in that situation,
every incremental room,

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you don't have to change the
prices on all the other ones

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causing this marginal revenue
curve to slope down faster.

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Instead, every incremental room,

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you get those dollars.

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And so in that situation,

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your demand curve is equal to
your marginal revenue curve.

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You're able to discriminate on prices.

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Let me write this.

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This is price discrimination.

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You're able to charge,
and price discrimination

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is a general term for
charging different customers,

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different consumers different rates,

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ideally based on their willingness to pay,

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and it might sound bad.

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In normal life, we don't like
discriminating against others,

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but price discrimination
is a very legitimate thing,

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and actually you will see it happen

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in things like the hotel industry,

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where they're going to try
to charge different prices

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to different people based
on their willingness to pay

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for essentially the same room.

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If you go stay in a hotel,

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it's very likely that the person

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in an identical room next to
you is paying a different rate.

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Airlines will also do it.

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Now they're not going to be
able to do it as perfectly

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as I just described with
this magical computer,

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but they'll do it where,
depending on how far ahead

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or whether you can return the ticket

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or cancel your reservation,

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you can get a different price,

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and the prices change over time.

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And so they're trying to capture as much

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of consumer's willingness
to pay as possible.

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But if we took this extreme situation

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where you're able to charge exactly

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everyone their willingness to pay,

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well then, what is going
to be the rational quantity

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for this profit maximizing
monopoly to produce?

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Well, once again, it would be

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where marginal cost
intersects marginal revenue,

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but the marginal revenue
curve is now the demand curve.

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So it'd be right over there.

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That's the quantity that
this monopoly would produce.

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And then, what's the price it would get?

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Pause this video and think about that.

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Well, you might be tempted
to just go horizontally here

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and say, okay, this is
the price it would get,

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like we did here.

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But remember, it's able
to get a different price

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for every consumer.

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So there isn't just one price,
like in this first example,

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that everyone is paying.

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So we can see the
quantity it is producing.

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You can see the average
total cost at that quantity,

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but the profit per room
is going to be dependent

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on what people are willing to pay.

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This first person is
going to pay a lot for it.

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They're not going to
get any consumer surplus

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in this extreme example,

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and so all of this is going
to accrue to the firm,

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and that's going to be the case

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for all of these consumers
in this extreme circumstance.

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And so now, you have a
fascinating situation.

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Notice, when this monopoly firm

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is able to do price discrimination,

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now, it's economic profit is far larger,

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economic profit.

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The consumer surplus shrunk
through price discrimination.

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In the extreme example, it disappeared.

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But you also see that this

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is actually allocatively efficient.

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That we are actually
producing at a quantity

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where marginal cost is
equal to marginal revenue.


Shutting down or exiting industry based on price

WEBVTT
Kind: captions
Language: en

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- [Instructor] We've spent
several videos already

00:00:02.090 --> 00:00:05.060
talking about graphs like you see here.

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This is the graph for a particular firm.

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Maybe it's making donuts

00:00:08.360 --> 00:00:09.690
so it's in the donut industry.

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And we can see how the marginal cost

00:00:11.630 --> 00:00:13.150
relates to the average variable cost

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and average total cost.

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We go into some depth several videos ago

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but we see that trend that marginal cost

00:00:18.800 --> 00:00:20.350
can trend down initially

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because as quantity increases

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each incremental unit could benefit

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from things like specialization.

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And then the marginal cost,

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the cost of each incremental unit

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as a function of quantity, could go up

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because of things like coordination costs.

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And then we've also seen

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how that relates to average variable cost,

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that while marginal cost

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is below average variable cost,

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every incremental unit is going

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to bring down the average variable cost.

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But then when marginal cost

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crosses average variable cost,

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well, now every incremental unit

00:00:48.240 --> 00:00:50.320
is going to bring up the
average variable cost.

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And the same thing happens

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once it crosses the average total cost.

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And of course the difference

00:00:54.740 --> 00:00:56.290
between, for any give quantity

00:00:56.290 --> 00:00:57.930
between the average total cost

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and the average variable cost,

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that is the average fixed cost.

00:01:01.480 --> 00:01:02.680
Now, with that out of the way

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we're going to think about how this firm

00:01:04.630 --> 00:01:07.640
would react under different
market conditions.

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We're going to assume that it's

00:01:08.700 --> 00:01:10.730
in a very competitive, or we could say

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a perfectly competitive market,

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and so it is a price taker.

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And so let's first imagine

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what would be a positive scenario

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for this firm?

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Let's imagine the price up here.

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So let's call this P sub-one

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and in the previous video we already said

00:01:28.760 --> 00:01:31.680
it would be rational for
a profit-maximizing firm

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to produce at a quantity

00:01:33.610 --> 00:01:35.880
where the marginal cost

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and the marginal revenue meet.

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If we're talking about
a competitive market,

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then this price right over here

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is not going to be a function

00:01:46.060 --> 00:01:47.330
of the firm's quantity,

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so that's why it's horizontal,

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and it would be the same thing

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as the marginal revenue.

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So, in this situation, at P sub-one,

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the firm would produce Q sub-one,

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and this is a good situation for the firm,

00:02:02.520 --> 00:02:04.960
because the price that it's getting

00:02:04.960 --> 00:02:08.000
is higher than its average total cost.

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And so there is going to be

00:02:10.230 --> 00:02:12.610
a nice amount of profit for this firm.

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The profit is going to be the price

00:02:15.010 --> 00:02:18.300
minus the average total
cost at that quantity

00:02:18.300 --> 00:02:20.930
times the actual quantity.

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So, because P one is greater

00:02:23.940 --> 00:02:26.880
than the average total cost,

00:02:26.880 --> 00:02:29.900
we have a situation where
the firm is profitable.

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Firm is profitable.

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It would want to stay in the market.

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But because you have a profitable firm

00:02:37.680 --> 00:02:39.070
in this market and you're likely to have

00:02:39.070 --> 00:02:41.010
many profitable firms in that market,

00:02:41.010 --> 00:02:43.350
it will probably attract entrance.

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Attract,

00:02:45.230 --> 00:02:46.300
attract entrance.

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Other people might say,

00:02:47.450 --> 00:02:50.090
hey, I want to make just as much money

00:02:50.090 --> 00:02:51.790
as this donut company right over here,

00:02:51.790 --> 00:02:53.530
then this firm, and so you'll probably

00:02:53.530 --> 00:02:55.430
have more and more
entrance into the market,

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which will probably reduce the prices.

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Now, they could reduce the prices

00:03:00.550 --> 00:03:03.100
until you get to a price that looks

00:03:03.100 --> 00:03:05.660
something like this.

00:03:05.660 --> 00:03:08.550
So I will call that P sub-two.

00:03:08.550 --> 00:03:10.420
Now, a profit-maximizing firm

00:03:10.420 --> 00:03:12.740
in this world would keep producing

00:03:12.740 --> 00:03:14.600
until the marginal cost is equal

00:03:14.600 --> 00:03:15.440
to the marginal revenue,

00:03:15.440 --> 00:03:16.920
which in this case is the price.

00:03:16.920 --> 00:03:18.600
And so this would be,

00:03:18.600 --> 00:03:20.240
my lines aren't completely straight there,

00:03:20.240 --> 00:03:21.430
but you get the idea.

00:03:21.430 --> 00:03:22.993
So that's Q sub-two.

00:03:23.830 --> 00:03:26.560
Now, in this situation, P sub-two

00:03:26.560 --> 00:03:29.690
is equal to the average total cost

00:03:29.690 --> 00:03:31.770
so the firm is breakeven.

00:03:31.770 --> 00:03:34.270
It's not running at a loss or a profit.

00:03:34.270 --> 00:03:35.510
So it is breakeven.

00:03:35.510 --> 00:03:38.110
And so here the firm is neutral

00:03:38.110 --> 00:03:39.770
about whether in the long run

00:03:39.770 --> 00:03:43.440
it stays in the market
or it exits the market.

00:03:43.440 --> 00:03:46.370
But you're no longer
likely attracting entrance.

00:03:46.370 --> 00:03:48.280
So no longer

00:03:49.210 --> 00:03:50.850
attracting,

00:03:50.850 --> 00:03:53.703
attracting entrance.

00:03:55.150 --> 00:03:56.790
But it does make sense for the firm

00:03:56.790 --> 00:04:00.070
to keep operating at this situation

00:04:00.070 --> 00:04:01.090
even in the long run,

00:04:01.090 --> 00:04:03.790
because it is at least breakeven.

00:04:03.790 --> 00:04:06.350
Now let's imagine another scenario.

00:04:06.350 --> 00:04:10.050
Let's imagine this price level.

00:04:10.050 --> 00:04:11.950
So for whatever reason the market price

00:04:11.950 --> 00:04:14.120
gets to that as we've talked about.

00:04:14.120 --> 00:04:16.560
A rational firm would be producing

00:04:16.560 --> 00:04:18.720
at Q sub-three.

00:04:18.720 --> 00:04:20.470
And at P sub-three right over here

00:04:20.470 --> 00:04:21.710
there's some interesting things.

00:04:21.710 --> 00:04:24.310
Because P sub-three is less

00:04:24.310 --> 00:04:26.610
than your average total cost,

00:04:26.610 --> 00:04:28.960
your firm is running at a loss.

00:04:28.960 --> 00:04:30.690
It's running at a loss here.

00:04:30.690 --> 00:04:32.833
So running, so firm,

00:04:33.810 --> 00:04:36.403
firm not profitable.

00:04:37.560 --> 00:04:39.090
Not profitable.

00:04:39.090 --> 00:04:39.923
Now, you might say,

00:04:39.923 --> 00:04:41.810
well, what is this firm likely to do?

00:04:41.810 --> 00:04:43.200
Would it just shut down?

00:04:43.200 --> 00:04:45.380
Well, in the short run it would not

00:04:45.380 --> 00:04:48.010
make sense for this firm to shut down,

00:04:48.010 --> 00:04:50.320
because the price that it's getting

00:04:50.320 --> 00:04:52.680
is still higher than its
average variable cost.

00:04:52.680 --> 00:04:54.730
In the short run, the fixed cost,

00:04:54.730 --> 00:04:56.260
they've already been spent.

00:04:56.260 --> 00:04:58.340
So, you might as well get as much

00:04:58.340 --> 00:05:01.820
incremental profit on
the margin as you can.

00:05:01.820 --> 00:05:04.230
And so as long as the price is higher

00:05:04.230 --> 00:05:06.460
than the average variable cost,

00:05:06.460 --> 00:05:08.530
well, outside of their fixed cost

00:05:08.530 --> 00:05:10.220
they're still making some money

00:05:10.220 --> 00:05:13.000
to make up those fixed costs.

00:05:13.000 --> 00:05:14.950
So you have two things going on.

00:05:14.950 --> 00:05:17.110
So, they would stay operating

00:05:17.110 --> 00:05:21.430
in the short run, stay operating,

00:05:21.430 --> 00:05:24.660
operating in the short run,

00:05:24.660 --> 00:05:25.650
short run.

00:05:25.650 --> 00:05:28.500
But what would this
firm do in the long run?

00:05:28.500 --> 00:05:30.210
Well, in the long run it makes no sense

00:05:30.210 --> 00:05:32.460
to have a, to be in a market

00:05:32.460 --> 00:05:34.040
where you can't make a profit.

00:05:34.040 --> 00:05:36.020
So in the long run it will exit.

00:05:36.020 --> 00:05:37.390
So, it will exit

00:05:38.230 --> 00:05:42.497
in the long run.

00:05:42.497 --> 00:05:44.350
And in general, the terminology

00:05:44.350 --> 00:05:45.957
when people are talking about "well,

00:05:45.957 --> 00:05:47.730
"do you start or stop in the short run,"

00:05:47.730 --> 00:05:49.377
they usually talk about "do you either

00:05:49.377 --> 00:05:51.550
"shut down or operate in the short run?"

00:05:51.550 --> 00:05:53.510
And in the long run we say hey,

00:05:53.510 --> 00:05:55.380
are you going to sell your factories,

00:05:55.380 --> 00:05:58.210
or somehow dismantle them,

00:05:58.210 --> 00:05:59.570
or are you gonna build new factories?

00:05:59.570 --> 00:06:00.840
That's all about exiting

00:06:00.840 --> 00:06:02.790
or entering the industry.

00:06:02.790 --> 00:06:04.500
And of course, you have another

00:06:04.500 --> 00:06:06.660
even worse scenario for this firm,

00:06:06.660 --> 00:06:09.130
which might be down here,

00:06:09.130 --> 00:06:11.310
where you have price sub-four.

00:06:11.310 --> 00:06:13.780
Here, in theory, this is where

00:06:13.780 --> 00:06:15.990
we intersect the marginal cost curve,

00:06:15.990 --> 00:06:17.740
Q sub-four.

00:06:17.740 --> 00:06:19.220
Now here it makes no sense

00:06:19.220 --> 00:06:20.860
for the company to operate at all.

00:06:20.860 --> 00:06:22.890
So, because P sub-four is less

00:06:22.890 --> 00:06:25.830
than the average total cost,

00:06:25.830 --> 00:06:29.010
you would want to exit in the long run,

00:06:29.010 --> 00:06:32.168
exit in long run, exit the market.

00:06:32.168 --> 00:06:33.810
Well, you wouldn't even
wait for that long,

00:06:33.810 --> 00:06:35.340
wait to sell your factories.

00:06:35.340 --> 00:06:37.060
Because P sub-four is less

00:06:37.060 --> 00:06:38.960
than your average variable cost,

00:06:38.960 --> 00:06:41.070
you would also just shut down,

00:06:41.070 --> 00:06:42.173
shut down,

00:06:43.090 --> 00:06:47.780
in the short run.

00:06:47.780 --> 00:06:48.860
So, a big picture.

00:06:48.860 --> 00:06:50.170
From a firm's point of view,

00:06:50.170 --> 00:06:52.100
you obviously want to be at P one

00:06:52.100 --> 00:06:53.470
where you make a profit,

00:06:53.470 --> 00:06:55.800
but you might attract entrance.

00:06:55.800 --> 00:06:57.840
At P sub-two, you as a firm

00:06:57.840 --> 00:06:59.300
in the long run are neutral

00:06:59.300 --> 00:07:01.040
versus exiting the market

00:07:01.040 --> 00:07:02.130
or entering the market,

00:07:02.130 --> 00:07:03.780
or other people entering the market.

00:07:03.780 --> 00:07:05.380
You're at breakeven.

00:07:05.380 --> 00:07:09.420
At P sub-three, in the long run,

00:07:09.420 --> 00:07:12.260
you'd want to exit because
you're not profitable,

00:07:12.260 --> 00:07:14.410
if the prices stay at P sub-three.

00:07:14.410 --> 00:07:17.080
Your price is below
your average total cost

00:07:17.080 --> 00:07:19.210
at the rational quantity to produce,

00:07:19.210 --> 00:07:22.160
so in the long run you would exit.

00:07:22.160 --> 00:07:25.550
But because P sub-three is greater

00:07:25.550 --> 00:07:27.390
than your average variable cost

00:07:27.390 --> 00:07:29.310
at the rational quantity,

00:07:29.310 --> 00:07:32.120
you would stay operating in the short run.

00:07:32.120 --> 00:07:34.290
And then, the last scenario, of course,

00:07:34.290 --> 00:07:36.160
is P sub-four where the price gets so low

00:07:36.160 --> 00:07:37.120
that just doesn't make sense

00:07:37.120 --> 00:07:39.163
to even operate another moment.


Factoring using polynomial division

WEBVTT
Kind: captions
Language: en

00:00:00.280 --> 00:00:02.600
- [Instructor] We are
told the polynomial p of x

00:00:02.600 --> 00:00:05.120
is equal to four x to the
third plus 19 x squared

00:00:05.120 --> 00:00:10.120
plus 19 x minus six has a
known factor of x plus two.

00:00:10.510 --> 00:00:15.150
Rewrite p of x as a
product of linear factors.

00:00:15.150 --> 00:00:18.430
So pause this video and see
if you can have a go at that.

00:00:18.430 --> 00:00:20.430
All right, now let's
work through it together.

00:00:20.430 --> 00:00:23.080
So if they didn't give
us this second piece

00:00:23.080 --> 00:00:25.270
of information that as a known factor of x

00:00:25.270 --> 00:00:29.930
plus two, this polynomial
would not be so easy to factor.

00:00:29.930 --> 00:00:31.890
But because we know we have a known factor

00:00:31.890 --> 00:00:33.940
of x plus two, I could divide that

00:00:33.940 --> 00:00:36.690
into our expression and
figure out what I have

00:00:36.690 --> 00:00:39.450
left over, and then see if
I can factor from there.

00:00:39.450 --> 00:00:40.570
So let's do that.

00:00:40.570 --> 00:00:45.570
Let's divide x plus two
into our polynomial.

00:00:46.710 --> 00:00:50.150
So it's four x to the third power plus 19

00:00:50.150 --> 00:00:54.370
x squared plus 19 x minus six.

00:00:54.370 --> 00:00:57.090
And we've done this
multiple times already.

00:00:57.090 --> 00:00:58.550
We look at the highest-degree terms.

00:00:58.550 --> 00:01:00.840
X goes into four x to the third.

00:01:00.840 --> 00:01:04.020
Four x squared times, I
put that in the x squared

00:01:04.020 --> 00:01:06.210
or the second-degree column.

00:01:06.210 --> 00:01:10.040
Four x squared times x
is four x to the third.

00:01:10.040 --> 00:01:15.040
Four x squared times two is
eight x, eight x squared.

00:01:15.240 --> 00:01:18.140
And then I wanna subtract
these from what I have up here.

00:01:18.140 --> 00:01:19.510
So I'll subtract.

00:01:19.510 --> 00:01:23.060
And then I'm going to be
left with 19 x squared

00:01:23.060 --> 00:01:26.230
minus eight x squared is 11 x squared.

00:01:26.230 --> 00:01:31.230
And then I will bring
down, bring down the 19 x.

00:01:31.530 --> 00:01:33.880
So plus 19 x.

00:01:33.880 --> 00:01:37.320
And so once again, look
at x and 11 x squared.

00:01:37.320 --> 00:01:40.830
X goes into 11, x squared 11 x times.

00:01:40.830 --> 00:01:43.760
So that's plus 11 x.

00:01:43.760 --> 00:01:46.930
11 x times x is 11 x squared.

00:01:46.930 --> 00:01:50.100
11 x times two is 22 x.

00:01:50.100 --> 00:01:53.380
Need to subtract these from
what we have in that teal color.

00:01:53.380 --> 00:01:57.270
And we are left with 19
minus 22 of something

00:01:57.270 --> 00:01:58.710
is negative three of that something.

00:01:58.710 --> 00:02:00.730
In this case, it's negative three x.

00:02:00.730 --> 00:02:05.200
And then we bring down that negative six.

00:02:05.200 --> 00:02:08.140
And then we look at, once again, at the x

00:02:08.140 --> 00:02:09.650
and the negative three x.

00:02:09.650 --> 00:02:13.460
X goes into negative three
x negative three times.

00:02:13.460 --> 00:02:17.710
And so negative three times
x is negative three x.

00:02:17.710 --> 00:02:20.220
Negative three times two is negative six.

00:02:20.220 --> 00:02:22.290
And then if we wanna
subtract what we have in red

00:02:22.290 --> 00:02:23.960
from what we have in magenta.

00:02:23.960 --> 00:02:26.170
So I could just multiply
them both by negative.

00:02:26.170 --> 00:02:29.750
And so everything just cancels
out and we have no remainder.

00:02:29.750 --> 00:02:32.680
And so we can rewrite p of x now.

00:02:32.680 --> 00:02:36.230
We can rewrite p of x as being equal to x

00:02:36.230 --> 00:02:40.080
plus two times all of this business,

00:02:40.080 --> 00:02:45.080
four x squared plus 11 x minus three.

00:02:45.180 --> 00:02:46.730
Now we're not done yet because we haven't

00:02:46.730 --> 00:02:49.310
expressed it as a product
of linear factors.

00:02:49.310 --> 00:02:52.210
This one over here is
linear but four x squared

00:02:52.210 --> 00:02:54.360
plus 11 x minus three,
that's still quadratic.

00:02:54.360 --> 00:02:56.740
So we have to factor that further.

00:02:56.740 --> 00:02:58.580
And let's see, there's a couple of ways

00:02:58.580 --> 00:02:59.413
we could approach it.

00:02:59.413 --> 00:03:01.270
We could use, well, we could try with this

00:03:01.270 --> 00:03:05.210
and the quadratic formula or
we could factor by grouping.

00:03:05.210 --> 00:03:07.190
And to factor by grouping,
and the whole reason

00:03:07.190 --> 00:03:08.700
we have to factor by grouping is we have

00:03:08.700 --> 00:03:11.430
a leading coefficient
here that is not one.

00:03:11.430 --> 00:03:14.020
And so we need to think of two numbers

00:03:14.020 --> 00:03:17.690
whose product is equal to
four times negative three.

00:03:17.690 --> 00:03:19.100
So we have to think of
two numbers, let's just

00:03:19.100 --> 00:03:19.933
call them a and b.

00:03:19.933 --> 00:03:23.180
A times b needs to be equal to four times

00:03:23.180 --> 00:03:25.150
negative three, which is negative 12.

00:03:25.150 --> 00:03:29.510
And a plus b needs to be equal to 11.

00:03:29.510 --> 00:03:32.630
And so the best that I
can think of, they have

00:03:32.630 --> 00:03:34.610
to be opposite signs 'cause their product

00:03:34.610 --> 00:03:35.780
is a negative.

00:03:35.780 --> 00:03:39.710
So If I had negative, if
I had, if I had positive

00:03:39.710 --> 00:03:41.450
12 and negative one, that works.

00:03:41.450 --> 00:03:44.760
If a is equal to negative
one and b is equal

00:03:44.760 --> 00:03:46.780
to positive 12, that works.

00:03:46.780 --> 00:03:49.210
And so what I wanna do is I wanna take

00:03:49.210 --> 00:03:52.450
this first-degree term
right over here, 11 x,

00:03:52.450 --> 00:03:56.250
and I wanna split it into a
12 x and a negative one x.

00:03:56.250 --> 00:03:57.600
So let's do that.

00:03:57.600 --> 00:04:01.615
So I can, let's just focus on this part,

00:04:01.615 --> 00:04:04.320
on this part right now
and then I'll put it all

00:04:04.320 --> 00:04:05.790
back together at the end.

00:04:05.790 --> 00:04:10.790
So I can rewrite all of this
business as four x squared.

00:04:12.600 --> 00:04:15.460
And instead of writing the 11 x there,

00:04:15.460 --> 00:04:17.990
I'm gonna use this blue
color, I'm gonna break

00:04:17.990 --> 00:04:19.810
it up as a 12 x.

00:04:19.810 --> 00:04:24.810
So plus 12 x and then minus one x.

00:04:24.930 --> 00:04:27.230
Notice these two still add up to 11 x.

00:04:27.230 --> 00:04:31.760
And then I have my minus three.

00:04:31.760 --> 00:04:34.750
And then, let's see,
out of these first two

00:04:34.750 --> 00:04:36.470
what can I factor out?

00:04:36.470 --> 00:04:38.810
Let's see, I can factor out a four x.

00:04:38.810 --> 00:04:41.070
So I can rewrite these first two as,

00:04:41.070 --> 00:04:43.500
and if this is unfamiliar
to you I encourage you

00:04:43.500 --> 00:04:46.240
to review factoring by
grouping on Khan Academy.

00:04:46.240 --> 00:04:48.890
So if we factor out a
four x that's going to be,

00:04:48.890 --> 00:04:51.870
it's going to be, we're
gonna be left with an x here.

00:04:51.870 --> 00:04:55.260
And then we're gonna be
left with a three over here.

00:04:55.260 --> 00:04:59.900
And then these second
two terms, if we factor

00:04:59.900 --> 00:05:02.820
out a negative one, so
I'll write negative one,

00:05:02.820 --> 00:05:06.300
times, we're going to
have an x plus three.

00:05:06.300 --> 00:05:09.950
And so then we can factor
out the x plus three.

00:05:09.950 --> 00:05:11.950
So let's do that.

00:05:11.950 --> 00:05:13.420
I'm running out of colors.

00:05:13.420 --> 00:05:16.980
So I factor out the x
plus three and I am left

00:05:16.980 --> 00:05:21.980
with x plus three times, times four x.

00:05:24.020 --> 00:05:25.370
And I'm gonna keep these colors the same

00:05:25.370 --> 00:05:27.110
so you know where I got 'em from.

00:05:27.110 --> 00:05:29.330
Four x minus one.

00:05:29.330 --> 00:05:32.200
This is a very colorful
solution that we have over here.

00:05:32.200 --> 00:05:33.060
And there you have it.

00:05:33.060 --> 00:05:36.240
I factored the second part
into these two factors.

00:05:36.240 --> 00:05:38.520
And so now I can put it all together.

00:05:38.520 --> 00:05:42.410
I can rewrite p of x as a
product of linear factors.

00:05:42.410 --> 00:05:47.410
P of x is equal to x plus
two times x plus three

00:05:48.440 --> 00:05:50.700
times four x minus one.

00:05:50.700 --> 00:05:52.410
Four x minus one.

00:05:52.410 --> 00:05:56.143
And we are done.


Introduction to polynomial division

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.480
- [Instructor] Earlier in
your algebraic careers,

00:00:02.480 --> 00:00:04.840
you learned how to multiply polynomials.

00:00:04.840 --> 00:00:09.840
So for example, if we had x
plus two times four x plus five,

00:00:12.410 --> 00:00:14.020
we learned that this is the same thing

00:00:14.020 --> 00:00:16.410
as really doing the
distributive property twice.

00:00:16.410 --> 00:00:20.720
But you can multiply x times
four x to get four x-squared.

00:00:20.720 --> 00:00:23.940
You can multiply x times
five to get five x.

00:00:23.940 --> 00:00:27.430
You can multiply two times
four x to get eight x.

00:00:27.430 --> 00:00:30.080
And you can multiply two
times five to get 10.

00:00:30.080 --> 00:00:32.790
If what I just did looks foreign to you,

00:00:32.790 --> 00:00:34.540
I encourage you to review

00:00:34.540 --> 00:00:37.460
multiplying binomials on Kahn Academy.

00:00:37.460 --> 00:00:39.880
But then this could be simplified as being

00:00:39.880 --> 00:00:44.880
four x-squared plus 13x plus 10.

00:00:44.890 --> 00:00:48.810
What we will now do as we
advance our algebraic careers

00:00:48.810 --> 00:00:50.990
is to think about how do
we go the other way around?

00:00:50.990 --> 00:00:54.140
How do we divide polynomials?

00:00:54.140 --> 00:00:57.320
So for example, we might
see an expression like this.

00:00:57.320 --> 00:01:02.320
Four x-squared plus 13x plus
10 divided by x plus two.

00:01:06.130 --> 00:01:08.470
Now, you might already
know from your knowledge

00:01:08.470 --> 00:01:10.730
of multiplication and
division that this is really

00:01:10.730 --> 00:01:12.840
reversing what we just did up here,

00:01:12.840 --> 00:01:16.280
that if x plus two times four x plus five

00:01:16.280 --> 00:01:19.980
is equal to this business,
then this business

00:01:19.980 --> 00:01:24.093
divided by x plus two should
be equal to four x plus five.

00:01:25.350 --> 00:01:27.290
And even if you didn't
have this information

00:01:27.290 --> 00:01:28.420
that we have up here,

00:01:28.420 --> 00:01:30.540
there's ways that you
would've approached this.

00:01:30.540 --> 00:01:32.590
One way is that you
could've tried to factor

00:01:32.590 --> 00:01:34.790
what we have here in the numerator.

00:01:34.790 --> 00:01:38.600
You could've said hey, let me
maybe factor this by grouping.

00:01:38.600 --> 00:01:39.780
And let's do that right over here.

00:01:39.780 --> 00:01:43.740
Four x-squared plus 13x plus 10.

00:01:43.740 --> 00:01:45.440
When you factor by grouping, you say hey,

00:01:45.440 --> 00:01:48.240
can I think of two numbers whose product

00:01:48.240 --> 00:01:49.550
is equal to four times 10?

00:01:49.550 --> 00:01:52.390
So I say a times b is equal to 40.

00:01:52.390 --> 00:01:54.830
And whose sum is equal to 13.

00:01:54.830 --> 00:01:58.330
So a plus b is equal to 13.

00:01:58.330 --> 00:02:00.810
And let's see what it could be.

00:02:00.810 --> 00:02:03.930
It could be eight and five.

00:02:03.930 --> 00:02:06.700
So a is equal to eight,
b is equal to five.

00:02:06.700 --> 00:02:08.800
This is just factoring by
grouping right over here.

00:02:08.800 --> 00:02:10.780
Once again, if this is unfamiliar to you,

00:02:10.780 --> 00:02:11.980
I encourage you to review

00:02:11.980 --> 00:02:13.800
factoring by grouping on Kahn Academy.

00:02:13.800 --> 00:02:15.980
And so we can break apart this 13x

00:02:15.980 --> 00:02:18.050
as an eight x and a five x.

00:02:18.050 --> 00:02:21.060
So we can rewrite this as four x-squared

00:02:21.990 --> 00:02:26.990
plus eight x plus five x plus 10, plus 10.

00:02:28.220 --> 00:02:31.060
And notice, it's just re,
undoing what we did up here.

00:02:31.060 --> 00:02:32.350
But we're assuming we don't even know

00:02:32.350 --> 00:02:35.080
about what we did up here,
that someone just gave us

00:02:35.080 --> 00:02:37.260
this quadratic and said factor it.

00:02:37.260 --> 00:02:40.010
And then here you could say
all right, for these first

00:02:40.010 --> 00:02:42.640
two terms, I could factor out a four x.

00:02:42.640 --> 00:02:45.430
So it becomes four x times x plus two.

00:02:45.430 --> 00:02:48.640
And then these second two terms,
I could factor out a five.

00:02:48.640 --> 00:02:51.770
So plus five times x plus two.

00:02:51.770 --> 00:02:53.450
And then I can factor out an x plus two.

00:02:53.450 --> 00:02:57.320
So it becomes, it becomes x plus two.

00:02:57.320 --> 00:02:59.260
I'll scroll down a little bit.

00:02:59.260 --> 00:03:03.433
It become x plus two, I'm
just factoring that out,

00:03:03.433 --> 00:03:08.433
times four x plus five, four x plus five,

00:03:09.660 --> 00:03:11.570
which is exactly what we had up there.

00:03:11.570 --> 00:03:14.560
And so you could rewrite this expression

00:03:14.560 --> 00:03:16.190
that involves division,

00:03:16.190 --> 00:03:19.060
sometimes called a
rational expression, as,

00:03:19.060 --> 00:03:24.060
we could rewrite this as x plus
two times four x plus five,

00:03:26.410 --> 00:03:29.710
all of that over x plus two.

00:03:29.710 --> 00:03:33.600
And as long as x does
not equal negative two,

00:03:33.600 --> 00:03:35.360
we can divide the
numerator and denominator

00:03:35.360 --> 00:03:37.810
by x plus two and we're going to be left

00:03:37.810 --> 00:03:40.500
with four x plus five.

00:03:40.500 --> 00:03:41.700
And we could constrain it.

00:03:41.700 --> 00:03:46.700
We could say for x not
equaling negative two.

00:03:48.250 --> 00:03:50.390
This is just a little bit of a primer.

00:03:50.390 --> 00:03:53.700
As we go deeper, we'll do many,
many, many examples of this.

00:03:53.700 --> 00:03:55.920
And we'll also see that
there's other techniques

00:03:55.920 --> 00:03:58.970
other than just factoring
this numerator over here.

00:03:58.970 --> 00:04:01.060
And we're going to do it with
higher degree polynomials,

00:04:01.060 --> 00:04:02.590
third degree polynomials.

00:04:02.590 --> 00:04:05.960
And we're gonna learn something
known as polynomial division

00:04:05.960 --> 00:04:07.540
where it's going to look an awful lot,

00:04:07.540 --> 00:04:09.200
and it's actually gonna
have a lot of similarities,

00:04:09.200 --> 00:04:11.680
with the long division
that you likely learned

00:04:11.680 --> 00:04:14.130
in probably fourth or fifth grade,

00:04:14.130 --> 00:04:17.010
where we're gonna take
expressions like x plus two

00:04:17.010 --> 00:04:19.470
and we're gonna divide it into

00:04:19.470 --> 00:04:23.730
four x-squared plus 13x plus 10.

00:04:23.730 --> 00:04:26.370
And instead of place value,
we'll have a new notion

00:04:26.370 --> 00:04:28.100
of place value which
is around which degree

00:04:28.100 --> 00:04:29.110
term you're thinking about.

00:04:29.110 --> 00:04:31.330
So we're gonna do things
like, and we're gonna do these

00:04:31.330 --> 00:04:33.840
completely in other
videos, hey, how many times

00:04:33.840 --> 00:04:35.430
does x go into four x-squared?

00:04:35.430 --> 00:04:37.790
Hey, it goes four x times.

00:04:37.790 --> 00:04:39.230
So you'd write the four x there

00:04:39.230 --> 00:04:41.160
and then you'd multiply
four x times x-squared.

00:04:41.160 --> 00:04:42.610
You get four x-squared.

00:04:42.610 --> 00:04:44.800
Four x times two is eight x.

00:04:44.800 --> 00:04:46.350
Then you subtract these.

00:04:46.350 --> 00:04:48.480
And then you keep going
just like you would

00:04:48.480 --> 00:04:49.720
typically do long division.

00:04:49.720 --> 00:04:52.030
13x minus eight x is five x.

00:04:52.030 --> 00:04:54.260
And then you bring that 10 down.

00:04:54.260 --> 00:04:55.740
And then you say hey, how many times

00:04:55.740 --> 00:04:56.617
does x go into five x?

00:04:56.617 --> 00:04:59.470
And you say hey, it goes five times.

00:04:59.470 --> 00:05:04.020
Five times x is equal to five x.

00:05:04.020 --> 00:05:06.560
Five times two is equal to 10.

00:05:06.560 --> 00:05:09.030
And then you subtract this

00:05:09.030 --> 00:05:11.260
and you're left with no remainder.

00:05:11.260 --> 00:05:13.040
And so notice, you just
saw when x plus two

00:05:13.040 --> 00:05:16.310
is divided into this,
you get four x plus five,

00:05:16.310 --> 00:05:18.270
just what we saw right over there.

00:05:18.270 --> 00:05:20.150
So we're gonna explore
these multiple techniques,

00:05:20.150 --> 00:05:21.560
including polynomial division.

00:05:21.560 --> 00:05:23.540
And we're also going to see what happens

00:05:23.540 --> 00:05:25.833
when you do have a remainder there.


Recognizing number pattern examples

WEBVTT
Kind: captions
Language: en

00:00:00.340 --> 00:00:02.970
- [Instructor] We are asked
which expression can we use

00:00:02.970 --> 00:00:05.700
to find the missing number in the pattern?

00:00:05.700 --> 00:00:08.210
So pause this video and
see if you can try this

00:00:08.210 --> 00:00:10.330
before we do this together.

00:00:10.330 --> 00:00:13.010
All right, now let's try this together.

00:00:13.010 --> 00:00:15.210
And the way I would tackle
it is I'll try to see,

00:00:15.210 --> 00:00:18.500
hey, can I figure out the
pattern between these numbers?

00:00:18.500 --> 00:00:20.430
And then I could try to
figure out the missing one.

00:00:20.430 --> 00:00:23.200
So to go from 36 to 45,

00:00:23.200 --> 00:00:27.060
it looks like I'm increasing
by nine, so I add nine.

00:00:27.060 --> 00:00:31.680
And it looks like to go from
63 to 72, we're adding nine.

00:00:31.680 --> 00:00:33.070
It's good to make sure that the pattern

00:00:33.070 --> 00:00:34.970
that you think is happening is happening

00:00:34.970 --> 00:00:36.920
across all of the numbers
that you're seeing.

00:00:36.920 --> 00:00:40.330
And to go from 72 to 81, it
looks like you're adding nine.

00:00:40.330 --> 00:00:41.710
So I'm pretty confident

00:00:41.710 --> 00:00:46.100
that to go from 45 to this blank
number, I have to add nine.

00:00:46.100 --> 00:00:50.450
And we see here indeed,
choice B is 45 plus nine.

00:00:50.450 --> 00:00:51.400
This number here,

00:00:51.400 --> 00:00:52.960
you don't even have to
figure out what it is.

00:00:52.960 --> 00:00:55.520
Some of you might realize it's 54.

00:00:55.520 --> 00:00:58.870
But this is just going to be
this number plus nine is going

00:00:58.870 --> 00:01:00.130
to go in the blank.

00:01:00.130 --> 00:01:02.420
So 45 plus nine.

00:01:02.420 --> 00:01:03.813
Let's do another example.

00:01:05.470 --> 00:01:07.280
So here's it's a little bit different.

00:01:07.280 --> 00:01:09.370
It says if the pattern continues,

00:01:09.370 --> 00:01:11.590
what will be the sixth
number in the pattern?

00:01:11.590 --> 00:01:14.500
Pause this video and
try to figure that out.

00:01:14.500 --> 00:01:16.900
All right, so first let's
figure out the pattern.

00:01:16.900 --> 00:01:20.270
To go from two to six, it
looks like we added four.

00:01:20.270 --> 00:01:23.760
Then we are adding four
again to go from six to 10.

00:01:23.760 --> 00:01:26.500
Then we're adding four
again to go from 10 to 14,

00:01:26.500 --> 00:01:28.410
so it seems like we're
adding four every time.

00:01:28.410 --> 00:01:31.830
And this is the first one,
two, three, four numbers

00:01:31.830 --> 00:01:32.680
in the pattern.

00:01:32.680 --> 00:01:34.770
So the fifth number in
the pattern is going

00:01:34.770 --> 00:01:38.410
to be 14 plus four,
which is going to be 18.

00:01:38.410 --> 00:01:39.900
And then the sixth number

00:01:39.900 --> 00:01:43.040
in the pattern is going to
be 18 plus four, which is 22.

00:01:43.040 --> 00:01:43.873
And that's what they want.

00:01:43.873 --> 00:01:46.210
They want the sixth number in the pattern.

00:01:46.210 --> 00:01:49.110
So 22 is choice D.

00:01:49.110 --> 00:01:50.360
Let's do another example.

00:01:51.510 --> 00:01:55.350
Here we're told the table
below shows the number

00:01:55.350 --> 00:01:59.020
of laps Carly ran around
the track each day.

00:01:59.020 --> 00:02:02.010
Which rule describes the pattern shown

00:02:02.010 --> 00:02:04.900
by the number of laps Carly ran?

00:02:04.900 --> 00:02:06.640
So like always, pause this video

00:02:06.640 --> 00:02:08.253
and see if you can work on this.

00:02:09.150 --> 00:02:12.120
All right, it seems like
every time a day goes by,

00:02:12.120 --> 00:02:16.640
the number of laps are going up by three.

00:02:16.640 --> 00:02:19.830
So if you wanted to
figure out the next day,

00:02:19.830 --> 00:02:24.240
let's say day six, you would
add three from day five.

00:02:24.240 --> 00:02:27.803
So the pattern here is to add three.


Patterns in hundreds chart

WEBVTT
Kind: captions
Language: en

00:00:00.220 --> 00:00:02.750
- [Instructor] So what
we have in this chart

00:00:02.750 --> 00:00:06.230
is all the numbers from
one to 100 organized

00:00:06.230 --> 00:00:07.500
in a fairly neat way.

00:00:07.500 --> 00:00:09.970
It's a somewhat intuitive
way to organize it

00:00:09.970 --> 00:00:12.190
where each row you have
10, so you go from one

00:00:12.190 --> 00:00:16.890
to 10, then 11 to 20, then
21 to 30, all the way to 100.

00:00:16.890 --> 00:00:18.960
And what we're gonna
look at is interesting

00:00:18.960 --> 00:00:21.340
patterns that might emerge from this.

00:00:21.340 --> 00:00:23.290
So if you look at what's highlighted here

00:00:23.290 --> 00:00:26.590
in this purplish color, what
numbers are highlighted there?

00:00:26.590 --> 00:00:29.080
Pause this video and think about it.

00:00:29.080 --> 00:00:32.400
Well, what's highlighted
are all of the even numbers.

00:00:32.400 --> 00:00:35.040
And you can see the even
numbers form these nice,

00:00:35.040 --> 00:00:38.420
neat pillars or columns on this chart.

00:00:38.420 --> 00:00:40.850
And we can look at that
and immediately start

00:00:40.850 --> 00:00:42.460
to see some patterns.

00:00:42.460 --> 00:00:45.960
For example, what numbers are always going

00:00:45.960 --> 00:00:49.090
to be in the ones place for an even number

00:00:49.090 --> 00:00:50.780
looking at this chart?

00:00:50.780 --> 00:00:53.040
Well, you can see in the
ones place you're either

00:00:53.040 --> 00:00:56.080
always going to have a
two in the ones place

00:00:57.060 --> 00:01:00.630
or you're going to have
a four or you're going

00:01:00.630 --> 00:01:03.590
to have a six or you're
going to have an eight

00:01:03.590 --> 00:01:06.150
or you're going to have a zero.

00:01:06.150 --> 00:01:08.540
So that ones place digit is always going

00:01:08.540 --> 00:01:10.680
to be an even number.

00:01:10.680 --> 00:01:12.023
Let's do another example.

00:01:13.020 --> 00:01:15.680
Here we've highlighted different numbers.

00:01:15.680 --> 00:01:17.580
So pause this video and
think about what's true

00:01:17.580 --> 00:01:19.980
about all of the numbers
that we've highlighted?

00:01:20.920 --> 00:01:24.220
Well, you might notice that
these are all multiples of five.

00:01:24.220 --> 00:01:28.900
Five, 10, 15, 20, 25, 30,
35, 40, so on and so forth.

00:01:28.900 --> 00:01:33.290
And so these form these
two columns on this chart.

00:01:33.290 --> 00:01:36.410
And here we can see very
clearly that multiples

00:01:36.410 --> 00:01:40.170
of five are either going to have a five

00:01:40.170 --> 00:01:44.240
in the ones place like
we have right over here.

00:01:44.240 --> 00:01:45.760
So they're either gonna have a five

00:01:45.760 --> 00:01:49.460
in the ones place or
they're gonna have a zero

00:01:49.460 --> 00:01:50.410
in the ones place.

00:01:50.410 --> 00:01:51.970
You might've realized
that before but you see

00:01:51.970 --> 00:01:55.850
it very clearly in these
two, you see it very

00:01:55.850 --> 00:01:58.573
clearly in these two columns.

00:01:59.750 --> 00:02:01.423
Let's do one other example.

00:02:02.490 --> 00:02:04.990
This one is really
interesting because it's not

00:02:04.990 --> 00:02:08.160
just one of those clean
column-type patterns.

00:02:08.160 --> 00:02:09.670
It looks like we started
one, and then we have

00:02:09.670 --> 00:02:11.995
this diagonal, then we go to 100.

00:02:11.995 --> 00:02:14.778
What's a pattern that
could describe how we go

00:02:14.778 --> 00:02:17.546
from one number to the
next, or another way

00:02:17.546 --> 00:02:20.170
of saying it, what's a rule for why

00:02:20.170 --> 00:02:22.350
we highlighted these numbers in purple?

00:02:22.350 --> 00:02:24.200
Pause the video and think about that.

00:02:25.180 --> 00:02:27.070
All right, well, one thing is if we go

00:02:27.070 --> 00:02:28.200
from one number to the next, you go

00:02:28.200 --> 00:02:31.560
from one to 10, we add nine.

00:02:31.560 --> 00:02:34.120
To go from 10 to 19, we add nine.

00:02:34.120 --> 00:02:37.150
To go from 19 to 28, we added nine.

00:02:37.150 --> 00:02:40.920
So each number is nine
plus the previous one.

00:02:40.920 --> 00:02:43.160
And if you go all the
way to 91, 91 plus nine

00:02:43.160 --> 00:02:44.840
is of course 100.

00:02:44.840 --> 00:02:46.280
Now, it's important to
realize these are not

00:02:46.280 --> 00:02:50.410
multiples of nine because we
started at one, not at zero.

00:02:50.410 --> 00:02:53.070
If you started at zero,
you go zero, nine, 18,

00:02:53.070 --> 00:02:56.140
so forth and so on until
you go nine, 18, 27,

00:02:56.140 --> 00:02:59.060
36, 45, 54, 63, 72, 81.

00:02:59.060 --> 00:03:01.240
Those would've been the multiples of nine.

00:03:01.240 --> 00:03:03.210
But everything got
shifted because we started

00:03:03.210 --> 00:03:04.410
at one, not at zero.

00:03:04.410 --> 00:03:06.780
So we go from one and then 10, 19, 28,

00:03:06.780 --> 00:03:10.570
all the way down this diagonal
and then we go back to 100.

00:03:10.570 --> 00:03:12.660
And so this is a really interesting
thing to think about it.

00:03:12.660 --> 00:03:17.130
These are all the
multiples of nine plus one

00:03:17.130 --> 00:03:18.610
is another way to think about it

00:03:18.610 --> 00:03:20.980
or this is if we started
at one and we keep

00:03:20.980 --> 00:03:23.350
adding nines, these are all the numbers

00:03:23.350 --> 00:03:24.810
that we would highlight.

00:03:24.810 --> 00:03:26.610
But you can see a pattern.

00:03:26.610 --> 00:03:29.460
Whenever you add nine
to a number, the value

00:03:29.460 --> 00:03:32.300
of the ones place decreases by one.

00:03:32.300 --> 00:03:34.420
You see that here as you
go down these diagonals.

00:03:34.420 --> 00:03:39.420
You go from nine, eight,
seven, six, five, four,

00:03:39.750 --> 00:03:41.950
three, two, one.

00:03:41.950 --> 00:03:44.280
And even up here, we started at zero

00:03:44.280 --> 00:03:46.350
but you don't have a lower digit than zero

00:03:46.350 --> 00:03:47.680
so it just goes back to nine.

00:03:47.680 --> 00:03:50.210
So it goes zero, then goes up to nine,

00:03:50.210 --> 00:03:51.970
then it keeps going
down, down, down, down,

00:03:51.970 --> 00:03:54.280
all the way until it gets to zero again

00:03:54.280 --> 00:03:57.230
and then it starts going
down from there again.

00:03:57.230 --> 00:03:59.570
So once again, interesting
patterns to look at.

00:03:59.570 --> 00:04:01.040
I encourage you to look
at a chart like this

00:04:01.040 --> 00:04:03.190
and think about what
patterns can you find?


Setting up 2 step expressions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.160
- [Instructor] My book is 58 pages.

00:00:03.160 --> 00:00:05.990
I have already read 13 pages.

00:00:05.990 --> 00:00:09.680
I plan to read five pages each day

00:00:09.680 --> 00:00:12.220
until I finish the book.

00:00:12.220 --> 00:00:15.990
Which equation could I use to find out

00:00:15.990 --> 00:00:20.990
how many days, d, it will take
to finish reading the book?

00:00:21.450 --> 00:00:24.580
So pause this video and see
if you can figure that out.

00:00:24.580 --> 00:00:25.630
Okay, so let's first think

00:00:25.630 --> 00:00:28.210
about the important
information they're giving us.

00:00:28.210 --> 00:00:32.160
They're giving us the number
of pages that are in the book.

00:00:32.160 --> 00:00:35.200
They're giving us how
much we've already read.

00:00:35.200 --> 00:00:37.000
We've already read 13 pages.

00:00:37.000 --> 00:00:40.110
And they also tell us
how much we plan to read.

00:00:40.110 --> 00:00:43.280
I plan to read five pages each day

00:00:43.280 --> 00:00:44.980
until the book is finished.

00:00:44.980 --> 00:00:47.620
And then we want an
equation that could use

00:00:47.620 --> 00:00:48.990
to find how many days.

00:00:48.990 --> 00:00:51.280
And d is how we're going to
represent the number of days,

00:00:51.280 --> 00:00:52.790
let me do it in the same color,

00:00:52.790 --> 00:00:54.650
it'll take to finish reading the book.

00:00:54.650 --> 00:00:58.450
All right, so if we knew how
many pages we have to read

00:00:58.450 --> 00:01:00.210
and we were divided by the fact

00:01:00.210 --> 00:01:02.970
that we're reading five pages per days,

00:01:02.970 --> 00:01:04.990
well, then that would give
us the number of days.

00:01:04.990 --> 00:01:08.080
So how many pages do we have
left to read in the book?

00:01:08.080 --> 00:01:10.700
Well, the book is 58 pages.

00:01:10.700 --> 00:01:15.310
We've already read 13, so we
could subtract out 13 of those.

00:01:15.310 --> 00:01:18.250
So this expression right over here,

00:01:18.250 --> 00:01:21.220
this is how many pages
we have left in the book.

00:01:21.220 --> 00:01:22.053
Let me write that down.

00:01:22.053 --> 00:01:26.210
How many pages left.

00:01:28.150 --> 00:01:30.310
And if we wanna figure out
how many days it's going

00:01:30.310 --> 00:01:31.530
to take to read these,

00:01:31.530 --> 00:01:34.800
we would divide this by how
many pages we read each day.

00:01:34.800 --> 00:01:37.820
And we see that we read
five pages each day.

00:01:37.820 --> 00:01:42.820
So if we divide this by five,
then this whole expression,

00:01:43.080 --> 00:01:44.790
let me do it in this color,

00:01:44.790 --> 00:01:46.820
this whole expression is going to tell us

00:01:46.820 --> 00:01:51.120
how many days left.

00:01:51.120 --> 00:01:53.600
How many days left.

00:01:53.600 --> 00:01:56.670
And so this is going to be equal to d

00:01:56.670 --> 00:01:58.640
'cause d is the number of days left

00:01:58.640 --> 00:02:00.340
to finish reading the book.

00:02:00.340 --> 00:02:01.840
Now, if we look at these choices,

00:02:01.840 --> 00:02:06.010
that's exactly what we have
right over here in choice C.

00:02:06.010 --> 00:02:08.060
We can make sure that these
others don't make sense.

00:02:08.060 --> 00:02:11.390
58 plus 13, no, that would be a situation

00:02:11.390 --> 00:02:13.240
where we're reading 13 pages more

00:02:13.240 --> 00:02:14.750
than the number of pages
that are in the book.

00:02:14.750 --> 00:02:15.740
That doesn't make sense.

00:02:15.740 --> 00:02:18.920
And that's happening for
both choice A and choice B.

00:02:18.920 --> 00:02:22.150
It would also not make
sense to multiply by five

00:02:22.150 --> 00:02:25.470
because it's not going to
take you five times more

00:02:25.470 --> 00:02:27.600
than the number of pages to read the book.

00:02:27.600 --> 00:02:28.790
You're going to divide by five.

00:02:28.790 --> 00:02:31.800
You're reading five pages every day.

00:02:31.800 --> 00:02:32.823
And we are done.


Examples thinking about multiplying even and odd numbers

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:03.250
- [Instructor] We are told
Liam multiplies two numbers

00:00:03.250 --> 00:00:05.756
and gets an even product.

00:00:05.756 --> 00:00:09.330
What could be true about
the numbers Liam multiplied?

00:00:09.330 --> 00:00:11.140
It says Choose 2 answers.

00:00:11.140 --> 00:00:13.340
So pause this video and
see if you can figure out

00:00:13.340 --> 00:00:16.580
which two of these could be true.

00:00:16.580 --> 00:00:18.410
All right, now let's do this together.

00:00:18.410 --> 00:00:20.460
And we have to think
about what could be true.

00:00:20.460 --> 00:00:21.800
They don't have to be true.

00:00:21.800 --> 00:00:23.641
It just has to be possible.

00:00:23.641 --> 00:00:27.107
Okay, now is it possible
that both numbers were even?

00:00:27.107 --> 00:00:32.107
Can you multiply two even
numbers and get an even product?

00:00:32.453 --> 00:00:33.330
Well, sure.

00:00:33.330 --> 00:00:35.580
And I can imagine examples
that get us there.

00:00:35.580 --> 00:00:39.690
If you multiply two times
four, which is equal to eight,

00:00:39.690 --> 00:00:43.240
you have an even times an
even that gets you an even.

00:00:43.240 --> 00:00:46.700
In fact, in general, when you
multiply two evens together,

00:00:46.700 --> 00:00:49.140
you will get an even product.

00:00:49.140 --> 00:00:54.140
And so this is definitely,
this definitely could be true.

00:00:54.710 --> 00:00:57.057
Both numbers were odd.

00:00:57.057 --> 00:01:00.400
You might remember that
an odd times an odd

00:01:00.400 --> 00:01:02.338
is always going to be an odd,

00:01:02.338 --> 00:01:04.850
or you could look at a few examples

00:01:04.850 --> 00:01:07.210
and feel pretty confident
that that's the case.

00:01:07.210 --> 00:01:11.930
If you multiply three times
five, that's equal to 15.

00:01:11.930 --> 00:01:14.485
Odd times odd is equal to an odd number.

00:01:14.485 --> 00:01:19.485
If you multiply seven
times nine, you get 63.

00:01:20.760 --> 00:01:23.830
Once again, odd times odd is equal to odd.

00:01:23.830 --> 00:01:27.027
So there's no way that you
can multiply two odd numbers

00:01:27.027 --> 00:01:29.640
and get an even product.

00:01:29.640 --> 00:01:31.680
So we would rule this out.

00:01:31.680 --> 00:01:35.743
One number was even
and one number was odd.

00:01:35.743 --> 00:01:37.330
Well, let's think about what happens

00:01:37.330 --> 00:01:40.019
when you multiply an even times an odd.

00:01:40.019 --> 00:01:44.225
If I multiply two times
three, that's equal to six.

00:01:44.225 --> 00:01:47.480
So you have an even,
I'll just put an e there,

00:01:47.480 --> 00:01:50.879
times an odd, is equal to an even.

00:01:50.879 --> 00:01:54.500
And so the two ways that
you can get an even product

00:01:54.500 --> 00:01:56.948
is either if you're multiplying
an even times an even

00:01:56.948 --> 00:02:00.510
or an even times an odd.

00:02:00.510 --> 00:02:04.130
Essentially, an even times
either an even or an odd

00:02:04.130 --> 00:02:06.260
is going to give you an even product.

00:02:06.260 --> 00:02:07.623
Let's do another example.

00:02:08.910 --> 00:02:11.640
We're asked which of the
following have an even product?

00:02:11.640 --> 00:02:13.450
It says, once again, Choose 2 answers.

00:02:13.450 --> 00:02:14.650
So pause this video again,

00:02:14.650 --> 00:02:16.300
and see if you can figure it out.

00:02:17.600 --> 00:02:19.602
Okay, 8 fives.

00:02:19.602 --> 00:02:24.600
So 8 fives is the same
thing as eight times five,

00:02:24.600 --> 00:02:27.320
and this is a situation where
we're multiplying an even,

00:02:27.320 --> 00:02:30.730
e for even, times an odd, which
we've already talked about

00:02:30.730 --> 00:02:34.600
is going to give us an even
result, an even product.

00:02:34.600 --> 00:02:37.370
You don't even have to know
that eight times five is 40,

00:02:37.370 --> 00:02:38.550
which is an even number.

00:02:38.550 --> 00:02:41.320
We have an even times an odd,

00:02:41.320 --> 00:02:43.270
which is going to give us an even product.

00:02:43.270 --> 00:02:46.370
So this will definitely
give us an even product.

00:02:46.370 --> 00:02:49.770
An odd number times an even number.

00:02:49.770 --> 00:02:51.070
Well, we've already talked about that.

00:02:51.070 --> 00:02:54.530
Any even, odd, or an even times an even

00:02:54.530 --> 00:02:56.470
is going to give you an even product.

00:02:56.470 --> 00:02:59.150
So we can fill that one in.

00:02:59.150 --> 00:03:02.640
And then five times three,
you have an odd times an odd,

00:03:02.640 --> 00:03:04.570
which is going to give you an odd number.

00:03:04.570 --> 00:03:07.020
Even if you didn't know
that this is going to be 15,

00:03:07.020 --> 00:03:08.630
you would know that an odd times an odd

00:03:08.630 --> 00:03:10.460
is always going to give
you an odd product,

00:03:10.460 --> 00:03:11.630
not an even product.

00:03:11.630 --> 00:03:13.413
So we would pick these first two.


2 step estimation word problems

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:02.670
- [Instructor] We are told
that a race car driver

00:00:02.670 --> 00:00:04.610
has 28 cars.

00:00:04.610 --> 00:00:06.850
Each car has four tires.

00:00:06.850 --> 00:00:09.990
He has to replace all
the tires on the cars.

00:00:09.990 --> 00:00:13.670
He has 22 tires right now.

00:00:13.670 --> 00:00:17.630
Estimate the total number
of tires he needs to buy.

00:00:17.630 --> 00:00:19.750
So pause this video and
see if you can do that.

00:00:19.750 --> 00:00:22.380
And the key word here is estimate.

00:00:22.380 --> 00:00:23.970
You don't have to figure out exactly.

00:00:23.970 --> 00:00:25.540
You just need to figure out roughly

00:00:25.540 --> 00:00:28.410
the total number of tires he needs to buy.

00:00:28.410 --> 00:00:30.300
All right, now let's work
through this together.

00:00:30.300 --> 00:00:34.030
And as I mentioned, the
key word here is estimate.

00:00:34.030 --> 00:00:36.980
And estimating is something
we'll do throughout our lives.

00:00:36.980 --> 00:00:39.710
I probably do it once a day at least.

00:00:39.710 --> 00:00:41.140
And when you're estimating,

00:00:41.140 --> 00:00:43.670
you want to make all the
numbers roughly right,

00:00:43.670 --> 00:00:46.220
but also think about,
hey, what do I need to,

00:00:46.220 --> 00:00:47.420
how do I need to them about them

00:00:47.420 --> 00:00:49.930
in order to make the
computation a little bit easier

00:00:49.930 --> 00:00:51.690
or a little bit friendlier?

00:00:51.690 --> 00:00:53.700
So if I wanted to figure out

00:00:53.700 --> 00:00:56.680
exactly how many tires he needed to buy,

00:00:56.680 --> 00:01:00.210
I would say, hey, he has 28 cars.

00:01:00.210 --> 00:01:02.740
He has 28 cars.

00:01:02.740 --> 00:01:05.320
Each car has four tires.

00:01:05.320 --> 00:01:07.730
So each car has four tires.

00:01:07.730 --> 00:01:09.530
So if you wanted to know the
total of number of tires,

00:01:09.530 --> 00:01:12.260
you would multiply 28 times four.

00:01:12.260 --> 00:01:13.560
And then it says he has to replace

00:01:13.560 --> 00:01:15.530
all the tires on the cars.

00:01:15.530 --> 00:01:16.660
And if he didn't have any,

00:01:16.660 --> 00:01:18.890
this is the number that
he would have to replace,

00:01:18.890 --> 00:01:21.410
but it says he has 22 tires right now.

00:01:21.410 --> 00:01:23.040
So if we were trying
to figure out exactly,

00:01:23.040 --> 00:01:25.050
you would say 28 times four,

00:01:25.050 --> 00:01:26.790
that's how many he needs to replace,

00:01:26.790 --> 00:01:29.740
but he already has 22,
so you subtract that out,

00:01:29.740 --> 00:01:32.860
and this whole number would
be the number he needs to buy.

00:01:32.860 --> 00:01:35.250
Now this calculation isn't so easy to do,

00:01:35.250 --> 00:01:38.340
especially if you're doing it
on a napkin or in your head.

00:01:38.340 --> 00:01:40.730
But that's why we're going to estimate.

00:01:40.730 --> 00:01:41.910
So if we're estimating, we say, hey,

00:01:41.910 --> 00:01:44.830
what's a friendlier number than 28?

00:01:44.830 --> 00:01:47.430
Well, 28 is pretty close to 30.

00:01:47.430 --> 00:01:51.930
So I could say, so if I say,
hey, 28 is pretty close to 30.

00:01:51.930 --> 00:01:54.860
So that's our way of making
the 28 a little bit friendlier.

00:01:54.860 --> 00:01:57.380
Four is already pretty easy to deal with.

00:01:57.380 --> 00:02:00.300
And then the 22, well,
that could be a lot easier

00:02:00.300 --> 00:02:03.100
to deal with, if instead of it being a 22,

00:02:03.100 --> 00:02:06.840
if that were, let's say that's
approximately equal to 20.

00:02:06.840 --> 00:02:09.590
So 22 is approximately, roughly equal,

00:02:09.590 --> 00:02:12.800
that's what these squiggly
equal signs mean, to 20.

00:02:12.800 --> 00:02:15.010
And so that simplifies
the calculation a lot.

00:02:15.010 --> 00:02:19.690
We could say that this thing
up here is roughly equal to,

00:02:19.690 --> 00:02:21.657
he roughly has 30 cars,

00:02:21.657 --> 00:02:25.151
and each of those cars need four tires,

00:02:25.151 --> 00:02:28.550
and then he's going to figure
out the total number of tires

00:02:28.550 --> 00:02:30.800
he needs to replace, but
then he subtracts out

00:02:30.800 --> 00:02:32.150
the number that he already has.

00:02:32.150 --> 00:02:34.520
So instead of subtracting out 22,

00:02:34.520 --> 00:02:37.580
we're gonna subtract out
roughly how many he already has,

00:02:37.580 --> 00:02:38.770
roughly 20.

00:02:38.770 --> 00:02:41.470
And this computation
right here is a lot easier

00:02:41.470 --> 00:02:42.590
to do in our head.

00:02:42.590 --> 00:02:47.497
Three times four is 12,
so 30 times four is 120.

00:02:48.500 --> 00:02:51.090
So this part right over here is 120.

00:02:51.090 --> 00:02:53.624
And then you subtract 20 from that.

00:02:53.624 --> 00:02:58.624
Well, 120 minus 20, well,
that is going to be 100.

00:02:59.640 --> 00:03:03.610
So he roughly needs, if I were
to estimate the total number

00:03:03.610 --> 00:03:07.623
of tires he needs to buy,
it would be about 100 tires.


Worked example of a profit maximization problem

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:02.860
- [Instructor] We're
told corn is used as food

00:00:02.860 --> 00:00:05.520
and as an input in the
production of ethanol,

00:00:05.520 --> 00:00:07.090
an alternative fuel.

00:00:07.090 --> 00:00:12.060
Assume corn is produced in a
perfectly competitive market.

00:00:12.060 --> 00:00:15.350
Draw correctly labeled side-by-side graphs

00:00:15.350 --> 00:00:19.720
for the corn market and a
representative corn farmer.

00:00:19.720 --> 00:00:22.310
On your graphs, show
each of the following:

00:00:22.310 --> 00:00:25.360
the equilibrium price and
quantity in the corn market,

00:00:25.360 --> 00:00:29.630
labeled P sub M and Q sub M,
respectfully, respectively;

00:00:29.630 --> 00:00:32.870
the profit-maximizing quantity of corn

00:00:32.870 --> 00:00:34.920
produced by the representative farmer

00:00:34.920 --> 00:00:39.060
earning zero economic
profit, labeled Q sub F.

00:00:39.060 --> 00:00:40.550
So, like always, pause this video,

00:00:40.550 --> 00:00:41.790
and see if you can do this on your own

00:00:41.790 --> 00:00:44.230
before we work through it together.

00:00:44.230 --> 00:00:45.870
All right, now let's
work through it together.

00:00:45.870 --> 00:00:50.320
So, we're going to do correctly
labeled side-by-side graphs.

00:00:50.320 --> 00:00:52.970
So, let me do, this is going to be my

00:00:52.970 --> 00:00:55.720
horizontal axis for the market,

00:00:55.720 --> 00:00:57.480
and then this is going to be

00:00:57.480 --> 00:01:01.130
the horizontal axis for the farmer.

00:01:01.130 --> 00:01:06.130
And this is going to be quantity
in the market, quantity,

00:01:07.240 --> 00:01:08.870
quantity, and then this is going to be

00:01:08.870 --> 00:01:13.800
quantity for the farmer, quantity.

00:01:13.800 --> 00:01:17.750
And then this is going to
be price in the market.

00:01:17.750 --> 00:01:19.290
And whatever the market price is,

00:01:19.290 --> 00:01:20.670
that's also going to be the price

00:01:20.670 --> 00:01:21.870
that the farmer has to take,

00:01:21.870 --> 00:01:24.380
because it says it's a
perfectly competitive market.

00:01:24.380 --> 00:01:26.963
So, the farmer's going
to be a price taker here.

00:01:27.890 --> 00:01:29.830
So, let me make these axes.

00:01:29.830 --> 00:01:33.320
So, this is price right over here,

00:01:33.320 --> 00:01:35.750
and this is price over here.

00:01:35.750 --> 00:01:38.070
So, first, let's draw the corn market.

00:01:38.070 --> 00:01:42.320
So, let me label this corn market.

00:01:42.320 --> 00:01:44.760
And we've done this
multiple times already.

00:01:44.760 --> 00:01:47.400
Our demand curve might
look something like this.

00:01:47.400 --> 00:01:49.393
This is our demand curve,

00:01:50.330 --> 00:01:54.330
as when price is high,
low quantity demanded;

00:01:54.330 --> 00:01:56.480
when is price is low,
high quantity demanded.

00:01:56.480 --> 00:01:58.580
And supply goes the other way around.

00:01:58.580 --> 00:02:02.200
So, our supply curve would
look something like this.

00:02:02.200 --> 00:02:05.490
And then this point,
this helps us figure out,

00:02:05.490 --> 00:02:08.010
this is going to be our equilibrium price.

00:02:08.010 --> 00:02:11.620
So, that's P sub M, P sub M.

00:02:11.620 --> 00:02:13.300
And then this is going to be

00:02:13.300 --> 00:02:18.300
our equilibrium quantity, so Q sub M.

00:02:18.400 --> 00:02:20.920
Now, this graph over here,

00:02:20.920 --> 00:02:23.370
we are going to draw the farmer.

00:02:23.370 --> 00:02:26.393
So, this is going to be the farmer,

00:02:27.640 --> 00:02:30.030
the farmer's firm right over here.

00:02:30.030 --> 00:02:31.740
So, the farmer's going
to be a price taker.

00:02:31.740 --> 00:02:34.160
So, whatever the equilibrium
price in the market,

00:02:34.160 --> 00:02:35.490
that is going to be the price

00:02:35.490 --> 00:02:38.130
that the farmer is going to have to take.

00:02:38.130 --> 00:02:39.940
That market price is going to be

00:02:39.940 --> 00:02:42.560
the farmer's marginal revenue.

00:02:42.560 --> 00:02:44.790
Now, they say the
profit-maximizing quantity

00:02:44.790 --> 00:02:47.400
of corn produced by the farmer,

00:02:47.400 --> 00:02:50.260
by the representative farmer earning zero

00:02:50.260 --> 00:02:53.200
economic profit, labeled Q sub F.

00:02:53.200 --> 00:02:55.600
So, we're gonna have some
quantity right over here.

00:02:55.600 --> 00:02:58.060
It is the profit-maximizing quantity,

00:02:58.060 --> 00:03:01.500
but it's also zero economic profit.

00:03:01.500 --> 00:03:04.250
So, the zero economic profit tells us

00:03:04.250 --> 00:03:06.340
that the price must be equal

00:03:06.340 --> 00:03:11.130
to the average total
cost at that quantity.

00:03:11.130 --> 00:03:13.260
So, I can make an average total cost curve

00:03:13.260 --> 00:03:15.113
that looks something like this.

00:03:16.470 --> 00:03:18.270
And I'm going to make its minimum point

00:03:18.270 --> 00:03:21.640
intersect that market price,

00:03:21.640 --> 00:03:24.890
because we know, from previous videos,

00:03:24.890 --> 00:03:27.490
that the profit-maximizing quantity

00:03:27.490 --> 00:03:30.110
happens where the marginal cost

00:03:30.110 --> 00:03:31.670
intersects the marginal revenue,

00:03:31.670 --> 00:03:33.830
which, in this case, would be the price

00:03:33.830 --> 00:03:35.510
that the farmer has to
take from the market.

00:03:35.510 --> 00:03:37.430
And we know that the marginal cost curve

00:03:37.430 --> 00:03:39.490
intersects the average total cost curve

00:03:39.490 --> 00:03:41.490
at this minimum point right over here.

00:03:41.490 --> 00:03:44.790
So, I could draw a marginal cost curve.

00:03:44.790 --> 00:03:47.320
It might look something like this.

00:03:47.320 --> 00:03:50.180
So, that is our marginal cost curve.

00:03:50.180 --> 00:03:52.200
And notice, the marginal cost curve

00:03:52.200 --> 00:03:54.850
intersects the average total
cost at that minimum point.

00:03:54.850 --> 00:03:57.710
We explain that in
multiple videos already.

00:03:57.710 --> 00:04:00.210
And we've explained in a previous video

00:04:00.210 --> 00:04:02.400
that the profit-maximizing quantity

00:04:02.400 --> 00:04:04.560
is the quantity at which the marginal cost

00:04:04.560 --> 00:04:06.730
and the marginal revenue meet.

00:04:06.730 --> 00:04:09.220
And the price is the marginal revenue.

00:04:09.220 --> 00:04:11.990
Beyond that point, every incremental unit,

00:04:11.990 --> 00:04:13.460
the corn farmer's gonna take a loss.

00:04:13.460 --> 00:04:17.150
It's gonna take 'em more
resources to produce that corn

00:04:17.150 --> 00:04:20.090
than they're going to be
able to get in the market.

00:04:20.090 --> 00:04:21.910
And we also mentioned that this has to be

00:04:21.910 --> 00:04:24.260
a situation of zero economic profit.

00:04:24.260 --> 00:04:27.970
So, the average total cost
has to be at that price,

00:04:27.970 --> 00:04:30.530
at that marginal revenue,
right at that point.

00:04:30.530 --> 00:04:34.860
So, this right over here would be our

00:04:34.860 --> 00:04:38.330
Q sub capital F.

00:04:38.330 --> 00:04:39.163
And we're done.


When to use multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:03.090
- [Instructor] We are
told there are 10 students

00:00:03.090 --> 00:00:04.750
in the poetry club.

00:00:04.750 --> 00:00:08.380
This week, each student wrote two poems.

00:00:08.380 --> 00:00:12.480
What does the expression
10 times two represent

00:00:12.480 --> 00:00:14.170
and they give us some choices.

00:00:14.170 --> 00:00:17.210
Pause this video and see
if you can work that out.

00:00:17.210 --> 00:00:19.390
All right, so there are 10 students

00:00:19.390 --> 00:00:23.810
in the poetry club and each
student wrote two poems.

00:00:23.810 --> 00:00:26.900
So I would use 10 times
two to figure out, hey,

00:00:26.900 --> 00:00:29.240
for each of those 10
students, if they each wrote

00:00:29.240 --> 00:00:34.240
two poems, then we have
10 times two total poems.

00:00:34.340 --> 00:00:37.270
And if we look at the
choices, that is choice C

00:00:37.270 --> 00:00:38.960
right over here.

00:00:38.960 --> 00:00:41.030
If we look at choice A,
the number of students,

00:00:41.030 --> 00:00:42.470
the number of students is 10.

00:00:42.470 --> 00:00:44.270
It wouldn't be 10 times two.

00:00:44.270 --> 00:00:46.650
And then the number of
poems each student wrote,

00:00:46.650 --> 00:00:48.940
well, that's the second
piece of information.

00:00:48.940 --> 00:00:50.270
They each wrote two poems.

00:00:50.270 --> 00:00:53.010
That wouldn't be 10 times two either.

00:00:53.010 --> 00:00:54.760
Let's do another one
of these, a little bit

00:00:54.760 --> 00:00:56.570
different this time.

00:00:56.570 --> 00:00:59.340
So now we're asked, let me scroll this

00:00:59.340 --> 00:01:02.670
over a little bit, which
problem can we solve

00:01:02.670 --> 00:01:04.780
with three times eight?

00:01:04.780 --> 00:01:07.240
So pause this video and have a go at it

00:01:07.240 --> 00:01:08.513
before we do it together.

00:01:09.440 --> 00:01:11.570
All right, now let's read these and see

00:01:11.570 --> 00:01:14.160
if you can solve it
with three times eight.

00:01:14.160 --> 00:01:16.130
Ellen has eight pieces of gum.

00:01:16.130 --> 00:01:17.150
Fair enough.

00:01:17.150 --> 00:01:19.600
She ate three pieces at school.

00:01:19.600 --> 00:01:21.260
Hope she didn't swallow the gum.

00:01:21.260 --> 00:01:22.750
That could be problematic.

00:01:22.750 --> 00:01:27.070
How many pieces of gum
does Ellen have left?

00:01:27.070 --> 00:01:29.460
So would you use multiplication for that?

00:01:29.460 --> 00:01:31.820
The way I would tackle
it, if she has eight

00:01:31.820 --> 00:01:35.080
pieces of gum and if she ate three

00:01:35.080 --> 00:01:36.970
and we wanna figure out how much gum does

00:01:36.970 --> 00:01:40.030
she have left, I'm assuming
outside of her body,

00:01:40.030 --> 00:01:42.370
well then I would
subtract three from eight.

00:01:42.370 --> 00:01:44.800
I wouldn't multiply eight by three.

00:01:44.800 --> 00:01:46.460
So I would rule this one out.

00:01:46.460 --> 00:01:47.590
That's not going to be my choice.

00:01:47.590 --> 00:01:49.050
I'm not going to solve this problem

00:01:49.050 --> 00:01:52.110
with multiplication or at
least by three times eight.

00:01:52.110 --> 00:01:53.980
Brendon has eight t-shirts.

00:01:53.980 --> 00:01:55.060
Fair enough.

00:01:55.060 --> 00:01:56.710
That's about how many I have.

00:01:56.710 --> 00:01:59.600
He goes to the store to
buy three more t-shirts.

00:01:59.600 --> 00:02:01.920
How many t-shirts does he have now?

00:02:01.920 --> 00:02:04.440
Would I solve that with multiplication?

00:02:04.440 --> 00:02:05.273
Well, let's see.

00:02:05.273 --> 00:02:07.320
The way I would solve it
is he has eight shirts

00:02:07.320 --> 00:02:09.010
and then he's buying three more.

00:02:09.010 --> 00:02:11.820
I would add three to
figure out how many total

00:02:11.820 --> 00:02:13.010
t-shirts he has.

00:02:13.010 --> 00:02:14.460
I wouldn't multiply by three.

00:02:14.460 --> 00:02:16.130
So I'd rule that out.

00:02:16.130 --> 00:02:18.520
So let's see how choice C is looking.

00:02:18.520 --> 00:02:21.760
There are eight chocolates in each box.

00:02:21.760 --> 00:02:25.080
Mike has three boxes of chocolates.

00:02:25.080 --> 00:02:27.370
How many chocolates are there?

00:02:27.370 --> 00:02:29.010
So this is interesting.

00:02:29.010 --> 00:02:31.960
So he has three boxes of chocolates.

00:02:31.960 --> 00:02:33.200
So let me draw those boxes.

00:02:33.200 --> 00:02:37.420
One box, two box, three
boxes of chocolate.

00:02:37.420 --> 00:02:38.950
And each of 'em have eight chocolates.

00:02:38.950 --> 00:02:42.390
One, two, three, four,
five, six, seven, eight.

00:02:42.390 --> 00:02:45.730
One, two, three, four,
five, six, seven, eight.

00:02:45.730 --> 00:02:49.820
One, two, three, four,
five, six, seven, eight.

00:02:49.820 --> 00:02:51.610
Well, yeah, if I wanna figure out how many

00:02:51.610 --> 00:02:55.030
total chocolates there are,
I have three groups of eight.

00:02:55.030 --> 00:02:58.990
I would solve that by
multiplying three times eight.

00:02:58.990 --> 00:03:01.540
So I like this choice a lot.

00:03:01.540 --> 00:03:03.153
Let's do another example.

00:03:04.810 --> 00:03:08.500
Here we are told Gerald is going to plant

00:03:08.500 --> 00:03:09.680
a garden this spring.

00:03:09.680 --> 00:03:11.150
That's a good thing to be doing.

00:03:11.150 --> 00:03:14.510
He plans to have seven plants in each row.

00:03:14.510 --> 00:03:16.490
There will be eight rows.

00:03:16.490 --> 00:03:19.160
Which expression can Gerald
use to find the total

00:03:19.160 --> 00:03:22.280
number of plants needed for his garden?

00:03:22.280 --> 00:03:25.650
Pause this video and see
if you can figure that out.

00:03:25.650 --> 00:03:26.980
All right, I'll do a
little bit of a drawing.

00:03:26.980 --> 00:03:28.530
I'll do it in green
'cause we're talking about

00:03:28.530 --> 00:03:29.500
rows of plants.

00:03:29.500 --> 00:03:33.510
So seven plants in each row
and there will be eight rows.

00:03:33.510 --> 00:03:38.200
So one, two, three,
four, five, six, seven.

00:03:38.200 --> 00:03:40.090
Seven plants in each row.

00:03:40.090 --> 00:03:42.660
And then we're going to have eight rows.

00:03:42.660 --> 00:03:45.040
So let me copy and paste that.

00:03:45.040 --> 00:03:49.793
So that's two rows.

00:03:50.740 --> 00:03:53.203
That's three rows.

00:03:54.080 --> 00:03:57.410
That is four rows.

00:03:57.410 --> 00:03:58.610
Getting there.

00:03:58.610 --> 00:04:00.103
Five rows.

00:04:02.010 --> 00:04:03.973
Six rows.

00:04:05.730 --> 00:04:07.483
Seven rows.

00:04:08.450 --> 00:04:13.350
And eight rows.

00:04:13.350 --> 00:04:15.060
So we don't have to actually calculate it

00:04:15.060 --> 00:04:16.450
but the way you could think about it

00:04:16.450 --> 00:04:18.800
is you have seven plants in each row

00:04:18.800 --> 00:04:21.890
and then you have eight rows of plants.

00:04:21.890 --> 00:04:23.340
Eight rows of plants.

00:04:23.340 --> 00:04:25.460
So if you wanna figure
out the total amount

00:04:25.460 --> 00:04:27.900
you would multiply eight times seven.

00:04:27.900 --> 00:04:29.950
And so let's see, that's this first choice

00:04:29.950 --> 00:04:30.783
right over here.

00:04:30.783 --> 00:04:32.900
You wouldn't multiply eight
times seven times seven.

00:04:32.900 --> 00:04:34.680
And eight plus seven
is definitely not going

00:04:34.680 --> 00:04:36.500
to give you the total
number of plants here.

00:04:36.500 --> 00:04:40.030
So yeah, we like using
the eight times seven.

00:04:40.030 --> 00:04:40.863
And we're done.


Order when multiplying commutative property of multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:01.960
- [Instructor] In this
video we're going to talk

00:00:01.960 --> 00:00:04.000
about one of the most important ideas

00:00:04.000 --> 00:00:06.950
in mathematics and that's
whether order matters

00:00:06.950 --> 00:00:09.440
when you multiply two numbers.

00:00:09.440 --> 00:00:13.720
So for example, is three times four

00:00:13.720 --> 00:00:17.360
the same thing as four times three?

00:00:17.360 --> 00:00:21.300
Are these two things equal to each other?

00:00:21.300 --> 00:00:23.390
And regardless of whether
these two are equal

00:00:23.390 --> 00:00:25.200
to each other, is it always the case

00:00:25.200 --> 00:00:26.640
that if I have some number times some

00:00:26.640 --> 00:00:29.530
other number, if I swap
the order am I going

00:00:29.530 --> 00:00:32.510
to get an equivalent number?

00:00:32.510 --> 00:00:35.260
Well, pause this video and see
if you can work through that.

00:00:35.260 --> 00:00:36.540
Try to think about that a little bit

00:00:36.540 --> 00:00:38.340
before we think through it together.

00:00:39.420 --> 00:00:41.780
Well, let's think through
this particular example.

00:00:41.780 --> 00:00:46.210
And we're going to do so with
the help of some angry cats.

00:00:46.210 --> 00:00:48.793
So we clearly see some angry cats here.

00:00:49.701 --> 00:00:51.150
Meow.

00:00:51.150 --> 00:00:52.890
Yes, they are angry.

00:00:52.890 --> 00:00:56.320
We could view this as
three groups of four.

00:00:56.320 --> 00:00:59.990
So this is one group right over here

00:00:59.990 --> 00:01:02.700
of angry cats, four angry cats.

00:01:02.700 --> 00:01:05.990
This is two groups of four angry cats.

00:01:05.990 --> 00:01:09.890
And this is three groups
of four angry cats

00:01:09.890 --> 00:01:12.920
if we view the first number here as groups

00:01:12.920 --> 00:01:14.580
of the second number.

00:01:14.580 --> 00:01:17.990
But we could also view it
as four groups of three.

00:01:17.990 --> 00:01:19.810
How would we do that?

00:01:19.810 --> 00:01:24.810
Well, we could have one
group of three angry cats.

00:01:24.910 --> 00:01:29.000
We can have two groups
of three angry cats.

00:01:29.000 --> 00:01:32.490
We can have three groups
of three angry cats.

00:01:32.490 --> 00:01:37.210
And we could of course have
four groups of three angry cats.

00:01:37.210 --> 00:01:39.200
So based on that, if you think of it,

00:01:39.200 --> 00:01:42.290
the first number as groups
of the second number,

00:01:42.290 --> 00:01:44.750
well, it seems like the
order doesn't matter.

00:01:44.750 --> 00:01:46.010
Another way you could think about it

00:01:46.010 --> 00:01:49.970
is here you have four
rows of three angry cats.

00:01:49.970 --> 00:01:54.720
You have one, two, three, four rows

00:01:54.720 --> 00:01:58.500
of one, two, three angry cats.

00:01:58.500 --> 00:02:00.620
And so to figure out how many total cats

00:02:00.620 --> 00:02:04.540
you have, you multiply four times three.

00:02:04.540 --> 00:02:08.220
But you could view this
same group of angry cats

00:02:08.220 --> 00:02:11.720
but just view it with a
slightly different perspective.

00:02:11.720 --> 00:02:14.830
So here we have our angry cats.

00:02:14.830 --> 00:02:18.190
And then let me rotate our angry cats,

00:02:18.190 --> 00:02:21.560
probably risking making
them a little bit angrier.

00:02:21.560 --> 00:02:22.993
Let me move them back in.

00:02:23.970 --> 00:02:28.250
And now, we could view this as three rows,

00:02:28.250 --> 00:02:31.900
one, two, three, of four cats in each row.

00:02:31.900 --> 00:02:33.910
So let me put all of these upright.

00:02:33.910 --> 00:02:38.180
So we have one, two,
three rows of one, two,

00:02:38.180 --> 00:02:39.900
three, four cats.

00:02:39.900 --> 00:02:41.290
Don't wanna look at those 'cause it might

00:02:41.290 --> 00:02:43.350
make us a little bit confused.

00:02:43.350 --> 00:02:47.100
And we're dealing with the
exact same number of cats.

00:02:47.100 --> 00:02:49.890
And so I'm only dealing
with three and four here

00:02:49.890 --> 00:02:52.580
but what you will see
is order doesn't matter

00:02:52.580 --> 00:02:54.800
when you are multiplying two numbers.

00:02:54.800 --> 00:02:57.800
And we could also see
that on a number line.

00:02:57.800 --> 00:03:01.050
And we could do that
with multiple examples.

00:03:01.050 --> 00:03:02.950
I'll keep a couple of angry cats here

00:03:02.950 --> 00:03:05.423
looking at us just to keep us in check.

00:03:06.290 --> 00:03:08.420
If we wanna think about three times four

00:03:08.420 --> 00:03:11.150
we could view it as four threes.

00:03:11.150 --> 00:03:15.967
So three, six, nine, 12.

00:03:17.650 --> 00:03:19.890
Or we could view it as three fours.

00:03:19.890 --> 00:03:23.980
Four, eight, and 12.

00:03:23.980 --> 00:03:25.700
And I focused a lot on three and four

00:03:25.700 --> 00:03:27.470
but we could do it with any two numbers

00:03:27.470 --> 00:03:29.460
that we're trying to multiply together.

00:03:29.460 --> 00:03:33.230
So let's say we wanted to do that with,

00:03:33.230 --> 00:03:37.610
I don't know, let's see
whether six times four,

00:03:37.610 --> 00:03:41.560
whether it's the same
thing as four times six.

00:03:41.560 --> 00:03:44.050
Well, you could view it as six fours.

00:03:44.050 --> 00:03:49.050
Four, eight, 12, 16, 20, and then 24.

00:03:54.470 --> 00:03:58.180
Or you could view it as four sixes.

00:03:58.180 --> 00:04:03.180
One six, 12, 18, and 24.

00:04:05.540 --> 00:04:08.120
So big takeaway, order doesn't matter

00:04:08.120 --> 00:04:10.010
when you are multiplying
numbers like this.

00:04:10.010 --> 00:04:11.190
And this is sometimes referred

00:04:11.190 --> 00:04:13.590
to as the commutative property.

00:04:13.590 --> 00:04:16.120
It's a fancy word but
it's really just saying

00:04:16.120 --> 00:04:18.170
that whether you're doing six times four

00:04:18.170 --> 00:04:20.760
or four times six the commutative property

00:04:20.760 --> 00:04:23.077
of multiplication says,
"Hey, those two things

00:04:23.077 --> 00:04:25.247
"are going to be equivalent."

00:04:26.480 --> 00:04:28.210
Meow.

00:04:28.210 --> 00:04:30.423
Yes, they are angry for being rotated.


Division as equal groupings

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:03.280
- [Instructor] So it looks
like we have some angry cats

00:00:03.280 --> 00:00:08.280
on our hands, meow, yeah they seem angry.

00:00:08.540 --> 00:00:11.010
And what we want to do is
think about how can we separate

00:00:11.010 --> 00:00:12.881
these angry cats 'cause the
only thing worse than an angry

00:00:12.881 --> 00:00:17.090
cat is 12 of them coordinating
potentially to take over

00:00:17.090 --> 00:00:17.960
the world.

00:00:17.960 --> 00:00:21.010
And so what we're going to
think about is can we divide

00:00:21.010 --> 00:00:25.383
these 12 angry cats into equal
groups and just for the sake

00:00:25.383 --> 00:00:29.130
of this video let's say we
want to divide them into three

00:00:29.130 --> 00:00:31.663
equal groups, how could we do that?

00:00:32.590 --> 00:00:37.580
Well let me see if I can do
it, let's see maybe this is one

00:00:37.580 --> 00:00:42.397
group right over here,
maybe this is another group

00:00:42.397 --> 00:00:47.290
right over there, and then
this is another group.

00:00:47.290 --> 00:00:49.943
Are those three equal groups?

00:00:50.970 --> 00:00:54.117
Let's see this top group has
four, this group has three,

00:00:54.117 --> 00:00:57.830
and this one has five, so no
these aren't three equal groups

00:00:57.830 --> 00:01:00.040
so that won't work it has
to be not just three groups,

00:01:00.040 --> 00:01:01.850
three equal groups.

00:01:01.850 --> 00:01:03.930
So let me see if I can
do something different.

00:01:03.930 --> 00:01:08.930
I could have these four in a
group and then I could have

00:01:09.099 --> 00:01:11.898
these four in a group, I think
this is going to work out

00:01:11.898 --> 00:01:13.480
this time.

00:01:13.480 --> 00:01:17.480
And then I could have
these four in a group.

00:01:17.480 --> 00:01:20.860
And as I just said out loud
all of these groups have

00:01:20.860 --> 00:01:23.480
exactly four angry cats in
them, which is a little bit

00:01:23.480 --> 00:01:25.740
more manageable and so this works out.

00:01:25.740 --> 00:01:28.500
I have just divided the
12 into three equal groups

00:01:28.500 --> 00:01:31.760
and how many cats are in
each of those equal groups?

00:01:31.760 --> 00:01:34.890
Well we can see very clearly
that there are four cats

00:01:34.890 --> 00:01:37.290
in each of those equal groups.

00:01:37.290 --> 00:01:40.230
So the way to think about
division in this example is you

00:01:40.230 --> 00:01:44.011
started with 12 and you
divide it into three equal,

00:01:44.011 --> 00:01:49.011
three equal groups and we
ended up with four cats,

00:01:50.034 --> 00:01:54.113
four cats per group.

00:01:55.050 --> 00:01:58.160
Now what if we wanted to
interpret it slightly differently?

00:01:58.160 --> 00:02:01.270
So let's bring in 12 more
angry cats which is always

00:02:01.270 --> 00:02:03.410
a dangerous thing to do but
we're doing it for the sake

00:02:03.410 --> 00:02:04.860
of learning so it's worth it.

00:02:06.372 --> 00:02:09.580
Meow, yep more angry cats.

00:02:09.580 --> 00:02:14.580
And now let's imagine 12
divided by three, 12 divided

00:02:14.950 --> 00:02:18.360
by three, but this time we're
not going to think about

00:02:18.360 --> 00:02:21.100
the three as three equal groups,
we're going to think about

00:02:21.100 --> 00:02:24.380
it as 12 divided into groups of three.

00:02:24.380 --> 00:02:28.200
So groups of three.

00:02:28.200 --> 00:02:31.400
So each of the equal groups
will have three in them

00:02:31.400 --> 00:02:35.500
and so then that's going to
tell us how many equal groups

00:02:35.500 --> 00:02:36.933
we're able to have.

00:02:37.963 --> 00:02:40.710
So let's do that, let's
create groups of three.

00:02:40.710 --> 00:02:44.270
So let's see that's a
group of three angry cats

00:02:44.270 --> 00:02:46.010
right over there.

00:02:46.010 --> 00:02:51.010
This is another group of three
angry cats, this is a group

00:02:51.520 --> 00:02:55.550
of three angry cats and then
this is a group of three

00:02:55.550 --> 00:02:56.820
angry cats.

00:02:56.820 --> 00:03:00.200
So how many equal groups of
three angry cats do I have?

00:03:00.200 --> 00:03:05.170
Well I have one, two, three,
and four equal groups.

00:03:05.170 --> 00:03:07.940
So if I want to take 12
angry cats and divide them

00:03:07.940 --> 00:03:12.270
into groups of three I
end up with four equal,

00:03:12.270 --> 00:03:15.430
four equal groups.

00:03:15.430 --> 00:03:18.030
So there you have it, there's
two different ways that

00:03:18.030 --> 00:03:20.020
we can imagine division.

00:03:20.020 --> 00:03:23.780
And the same exact division
expression 12 divided by three,

00:03:23.780 --> 00:03:26.850
you could do it as 12 being
divided into three equal

00:03:26.850 --> 00:03:31.850
groups and then the answer
would be how many things

00:03:32.040 --> 00:03:32.873
per group.

00:03:32.873 --> 00:03:35.910
Or you could say hey 12
divided into groups of three

00:03:35.910 --> 00:03:38.893
and then that would end
up with four equal groups.