Introduction to the Black-Scholes formula

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Kind: captions
Language: en

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Voiceover: We're now gonna talk about

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probably the most famous
formula in all of finance,

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and that's the Black-Scholes Formula,

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sometimes called the
Black-Scholes-Merton Formula,

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and it's named after these gentlemen.

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This right over here is Fischer Black.

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This is Myron Scholes.

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They really laid the
foundation for what led to

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the Black-Scholes Model and
the Black-Scholes Formula

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and that's why it has their name.

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This is Bob Merton, who really
took what Black-Scholes did

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and took it to another level

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to really get to our
modern interpretations

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of the Black-Scholes Model
and the Black-Scholes Formula.

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All three of these
gentlemen would have won

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the Nobel Prize in Economics,

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except for the unfortunate fact

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that Fischer Black passed away
before the award was given,

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but Myron Scholes and Bob Merton

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did get the Nobel Prize for their work.

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The reason why this is such a big deal,

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why it is Nobel Prize worthy,

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and, actually, there's many reasons.

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I could do a whole
series of videos on that,

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is that people have been
trading stock options,

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or they've been trading options
for a very, very, very long time.

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They had been trading them,

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they had been buying them,

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they had been selling them.

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It was a major part of
financial markets already,

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but there was no really good way

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of putting our mathematical minds around

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how to value an option.

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People had a sense of the
things that they cared about,

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and I would assume
especially options traders

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had a sense of the things
that they cared about

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when they were trading options,

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but we really didn't have an
analytical framework for it,

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and that's what the
Black-Scholes Formula gave us.

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Let's just, before we dive into
this seemingly hairy formula,

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but the more we talk about it,

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hopefully it'll start
to seem a lot friendlier

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than it looks right now.

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Let's start to get an intuition

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for the things that we would care about

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if we were thinking about
the price of a stock option.

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You would care about the stock price.

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You would care about the exercise price.

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You would especially care
about how much higher or lower

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the stock price is relative
to the exercise price.

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You would care about the
risk-free interest rate.

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The risk-free interest
rate keeps showing up

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when we think about taking a
present value of something,

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If we want to discount the value
of something back to today.

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You would, of course, think
about how long do I have

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to actually exercise this option?

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Finally, this might look a
little bit bizarre at first,

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but we'll talk about it in a second.

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You would care about how
volatile that stock is,

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and we measure volatility
as a standard deviation

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of log returns for that security.

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That seems very fancy,

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and we'll talk about that in
more depth in future videos,

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but at just an intuitive level,

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just think about 2 stocks.

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So let's say that this is
stock 1 right over here,

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and it jumps around,

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and I'll make them go flat,

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just so we make no judgment

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about whether it's a good investment.

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You have one stock that kind of does that,

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and then you have another stock.

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Actually, I'll draw them on the same,

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so let's say that is stock 1,

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and then you have a
stock 2 that does this,

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it jumps around all over the place.

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So this green one right
over here is stock 2.

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You could imagine stock 2

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just in the way we use the word
'volatile' is more volatile.

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It's a wilder ride.

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Also, if you were looking at
how dispersed the returns are

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away from their mean, you see it has,

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the returns have more dispersion.

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It'll have a higher standard deviation.

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So, stock 2 will have a higher volatility,

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or a higher standard deviation
of logarithmic returns,

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and in a future video, we'll talk about

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why we care about log returns,

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Stock 1 would have a lower volatility,

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so you can imagine,
options are more valuable

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when you're dealing with,

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or if you're dealing with a
stock that has higher volatility,

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that has higher sigma like this,

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this feels like it would drive
the value of an option up.

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You would rather have an option

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when you have something like this,

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because, look, if you're owning the stock,

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man, you have to go after,
this is a wild ride,

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but if you have the option,
you could ignore the wildness,

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and then it might actually make,

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and then you could exercise the option

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if it seems like the right time to do it.

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So it feels like, if you
were just trading it,

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that the more volatile something is,

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the more valuable an
option would be on that.

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Now that we've talked about this,

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let's actually look at
the Black-Scholes Formula.

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The variety that I have right over here,

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this is for a European call option.

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We could do something very
similar for a European put option,

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so this is right over here
is a European call option,

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and remember, European call option,

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it's mathematically simpler
than an American call option

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in that there's only one time
at which you can exercise it

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on the exercise date.

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On an American call option,

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you can exercise it an any point.

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With that said, let's try to
at least intuitively dissect

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the Black-Scholes Formula a little bit.

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So the first thing you have here,

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you have this term that involved
the current stock price,

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and then you're multiplying
it times this function

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that's taking this as an input,

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and this as how we define that input,

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and then you have minus the exercise price

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discounted back, this discounts
back the exercise price,

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times that function again,

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and now that input is slightly different

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into that function.

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Just so that we have a
little bit of background

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about what this function N is,

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N is the cumulative distribution function

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for a standard, normal distribution.

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I know that seems, might
seem a little bit daunting,

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but you can look at the
statistics playlist,

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and it shouldn't be that bad.

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This is essentially saying for
a standard, normal distribution,

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the probability that your
random variable is less than

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or equal to x,

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and another way of thinking about that,

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if that sounds a little,

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and it's all explained in
our statistics play list

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if that was confusing,

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but if you want to think about
it a little bit mathematically,

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you also know that this is going to be,

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it's a probability.

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It's always going to be greater than zero,

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and it is going to be less than one.

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With that out of the way,

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let's think about what
these pieces are telling us.

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This, right over here,

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is dealing with, it's
the current stock price,

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and it's being weighted by
some type of a probability,

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and so this is, essentially,
one way of thinking about it,

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in very rough terms, is this
is what you're gonna get.

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You're gonna get the stock,

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and it's kind of being
weighted by the probability

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that you're actually
going to do this thing,

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and I'm speaking in very rough terms,

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and then this term right
over here is what you pay.

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This is what you pay.

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This is your exercise
price discounted back,

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somewhat being weighted,

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and I'm speaking, once again,

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I'm hand-weaving a lot of the mathematics,

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by like are we actually
going to do this thing?

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Are we actually going
to exercise our option?

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That makes sense right over there,

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and it makes sense if the
stock price is worth a lot more

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than the exercise price,

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and if we're definitely going to do this,

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let's say that D1 and D2 are
very, very large numbers,

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we're definitely going to do
this at some point in time,

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that it makes sense that
the value of the call option

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would be the value of the
stock minus the exercise price

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discounted back to today.

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This right over here,
this is the discounting,

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kind of giving us the present
value of the exercise price.

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We have videos on discounting
and present value,

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if you find that a little bit daunting.

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It also makes sense that the more,

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the higher the stock price is,

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so we see that right over here,

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relative to the exercise price,

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the more that the option would be worth,

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it also makes sense that
the higher the stock price

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relative to the exercise price,

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the more likely that we will
actually exercise the option.

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You see that in both of
these terms right over here.

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You have the ratio of the stock
price to the exercise price.

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A ratio of the stock price
to the exercise price.

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We're taking a natural log of it,

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but the higher this ratio
is, the larger D1 or D2 is,

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so that means the larger the input

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into the cumulative
distribution function is,

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which means the higher
probabilities we're gonna get,

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and so it's a higher chance
we're gonna exercise this price,

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and it makes sense that then

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this is actually going to have some value.

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So that makes sense,

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the relationship between the stock price

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and the exercise price.

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The other thing I will focus on,

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because this tends to be a deep focus

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of people who operate with options,

00:08:18.698 --> 00:08:20.043
is the volatility.

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We already had an intuition,

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that the higher the volatility,

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the higher the option price,

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so let's see where this factors
into this equation, here.

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We don't see it at this first level,

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but it definitely factors into D1 and D2.

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In D1, the higher your standard
deviation of your log returns,

00:08:38.185 --> 00:08:39.610
so the higher sigma,

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we have a sigma in the
numerator and the denominator,

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but in the numerator, we're squaring it.

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So a higher sigma will make D1 go up,

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so sigma goes up, D1 will go up.

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Let's think about what's happening here.

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Well, here we have a sigma.

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It's still squared. It's in the numerator,

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but we're subtracting it.

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This is going to grow faster than this,

00:09:02.049 --> 00:09:03.497
but we're subtracting it now,

00:09:03.497 --> 00:09:08.321
so for D2, a higher sigma
is going to make D2 go down

00:09:08.321 --> 00:09:09.739
because we are subtracting it.

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This will actually make,

00:09:11.337 --> 00:09:12.869
can we actually say this is going to make,

00:09:12.869 --> 00:09:14.577
a higher sigma's going to make the value

00:09:14.577 --> 00:09:16.756
of our call option higher.

00:09:16.756 --> 00:09:18.124
Well, let's look at it.

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If the value of our sigma goes up,

00:09:20.576 --> 00:09:21.985
then D1 will go up,

00:09:21.985 --> 00:09:25.008
then this input, this input goes up.

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If that input goes up,

00:09:26.260 --> 00:09:28.650
our cumulative distribution
function of that input

00:09:28.650 --> 00:09:30.870
is going to go up, and so this term,

00:09:30.870 --> 00:09:35.226
this whole term is gonna
drive this whole term up.

00:09:35.226 --> 00:09:36.489
Now, what's going to happen here.

00:09:36.489 --> 00:09:38.587
Well, if D2 goes down,

00:09:38.587 --> 00:09:41.024
then our cumulative distribution
function evaluated there

00:09:41.024 --> 00:09:42.490
is going to go down,

00:09:42.490 --> 00:09:45.278
and so this whole thing
is going to be lower

00:09:45.278 --> 00:09:47.889
and so we're going to have to pay less.

00:09:47.889 --> 00:09:49.159
If we get more and pay less,

00:09:49.159 --> 00:09:50.803
and I'm speaking in very hand-wavy terms,

00:09:50.803 --> 00:09:52.293
but this is just to understand

00:09:52.293 --> 00:09:55.158
that this is as intuitively
daunting as you might think,

00:09:55.158 --> 00:09:56.957
but it looks definitively,

00:09:56.957 --> 00:09:59.074
that if the standard deviation,

00:09:59.074 --> 00:10:01.298
if the standard deviation
of our log returns

00:10:01.298 --> 00:10:02.803
or if our volatility goes up,

00:10:02.803 --> 00:10:05.088
the value of our call option,

00:10:05.088 --> 00:10:08.411
the value of our European
call option goes up.

00:10:08.411 --> 00:10:10.519
Likewise, using the same logic,

00:10:10.519 --> 00:10:13.375
if our volatility were to be lower,

00:10:13.375 --> 00:10:18.312
then the value of our
call option would go down.

00:10:18.312 --> 00:10:19.419
I'll leave you there.

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In future videos, we'll think about this

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in a little bit more depth.


Slope and triangle similarity 4

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Kind: captions
Language: en

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Which of the following triangles
are similar to triangle ABC?

00:00:04.850 --> 00:00:06.380
So let's just review
what similarity

00:00:06.380 --> 00:00:08.580
means when we're
talking about triangles.

00:00:08.580 --> 00:00:10.470
So let's say this
is one triangle.

00:00:10.470 --> 00:00:12.180
And I'll assume it's
a right triangle,

00:00:12.180 --> 00:00:15.190
although what I'm about to
say is true for any triangle.

00:00:15.190 --> 00:00:17.290
And let's say this
is another one.

00:00:17.290 --> 00:00:20.350
You have similar triangles if
all of the corresponding angles

00:00:20.350 --> 00:00:21.270
are equivalent.

00:00:21.270 --> 00:00:23.450
So that angle is
equivalent to that angle.

00:00:23.450 --> 00:00:25.990
And I drew that
as a right angle.

00:00:25.990 --> 00:00:29.010
Let's say that this angle
is equivalent to this angle.

00:00:29.010 --> 00:00:32.689
And let's say that this angle
is equivalent to that angle.

00:00:32.689 --> 00:00:35.240
Then these two
triangles are similar.

00:00:35.240 --> 00:00:37.120
They are not
necessarily congruent.

00:00:37.120 --> 00:00:39.970
They don't necessarily
have all of the same sides.

00:00:39.970 --> 00:00:41.980
But they do have all
of the same angles.

00:00:41.980 --> 00:00:43.688
And when you have all
of the same angles,

00:00:43.688 --> 00:00:47.010
the ratios between corresponding
sides is going to be constant.

00:00:47.010 --> 00:00:50.690
So for example, if this side
has length 2 and this side

00:00:50.690 --> 00:00:53.720
has length 1, you
have a 2 to 1 ratio

00:00:53.720 --> 00:00:56.576
between the second longest
side and the shortest side.

00:00:56.576 --> 00:00:58.700
Then you're also going to
have a 2 to 1 ratio here.

00:00:58.700 --> 00:01:00.720
So maybe this has length 3.

00:01:00.720 --> 00:01:02.542
Then this one would be 1.5.

00:01:02.542 --> 00:01:06.800
3 to 1.5 is the same
thing as 2 to 1.

00:01:06.800 --> 00:01:08.710
Now, let's think
about the triangles

00:01:08.710 --> 00:01:10.220
that they are proposing.

00:01:10.220 --> 00:01:13.805
So the first one
is triangle CDE.

00:01:21.880 --> 00:01:28.370
Just like triangle ABC, this is
going to be a right triangle.

00:01:28.370 --> 00:01:29.420
How do I know?

00:01:29.420 --> 00:01:31.780
Well, side CB is
perfectly horizontal,

00:01:31.780 --> 00:01:34.190
and side AB is perfectly
vertical-- same thing.

00:01:34.190 --> 00:01:38.332
Side DE is perfectly
horizontal, and side CE

00:01:38.332 --> 00:01:40.580
is perfectly vertical.

00:01:40.580 --> 00:01:43.940
Now, you might realize that
you have the same line acting

00:01:43.940 --> 00:01:48.160
as a hypotenuse of both
of these triangles.

00:01:48.160 --> 00:01:51.430
So you have the same
line right over here.

00:01:51.430 --> 00:01:54.480
And if we wanted to prove that
the corresponding angles are

00:01:54.480 --> 00:01:56.170
equivalent, we just
have to break out

00:01:56.170 --> 00:01:58.020
a little bit of our geometry.

00:01:58.020 --> 00:02:00.080
We see that we have
these two vertical lines.

00:02:00.080 --> 00:02:02.870
So CE, this line segment
CE, you could view it

00:02:02.870 --> 00:02:06.300
as part of a larger line
that goes like this.

00:02:06.300 --> 00:02:08.020
It is parallel to AB.

00:02:11.940 --> 00:02:14.130
So if you view these
as two parallel lines--

00:02:14.130 --> 00:02:17.700
so that is parallel
to this-- and you

00:02:17.700 --> 00:02:20.260
have a transversal
right over here,

00:02:20.260 --> 00:02:24.020
then corresponding angles
are going to be equivalent.

00:02:24.020 --> 00:02:27.220
So this angle right
over here is going

00:02:27.220 --> 00:02:30.150
to be equivalent to that
angle right over there.

00:02:30.150 --> 00:02:34.920
Similarly-- I have trouble
saying that word-- segment

00:02:34.920 --> 00:02:41.200
DE right over here
is parallel to BC.

00:02:41.200 --> 00:02:43.990
And so if you, once again,
look at the blue line

00:02:43.990 --> 00:02:46.460
as a transversal,
this angle is going

00:02:46.460 --> 00:02:47.810
to be equivalent to that angle.

00:02:47.810 --> 00:02:50.680
So all of their corresponding
angles are equivalent.

00:02:50.680 --> 00:02:53.860
So these are definitely
similar triangles.

00:02:53.860 --> 00:02:56.370
Now, I went through that
fairly long-winded geometry

00:02:56.370 --> 00:02:57.790
way of thinking about it.

00:02:57.790 --> 00:02:59.860
But there's an easier
way to think about it.

00:02:59.860 --> 00:03:06.920
Any time you have a line and
you construct a right angle,

00:03:06.920 --> 00:03:08.420
where the base of
the right triangle

00:03:08.420 --> 00:03:11.490
is essentially your change in
x between these two points.

00:03:11.490 --> 00:03:15.490
So if you think about going
from point C to point A.

00:03:15.490 --> 00:03:18.520
And if you first just want to
have a change in x-- so this

00:03:18.520 --> 00:03:20.990
is our change in x right
over there-- and then

00:03:20.990 --> 00:03:26.370
you have a change in y-- so
this is our change in y--

00:03:26.370 --> 00:03:28.920
you're going to construct
a right triangle.

00:03:28.920 --> 00:03:31.070
And the ratio between
these two sides

00:03:31.070 --> 00:03:32.900
is always going to be the same.

00:03:32.900 --> 00:03:36.030
So delta y, the change in
y, this triangle-- this

00:03:36.030 --> 00:03:38.250
is the Greek letter
delta, just a shorthand

00:03:38.250 --> 00:03:41.430
for saying change in y-- which
is the length of this side,

00:03:41.430 --> 00:03:46.260
over your change in x is
always going to be constant

00:03:46.260 --> 00:03:51.530
when you're dealing with a line.

00:03:51.530 --> 00:03:54.010
So any triangle that's
constructed this way

00:03:54.010 --> 00:03:57.180
from this line, the ratio
between the corresponding sides

00:03:57.180 --> 00:03:58.330
are going to be equivalent.

00:03:58.330 --> 00:04:00.219
And they're going to
be similar triangles.

00:04:00.219 --> 00:04:01.760
So that would have
been an easier way

00:04:01.760 --> 00:04:03.230
to think about this
right over here.

00:04:03.230 --> 00:04:05.271
Now let's look at the
other options they give us.

00:04:05.271 --> 00:04:16.190
DFG-- well, there's not really
a lot that we can say about DFG.

00:04:16.190 --> 00:04:18.959
We're not really using the
line anymore as a hypotenuse,

00:04:18.959 --> 00:04:22.140
so we're not going to throw
that in there as being similar.

00:04:22.140 --> 00:04:24.350
We could actually look at
the lengths and all of that

00:04:24.350 --> 00:04:27.660
if we wanted, but we'd
see that it's not similar.

00:04:27.660 --> 00:04:34.980
CDG-- let me clean
this up a little bit.

00:04:34.980 --> 00:04:39.000
CDG I'll do in-- let's
see, I've already

00:04:39.000 --> 00:04:40.250
used-- I'll do it in blue.

00:04:40.250 --> 00:04:44.734
So CDG-- this is not
a right triangle.

00:04:44.734 --> 00:04:46.650
This isn't just a change
in x right over here.

00:04:46.650 --> 00:04:48.160
This isn't a horizontal line.

00:04:48.160 --> 00:04:50.360
And this one right over
here is not a vertical line.

00:04:50.360 --> 00:04:52.349
So once again, this
is not similar.

00:04:52.349 --> 00:04:54.140
In fact, this is not
even a right triangle.

00:04:54.140 --> 00:04:56.740
So we'll mark both of these out.

00:04:56.740 --> 00:04:57.670
And then they say ADF.

00:05:00.670 --> 00:05:08.460
Well, once again, ADF, it's a
right triangle between D and F.

00:05:08.460 --> 00:05:14.460
It's your change in x going
from D-- so D is at negative 2.

00:05:14.460 --> 00:05:15.660
A is at 3.

00:05:15.660 --> 00:05:18.100
This right over here
is your change in x.

00:05:18.100 --> 00:05:20.270
You literally had to go 5.

00:05:20.270 --> 00:05:22.130
This length right
over here is 5.

00:05:22.130 --> 00:05:25.540
And then this right over
here is now vertical line.

00:05:25.540 --> 00:05:27.130
This is a right triangle.

00:05:27.130 --> 00:05:30.509
This is your change in y.

00:05:30.509 --> 00:05:32.050
And these two ratios
are always going

00:05:32.050 --> 00:05:35.300
to be constant when your
hypotenuse is a line.

00:05:35.300 --> 00:05:38.040
So once again, this is just
one of those classic right

00:05:38.040 --> 00:05:40.140
triangles constructed
from the line where

00:05:40.140 --> 00:05:42.020
the base is the change in x.

00:05:42.020 --> 00:05:43.590
This height is the change in y.

00:05:43.590 --> 00:05:45.520
The hypotenuse is
the line itself.

00:05:45.520 --> 00:05:47.240
These are going to be similar.

00:05:47.240 --> 00:05:50.077
So this is definitely
a similar triangle.

00:05:50.077 --> 00:05:52.160
And now let's actually
input it, just to make sure

00:05:52.160 --> 00:05:53.810
that we got the right answer.

00:05:53.810 --> 00:06:00.130
So I'm saying that and that are
similar triangles to triangle

00:06:00.130 --> 00:06:01.544
ABC.


Slope and triangle similarity 3

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:06.580
Use the points 0 comma 2,
2 comma 8, and x comma y,

00:00:06.580 --> 00:00:09.387
on the blue line in the graph
below to complete the equation.

00:00:09.387 --> 00:00:11.220
And they have this first
equation, y minus 2

00:00:11.220 --> 00:00:13.020
over x is equal to blank.

00:00:13.020 --> 00:00:14.520
And then they say
write the equation

00:00:14.520 --> 00:00:18.520
of the line in the form
y equals mx plus b.

00:00:18.520 --> 00:00:21.325
So we have a line here and
they give us two exact points,

00:00:21.325 --> 00:00:22.950
and then they tell
us one general point

00:00:22.950 --> 00:00:24.200
in terms of x comma y.

00:00:24.200 --> 00:00:26.200
Although the way they've
drawn it on the diagram

00:00:26.200 --> 00:00:29.310
it looks like it's the point
negative 3 comma negative 7,

00:00:29.310 --> 00:00:31.450
but we'll assume
that we don't know

00:00:31.450 --> 00:00:33.930
what those are, that those
are just two general points.

00:00:33.930 --> 00:00:37.720
And what we want to do is
figure out what this ratio is.

00:00:37.720 --> 00:00:40.300
And then we need to figure
out the equation of the line.

00:00:40.300 --> 00:00:41.800
Well, a good place
to start is just

00:00:41.800 --> 00:00:45.080
figure out the common ratio
between our change in y

00:00:45.080 --> 00:00:48.687
and our change in x between
any two points in the line.

00:00:48.687 --> 00:00:49.770
So let's think about that.

00:00:49.770 --> 00:00:54.260
If I were to go from point 0
comma to 2, to point 2 comma 8,

00:00:54.260 --> 00:00:56.300
what's going to
be my change in x?

00:00:56.300 --> 00:00:59.450
So my change in x
is just like that.

00:00:59.450 --> 00:01:02.821
So my change in x
is equal to-- well

00:01:02.821 --> 00:01:06.270
to go from x equals
0 to x equals 2,

00:01:06.270 --> 00:01:09.829
you had to go up two
in the x direction,

00:01:09.829 --> 00:01:11.870
or you have to go to the
right two, I should say,

00:01:11.870 --> 00:01:13.280
in the x direction.

00:01:13.280 --> 00:01:15.290
And our change in y--
well, we're going from y

00:01:15.290 --> 00:01:18.240
equals 2 to y equals 8.

00:01:18.240 --> 00:01:21.170
So our change in y, if you're
going to go from 2 to 8,

00:01:21.170 --> 00:01:23.780
your change in y--
remember triangle y,

00:01:23.780 --> 00:01:26.470
this literally is the Greek
letter for delta, delta y,

00:01:26.470 --> 00:01:28.820
is just a shorthand for
saying your change in y--

00:01:28.820 --> 00:01:31.870
well I went from 2
to 8 is equal to 6.

00:01:31.870 --> 00:01:34.340
So if you construct a
right triangle like this,

00:01:34.340 --> 00:01:37.830
this side's length is 6,
this side's length is 2,

00:01:37.830 --> 00:01:41.110
and the ratio of this
height to this base

00:01:41.110 --> 00:01:44.330
is going to be the
slope of your line.

00:01:44.330 --> 00:01:56.050
So delta y over delta x over
delta x is equal to 6 over 2,

00:01:56.050 --> 00:01:59.600
which is equal to 3.

00:01:59.600 --> 00:02:01.630
So what do we do with this?

00:02:01.630 --> 00:02:05.870
How does this answer
either of those questions?

00:02:05.870 --> 00:02:09.970
Well, this ratio, your change
in y-- your change in height--

00:02:09.970 --> 00:02:11.970
versus your change in the
base right over here--

00:02:11.970 --> 00:02:13.590
versus your change
in x-- is going

00:02:13.590 --> 00:02:16.150
to be constant for any
right triangle constructed

00:02:16.150 --> 00:02:18.696
this way on this line.

00:02:18.696 --> 00:02:21.070
So for example, when you're
going-- this right over here,

00:02:21.070 --> 00:02:22.610
y minus 2 over x.

00:02:22.610 --> 00:02:24.840
Let's think about it
a little bit more.

00:02:24.840 --> 00:02:30.100
If I were to go from the point
x comma y to the point right

00:02:30.100 --> 00:02:33.900
here, which is the point 0 comma
2, what would my change in x

00:02:33.900 --> 00:02:34.790
be?

00:02:34.790 --> 00:02:39.560
So my change in x is going
to be equal to-- well

00:02:39.560 --> 00:02:51.150
if I end up at x equals 0,
and I started at x, my change

00:02:51.150 --> 00:02:52.989
in x is going to be 0 minus x.

00:02:52.989 --> 00:02:55.530
And if that doesn't make a lot
of sense, just think about it.

00:02:55.530 --> 00:02:58.890
If x is negative 3,
then our change in x

00:02:58.890 --> 00:03:01.720
is 0 minus negative
3, which is 3.

00:03:01.720 --> 00:03:05.320
The length of the side of the
triangle right over here is 3.

00:03:05.320 --> 00:03:07.690
Now, let's think about the
height, our change in y.

00:03:10.780 --> 00:03:13.390
Well, our change in y--
we end up at a y equals 2,

00:03:13.390 --> 00:03:19.657
so it's going to be 2 minus
whatever our original y is.

00:03:19.657 --> 00:03:21.990
And it looks like, if you
just eyeball it at this point,

00:03:21.990 --> 00:03:23.830
it looks like it's
at about negative 7.

00:03:23.830 --> 00:03:27.210
And if you have 2 minus
negative 7 you get 9,

00:03:27.210 --> 00:03:30.670
and this indeed has length 9.

00:03:30.670 --> 00:03:33.470
So this ratio, delta
y over delta x,

00:03:33.470 --> 00:03:36.760
should be constant for any
right triangle constructed

00:03:36.760 --> 00:03:40.760
like this where the
hypotenuse is the line itself.

00:03:40.760 --> 00:03:47.040
So delta x over delta y-- or I
should say delta y over delta

00:03:47.040 --> 00:03:51.380
x, the ratio of this
side to this side,

00:03:51.380 --> 00:03:52.700
is always going to be the same.

00:03:52.700 --> 00:03:54.150
These are all similar triangles.

00:03:54.150 --> 00:03:56.566
It's going to have the same
ratio as this right over here.

00:03:56.566 --> 00:03:59.920
So change in y is 2 minus y.

00:03:59.920 --> 00:04:02.830
Change in x is 0 minus x.

00:04:02.830 --> 00:04:04.440
This is going to
have the same ratio.

00:04:04.440 --> 00:04:08.834
It's going to be 6 to
2, or 3 to 1, or just 3.

00:04:08.834 --> 00:04:10.250
Now you're saying,
wait, Sal, that

00:04:10.250 --> 00:04:11.540
still doesn't
answer my question.

00:04:11.540 --> 00:04:13.070
Even if I get this
0 out of the way

00:04:13.070 --> 00:04:17.890
I could write 2 minus y is
equal to negative x, or 2 minus

00:04:17.890 --> 00:04:21.550
y over negative x is equal to 3.

00:04:21.550 --> 00:04:24.957
But how does this get us to
this ratio right over here?

00:04:24.957 --> 00:04:27.290
Well, if you multiply this
numerator and the denominator

00:04:27.290 --> 00:04:30.820
by a negative sign-- so if you
just multiply the numerator

00:04:30.820 --> 00:04:32.569
and the denominator
by a negative sign--

00:04:32.569 --> 00:04:34.610
as long as you're doing
the same to the numerator

00:04:34.610 --> 00:04:36.110
and denominator,
you're not changing

00:04:36.110 --> 00:04:37.410
the value of this expression.

00:04:37.410 --> 00:04:40.040
In the numerator
you'll get y minus 2,

00:04:40.040 --> 00:04:43.604
and in the denominator negative
1 times negative x is x.

00:04:43.604 --> 00:04:45.520
Well, that's still going
to be the same ratio.

00:04:45.520 --> 00:04:48.170
That's still going to be 3.

00:04:48.170 --> 00:04:50.980
So this right over
here is going to be 3.

00:04:50.980 --> 00:04:53.100
And then they say write
the equation in the form y

00:04:53.100 --> 00:04:55.120
equals mx plus b.

00:04:55.120 --> 00:05:00.090
So when you have an equation
in the form y equals mx plus b,

00:05:00.090 --> 00:05:02.190
this term right
over here, this is

00:05:02.190 --> 00:05:03.760
that ratio we've
been dealing with.

00:05:03.760 --> 00:05:05.440
This says, hey,
every time I increase

00:05:05.440 --> 00:05:09.400
x by some amount how much
am I increasing y by?

00:05:09.400 --> 00:05:11.960
This is your slope
here, and so we

00:05:11.960 --> 00:05:19.110
know that this is going to
be of the form y is equal

00:05:19.110 --> 00:05:21.840
to-- I want to do it
in that black color--

00:05:21.840 --> 00:05:27.130
y is equal to our
slope, for this line is

00:05:27.130 --> 00:05:30.470
3 is equal to 3x plus b.

00:05:30.470 --> 00:05:32.520
And now we have to
figure out what b is.

00:05:32.520 --> 00:05:36.029
And what helps us in
this is to recognize

00:05:36.029 --> 00:05:37.570
that they've given
us several points.

00:05:37.570 --> 00:05:41.070
They tell us what happens--
well, when x is 0, y is 2.

00:05:41.070 --> 00:05:42.680
When x is 2, y is 8.

00:05:42.680 --> 00:05:45.205
So we can use any one of
those points to solve for b.

00:05:45.205 --> 00:05:47.520
I like using x
equals 0, y equals 2

00:05:47.520 --> 00:05:50.330
because it cancels this
term out very nicely.

00:05:50.330 --> 00:05:53.730
So we know that when x
is equal to 0, y is 2.

00:05:53.730 --> 00:05:59.960
Or we could say when y is 2, or
when x is equal to 0, y is 2.

00:05:59.960 --> 00:06:04.500
So 3 times 0 plus b is
going to be equal to 2.

00:06:04.500 --> 00:06:06.520
Well, that's just a
0, and you're just

00:06:06.520 --> 00:06:10.780
left with b is equal to
2, or 2 is equal to b.

00:06:10.780 --> 00:06:11.490
So we're done.

00:06:11.490 --> 00:06:17.330
It's y equals 3x plus 2.

00:06:17.330 --> 00:06:21.570
So let's fill that in
on the actual exercise.

00:06:21.570 --> 00:06:25.140
So this ratio is 3.

00:06:25.140 --> 00:06:29.600
It's the ratio between a change
in height over our change in x,

00:06:29.600 --> 00:06:31.880
that's constant,
across the line when

00:06:31.880 --> 00:06:33.550
you construct a right triangle.

00:06:33.550 --> 00:06:35.383
They're all going to
be similar they're, all

00:06:35.383 --> 00:06:37.180
going to have that same ratio.

00:06:37.180 --> 00:06:40.850
And then the equation of the
line, in that form, it's y

00:06:40.850 --> 00:06:43.390
equals that change
in y over change

00:06:43.390 --> 00:06:44.990
in x, that constant ratio.

00:06:44.990 --> 00:06:50.720
Which is defined as
slope, 3x plus 2.

00:06:50.720 --> 00:06:53.400
This 2, which is where
you intersect the y,

00:06:53.400 --> 00:06:56.460
it is the value you
get when x equals 0.

00:06:56.460 --> 00:06:59.780
That's often called the
y-intercept because, you see,

00:06:59.780 --> 00:07:03.640
when x is equal to 0 that's
when you intersect the y-axis.

00:07:03.640 --> 00:07:05.860
So y equals 3x plus
2, and you could

00:07:05.860 --> 00:07:11.090
verify that any of these points
will satisfy this equation.

00:07:11.090 --> 00:07:13.030
And we got it right.


Slope and triangle similarity 2

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:05.460
Which of the following triangles
are similar to triangle ABC?

00:00:05.460 --> 00:00:08.260
So this is triangle ABC.

00:00:08.260 --> 00:00:10.280
Before we even look
at these options,

00:00:10.280 --> 00:00:13.560
let's remind ourselves of what
it means for two triangles

00:00:13.560 --> 00:00:14.280
to be similar.

00:00:14.280 --> 00:00:16.215
There's multiple ways
of thinking about it.

00:00:16.215 --> 00:00:19.600
The classical definition
of similar triangles

00:00:19.600 --> 00:00:22.710
is that all the corresponding
angles are equivalent.

00:00:22.710 --> 00:00:26.890
So for example, if these
two are similar triangles,

00:00:26.890 --> 00:00:29.360
then if this angle
corresponds to that one,

00:00:29.360 --> 00:00:31.450
then these two angles
are going to be the same.

00:00:31.450 --> 00:00:33.060
If this angle
corresponds to this one,

00:00:33.060 --> 00:00:35.400
then these two angles
are going to be the same.

00:00:35.400 --> 00:00:39.032
And if this angle
corresponds to this one,

00:00:39.032 --> 00:00:40.990
then these two angles
are going to be the same,

00:00:40.990 --> 00:00:42.450
and these are similar.

00:00:42.450 --> 00:00:44.991
And the reason why similarity
is useful is because that tells

00:00:44.991 --> 00:00:46.820
you the ratio between
corresponding sides

00:00:46.820 --> 00:00:48.570
are going to be the same.

00:00:48.570 --> 00:00:51.820
So for example, if this side
has length 2 and this side

00:00:51.820 --> 00:00:54.660
has length 1, then the ratio
of this side to this side

00:00:54.660 --> 00:00:55.850
is 2 to 1.

00:00:55.850 --> 00:00:58.730
The corresponding sides here
are this side and this side.

00:00:58.730 --> 00:01:01.580
So they would also have
to have a 2 to 1 ratio.

00:01:01.580 --> 00:01:04.786
So for example, this could
be 4, and this could be 2-- 2

00:01:04.786 --> 00:01:05.660
to 1 ratio.

00:01:05.660 --> 00:01:08.050
4/2 is the same thing as 2/1.

00:01:08.050 --> 00:01:11.670
Or it could be 7 to 3 and 1/2.

00:01:11.670 --> 00:01:16.030
Or it could be 10 to 5.

00:01:16.030 --> 00:01:19.230
It would always have a 2 to
1 ratio if this had a 2 to 1

00:01:19.230 --> 00:01:21.740
ratio and if these
are similar triangles.

00:01:21.740 --> 00:01:23.512
Now, in the context
of a line and slope,

00:01:23.512 --> 00:01:25.220
I'll think about
another one, another way

00:01:25.220 --> 00:01:27.000
of thinking about similarity.

00:01:27.000 --> 00:01:31.580
If you have two
right triangles--

00:01:31.580 --> 00:01:34.620
so let's say this is one right
triangle right over here--

00:01:34.620 --> 00:01:37.090
so if you have one right
triangle right over there

00:01:37.090 --> 00:01:43.200
and you have another right
triangle just like that,

00:01:43.200 --> 00:01:48.740
if you have two right triangles
and the ratio of the two

00:01:48.740 --> 00:01:50.954
sides on either side
of the right angle

00:01:50.954 --> 00:01:53.370
correspond to each other-- so
if the ratio of this magenta

00:01:53.370 --> 00:01:59.460
side to this green side is
the same in this triangle

00:01:59.460 --> 00:02:06.400
as it is in this triangle--
then we are similar.

00:02:06.400 --> 00:02:09.110
So for example, if these
are two right triangles

00:02:09.110 --> 00:02:14.240
and if this is a, let's
say, this side is length 6

00:02:14.240 --> 00:02:19.430
and this side is length 9
and this side is length 2

00:02:19.430 --> 00:02:21.810
and this side is
length 3, these are

00:02:21.810 --> 00:02:23.500
going to be similar triangles.

00:02:23.500 --> 00:02:25.860
You have an angle
right over here.

00:02:25.860 --> 00:02:29.390
And the ratio of 6/2 is
the same thing as 3/1.

00:02:29.390 --> 00:02:32.890
The ratio of 9/3 is the
same thing as 3 to 1.

00:02:32.890 --> 00:02:33.905
These are similar.

00:02:38.780 --> 00:02:40.780
Now, why am I telling
you that in the context

00:02:40.780 --> 00:02:42.400
of this right over here?

00:02:42.400 --> 00:02:45.570
Well, we have a line and
you connect any two points

00:02:45.570 --> 00:02:48.750
on the line, let's say
point c to point a,

00:02:48.750 --> 00:02:51.470
the ratio of your change
in y over your change

00:02:51.470 --> 00:02:54.600
in x is always going
to be constant.

00:02:54.600 --> 00:02:58.080
And when you have a trying to
constructed this way, where you

00:02:58.080 --> 00:03:00.700
have your change in x is
going perfectly horizontal

00:03:00.700 --> 00:03:03.460
and your change in y is
going perfectly vertical,

00:03:03.460 --> 00:03:05.840
it's going to be
a right triangle.

00:03:05.840 --> 00:03:08.300
And so any right
triangle you construct

00:03:08.300 --> 00:03:10.020
between two points
on the line this way,

00:03:10.020 --> 00:03:12.560
they're going to be
similar, because your ratio

00:03:12.560 --> 00:03:16.260
between your change in y--
notice if you go from point c

00:03:16.260 --> 00:03:20.022
to point a, you can point
c is at 3, point a is at 9.

00:03:20.022 --> 00:03:21.480
This is literally
your change in y.

00:03:21.480 --> 00:03:24.160
Going from 3 to 9,
you increased by 6.

00:03:24.160 --> 00:03:29.040
Your change in y is equal
to 6 right over here.

00:03:29.040 --> 00:03:31.880
And in the horizontal direction,
this base of this triangle

00:03:31.880 --> 00:03:34.550
right over here, this
is your change in x.

00:03:34.550 --> 00:03:37.920
c is that point at
your x value is at 1.

00:03:37.920 --> 00:03:40.260
a, your x value is 3.

00:03:40.260 --> 00:03:43.940
So you went, your change
in x is equal to 2.

00:03:43.940 --> 00:03:45.710
And any right
triangle you construct

00:03:45.710 --> 00:03:48.550
this way between any two
points on a line, this ratio,

00:03:48.550 --> 00:03:51.310
your change in y over your
change in x, which in this case

00:03:51.310 --> 00:03:54.340
is 6/2, is going to be constant.

00:03:54.340 --> 00:03:56.215
And since that ratio is
going to be constant,

00:03:56.215 --> 00:03:59.010
it's a right triangle, any
two triangles constructed

00:03:59.010 --> 00:04:01.690
in this way with the line
between two points on the line,

00:04:01.690 --> 00:04:03.970
they are going to be similar.

00:04:03.970 --> 00:04:06.440
Now, let's look at our options.

00:04:06.440 --> 00:04:16.490
So triangle ADG-- well,
once again, the hypotenuse,

00:04:16.490 --> 00:04:20.420
just like triangle ABC, the
hypotenuse is still our line.

00:04:20.420 --> 00:04:22.290
We are essentially
thinking about the base

00:04:22.290 --> 00:04:27.000
of the triangle is the
change in x between going

00:04:27.000 --> 00:04:32.000
from between our x value
for d and our x value for a.

00:04:32.000 --> 00:04:36.470
And this height right over here
or this side right over here,

00:04:36.470 --> 00:04:41.310
this is our change in y going
from y equals negative 3 to y

00:04:41.310 --> 00:04:42.300
equals 9.

00:04:42.300 --> 00:04:44.060
So this is our change in y.

00:04:44.060 --> 00:04:46.810
We know that your change
in y over your change in x

00:04:46.810 --> 00:04:48.870
is going to be
constant for a line.

00:04:48.870 --> 00:04:50.340
This is a right triangle.

00:04:50.340 --> 00:04:53.742
So these two triangles are
definitely going to be similar.

00:04:53.742 --> 00:04:56.200
And if you want to break out
a little bit of your geometry,

00:04:56.200 --> 00:04:59.920
it's not hard to prove that
the corresponding angles are

00:04:59.920 --> 00:05:02.160
equivalent, using a little
bit of your parallel lines

00:05:02.160 --> 00:05:05.180
and transversals of
parallel lines knowledge.

00:05:05.180 --> 00:05:07.002
Now, let's look at triangle BEF.

00:05:10.977 --> 00:05:13.060
Well, this is a right
triangle, but its hypotenuse

00:05:13.060 --> 00:05:16.135
does not sit on--
so triangle BBEBEF.

00:05:26.840 --> 00:05:29.110
So this triangle is
a right triangle.

00:05:29.110 --> 00:05:30.980
But its hypotenuse does
not sit on the line.

00:05:30.980 --> 00:05:32.813
Now, it could be similar,
but you could even

00:05:32.813 --> 00:05:35.260
see that this line right here
has a different inclination.

00:05:35.260 --> 00:05:36.400
It has a different slope.

00:05:36.400 --> 00:05:38.830
So that's a pretty good
sign that these are not

00:05:38.830 --> 00:05:40.537
going to be similar triangles.

00:05:40.537 --> 00:05:42.120
And if you really
wanted to verify it,

00:05:42.120 --> 00:05:44.484
you could look at the
ratio of your change in y

00:05:44.484 --> 00:05:45.400
over your change in x.

00:05:45.400 --> 00:05:47.066
And in fact, let's
just do that for fun,

00:05:47.066 --> 00:05:50.430
just to prove to ourselves
that it's not similar.

00:05:50.430 --> 00:05:54.140
To go from F to B, you're
going from negative 6

00:05:54.140 --> 00:05:57.460
to 3-- 3 minus negative 6.

00:05:57.460 --> 00:06:01.467
I'll just say this length right
over here, BF, is equal to 9.

00:06:01.467 --> 00:06:03.300
I'm not going to say
change in y over change

00:06:03.300 --> 00:06:05.947
in x, because we're not going
between two points on the line.

00:06:05.947 --> 00:06:08.030
I'm just going to look at
the length of this line.

00:06:08.030 --> 00:06:10.420
The length of this line is 9.

00:06:10.420 --> 00:06:13.450
The length of this line
right over here-- let's see,

00:06:13.450 --> 00:06:15.950
we're going from
negative 1 to 3-- is 4.

00:06:15.950 --> 00:06:18.930
So the ratio of 9
to 4 is definitely

00:06:18.930 --> 00:06:23.320
not the same as the ratio
of 6 to 2 or 3 to 1.

00:06:23.320 --> 00:06:27.340
So we can rule BEF out.

00:06:27.340 --> 00:06:28.450
Now, let's look at EFG.

00:06:31.119 --> 00:06:33.160
Well, once again, this
isn't sitting on the line.

00:06:33.160 --> 00:06:35.118
That doesn't necessarily
mean it's not similar.

00:06:35.118 --> 00:06:37.940
But you can even see the
inclination of its hypotenuse

00:06:37.940 --> 00:06:40.560
is very different than
the line right over here.

00:06:40.560 --> 00:06:44.540
And you see the
ratio of these sides.

00:06:44.540 --> 00:06:47.410
You have a side length
3, side length 4.

00:06:47.410 --> 00:06:49.860
The ratio of 3 to 4 is
definitely not 6 to 2.

00:06:49.860 --> 00:06:52.900
If this was somehow oriented
different, the ratio of 4 to 3

00:06:52.900 --> 00:06:55.970
is also not the
ratio of 6 to 2--

00:06:55.970 --> 00:06:58.620
I guess this is the 6 to 200
the 6 to 2 or the 3 to 1.

00:06:58.620 --> 00:07:01.020
So we can rule this
one out as well.

00:07:01.020 --> 00:07:09.780
And then finally, triangle
BDG-- well, once again-- you're

00:07:09.780 --> 00:07:13.500
tired of me saying
this-- triangle BDG,

00:07:13.500 --> 00:07:17.310
its hypotenuse does
not sit on the line.

00:07:17.310 --> 00:07:19.610
Its hypotenuse has a very
different inclination

00:07:19.610 --> 00:07:22.170
or a very different slope
than the line itself.

00:07:22.170 --> 00:07:24.740
So this is not going to
be a similar triangle.

00:07:24.740 --> 00:07:25.800
You can often eyeball it.

00:07:25.800 --> 00:07:28.890
The similar triangle
will look the same.

00:07:28.890 --> 00:07:31.250
It'll have the same proportions.

00:07:31.250 --> 00:07:33.520
They don't necessarily have
to have different sides,

00:07:33.520 --> 00:07:34.730
different sizes.

00:07:34.730 --> 00:07:36.220
But they can have
different sizes.

00:07:36.220 --> 00:07:38.015
One is a scaled up
version of the other.

00:07:38.015 --> 00:07:40.140
And you can see here, this
triangle right over here

00:07:40.140 --> 00:07:42.580
looks very different
than triangle ABC.

00:07:42.580 --> 00:07:45.000
And you can even verify
that the ratio of this side

00:07:45.000 --> 00:07:50.730
right over here to this side
is very different than 3 to 1.

00:07:50.730 --> 00:07:52.780
So we can rule this
one out as well.

00:07:52.780 --> 00:07:54.810
So let's actually input
that and make sure

00:07:54.810 --> 00:07:58.500
that we actually got
the right answer.

00:07:58.500 --> 00:08:02.430
So let me input it right there.

00:08:02.430 --> 00:08:04.990
And we got the right answer.


Similar triangles to prove that the slope is constant for a line

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:04.860
We tend to be told in algebra
class that if we have a line,

00:00:04.860 --> 00:00:08.640
our line will have a
constant rate of change of y

00:00:08.640 --> 00:00:09.551
with respect to x.

00:00:09.551 --> 00:00:11.050
Or another way of
thinking about it,

00:00:11.050 --> 00:00:13.680
that our line will have
a constant inclination,

00:00:13.680 --> 00:00:16.920
or that our line will
have a constant slope.

00:00:16.920 --> 00:00:21.190
And our slope is literally
defined as your change

00:00:21.190 --> 00:00:24.650
in y-- this triangle is
the Greek letter delta.

00:00:24.650 --> 00:00:26.580
It's a shorthand
for "change in."

00:00:26.580 --> 00:00:29.960
It means change in y--
delta y means change in y--

00:00:29.960 --> 00:00:31.640
over change in x.

00:00:31.640 --> 00:00:34.500
And if you're dealing with
a line, this right over here

00:00:34.500 --> 00:00:40.630
is constant for a line.

00:00:40.630 --> 00:00:42.270
What I want to do
in this video is

00:00:42.270 --> 00:00:46.270
to actually prove that
using similar triangles

00:00:46.270 --> 00:00:48.420
from geometry.

00:00:48.420 --> 00:00:51.850
So let's think about
2 sets of 2 points.

00:00:51.850 --> 00:00:54.290
So let's say that's
a point there.

00:00:54.290 --> 00:00:55.980
Let me do it in a
different color.

00:00:55.980 --> 00:00:58.930
Let me start at this point.

00:00:58.930 --> 00:01:01.160
And let me end up at that point.

00:01:01.160 --> 00:01:04.560
So what is our change in
x between these 2 points?

00:01:04.560 --> 00:01:07.860
So this point's x value
is right over here.

00:01:07.860 --> 00:01:12.020
This point's x value
is right over here.

00:01:12.020 --> 00:01:19.090
So our change in x is going
to be that right over there.

00:01:19.090 --> 00:01:21.590
And what's our change in y?

00:01:21.590 --> 00:01:23.930
Well, this point's y
value is right over here.

00:01:23.930 --> 00:01:25.730
This point's y value's
right over here.

00:01:25.730 --> 00:01:29.220
So this height or this
height is our change in y.

00:01:29.220 --> 00:01:34.650
So that is our change in y.

00:01:34.650 --> 00:01:36.590
Now, let's look
at 2 other points.

00:01:36.590 --> 00:01:40.379
Let's say I have this point
and this point right over here.

00:01:40.379 --> 00:01:41.670
And let's do the same exercise.

00:01:41.670 --> 00:01:43.280
What's the change in x?

00:01:43.280 --> 00:01:44.050
Well, let's see.

00:01:44.050 --> 00:01:47.414
If we're going-- this
point's x value's here.

00:01:47.414 --> 00:01:48.580
This point's x value's here.

00:01:48.580 --> 00:01:51.390
So if we start here
and we go this far,

00:01:51.390 --> 00:01:55.030
this would be the change
in x between this point

00:01:55.030 --> 00:01:56.280
and this point.

00:01:56.280 --> 00:01:59.696
And this is going
to be the change--

00:01:59.696 --> 00:02:01.320
let me do that in
the same green color.

00:02:01.320 --> 00:02:03.653
So this is going to be the
change in x between those two

00:02:03.653 --> 00:02:04.210
points.

00:02:04.210 --> 00:02:07.770
And our change in y, well,
this y value is here.

00:02:07.770 --> 00:02:09.340
This y value's up here.

00:02:09.340 --> 00:02:13.306
So our change in y is going
to be that right over here.

00:02:13.306 --> 00:02:14.680
So what I need to
show-- I'm just

00:02:14.680 --> 00:02:15.980
picking 2 arbitrary points.

00:02:15.980 --> 00:02:19.210
I need to show that the
ratio of this change in y

00:02:19.210 --> 00:02:20.930
to this change of
x is going to be

00:02:20.930 --> 00:02:23.120
the same as the ratio
of this change in y

00:02:23.120 --> 00:02:24.330
to this change of x.

00:02:24.330 --> 00:02:27.620
Or the ratio of this purple
side to this green side

00:02:27.620 --> 00:02:29.350
is going to be the
same as the ratio

00:02:29.350 --> 00:02:31.180
of this purple side
to this green side.

00:02:31.180 --> 00:02:35.370
Remember, I'm just picking 2
sets of arbitrary points here.

00:02:35.370 --> 00:02:38.620
And the way that I will show
it is through similarity.

00:02:38.620 --> 00:02:42.080
If I can show that this triangle
is similar to this triangle,

00:02:42.080 --> 00:02:44.010
then we are all set up.

00:02:44.010 --> 00:02:47.310
And just as a reminder
of what similarity is,

00:02:47.310 --> 00:02:50.060
2 triangles are similar--
and there's multiple ways

00:02:50.060 --> 00:02:54.030
of thinking about
it-- if and only

00:02:54.030 --> 00:02:56.450
if all corresponding--
or I should say,

00:02:56.450 --> 00:02:59.285
all three angles are the
same, or are congruent.

00:03:01.445 --> 00:03:02.570
And let me be careful here.

00:03:02.570 --> 00:03:04.710
They don't have to be
the same exact angle.

00:03:04.710 --> 00:03:07.490
The corresponding angles
have to be the same.

00:03:07.490 --> 00:03:15.590
So corresponding--
I always misspell

00:03:15.590 --> 00:03:24.040
it-- angles are
going to be equal.

00:03:24.040 --> 00:03:26.810
Or we could say
they are congruent.

00:03:26.810 --> 00:03:32.520
So for example, if I have
this triangle right over here.

00:03:32.520 --> 00:03:37.600
And this is 30, this
is 60, and this is 90.

00:03:37.600 --> 00:03:40.694
And then I have this
triangle right over here.

00:03:40.694 --> 00:03:42.610
I'll try to draw it--
so I have this triangle,

00:03:42.610 --> 00:03:45.760
where this is 30 degrees,
this is 60 degrees,

00:03:45.760 --> 00:03:47.336
and this is 90 degrees.

00:03:47.336 --> 00:03:49.210
Even though their side
lengths are different,

00:03:49.210 --> 00:03:50.890
these are going to
be similar triangles.

00:03:50.890 --> 00:03:53.098
They're essentially scaled
up versions of each other.

00:03:53.098 --> 00:03:55.590
All the corresponding
angles-- 60 corresponds

00:03:55.590 --> 00:03:58.000
to this 60, 30
corresponds to this 30,

00:03:58.000 --> 00:03:59.580
and 90 corresponds to this 1.

00:03:59.580 --> 00:04:01.744
So these 2 triangles
are similar.

00:04:01.744 --> 00:04:03.410
And what's neat about
similar triangles,

00:04:03.410 --> 00:04:06.020
if you can establish that
2 triangles are similar,

00:04:06.020 --> 00:04:07.811
then the ratio between
corresponding sides

00:04:07.811 --> 00:04:08.810
is going to be the same.

00:04:08.810 --> 00:04:11.260
So if these 2 are
similar, then the ratio

00:04:11.260 --> 00:04:15.280
of this side to
this side is going

00:04:15.280 --> 00:04:18.829
to be the same as
the ratio of-- let

00:04:18.829 --> 00:04:25.592
me do that pink color--
this side to this side.

00:04:25.592 --> 00:04:27.050
And so you can see
why that will be

00:04:27.050 --> 00:04:29.974
useful in proving that the
slope is constant here,

00:04:29.974 --> 00:04:31.390
because all we
have to do is look.

00:04:31.390 --> 00:04:33.140
If these 2 triangles
are similar,

00:04:33.140 --> 00:04:35.270
then the ratio between
corresponding sides

00:04:35.270 --> 00:04:36.970
is always going to be the same.

00:04:36.970 --> 00:04:39.040
We've picked 2 arbitrary
sets of points.

00:04:39.040 --> 00:04:41.470
Then this would be true,
really, for any 2 arbitrary set

00:04:41.470 --> 00:04:42.553
of points across the line.

00:04:42.553 --> 00:04:45.320
It would be true
for the entire line.

00:04:45.320 --> 00:04:47.530
So let's try to
prove similarity.

00:04:47.530 --> 00:04:49.530
So the first thing we
know is that both of these

00:04:49.530 --> 00:04:50.530
are right triangles.

00:04:50.530 --> 00:04:52.880
These green lines are
perfectly horizontal.

00:04:52.880 --> 00:04:55.980
These purple lines
are perfectly vertical

00:04:55.980 --> 00:04:58.160
because the green
lines literally

00:04:58.160 --> 00:04:59.720
go in the horizontal direction.

00:04:59.720 --> 00:05:01.750
The purple lines go in
the vertical direction.

00:05:01.750 --> 00:05:04.180
So let me make sure
that we mark that.

00:05:04.180 --> 00:05:06.040
So we know that these
are both right angles.

00:05:06.040 --> 00:05:08.510
So we have 1 corresponding
angle that is congruent.

00:05:08.510 --> 00:05:10.382
Now we have to show
that the other ones are.

00:05:10.382 --> 00:05:11.840
And we can show
that the other ones

00:05:11.840 --> 00:05:13.810
are using our knowledge
of parallel lines

00:05:13.810 --> 00:05:15.320
and transversals.

00:05:15.320 --> 00:05:17.190
Let's look at these
2 green lines.

00:05:17.190 --> 00:05:18.280
So I'll continue them.

00:05:18.280 --> 00:05:20.680
These are line segments,
but if we view them as lines

00:05:20.680 --> 00:05:23.310
and we just continue
them, on and on and on.

00:05:23.310 --> 00:05:25.510
So let me do that,
just like here.

00:05:25.510 --> 00:05:28.530
So this line is clearly
parallel to that 1.

00:05:28.530 --> 00:05:31.860
They essentially are
perfectly horizontal.

00:05:31.860 --> 00:05:35.910
And now you can view our
orange line as a transversal.

00:05:35.910 --> 00:05:37.870
And if you view it
as a transversal,

00:05:37.870 --> 00:05:41.620
then you know that this angle
corresponds to this angle.

00:05:41.620 --> 00:05:44.310
And we know from transversals
of parallel lines

00:05:44.310 --> 00:05:47.150
that corresponding
angles are congruent.

00:05:47.150 --> 00:05:49.810
So this angle is going to
be congruent to that angle

00:05:49.810 --> 00:05:51.200
right over there.

00:05:51.200 --> 00:05:53.400
Now, we make a very similar
argument for this angle,

00:05:53.400 --> 00:05:55.470
but now we use the
2 vertical lines.

00:05:55.470 --> 00:05:59.340
We know that this segment, we
could continue it as a line.

00:05:59.340 --> 00:06:01.740
So we could continue it,
if we wanted, as a line,

00:06:01.740 --> 00:06:04.067
so just like that,
a vertical line.

00:06:04.067 --> 00:06:06.150
And we could continue this
one as a vertical line.

00:06:06.150 --> 00:06:07.720
We know that these
are both vertical.

00:06:07.720 --> 00:06:09.386
They're just measuring--
they're exactly

00:06:09.386 --> 00:06:11.870
in the y direction,
the vertical direction.

00:06:11.870 --> 00:06:17.110
So this line is parallel to
this line right over here.

00:06:17.110 --> 00:06:22.040
Once again, our orange line
is a transversal of it.

00:06:22.040 --> 00:06:27.215
And this angle corresponds to
this angle right over here.

00:06:27.215 --> 00:06:28.090
And there we have it.

00:06:28.090 --> 00:06:28.840
They're congruent.

00:06:28.840 --> 00:06:31.490
Corresponding angles of the
transversal of 2 parallel lines

00:06:31.490 --> 00:06:32.150
are congruent.

00:06:32.150 --> 00:06:33.914
We learned that
in geometry class.

00:06:33.914 --> 00:06:34.830
And there you have it.

00:06:34.830 --> 00:06:36.520
All of the
corresponding-- this angle

00:06:36.520 --> 00:06:37.970
is congruent to this angle.

00:06:37.970 --> 00:06:39.620
This angle is congruent
to that angle.

00:06:39.620 --> 00:06:41.590
And then both of
these are 90 degrees.

00:06:41.590 --> 00:06:43.365
So both of these are
similar triangles.

00:06:45.826 --> 00:06:47.200
Just let me write
that down so we

00:06:47.200 --> 00:06:53.170
know that these are
both similar triangles.

00:06:53.170 --> 00:06:56.110
And now we can use the
common ratio of both sides.

00:06:56.110 --> 00:07:01.620
So for example, if we
called this side length a.

00:07:01.620 --> 00:07:04.870
And we said that this
side has length b.

00:07:04.870 --> 00:07:07.430
And we said this
side has length c.

00:07:07.430 --> 00:07:09.600
And this side has length d.

00:07:09.600 --> 00:07:12.330
We know for a fact
that the ratio,

00:07:12.330 --> 00:07:14.130
because these are
similar triangles,

00:07:14.130 --> 00:07:19.450
between corresponding
sides, the ratio of a to b

00:07:19.450 --> 00:07:22.470
is going to be equal
to the ratio of c to d.

00:07:29.090 --> 00:07:31.290
And that ratio is
literally the definition

00:07:31.290 --> 00:07:34.150
of slope, your change in
y over your change in x.

00:07:34.150 --> 00:07:38.120
And this is constant
because any right triangles

00:07:38.120 --> 00:07:40.050
that you generate
between these two points,

00:07:40.050 --> 00:07:43.436
we've just shown that they
are going to be similar.

00:07:43.436 --> 00:07:45.060
And if they are
similar, then the ratio

00:07:45.060 --> 00:07:47.730
of the length of
this vertical line

00:07:47.730 --> 00:07:50.300
segment to this horizontal
line segment is constant.

00:07:50.300 --> 00:07:52.100
That is the definition of slope.

00:07:52.100 --> 00:07:55.870
So the slope is
constant for a line.


Slope and triangle similarity 1

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:03.570
Use the graph below
to decide which

00:00:03.570 --> 00:00:05.242
of the following
equations are true.

00:00:05.242 --> 00:00:06.950
So before I even look
at these equations,

00:00:06.950 --> 00:00:08.900
let's think about
what's going on here.

00:00:08.900 --> 00:00:11.490
So I have a line.

00:00:11.490 --> 00:00:14.730
And then we've defined all of
these right triangles, where

00:00:14.730 --> 00:00:18.320
the hypotenuse of the right
triangles are the line.

00:00:18.320 --> 00:00:21.390
And you could view
this right triangle,

00:00:21.390 --> 00:00:25.200
say this one in red, as
really, side C is our change

00:00:25.200 --> 00:00:28.180
in y, our vertical change
as we go from this point

00:00:28.180 --> 00:00:29.250
to this point.

00:00:29.250 --> 00:00:33.390
And side D right over here
is our horizontal change.

00:00:33.390 --> 00:00:35.290
Notice we go D to the right.

00:00:35.290 --> 00:00:37.540
And then we go C up.

00:00:37.540 --> 00:00:40.700
If you're starting at this
point and going to that point,

00:00:40.700 --> 00:00:43.200
side B is your
horizontal change.

00:00:43.200 --> 00:00:45.820
And side A is your
vertical change.

00:00:45.820 --> 00:00:48.070
And the one thing
we know about lines

00:00:48.070 --> 00:00:51.120
is, if you construct these
right triangles like this,

00:00:51.120 --> 00:00:54.090
your change in y over
your change in x,

00:00:54.090 --> 00:00:57.790
the ratio between those two is
always going to be constant.

00:00:57.790 --> 00:00:59.825
We call that ratio the slope.

00:00:59.825 --> 00:01:01.200
We also know that
these are going

00:01:01.200 --> 00:01:03.110
to be similar triangles,
that the ratio

00:01:03.110 --> 00:01:05.920
between corresponding sides
is going to be the same.

00:01:05.920 --> 00:01:07.830
So these are obviously
right triangles.

00:01:07.830 --> 00:01:09.009
These are horizontal lines.

00:01:09.009 --> 00:01:10.050
These are vertical lines.

00:01:10.050 --> 00:01:12.950
And all of these are
similar triangles.

00:01:12.950 --> 00:01:14.750
And because they're
all similar, the ratio

00:01:14.750 --> 00:01:17.790
between this vertical side
and this base right over here,

00:01:17.790 --> 00:01:19.680
is always going to be the same.

00:01:19.680 --> 00:01:23.340
So for example, we know
that the ratio of A to B

00:01:23.340 --> 00:01:31.180
is going to be equal to the
ratio of C to D. Now let's work

00:01:31.180 --> 00:01:33.850
through some of this other
stuff right over here.

00:01:33.850 --> 00:01:39.440
So they say y plus 1 over x plus
2 is equal to the ratio of A

00:01:39.440 --> 00:01:41.490
to B. Let's see if that's right.

00:01:41.490 --> 00:01:44.490
So this is the point, x, y.

00:01:44.490 --> 00:01:46.702
So what's y plus 1?

00:01:46.702 --> 00:01:49.970
Well, y plus 1 is
literally the same thing

00:01:49.970 --> 00:01:52.230
as the length C right over here.

00:01:52.230 --> 00:01:55.450
Notice this height-- let
me do this in a new color--

00:01:55.450 --> 00:01:59.410
this height right
over here, this is y.

00:01:59.410 --> 00:02:03.040
But if you add 1, you get
all the way down here.

00:02:03.040 --> 00:02:07.380
So this right over
here is y plus 1.

00:02:07.380 --> 00:02:12.110
And then this base right
over here, once again,

00:02:12.110 --> 00:02:13.910
you see this is x plus 2.

00:02:13.910 --> 00:02:17.284
This length right
over here, let me

00:02:17.284 --> 00:02:20.670
see if you can see if, this
length right here, that is x.

00:02:20.670 --> 00:02:25.620
You start at x equals 0,
you go x to the right,

00:02:25.620 --> 00:02:26.930
and you get right over there.

00:02:26.930 --> 00:02:28.805
But if you want to get
all the way over here,

00:02:28.805 --> 00:02:30.630
you have to add 2.

00:02:30.630 --> 00:02:35.410
So this right over
here is x plus 2.

00:02:35.410 --> 00:02:38.910
And we know the ratio of
this height to this base

00:02:38.910 --> 00:02:41.950
right over here is always
going to be the same.

00:02:41.950 --> 00:02:48.730
So the ratio of y
plus 1 to x plus 2

00:02:48.730 --> 00:02:53.200
is going to be equal
to the ratio of C

00:02:53.200 --> 00:03:00.687
to D, which is the same
thing as the ratio as A to B.

00:03:00.687 --> 00:03:02.770
Now, that's what they're
saying right over here. y

00:03:02.770 --> 00:03:10.400
plus 1 over x plus 2 needs
to be equal to A/B. So

00:03:10.400 --> 00:03:13.050
that one's right.

00:03:13.050 --> 00:03:16.280
Here they've swapped the two.

00:03:16.280 --> 00:03:19.420
And now they're mixing
A and D together.

00:03:19.420 --> 00:03:20.970
Now, this is kind
of a strange thing

00:03:20.970 --> 00:03:25.760
that they're doing, looking
at the ratio of A to D.

00:03:25.760 --> 00:03:28.170
You could swap these
two and say, well,

00:03:28.170 --> 00:03:33.010
what's the ratio of x plus
2 to y plus 1 going to be.

00:03:33.010 --> 00:03:34.610
If this ratio is
always constant,

00:03:34.610 --> 00:03:37.680
then the reciprocal
ratio is always

00:03:37.680 --> 00:03:39.780
going to be constant as well.

00:03:39.780 --> 00:03:43.110
We could call that
1 over the slope.

00:03:43.110 --> 00:03:47.137
So x plus 2 over y
plus 1 would need

00:03:47.137 --> 00:03:48.970
to be equal to the
reciprocal of this, which

00:03:48.970 --> 00:03:54.687
would be D/C, which would be
the reciprocal of this, or B/A.

00:03:54.687 --> 00:03:55.520
Do we see that here?

00:03:55.520 --> 00:03:55.710
No.

00:03:55.710 --> 00:03:56.709
They've mixed it all up.

00:03:56.709 --> 00:03:58.910
They've mixed the A
and the D together.

00:03:58.910 --> 00:04:03.100
x plus 2 over y plus 1
is not the same thing

00:04:03.100 --> 00:04:06.480
as A/D. We're not even
using corresponding sides.

00:04:06.480 --> 00:04:07.580
We're mixing up triangles.

00:04:07.580 --> 00:04:09.600
So this is not the case.

00:04:09.600 --> 00:04:14.850
Now, they say the
ratio of y/x needs

00:04:14.850 --> 00:04:19.390
to be the same as the ratio
of C/D. So what's y/x?

00:04:19.390 --> 00:04:25.820
So y is this length
right over here.

00:04:25.820 --> 00:04:27.580
That what I did in black.

00:04:27.580 --> 00:04:29.080
That is y.

00:04:29.080 --> 00:04:30.580
I don't want to
cover anything else.

00:04:30.580 --> 00:04:32.590
And what is x?

00:04:32.590 --> 00:04:35.930
x is this length
right over here.

00:04:35.930 --> 00:04:38.300
So you could construct
a right triangle

00:04:38.300 --> 00:04:40.310
that looks something like this.

00:04:40.310 --> 00:04:45.290
But notice, this is
x right over here.

00:04:45.290 --> 00:04:48.726
Its hypotenuse is not the line.

00:04:48.726 --> 00:04:50.100
It's not a
essentially our change

00:04:50.100 --> 00:04:52.560
in x between some starting
point and some ending point.

00:04:52.560 --> 00:04:54.070
We're starting at the origin.

00:04:54.070 --> 00:04:56.030
And we're getting 2xy.

00:04:56.030 --> 00:04:58.340
It has a different hypotenuse
than the actual line.

00:04:58.340 --> 00:05:00.790
So this triangle is
not similar to all

00:05:00.790 --> 00:05:02.620
of these other triangles.

00:05:02.620 --> 00:05:06.950
So the ratio of y/x is not going
to be the same as the ratio

00:05:06.950 --> 00:05:11.370
as C/D. So this is not
going to be the case.

00:05:11.370 --> 00:05:13.580
If this line went through
the origin, then maybe

00:05:13.580 --> 00:05:15.800
we could talk a little
further about that.

00:05:15.800 --> 00:05:18.580
But this is definitely
not going to be the case.

00:05:18.580 --> 00:05:19.830
Now let's look at this.

00:05:19.830 --> 00:05:23.051
C/D, the ratio of this
entire height-- and I

00:05:23.051 --> 00:05:24.800
know I'm using too
many colors right here,

00:05:24.800 --> 00:05:27.200
it's hard to see-- so C
is this entire length.

00:05:27.200 --> 00:05:31.750
The ratio of C to D, so
this is our slope, the ratio

00:05:31.750 --> 00:05:35.090
of our rise over our run,
the ratio of our change in y

00:05:35.090 --> 00:05:36.690
over our change in x.

00:05:36.690 --> 00:05:44.170
So the ratio of C to D is
the same, is equal to A/B.

00:05:44.170 --> 00:05:47.040
Well, that's the
definition of our slope.

00:05:47.040 --> 00:05:49.980
This ratio is constant when
we're dealing with a line.

00:05:49.980 --> 00:05:53.186
So this is definitely true.

00:05:53.186 --> 00:05:55.060
Now let's actually input
it into the exercise

00:05:55.060 --> 00:05:56.540
and make sure that we've got it.

00:05:59.890 --> 00:06:02.120
We have that one,
we have that one.

00:06:02.120 --> 00:06:03.400
And we can check our answer.

00:06:03.400 --> 00:06:05.520
And we got it right.


Comparing proportional relationships

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:02.700
Choose all the
relationships that

00:00:02.700 --> 00:00:08.210
are described by the following
equation-- y is equal to 7/8 x.

00:00:08.210 --> 00:00:10.350
All right, let's look
at these relationships.

00:00:10.350 --> 00:00:13.780
Relationship A-- a
bouquet of 8 daisies

00:00:13.780 --> 00:00:16.239
cost $7 at Flora's Flowers.

00:00:16.239 --> 00:00:19.150
A bouquet of 12
daisies costs $11.

00:00:19.150 --> 00:00:20.870
The equation describes
the relationship

00:00:20.870 --> 00:00:23.800
between the number of
daisies, x, and their price

00:00:23.800 --> 00:00:26.000
at Flora's Flowers, y.

00:00:26.000 --> 00:00:28.080
So let's look at scenario A.

00:00:28.080 --> 00:00:31.390
So our x is the
number of daisies,

00:00:31.390 --> 00:00:36.580
and our y is their price.

00:00:36.580 --> 00:00:41.980
So when we get 8 daisies,
they say it costs us $7.

00:00:41.980 --> 00:00:47.470
When we get 12 daisies, they
tell us that it cost $11.

00:00:47.470 --> 00:00:50.150
So let's see whether this
relationship right over here

00:00:50.150 --> 00:00:53.270
is the same as this
relationship right over here.

00:00:53.270 --> 00:00:56.700
So if I were to take-- so let
me write this right over here.

00:00:56.700 --> 00:00:59.760
So here, it says y needs
to be equal to 7/8 x.

00:00:59.760 --> 00:01:03.090
So let's figure out
what's 7/8 x is equal to.

00:01:05.660 --> 00:01:09.092
So if I take 7/8 times 8--
well, that's going to be 7.

00:01:09.092 --> 00:01:10.050
Let me write this down.

00:01:10.050 --> 00:01:14.540
7 over 8 times 8 is equal to 7.

00:01:14.540 --> 00:01:15.550
So this checks.

00:01:15.550 --> 00:01:18.840
7/8 x is the same as the
price that they gave us.

00:01:18.840 --> 00:01:21.280
So so far, these two
seem consistently,

00:01:21.280 --> 00:01:23.680
at least based on
this first data point.

00:01:23.680 --> 00:01:25.060
Now let's look at 7/8 of 12.

00:01:29.810 --> 00:01:32.840
Let's see, if we divide
12 and 8 each by 4,

00:01:32.840 --> 00:01:38.600
this is going to be the same
thing as 7 times 3 over 2,

00:01:38.600 --> 00:01:46.320
which is the same thing as 21
over 2, which is equal to 10.5.

00:01:46.320 --> 00:01:51.740
So if y were equal to 7/8 x,
12 daisies would cost $10.50.

00:01:51.740 --> 00:01:55.220
But we just learned that
12 daisies cost $11.

00:01:55.220 --> 00:02:00.710
So these two things aren't
equal to each other.

00:02:00.710 --> 00:02:03.040
So this is not the
same relationship.

00:02:03.040 --> 00:02:07.320
In this scenario, y
does not equals 7/8x,

00:02:07.320 --> 00:02:10.130
so we can rule this one out.

00:02:10.130 --> 00:02:14.540
Let's look at relationship
B. There's a 12.5% discount

00:02:14.540 --> 00:02:16.620
for every item at Crazy Joe's.

00:02:16.620 --> 00:02:18.880
The equation describes
a relationship

00:02:18.880 --> 00:02:21.820
between a given item's
price before the discount

00:02:21.820 --> 00:02:23.840
and after the discount.

00:02:23.840 --> 00:02:27.290
So here, they're
giving us a percentage.

00:02:27.290 --> 00:02:29.140
And this equation
right over here

00:02:29.140 --> 00:02:30.450
it's written as a fraction.

00:02:30.450 --> 00:02:32.545
So let me at least convert
this into a fraction,

00:02:32.545 --> 00:02:34.670
and see what kind of a
fractional discount this is.

00:02:34.670 --> 00:02:36.420
And maybe I can
construct an equation,

00:02:36.420 --> 00:02:39.270
and see if that equation is
the same as this one here.

00:02:39.270 --> 00:02:53.410
So 12.5% is the same
thing as 12.5 over 100.

00:02:53.410 --> 00:02:56.430
And if we multiply the
numerator and denominator by 10,

00:02:56.430 --> 00:03:01.620
that's the same thing
as 125 over 1,000.

00:03:01.620 --> 00:03:04.450
Now let's see if we can
simplify this a little bit.

00:03:04.450 --> 00:03:11.641
If we divide the
numerator-- well,

00:03:11.641 --> 00:03:13.890
actually, it looks like
they're both divisible by 125.

00:03:13.890 --> 00:03:16.640
We divide the numerator
by 125, we get 1.

00:03:16.640 --> 00:03:19.420
We divide the denominator
by 125, we get 8.

00:03:19.420 --> 00:03:22.580
8 times 125 is 800 plus
200, which is 1,000.

00:03:22.580 --> 00:03:24.730
So this is the
same thing as 1/8.

00:03:24.730 --> 00:03:28.210
So really, there is a 1/8
discount for every item

00:03:28.210 --> 00:03:29.890
at Crazy Joe's.

00:03:29.890 --> 00:03:31.610
The equation describes
a relationship

00:03:31.610 --> 00:03:34.930
between a given item's
price before the discount

00:03:34.930 --> 00:03:36.599
and the price
after the discount.

00:03:36.599 --> 00:03:37.890
So let's just think about that.

00:03:37.890 --> 00:03:40.360
The price after
the discount, based

00:03:40.360 --> 00:03:42.030
on the information
they gave here,

00:03:42.030 --> 00:03:43.600
is going to be
equal to the price

00:03:43.600 --> 00:03:47.230
before the discount
minus the discount.

00:03:47.230 --> 00:03:50.780
So minus 1/8 of
the original price.

00:03:50.780 --> 00:03:52.750
Well, what's x minus 1/8 x?

00:03:52.750 --> 00:03:56.810
Or another way of thinking
about, what's 1x minus 1/8 x?

00:03:56.810 --> 00:04:00.800
Well, that's just
going to be 7/8 x?

00:04:00.800 --> 00:04:02.860
So this relationship
is the exact same

00:04:02.860 --> 00:04:04.160
as this one over here.

00:04:04.160 --> 00:04:08.370
The price after the discount
is 7/8 of the original price.

00:04:08.370 --> 00:04:10.640
So we can feel good
about this one.

00:04:10.640 --> 00:04:13.030
These two are the same.

00:04:13.030 --> 00:04:18.149
Let's look at scenario C. And
I will do scenario C in orange.

00:04:18.149 --> 00:04:20.670
A ticket to the paint
ball arena cost $7,

00:04:20.670 --> 00:04:22.430
and comes with 8 paint balls.

00:04:22.430 --> 00:04:25.106
Every extra paint ball costs $1.

00:04:25.106 --> 00:04:26.730
The equation describes
the relationship

00:04:26.730 --> 00:04:29.494
between the number of paint
balls x and their price y.

00:04:29.494 --> 00:04:30.410
So let's look at this.

00:04:30.410 --> 00:04:32.600
We'll do something similar
to what we did up here.

00:04:32.600 --> 00:04:36.580
So we'll have x, which is
the number of paint balls.

00:04:36.580 --> 00:04:39.340
And then we have y,
which is their price.

00:04:39.340 --> 00:04:42.070
And then we will
think about what 7/8

00:04:42.070 --> 00:04:43.720
x would have been equal to.

00:04:43.720 --> 00:04:46.940
And let's see if y
is equal to 7/8 x.

00:04:46.940 --> 00:04:49.059
So when we get 8
paint balls-- so

00:04:49.059 --> 00:04:50.350
x is the number of paint balls.

00:04:50.350 --> 00:04:54.270
So when we get 8 paint
balls, the price is 7.

00:04:54.270 --> 00:04:59.410
And if we take 7/8 x-- so
7/8 times 8 is indeed 7.

00:04:59.410 --> 00:05:01.220
So in this scenario,
for this data point,

00:05:01.220 --> 00:05:04.567
it is true that y
is equal to 7 x.

00:05:04.567 --> 00:05:06.400
Now let's get a couple
of other data points.

00:05:06.400 --> 00:05:07.941
And we might have
to construct these.

00:05:07.941 --> 00:05:09.940
So every extra
paint ball costs $1.

00:05:09.940 --> 00:05:11.477
So if we have one
extra paint ball--

00:05:11.477 --> 00:05:13.060
so we have 9 paint
balls-- we're going

00:05:13.060 --> 00:05:14.268
to have to pay 1 more dollar.

00:05:14.268 --> 00:05:16.200
We're going to have to pay $8.

00:05:16.200 --> 00:05:17.920
And so, what's 7/8 x here?

00:05:17.920 --> 00:05:19.901
So 7/8 times 9?

00:05:19.901 --> 00:05:21.525
Well, we don't even
have to compute it.

00:05:21.525 --> 00:05:24.770
This is going to be something
that's going to be like 63/8.

00:05:24.770 --> 00:05:27.080
64/8 would be 8.

00:05:27.080 --> 00:05:30.650
So this is a little
bit less than 8.

00:05:30.650 --> 00:05:34.870
But the important point is
that 8 is not equal to 63/8.

00:05:34.870 --> 00:05:39.310
y does not equal 7/8 x in
this scenario right over here.

00:05:39.310 --> 00:05:42.670
So this equation does not
describe all of these data

00:05:42.670 --> 00:05:44.930
points, so we can rule that out

00:05:44.930 --> 00:05:46.890
Finally, they say the
equation corresponds

00:05:46.890 --> 00:05:48.900
to the graph following graph.

00:05:48.900 --> 00:05:53.820
So let's see if y equals 7
x corresponds to this graph

00:05:53.820 --> 00:05:55.430
right over here.

00:05:55.430 --> 00:05:58.426
So what we're really
saying is that for any x,

00:05:58.426 --> 00:06:00.550
y is going to be 7/8 of that?

00:06:00.550 --> 00:06:03.140
And the easiest one is to
see when x is equal to 8.

00:06:03.140 --> 00:06:06.810
When x is equal to
7/8, x would be 7.

00:06:06.810 --> 00:06:10.200
When x is 0, 7/8 of 0 will be 0.

00:06:10.200 --> 00:06:14.254
And so, linear relationship--
if we have a line,

00:06:14.254 --> 00:06:15.670
and if it goes
through the origin,

00:06:15.670 --> 00:06:17.360
it is a proportional
relationship--

00:06:17.360 --> 00:06:21.090
that the ratio between
the x and the y-axis

00:06:21.090 --> 00:06:25.340
is always going to be the same,
or between the y and the x

00:06:25.340 --> 00:06:28.070
values is also always
going to be the same.

00:06:28.070 --> 00:06:30.150
And we know that that
ratio is the correct one.

00:06:30.150 --> 00:06:34.400
The ratio of y to x
throughout this line is 7/8.

00:06:34.400 --> 00:06:36.370
Here, it literally is--
our numbers are 7/8.

00:06:36.370 --> 00:06:38.310
Well, we can even
take other points.

00:06:38.310 --> 00:06:44.080
The ratio of 3 and 1/2 to 4 is
still the same thing as 7/8.

00:06:44.080 --> 00:06:45.950
So this relationship,
relationship

00:06:45.950 --> 00:06:48.620
D-- the relationship
described by the graph--

00:06:48.620 --> 00:06:52.170
does coincide with the
relationships described

00:06:52.170 --> 00:06:53.020
by this equation.

00:06:53.020 --> 00:06:55.532
So I would go with B and D.


Writing an equation for a proportional relationship (example)

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:05.840
I scream, you scream, we
all scream for ice cream.

00:00:05.840 --> 00:00:08.400
The following table
describes the relationship

00:00:08.400 --> 00:00:11.670
between the number of
scoops in an ice cream cone,

00:00:11.670 --> 00:00:13.380
represented by x.

00:00:13.380 --> 00:00:17.410
So this is the number of
scoops in an ice cream cone.

00:00:17.410 --> 00:00:22.470
So that's x, and the price of
the cone, represented by y.

00:00:22.470 --> 00:00:24.450
I'll do y in purple.

00:00:24.450 --> 00:00:28.080
Write the equation that
describes this relationship.

00:00:28.080 --> 00:00:28.660
So let's see.

00:00:28.660 --> 00:00:32.229
When x is 0, y is 0.

00:00:32.229 --> 00:00:35.320
When x is 1, y is 1 and 3/4.

00:00:35.320 --> 00:00:37.320
So let me write this as
an improper fraction,

00:00:37.320 --> 00:00:38.990
just so I can
visualize it better.

00:00:38.990 --> 00:00:44.050
So this is 4/4 plus 3/4,
which is equal to 7/4.

00:00:44.050 --> 00:00:46.946
When x is 2, y is 3 and 1/2.

00:00:46.946 --> 00:00:48.320
So let me see if
I can write this

00:00:48.320 --> 00:00:49.694
in a little bit
of a clearer way.

00:00:49.694 --> 00:00:53.460
So 2 times 3 is 6,
plus 1 is 7, so this

00:00:53.460 --> 00:00:59.530
is 7/2-- which is the
same thing as 14 over 4.

00:00:59.530 --> 00:01:01.600
And then here we
have, when x is 3,

00:01:01.600 --> 00:01:04.390
y is equal to-- so 5 and
1/4-- if I would write it

00:01:04.390 --> 00:01:08.850
as an improper fraction-- 4
times 5 is 20, plus 1 is 21.

00:01:08.850 --> 00:01:12.010
So this is equal to 21 over 4.

00:01:12.010 --> 00:01:15.080
And then finally, if we were to
write this as something over 4,

00:01:15.080 --> 00:01:17.320
this is equal to 28 over 4.

00:01:17.320 --> 00:01:19.610
7 is the same
thing as 28 over 4.

00:01:19.610 --> 00:01:23.340
So you see that this is a
proportional relationship.

00:01:23.340 --> 00:01:25.910
The ratio between y and x.

00:01:25.910 --> 00:01:27.120
So let me write this.

00:01:27.120 --> 00:01:34.470
The ratio between y and
x is always equal to 7/4.

00:01:34.470 --> 00:01:37.240
Notice here, y is 7/4 of x.

00:01:40.380 --> 00:01:42.030
7/4-- it's a bigger number.

00:01:42.030 --> 00:01:44.410
Or you could say 1 and 3/4 of x.

00:01:44.410 --> 00:01:45.600
So let me make that clear.

00:01:45.600 --> 00:01:49.960
So y over x is equal to 7/4.

00:01:49.960 --> 00:01:52.910
Or, we can say that
y is always 7/4 of x.

00:01:52.910 --> 00:01:56.500
We can multiply both
sides by x, if we like.

00:01:56.500 --> 00:01:58.810
So if we multiply
both sides by x,

00:01:58.810 --> 00:02:08.400
we get y is equal
to 7/4 times x.

00:02:08.400 --> 00:02:09.610
And you see it here.

00:02:09.610 --> 00:02:16.010
When x is 4, 7/4 times 4 is 7.

00:02:16.010 --> 00:02:18.820
When x is 0, y is 0.

00:02:18.820 --> 00:02:23.400
When x is 3, 7/4
times 3 is 21 over 4,

00:02:23.400 --> 00:02:25.700
which is the same
thing as 5 and 1/4.

00:02:25.700 --> 00:02:26.360
So there we go.

00:02:26.360 --> 00:02:27.550
And let me input
it, just to make

00:02:27.550 --> 00:02:28.674
sure we can input it right.

00:02:28.674 --> 00:02:30.535
So y is equal to 7/4 x.

00:02:35.200 --> 00:02:43.000
We would just write y
is equal to 7/4 times x.

00:02:43.000 --> 00:02:44.380
And let's check our answer.

00:02:44.380 --> 00:02:46.860
And we got it right.


Identifying a proportional relationship from a table (example)

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.390
The following table
describes the relationship

00:00:03.390 --> 00:00:05.580
between the number of
servings of spaghetti

00:00:05.580 --> 00:00:08.119
bolognese-- I don't know
if I'm pronouncing that--

00:00:08.119 --> 00:00:10.380
or bolognese, and the
number of tomatoes

00:00:10.380 --> 00:00:12.110
needed to prepare them.

00:00:12.110 --> 00:00:14.330
Test the ratios for
equivalents, and determine

00:00:14.330 --> 00:00:16.510
whether the relationship
is proportional.

00:00:16.510 --> 00:00:18.415
Well, you have a
proportional relationship

00:00:18.415 --> 00:00:20.790
between the number of servings
and the number of tomatoes

00:00:20.790 --> 00:00:22.656
is if the ratio of
the number of servings

00:00:22.656 --> 00:00:24.530
to the number of tomatoes
is always the same.

00:00:24.530 --> 00:00:26.238
Or if the ratio of
the number of tomatoes

00:00:26.238 --> 00:00:28.300
to the number of servings
is always the same.

00:00:28.300 --> 00:00:31.810
So let's just think about the
ratio of the number of tomatoes

00:00:31.810 --> 00:00:32.980
to the number of servings.

00:00:32.980 --> 00:00:37.210
So it's 10 to 6, which is
the same thing as 5 to 3.

00:00:37.210 --> 00:00:39.450
So here the ratio is 5 to 3.

00:00:39.450 --> 00:00:44.860
15 to 9, if you divide both
of these by 3, you get 5 to 3.

00:00:44.860 --> 00:00:46.550
So it's the same ratio.

00:00:46.550 --> 00:00:51.560
15 to 25, if you divide both
of these by 5, you get 5 to 3.

00:00:51.560 --> 00:00:53.700
So based on this data,
it looks like the ratio

00:00:53.700 --> 00:00:56.200
between the number of tomatoes
and the number of servings

00:00:56.200 --> 00:00:57.660
is always constant.

00:00:57.660 --> 00:01:01.890
So yes, this relationship
is proportional.


Analyzing and identifying proportional relationships ex1

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.970
The following tape diagram
describes the relationship

00:00:03.970 --> 00:00:07.310
between the amount of red paint
and the amount of blue paint

00:00:07.310 --> 00:00:11.230
needed in order to make
Plummy Purple paint mix.

00:00:11.230 --> 00:00:14.410
So they show us here that
for every 4 parts red,

00:00:14.410 --> 00:00:16.510
you need 3 parts blue.

00:00:16.510 --> 00:00:19.950
Or you could say that the
ratio of red to blue is 4 to 3

00:00:19.950 --> 00:00:24.440
or that the ratio of
blue to red is 3 to 4.

00:00:24.440 --> 00:00:28.850
Now, they tell us, suppose
the x-axis represents

00:00:28.850 --> 00:00:31.840
the number of red paint
cans and the y-axis

00:00:31.840 --> 00:00:33.760
represents the number
of blue paint cans.

00:00:33.760 --> 00:00:35.180
So let's scroll
down a little bit.

00:00:35.180 --> 00:00:37.470
So the x-axis is
the number of red.

00:00:37.470 --> 00:00:39.850
The y-axis is the
number of blue.

00:00:39.850 --> 00:00:42.850
First, find two sets
of corresponding values

00:00:42.850 --> 00:00:45.230
of x and y and graph them.

00:00:45.230 --> 00:00:49.630
Then determine whether the
relationship is proportional.

00:00:49.630 --> 00:00:52.750
So we know if we have 4 parts
red, we need 3 parts blue.

00:00:52.750 --> 00:00:53.720
So let me graph that.

00:00:56.340 --> 00:00:58.850
So if we have 4 parts
red-- that's that axis--

00:00:58.850 --> 00:01:00.720
we're going to
have 3 parts blue.

00:01:00.720 --> 00:01:02.480
And what if we had 8 parts red?

00:01:02.480 --> 00:01:05.069
If we had 8 parts red, well,
that's 2 groups of four.

00:01:05.069 --> 00:01:08.670
Then we're going to need 2
groups of 3 for the blue.

00:01:08.670 --> 00:01:13.710
Once again, the ratio of
blue to red is always 3 to 4.

00:01:13.710 --> 00:01:16.676
Here, literally, you
have 3 blues and 4 reds.

00:01:16.676 --> 00:01:17.800
So you have a 3 to 4 ratio.

00:01:17.800 --> 00:01:20.760
Here, you have 6 parts
blue, 8 parts red.

00:01:20.760 --> 00:01:25.330
The ratio of 6 to 8
is the same as 3 to 4.

00:01:25.330 --> 00:01:27.280
Now, the question is,
is this proportional?

00:01:27.280 --> 00:01:29.550
So you have a
proportional relationship

00:01:29.550 --> 00:01:33.589
if the ratio between these two
variables is always the same.

00:01:33.589 --> 00:01:35.380
And we already said,
the ratio between them

00:01:35.380 --> 00:01:36.200
is always the same.

00:01:36.200 --> 00:01:41.530
The ratio of blue to red is
always going to be 3 to 4.

00:01:41.530 --> 00:01:44.660
Another sign that this is
a proportional relationship

00:01:44.660 --> 00:01:46.420
is that it's a line.

00:01:46.420 --> 00:01:48.240
And not only is it
a line, but it's

00:01:48.240 --> 00:01:49.905
a line that goes
through the origin.

00:01:49.905 --> 00:01:50.780
And that makes sense.

00:01:50.780 --> 00:01:55.070
If I have 0 red, well, I'm
going to have 3/4 of that blue.

00:01:55.070 --> 00:01:58.700
Well, 3/4 times 0 is
still going to be 0.

00:01:58.700 --> 00:02:01.377
So this is a proportional
relationship.

00:02:01.377 --> 00:02:02.460
We could check our answer.

00:02:02.460 --> 00:02:03.290
Yes, it is.

00:02:03.290 --> 00:02:04.760
And let me explain
that last point

00:02:04.760 --> 00:02:06.580
in a little bit more detail.

00:02:06.580 --> 00:02:10.250
So as we said, the
ratio-- and I'm

00:02:10.250 --> 00:02:13.100
going to do this in the
right colors-- of blue

00:02:13.100 --> 00:02:16.657
to-- I don't have an exact red.

00:02:16.657 --> 00:02:17.490
I'll do this orange.

00:02:17.490 --> 00:02:18.323
That's close enough.

00:02:18.323 --> 00:02:20.960
The ratio of blue
to red is always

00:02:20.960 --> 00:02:26.470
going to be equal to 3 to 4.

00:02:26.470 --> 00:02:30.340
And so if you multiply both
sides by the number of red,

00:02:30.340 --> 00:02:32.510
you could say that the
amount you have in blue

00:02:32.510 --> 00:02:35.910
is always going to be 3/4 the
amount that you have in red.

00:02:39.940 --> 00:02:41.020
You can also verify this.

00:02:41.020 --> 00:02:45.490
When R is 4-- if you have
a 4 here-- 3/4 of 4 is 3.

00:02:45.490 --> 00:02:47.430
Blue is 3.

00:02:47.430 --> 00:02:50.410
And so if R is a 0, if this
is really proportional,

00:02:50.410 --> 00:02:54.140
then this is also going to be 0.

00:02:54.140 --> 00:02:57.620
A proportional relationship will
always go through the origin.

00:02:57.620 --> 00:02:59.130
If you don't go
through the origin,

00:02:59.130 --> 00:03:00.710
it's not a proportional
relationship.

00:03:00.710 --> 00:03:04.004
If it's not linear, it's not
a proportional relationship.


Analyzing and identifying proportional relationships ex2

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:03.640
For every 3 hours
of computer games,

00:00:03.640 --> 00:00:06.390
Sandy has 2 hours of homework.

00:00:06.390 --> 00:00:08.970
For every 7 hours
of computer games,

00:00:08.970 --> 00:00:11.350
Madison has 4 hours of homework.

00:00:11.350 --> 00:00:13.990
I guess they have to do a
certain amount of homework

00:00:13.990 --> 00:00:17.240
for every amount of computer
games that they play.

00:00:17.240 --> 00:00:19.510
The following graphs
describe the relationship

00:00:19.510 --> 00:00:23.570
between the number of hours of
computer games on the x-axis

00:00:23.570 --> 00:00:27.300
and the number of hours
of homework on the y-axis.

00:00:27.300 --> 00:00:30.690
Which point can tell us how
many hours of homework Sandy

00:00:30.690 --> 00:00:34.380
had if she played computer
games for 5 hours?

00:00:34.380 --> 00:00:36.270
So we have to pick which
one of these graphs

00:00:36.270 --> 00:00:42.460
describe the relationship for
Sandy between computer games--

00:00:42.460 --> 00:00:45.160
that's what's going on on
the x-axis-- and homework.

00:00:45.160 --> 00:00:48.410
So they tell us for Sandy, that
for every 3 hours of computer

00:00:48.410 --> 00:00:50.900
games, she has 2
hours of homework.

00:00:50.900 --> 00:00:54.790
So for every 3 hours of
computer games on the x-axis,

00:00:54.790 --> 00:00:57.170
she has 2 hours of homework.

00:00:57.170 --> 00:00:59.710
So it looks like that
point is sitting right here

00:00:59.710 --> 00:01:01.480
on this red line.

00:01:01.480 --> 00:01:03.800
And this red line,
every time the x-axis

00:01:03.800 --> 00:01:07.790
increases by 3-- which means 3
more hours of computer games--

00:01:07.790 --> 00:01:09.180
it increases by 2.

00:01:09.180 --> 00:01:10.680
So there's 2 more
hours of homework.

00:01:10.680 --> 00:01:13.650
So this red line
describes the relationship

00:01:13.650 --> 00:01:16.310
between hours of
computer games and hours

00:01:16.310 --> 00:01:19.130
of homework for Sandy.

00:01:19.130 --> 00:01:21.030
So we have to look
at the red graph.

00:01:21.030 --> 00:01:24.300
And so they're saying, if Sandy
played 5 hours of computer

00:01:24.300 --> 00:01:26.870
games, so 5 hours of
computer games for Sandy,

00:01:26.870 --> 00:01:28.210
you go to the red graph.

00:01:28.210 --> 00:01:30.550
You get to point C.
So this describes it.

00:01:30.550 --> 00:01:33.510
It looks like it's a little
over 3 hours of homework,

00:01:33.510 --> 00:01:36.840
if she spends 5 hours
playing computer games.

00:01:36.840 --> 00:01:40.730
Pick choice C. And it is right.


Adding decimals word problem

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:05.490
On Saturday, William's
parents gave birth to twins

00:00:05.490 --> 00:00:08.189
and named them
Nadia and Vanessa.

00:00:08.189 --> 00:00:14.030
When they were first born,
Nadia weighed 7.27 pounds

00:00:14.030 --> 00:00:21.720
and was 21.5 inches tall, and
Vanessa weighed 8.34 pounds.

00:00:21.720 --> 00:00:24.950
How much did the
babies weigh in total?

00:00:24.950 --> 00:00:28.980
So they tell us Nadia weighed
7.27 and Vanessa weighed 8.34.

00:00:28.980 --> 00:00:30.360
We have to add those up.

00:00:30.360 --> 00:00:32.009
And really, they
just gave us this.

00:00:32.009 --> 00:00:35.019
The length of Nadia at birth
really is a distraction,

00:00:35.019 --> 00:00:36.560
just so that we just
don't mindlessly

00:00:36.560 --> 00:00:37.760
add any numbers we see.

00:00:37.760 --> 00:00:40.360
So this is really
unnecessary information

00:00:40.360 --> 00:00:41.730
just to distract us.

00:00:41.730 --> 00:00:45.070
So we need to add Nadia's
birth weight plus Vanessa's.

00:00:45.070 --> 00:00:47.260
So it's 7.27.

00:00:47.260 --> 00:00:53.170
So 7.27 plus 8.34.

00:00:53.170 --> 00:00:55.460
And it's always important
that we line up the decimal.

00:00:55.460 --> 00:00:57.001
And I'd like to do
the decimal first.

00:00:57.001 --> 00:01:00.100
So it's 8.34.

00:01:00.100 --> 00:01:03.140
And we'll just add these two up.

00:01:03.140 --> 00:01:09.410
So 7 plus 4, and this is
really 7/100 plus 4/100

00:01:09.410 --> 00:01:14.950
is 11/100, which is the same
thing as 1/100 and 1/10.

00:01:14.950 --> 00:01:19.320
1/10 plus 2/10
plus 3/10 is 6/10.

00:01:19.320 --> 00:01:21.780
We've got our decimal
sign right over there.

00:01:21.780 --> 00:01:24.790
And then 7 plus 8 is
15, or you could even

00:01:24.790 --> 00:01:28.000
say it's five 1's and one 10.

00:01:28.000 --> 00:01:28.710
And we're done.

00:01:28.710 --> 00:01:32.890
They weighed 15.61
pounds in total.