Writing expressions with exponents

WEBVTT
Kind: captions
Language: en

00:00:00.334 --> 00:00:01.449
- [Voiceover] What I
want to do in this video

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is give ourselves some
practice constructing

00:00:03.932 --> 00:00:05.476
expressions with exponents

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to solve real problems.

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So let's start looking
at these real problems.

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So this first one says the surface area

00:00:12.326 --> 00:00:15.007
of a cube is equal to the sum of the areas

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of its six sides, and that makes sense

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if I were to draw a cube here.

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If I were to draw a cube,

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it has six sides, it has the three sides

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we see right over here,

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one, two, three, and
it has the three sides

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that we don't see, if we were

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to make this a transparent cube

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then we could count those as well,

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so the one in the back here,

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one in the back there, and then
the bottom right over here.

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So you have six sides of
a cube, we believe that,

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and it says, Spencer has
two cube shaped blocks

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that he wants to paint.

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One cube has sides of length four units.

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One cube has sides of length four units,

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so let's say that this is
this one right over here.

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So it's a cube, so all of these dimensions

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are four units.

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The other cube has lengths
of sides 2.5 units,

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so the other cube is going
to look something like this.

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The other cube looks something like this,

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and the sides are 2.5, 2.5, 2.5.

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Write a numerical expression

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with exponents for the total surface area

00:01:16.320 --> 00:01:18.340
he has to paint.

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Well, there's six sides here,

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and what's the area of
each of the six sides?

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So there's six sides like this,

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so there's going to be six times

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the area of each side, and what's the area

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of each side?

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Well, each side is going
to be four by four,

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each side is a four by four square.

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So each side is going to have an area

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of four squared.

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So the surface area of the six sides

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of this cube, is six times four squared.

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Once again, the surface area,

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or the area of each side,

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is going to be four squared,

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and there's six of them,

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so that's the surface area of this cube,

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and then the same logic
for the surface area

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of this cube.

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The area of each of the sides

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is going to be 2.5 times 2.5,

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or we could write that as
2.5 to the second power,

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and there's going to be
six sides to the cube,

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so it's going to be six times 2.5 squared,

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so a numerical expression

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with exponents for the total
surface area he has to paint,

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well that's the surface
area of both cubes,

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this is the surface
area of the yellow cube,

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this is the surface area of the blue cube,

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and we are done.

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And just as a reminder on a little bit

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of the notation I'm using,

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this is not a decimal, this is higher

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than a decimal, it's kind
of halfway up the numbers,

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and when I write it like that,

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that's a way of writing multiplication,

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and you're going to see that more and more

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as you go into algebra.

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I could have also written it this way,

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I could have written it as six times four

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squared, plus, six times

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2.5 squared.

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This would have been
another way to write it,

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and here, when you just
write another number

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next to the parentheses,

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it means multiply by
whatever's in the parentheses,

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so this is six times four squared,

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this is six times four squared.

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So while we're constructing
these exponent expressions

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it's also good to get used
to some of this notation.

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So let's do another one,

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in fact, let's do two more.

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Boom Town was established in a remote part

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of the world in 2010.

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Its population has tripled,

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let me underline that,

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its population has tripled every year

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since its founding.

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If there were 1,050 people

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in Boom Town when it was established,

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let me write that down.

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So there's 1,050 people in Boom Town

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when it was established,

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what would be the
population after six years?

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So remember, after every year it triples.

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So after the first year,

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we're going to multiply it by three,

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we're going to have three
times the population.

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The second year, multiply it by three,

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third year, fourth year,

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fifth year, and sixth year.

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Or, maybe the easier way,

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or more compact way of writing this is,

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its population after six years

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is going to be its initial population,

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times three to the sixth power.

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We have six, we have six threes

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that we're multiplying together.

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Times three to the sixth power,

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or, just as we just talked about,

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the initial population, 1,050 people,

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times three to the sixth power.

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And the whole point here

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is not to evaluate these things,

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but just to construct the expression,

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to know that we could evaluate this,

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and actually figure out the exact answer,

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but all they're asking us for

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is write a numerical
expression with exponents

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to describe the population of Boom Town

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after six years, and we just did that.

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Let's do one more of these,

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this is strangely fun (chuckles)

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So we have a square wall with sides

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of length four meters.

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Let's draw our square wall with sides

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of length four meters.

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So it's a square wall.

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I'm sensing a lot of
squares in this exercise

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or these exercises.

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So I have a square wall

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with sides of length four meters,

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so this is four meters,

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and this is four meters.

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And it has three square windows.

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It has three square windows,

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let me do this in a different color.

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It has three square windows

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each with sides of length 1.5 meters.

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So it might look like this.

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One square window,

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another square window,

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and another square window,

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and these are each 1.5 by 1.5.

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Lena, or Lena, wants to paper the wall,

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so she needs to determine the area

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of the wall that does
not include the windows.

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Write a numerical
expression with exponents

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that describes the area

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in square meters of the wall
that Lena wants to paper.

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So that's going to be all this stuff

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that I am shading in blue.

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Essentially the area of the larger wall,

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minus the area of those three windows.

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And, so what's that going to be?

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Well, the area of the larger wall

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is going to be four times four,

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we can just write that as
four to the second power

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or four squared.

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The reason why people even call it

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four squared, is because that's how

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you figure out the area of a square.

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Four squared, and then
you want to subtract out

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the area of these three windows.

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So there's three windows,

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each of them have an area of 1.5 squared.

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This area is 1.5 squared,

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1.5 times 1.5, this area

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is 1.5 times 1.5,

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this area is 1.5 times 1.5.

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So you get four squared minus

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three times 1.5 squared.

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And there you go, that's going

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to be the area that she wants to paper.


Rewriting an exponential expression in a hairier way

WEBVTT
Kind: captions
Language: en

00:00:00.372 --> 00:00:01.873
- [Voiceover] What I hope
to do in this video is,

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start with a exponential expression

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that's in a fairly straightforward form

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and then turn it into
one in a hairier form.

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And let's actually just do it.

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I'll show the initial expression

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and then the form that I want
it in, or that we want it in.

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And then we can talk a little bit about,

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why would ever actually wanna do that?

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So let's say my expression is 1/32

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times two to the t power.

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So this is fairly straightforward
exponential expression.

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But let's say we want it,

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we want it in the form

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A time B,

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and this is where it's gonna get hairy,

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A time B to the t over 10 power,

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minus one.

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And so, you're probably
immediately saying,

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"Why would I ever wanna take something

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"nice and simple like this

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"and turn it into this beastly
thing right over here?"

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And the answer is, when
you get into higher math

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and you start doing your
physics and your chemistry,

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you're going to see, maybe, you know,

00:00:56.013 --> 00:00:57.296
you got a result like this,

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but then you look in your textbook,

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or your professor, it
has a result like this.

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Just wanna know, "How do I
transfer from this to this?"

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Or actually, sometimes when you transfer

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to a form like this ...

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Obviously, I've just arbitrarily
written this form here.

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But sometimes when you write it

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in another form that might
even be a little hairier,

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it can give you an intuition
on the underlying processes

00:01:16.595 --> 00:01:19.987
that that expression
is trying to describe.

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So if you can take that
on a leap of faith,

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let's actually try to do it.

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And, at a minimum, it's
gonna make you a lot better

00:01:24.843 --> 00:01:26.774
at exponent properties.

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So see if you can rewrite
this in this form.

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So I'm assuming you took a go at it.

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So let's try to do it together.

00:01:33.793 --> 00:01:36.069
So the first thing,

00:01:36.069 --> 00:01:40.849
the first thing that I might wanna do is,

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well, let's see if we can ...

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Let's see. What would I wanna do?

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The first thing I wanna do is,

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take this t and get it into a t ...

00:01:51.064 --> 00:01:53.987
let's get it into a t over 10.

00:01:53.987 --> 00:01:55.551
So, to do that,

00:01:55.551 --> 00:01:57.705
we essentially just need to multiply by 10

00:01:57.705 --> 00:01:59.168
and divide by 10.

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So let's multiply by 10,

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and then also divide by 10.

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Then we haven't changed the value up here.

00:02:07.254 --> 00:02:09.428
So we can rewrite this,

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we can rewrite this as

00:02:12.069 --> 00:02:15.022
1/32

00:02:15.022 --> 00:02:19.110
times two

00:02:19.110 --> 00:02:20.402
to the ...

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So let me circle t ...

00:02:23.512 --> 00:02:25.258
Do this in a different color.

00:02:25.258 --> 00:02:30.235
T over 10,

00:02:30.235 --> 00:02:32.553
t over 10

00:02:32.553 --> 00:02:34.503
times 10.

00:02:34.503 --> 00:02:38.405
Times 10.

00:02:38.405 --> 00:02:39.643
All right. So we got a

00:02:39.643 --> 00:02:42.732
t over 10 over here.

00:02:42.732 --> 00:02:44.397
But then I have this times 10.

00:02:44.397 --> 00:02:46.856
So how do I deal with this?

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Well, one thing that I can do ...

00:02:48.238 --> 00:02:50.269
Let me actually just write
this the other way around.

00:02:50.269 --> 00:02:51.793
Let me write it as,

00:02:51.793 --> 00:02:53.621
let me write it as

00:02:53.621 --> 00:02:57.117
10 times t over 10.

00:02:57.117 --> 00:02:59.249
10 times

00:02:59.249 --> 00:03:01.404
t over 10.

00:03:01.404 --> 00:03:02.967
So hopefully what I just did here

00:03:02.967 --> 00:03:04.207
isn't a huge stretch here.

00:03:04.207 --> 00:03:06.990
I just literally multiplied and divide by,

00:03:06.990 --> 00:03:09.021
divided by 10, times this t over 10.

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But when I write it this way,

00:03:10.688 --> 00:03:13.024
an exponent property
might jump out at you.

00:03:13.024 --> 00:03:15.665
If I have, if I have

00:03:15.665 --> 00:03:17.271
a to the b,

00:03:17.271 --> 00:03:19.163
and then I raise that to the c,

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that's going to be a

00:03:20.301 --> 00:03:22.211
to the bc.

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Or, another way around,

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a to the bc is going to be a to the b

00:03:25.169 --> 00:03:27.281
to the c.

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And so, this piece right over here,

00:03:30.817 --> 00:03:32.585
I can rewrite it as

00:03:32.585 --> 00:03:34.311
two to the 10th,

00:03:34.311 --> 00:03:35.988
and then raise that

00:03:35.988 --> 00:03:38.832
to the t over 10 power.

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To the t over 10 power.

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Once again, two to the 10th

00:03:42.997 --> 00:03:44.745
and then raise that to the t over 10,

00:03:44.745 --> 00:03:45.943
that's gonna be the same thing

00:03:45.943 --> 00:03:49.457
as two to the 10 times t over 10.

00:03:49.457 --> 00:03:52.728
And of course, we still have
the one over 32 over here.

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One over,

00:03:55.208 --> 00:03:56.955
one over 32.

00:03:56.955 --> 00:03:58.012
I'm tempted to write that

00:03:58.012 --> 00:04:00.309
as two to the negative fifth power,

00:04:00.309 --> 00:04:02.543
but I won't do that just yet.

00:04:02.543 --> 00:04:05.063
So let's see. What's
two to the 10th power?

00:04:05.063 --> 00:04:06.810
Actually, let's just, let's just keep it,

00:04:06.810 --> 00:04:08.820
let's just keep it as
two to the 10th power,

00:04:08.820 --> 00:04:10.202
just for simplicity right now.

00:04:10.202 --> 00:04:11.154
Later we can,

00:04:11.154 --> 00:04:13.553
you might know that that's gonna be 1,024.

00:04:13.553 --> 00:04:15.682
But let's just, let's
see what else we can do.

00:04:15.682 --> 00:04:17.764
So we know this is going
to be some num- ...

00:04:17.764 --> 00:04:19.876
Actually, let me just write out as 1,024.

00:04:19.876 --> 00:04:22.070
So we have one over 32

00:04:22.070 --> 00:04:26.256
times 1,024

00:04:26.256 --> 00:04:28.571
to the t over 10,

00:04:28.571 --> 00:04:30.585
to the t over 10 power.

00:04:30.585 --> 00:04:31.883
So it seems like we're getting close.

00:04:31.883 --> 00:04:34.495
If there was no minus one here,

00:04:34.495 --> 00:04:35.958
we're essentially done.

00:04:35.958 --> 00:04:37.258
But now there's this minus one.

00:04:37.258 --> 00:04:38.977
So how do we deal with that?

00:04:38.977 --> 00:04:41.273
Well, we can do a
similar type of strategy.

00:04:41.273 --> 00:04:42.999
We can subtract one,

00:04:42.999 --> 00:04:44.930
and then we could add,

00:04:44.930 --> 00:04:46.250
and then we could add one.

00:04:46.250 --> 00:04:47.895
Then we're not actually
changing the value.

00:04:47.895 --> 00:04:50.090
Just as we multiplied
by 10 and divide by 10,

00:04:50.090 --> 00:04:51.797
we're not changing the value up here.

00:04:51.797 --> 00:04:53.829
If you subtract one and
add one to the exponent,

00:04:53.829 --> 00:04:55.859
you're not changing its value.

00:04:55.859 --> 00:04:57.546
And so, what is this going to be?

00:04:57.546 --> 00:04:59.699
We wanna leave this minus one here.

00:04:59.699 --> 00:05:01.427
But we wanna get rid of,

00:05:01.427 --> 00:05:04.189
we wanna get rid of this plus one somehow.

00:05:04.189 --> 00:05:06.404
And here, we just have to remind ourselves

00:05:06.404 --> 00:05:09.656
that, if we have a to the b

00:05:09.656 --> 00:05:13.131
times a to the c,

00:05:13.131 --> 00:05:14.430
that's going to be equal to

00:05:14.430 --> 00:05:16.827
a to the b plus c.

00:05:16.827 --> 00:05:19.298
If you have the same base, multiplied,

00:05:19.298 --> 00:05:21.235
same base raised to different exponents

00:05:21.235 --> 00:05:22.109
and you multiply them,

00:05:22.109 --> 00:05:23.857
you could just add the exponents.

00:05:23.857 --> 00:05:25.502
And so you could also
go the other way around.

00:05:25.502 --> 00:05:27.472
If you have a to the b plus c,

00:05:27.472 --> 00:05:29.138
you could break it up into a to the b

00:05:29.138 --> 00:05:31.048
times a to the c.

00:05:31.048 --> 00:05:34.096
So, this business right over here,

00:05:34.096 --> 00:05:36.005
this business right over here,

00:05:36.005 --> 00:05:38.603
this is 1,024 to the

00:05:38.603 --> 00:05:40.407
t over 10 minus one,

00:05:40.407 --> 00:05:42.298
plus one.

00:05:42.298 --> 00:05:44.431
So we can break this up as,

00:05:44.431 --> 00:05:46.544
we can break this up as,

00:05:46.544 --> 00:05:48.169
1,024,

00:05:48.169 --> 00:05:51.390
1,024 to the t

00:05:51.390 --> 00:05:53.748
over 10 minus one,

00:05:53.748 --> 00:05:55.210
that's this part here,

00:05:55.210 --> 00:05:58.013
and then times 1,024 to the one.

00:05:58.013 --> 00:05:59.090
Times ...

00:05:59.090 --> 00:06:00.654
Let me make this in a different color.

00:06:00.654 --> 00:06:02.930
So, let's see green.

00:06:02.930 --> 00:06:04.048
So this right over here.

00:06:04.048 --> 00:06:07.847
So times 1,024

00:06:07.847 --> 00:06:09.309
to the one power.

00:06:09.309 --> 00:06:11.463
That's this one power right over here.

00:06:11.463 --> 00:06:13.068
And of course, we still have

00:06:13.068 --> 00:06:14.816
the one over,

00:06:14.816 --> 00:06:17.091
the one over 32.

00:06:17.091 --> 00:06:18.533
All right. So now we're really close.

00:06:18.533 --> 00:06:21.007
We have the 1,024 to
the t over 10 minus one,

00:06:21.007 --> 00:06:22.676
we have t over 10 minus 1.

00:06:22.676 --> 00:06:24.340
And now we just have to simplify.

00:06:24.340 --> 00:06:25.132
We can rewrite.

00:06:25.132 --> 00:06:26.718
This is going to be equal to,

00:06:26.718 --> 00:06:28.098
this is going to be equal to,

00:06:28.098 --> 00:06:29.845
we can just bring the 1,024,

00:06:29.845 --> 00:06:32.102
1,024 to the first
power, that's just 1,024.

00:06:32.102 --> 00:06:35.593
So that's going to be 1,024

00:06:35.595 --> 00:06:37.038
over ...

00:06:37.038 --> 00:06:39.009
I can put all these commas here if I like.

00:06:39.009 --> 00:06:42.317
1,024 over this 32.

00:06:42.317 --> 00:06:45.038
Let me do that magenta color.

00:06:45.038 --> 00:06:47.334
Over this 32

00:06:47.334 --> 00:06:48.776
times ...

00:06:48.776 --> 00:06:49.955
home stretch.

00:06:49.955 --> 00:06:51.438
Times,

00:06:51.438 --> 00:06:54.790
times 1,024

00:06:54.790 --> 00:06:56.946
to the t over,

00:06:56.946 --> 00:06:59.464
to the t over 10

00:06:59.464 --> 00:07:01.048
minus one power.

00:07:01.048 --> 00:07:02.999
And now we could just simplify this.

00:07:02.999 --> 00:07:05.010
You might recognize 1,024,

00:07:05.010 --> 00:07:06.321
we already saw, that was the same thing

00:07:06.321 --> 00:07:08.070
as two to the 10th power.

00:07:08.070 --> 00:07:12.396
32 is the same thing as
two to the fifth power.

00:07:12.396 --> 00:07:14.967
So two to the 10th divided
by two to the fifth ...

00:07:14.967 --> 00:07:17.467
Actually, this is another
exponent property at play here,

00:07:17.467 --> 00:07:19.214
although you could just
divide the numbers.

00:07:19.214 --> 00:07:22.139
If you have a to the b

00:07:22.139 --> 00:07:23.440
over a to the c,

00:07:23.440 --> 00:07:24.579
this is going to be equal to

00:07:24.579 --> 00:07:27.240
a to the b minus c.

00:07:27.240 --> 00:07:28.234
So this is going to be

00:07:28.234 --> 00:07:30.206
two to the 10 minus five,

00:07:30.206 --> 00:07:32.268
over this whole thing.

00:07:32.268 --> 00:07:34.525
This whole thing right over here.

00:07:34.525 --> 00:07:35.805
And this thing ...

00:07:35.805 --> 00:07:37.572
See, I wanted to do that
in a different color ...

00:07:37.572 --> 00:07:39.645
This thing is just

00:07:39.645 --> 00:07:42.245
going to be two to the fifth power,

00:07:42.245 --> 00:07:43.483
or 32.

00:07:43.483 --> 00:07:45.231
So this is going to be 32

00:07:45.231 --> 00:07:47.618
times 1,024.

00:07:48.811 --> 00:07:51.016
We were in the home stretch here.

00:07:51.016 --> 00:07:53.819
1,024

00:07:53.819 --> 00:07:56.765
to the t over 10 minus one.

00:07:56.765 --> 00:07:57.740
So once again,

00:07:57.740 --> 00:07:59.002
normally in our lives,

00:07:59.002 --> 00:08:00.544
we like to make things simpler,

00:08:00.544 --> 00:08:01.986
and I'm a big advocate of that.

00:08:01.986 --> 00:08:03.390
It's a good life philosophy.

00:08:03.390 --> 00:08:04.974
But this is a case where we really did

00:08:04.974 --> 00:08:06.112
make it more complicated.

00:08:06.112 --> 00:08:07.939
We started with 1/32

00:08:07.939 --> 00:08:09.850
times two to the t ...

00:08:09.850 --> 00:08:10.624
Actually, we could have ...

00:08:10.624 --> 00:08:12.166
Well, there's other ways we
could have written that ...

00:08:12.166 --> 00:08:13.404
And we turned it into this thing

00:08:13.404 --> 00:08:15.498
with this somewhat hairier exponent.

00:08:15.498 --> 00:08:17.347
But it's a useful skill to have

00:08:17.347 --> 00:08:19.276
because you might get a result like

00:08:19.276 --> 00:08:20.477
what we originally started with,

00:08:20.477 --> 00:08:22.792
and then someone else might
get a result like this,

00:08:22.792 --> 00:08:23.725
and it's very important to realize,

00:08:23.725 --> 00:08:25.669
"Hey, you actually go the same result.

00:08:25.669 --> 00:08:27.132
"They're just different ways of expressing

00:08:27.132 --> 00:08:30.208
"the same exponential expression."


Modeling mosquitos with quadratics

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.133
- [Voiceover] The number of
mosquitos in millions, M,

00:00:03.133 --> 00:00:04.737
so M is the number of
mosquitos in millions.

00:00:04.737 --> 00:00:06.895
In Brooklyn, New York
it depends on the June

00:00:06.895 --> 00:00:09.287
rainfall in centimeters, R.

00:00:09.287 --> 00:00:11.888
So R is the amount of rainfall
measured in centimeters.

00:00:11.888 --> 00:00:14.094
And can be modeled by the function

00:00:14.094 --> 00:00:16.160
M, so the number of mosquitos in millions,

00:00:16.160 --> 00:00:18.667
is equal to negative R times R minus four,

00:00:18.667 --> 00:00:22.266
where R, once again, is the
rainfall in centimeters,

00:00:22.266 --> 00:00:23.567
I guess in June.

00:00:23.567 --> 00:00:24.727
Alright, fair enough.

00:00:24.727 --> 00:00:26.261
Now let's see if we can
answer these questions.

00:00:26.261 --> 00:00:29.651
And like always, I encourage
you to get inspired

00:00:29.651 --> 00:00:32.576
and try to answer these
questions on your own.

00:00:32.576 --> 00:00:34.225
(laughing) Alright, first one.

00:00:34.225 --> 00:00:37.934
How many centimeters of
rainfall will there be...

00:00:39.020 --> 00:00:40.784
After how many centimeters of rainfall

00:00:40.784 --> 00:00:44.291
will there be a population
of zero million mosquitos?

00:00:44.291 --> 00:00:46.218
Or will there be just no mosquitos?

00:00:46.218 --> 00:00:47.866
Zero mosquitos.

00:00:47.866 --> 00:00:49.747
Well let's just rewrite this function.

00:00:49.747 --> 00:00:51.721
So our population of mosquitos,

00:00:51.721 --> 00:00:53.554
it actually just helps
my brain to rewrite it,

00:00:53.554 --> 00:00:55.436
to kind of process what's going on.

00:00:55.436 --> 00:00:57.989
My population of mosquitos in millions

00:00:57.989 --> 00:01:02.377
is going to be negative
R times R minus four,

00:01:02.377 --> 00:01:04.909
where R is the number of,

00:01:04.909 --> 00:01:07.649
the number of centimeters of rainfall.

00:01:07.649 --> 00:01:09.831
So in order to have a
zero million population,

00:01:09.831 --> 00:01:12.176
so that means M is going to
have to be equal to zero.

00:01:12.176 --> 00:01:15.208
So we have to figure out at what Rs,

00:01:15.208 --> 00:01:18.784
at what Rs is M equal to zero?

00:01:18.784 --> 00:01:21.012
So we just have to solve
this right over here.

00:01:21.012 --> 00:01:23.055
Well in order for this
expression right over here

00:01:23.055 --> 00:01:23.927
to be equal to zero, notice it's

00:01:23.927 --> 00:01:25.715
the product of two expressions.

00:01:25.715 --> 00:01:27.736
We are multiplying, you could view it as

00:01:27.736 --> 00:01:32.402
we're multiplying negative
R times R minus four.

00:01:32.402 --> 00:01:34.353
Let me do R minus four
in a different color.

00:01:34.353 --> 00:01:36.187
Times R minus four.

00:01:36.187 --> 00:01:38.857
So if you're multiplying two
expressions and you get zero,

00:01:38.857 --> 00:01:41.645
one or both of them need
to be equal to zero.

00:01:41.645 --> 00:01:42.968
So let's see how that happens.

00:01:42.968 --> 00:01:46.334
So either negative R is
going to be equal to zero,

00:01:46.334 --> 00:01:49.295
negative R is equal to zero, or

00:01:50.304 --> 00:01:54.335
or you're going to have R
minus four is equal to zero.

00:01:54.335 --> 00:01:57.098
Or R minus four is equal to zero.

00:01:57.098 --> 00:01:58.816
So how does negative R equal zero?

00:01:58.816 --> 00:02:00.279
We divide both sides by negative one

00:02:00.279 --> 00:02:01.719
or multiply both sides by negative one

00:02:01.719 --> 00:02:04.435
and you get r would need
to be equal to zero.

00:02:04.435 --> 00:02:06.456
How does R minus four equal zero?

00:02:06.456 --> 00:02:07.848
Well you add four to both sides,

00:02:07.848 --> 00:02:11.284
you see that R would
have to be equal to four.

00:02:11.284 --> 00:02:15.046
So it either is no rainfall,
zero centimeters of rainfall,

00:02:15.046 --> 00:02:17.067
or four centimeters of rainfall,

00:02:17.067 --> 00:02:19.691
you get a population of zero mosquitos.

00:02:19.691 --> 00:02:20.549
Now why does that,

00:02:20.549 --> 00:02:22.268
let's just think about
whether that makes sense.

00:02:22.268 --> 00:02:24.148
Of whether this is a good model for

00:02:24.148 --> 00:02:26.424
for the number of mosquitos.

00:02:26.424 --> 00:02:29.141
Well we always imagine mosquitos
need some standing water,

00:02:29.141 --> 00:02:30.998
so if you have no rainfall,
there's not going to be

00:02:30.998 --> 00:02:32.624
any place for them to breed.

00:02:32.624 --> 00:02:34.853
There's not going to be
any standing water, maybe.

00:02:34.853 --> 00:02:36.966
And the way this model seems, it's like

00:02:36.966 --> 00:02:38.544
well if you have enough rainfall,

00:02:38.544 --> 00:02:40.518
so some rain is gonna
help the mosquitos breed,

00:02:40.518 --> 00:02:42.144
but then if you have enough rainfall,

00:02:42.144 --> 00:02:44.164
they're just gonna get I
guess washed out someplace.

00:02:44.164 --> 00:02:46.090
i guess raindrops are
gonna hit them (laughing)

00:02:46.090 --> 00:02:47.461
as they fly, I don't know what the,

00:02:47.461 --> 00:02:49.852
what's actually happening
at the mosquito scale,

00:02:49.852 --> 00:02:51.803
but it's implying that if
there's a lot of rainfall

00:02:51.803 --> 00:02:53.683
it would actually be hard.

00:02:53.683 --> 00:02:55.100
I guess you don't have
standing water any more.

00:02:55.100 --> 00:02:57.398
The water is moving
around and is overflowing

00:02:57.398 --> 00:02:59.326
and it's going into the
gutter and whatever else.

00:02:59.326 --> 00:03:01.827
It would make it vey hard
for the mosquitos to breed.

00:03:02.640 --> 00:03:06.008
What is the maximum possible
number of mosquitos?

00:03:06.008 --> 00:03:08.236
So the very idea that
they're asking us a maximum

00:03:08.236 --> 00:03:11.395
implies that this quadratic over here,

00:03:11.395 --> 00:03:12.904
and it is a quadratic and we'll see that

00:03:12.904 --> 00:03:15.063
more clearly in a few seconds,

00:03:15.063 --> 00:03:18.639
is shaped like this in order
to have a maximum point.

00:03:18.639 --> 00:03:19.707
Because if it was shaped like this,

00:03:19.707 --> 00:03:21.100
it would have a minimum point,

00:03:21.100 --> 00:03:22.842
it wouldn't have any one maximum.

00:03:22.842 --> 00:03:25.001
And because we have
this negative out here,

00:03:25.001 --> 00:03:26.836
that makes us pretty confident

00:03:26.836 --> 00:03:29.924
it is going to be a
downward opening parabola.

00:03:29.924 --> 00:03:32.547
But to understand what
this maximum point is,

00:03:32.547 --> 00:03:34.777
let's write it in vertex form.

00:03:34.777 --> 00:03:36.959
Just as a reminder what
vertex form would be,

00:03:36.959 --> 00:03:39.675
it would be writing it in the form of M

00:03:39.675 --> 00:03:41.975
is equal to

00:03:41.975 --> 00:03:45.639
A times R minus

00:03:45.639 --> 00:03:48.355
B squared, plus C.

00:03:48.355 --> 00:03:49.888
This right over here would be vertex form

00:03:49.888 --> 00:03:51.977
and it's very useful
because you get a sense

00:03:51.977 --> 00:03:54.253
of when you hit a
minimum or maximum point.

00:03:54.253 --> 00:03:55.925
If thing right over here is negative,

00:03:55.925 --> 00:03:58.385
then the maximum value,
because this thing can only be

00:03:58.385 --> 00:04:01.032
non-positive, the maximum
value would be this,

00:04:01.032 --> 00:04:02.821
and then we could figure out
when does this equal zero?

00:04:02.821 --> 00:04:04.863
And that's when you
hit that maximum value.

00:04:04.863 --> 00:04:06.814
And if any of this seems foreign,

00:04:06.814 --> 00:04:08.440
I encourage you to watch the videos

00:04:08.440 --> 00:04:09.716
on vertex form.

00:04:09.716 --> 00:04:12.898
And I'm about to complete
the square on this character

00:04:12.898 --> 00:04:14.361
in order to get here, so I encourage you

00:04:14.361 --> 00:04:16.885
to also watch the videos
on completing the square.

00:04:17.607 --> 00:04:18.976
So let's do that.

00:04:18.976 --> 00:04:20.788
So let's start with this expression

00:04:20.788 --> 00:04:22.528
and I'm gonna start by,

00:04:22.528 --> 00:04:24.456
actually, let's start by,

00:04:24.456 --> 00:04:27.149
I'm gonna just distribute the R.

00:04:27.149 --> 00:04:28.786
So if you have M

00:04:29.588 --> 00:04:31.806
is equal to, I'm gonna keep
the negative out front,

00:04:31.806 --> 00:04:33.060
but I'm gonna distribute the R.

00:04:33.060 --> 00:04:35.522
R times R is R squared.

00:04:35.522 --> 00:04:38.842
R times negative four is minus four R.

00:04:38.842 --> 00:04:40.304
And I want to complete the square.

00:04:40.304 --> 00:04:42.302
And once again, I encourage
you to watch those videos

00:04:42.302 --> 00:04:44.949
if what I'm about to do seems like voodoo,

00:04:44.949 --> 00:04:46.271
but to complete the square I want

00:04:46.271 --> 00:04:47.317
to add and subtract the same thing.

00:04:47.317 --> 00:04:48.687
I don't want to actually change the value

00:04:48.687 --> 00:04:49.894
of the expression.

00:04:49.894 --> 00:04:51.195
And so I'll look at this coefficient,

00:04:51.195 --> 00:04:53.679
this first degree coefficient
here, it's negative four.

00:04:53.679 --> 00:04:55.955
Half of that is negative two.

00:04:55.955 --> 00:04:58.834
If I square it, I want to add four.

00:04:58.834 --> 00:05:00.436
I want to,

00:05:00.436 --> 00:05:02.989
I want to add four.

00:05:02.989 --> 00:05:05.259
I want to add,

00:05:05.259 --> 00:05:07.396
let me make it a different color.

00:05:07.396 --> 00:05:10.013
I want to add four.

00:05:10.665 --> 00:05:12.545
Now once again, I can't just go around

00:05:12.545 --> 00:05:14.774
adding numbers to expressions and assuming

00:05:14.774 --> 00:05:16.214
the equality still holds.

00:05:16.214 --> 00:05:18.555
I have to add and subtract the same thing.

00:05:18.555 --> 00:05:21.852
While you might be tempted
to write a minus four

00:05:21.852 --> 00:05:23.291
out here, because hey, I added four

00:05:23.291 --> 00:05:24.592
and then I subtract four.

00:05:24.592 --> 00:05:26.658
But remember, when I add this
four in this parentheses,

00:05:26.658 --> 00:05:28.168
i have this negative out front.

00:05:28.168 --> 00:05:29.863
If I were to distribute this negative,

00:05:29.863 --> 00:05:31.860
I essentially, this is a minus four.

00:05:31.860 --> 00:05:33.716
So if I wanted to add them out to be zero,

00:05:33.716 --> 00:05:35.527
distribute this negative,
this would be a negative four.

00:05:35.527 --> 00:05:37.200
If I wanted to cancel out
this, this needs to be

00:05:37.200 --> 00:05:38.732
a positive four.

00:05:38.732 --> 00:05:40.497
So once again, when
you look superficially,

00:05:40.497 --> 00:05:42.378
it looks like I just added four twice,

00:05:42.378 --> 00:05:43.214
but I didn't.

00:05:43.214 --> 00:05:44.746
this right over here since
it has this negative sign,

00:05:44.746 --> 00:05:45.884
is a negative four.

00:05:45.884 --> 00:05:47.532
Negative four plus four is zero.

00:05:47.532 --> 00:05:50.713
I have not changed the
value of this expression.

00:05:50.713 --> 00:05:53.104
But anyway, now that I've done that,

00:05:53.104 --> 00:05:55.124
this piece right over here,

00:05:55.124 --> 00:05:56.472
the whole reason why I added four

00:05:56.472 --> 00:05:58.028
is to make it a perfect square,

00:05:58.028 --> 00:06:02.579
this is now the same thing as R minus two,

00:06:02.579 --> 00:06:05.457
R minus two squared.

00:06:05.457 --> 00:06:07.362
I have the negative out front.

00:06:07.362 --> 00:06:11.658
We have M equals negative
R minus two squared

00:06:11.658 --> 00:06:13.236
plus four.

00:06:13.236 --> 00:06:14.444
Plus four.

00:06:14.444 --> 00:06:17.578
And just like that, I've
written it in vertex form.

00:06:17.578 --> 00:06:19.552
So what's the maximum number of mosquitos?

00:06:19.552 --> 00:06:21.618
Well let's just look at the structure

00:06:21.618 --> 00:06:23.823
of this expression a little bit.

00:06:23.823 --> 00:06:25.241
This piece,

00:06:25.241 --> 00:06:27.238
this piece right over here,

00:06:27.238 --> 00:06:28.004
that thing,

00:06:28.004 --> 00:06:29.860
actually, let me not do it in
the same color as the four,

00:06:29.860 --> 00:06:31.742
let me do it in this orangish color.

00:06:31.742 --> 00:06:35.132
This piece right over
here is always going to be

00:06:35.132 --> 00:06:36.897
less than or equal to zero.

00:06:36.897 --> 00:06:38.335
So that's always going to be less than

00:06:38.335 --> 00:06:39.288
or equal to zero.

00:06:39.288 --> 00:06:41.738
It's always going to be non-positive.

00:06:41.738 --> 00:06:46.486
And so this whole expression
hits a maximum value

00:06:46.486 --> 00:06:48.344
when this thing is zero,

00:06:48.344 --> 00:06:49.363
because at any other point,

00:06:49.363 --> 00:06:52.219
it's gong to just take away
from this right over here.

00:06:52.219 --> 00:06:54.448
So you hit a maximum value when this

00:06:54.448 --> 00:06:55.886
is equal to zero.

00:06:55.886 --> 00:07:00.019
When negative R minus two
squared is equal to zero.

00:07:00.019 --> 00:07:02.040
Well that's going to happen when,

00:07:02.040 --> 00:07:05.337
well you can see when R
minus two is equal to zero.

00:07:05.337 --> 00:07:07.706
Or when R is equal to two.

00:07:07.706 --> 00:07:11.234
And when R is equal to two,
what is M going to be equal to?

00:07:11.234 --> 00:07:13.440
Well when R is equal to two, this is zero

00:07:13.440 --> 00:07:15.182
and M is equal to four.

00:07:15.182 --> 00:07:18.317
R equals two and M is equal to four.

00:07:18.317 --> 00:07:20.128
So what's the maximum possible number

00:07:20.128 --> 00:07:21.382
of mosquitos?

00:07:21.382 --> 00:07:24.656
Well M equals four, it's gonna
be four million mosquitos.

00:07:24.656 --> 00:07:27.023
Remember, M is measuring in millions.

00:07:27.023 --> 00:07:28.859
Four million mosquitos.

00:07:28.859 --> 00:07:31.877
And then they ask us after how
many centimeters of rainfall

00:07:31.877 --> 00:07:34.199
will the maximum number
of mosquitos occur?

00:07:34.199 --> 00:07:35.824
Well, we figured it out.

00:07:35.824 --> 00:07:37.171
After two.

00:07:37.171 --> 00:07:38.332
After,

00:07:38.332 --> 00:07:40.515
after two centimeters.

00:07:40.515 --> 00:07:42.465
Two centimeters of rainfall.

00:07:42.465 --> 00:07:43.974
So if we go back to our,

00:07:43.974 --> 00:07:46.064
if we go back to our model,

00:07:46.064 --> 00:07:49.244
at zero rainfall you have no mosquitos,

00:07:49.244 --> 00:07:50.569
then as it rains more and more,

00:07:50.569 --> 00:07:51.660
you have more and more mosquitos.

00:07:51.660 --> 00:07:52.773
I guess more and more standing water.

00:07:52.773 --> 00:07:54.260
You have a maximum number of mosquitos

00:07:54.260 --> 00:07:56.141
at two centimeters of rainfall.

00:07:56.141 --> 00:07:58.091
At two centimeters we have
four million mosquitos,

00:07:58.091 --> 00:08:00.878
but after that I guess the
rain starts washing away

00:08:00.878 --> 00:08:02.572
the mosquito eggs or something

00:08:02.572 --> 00:08:04.338
and you start having
fewer and fewer mosquitos

00:08:04.338 --> 00:08:07.007
all the way until you get to
four centimeters of rainfall.

00:08:07.007 --> 00:08:09.236
And I guess after four
centimeters of rainfall,

00:08:09.236 --> 00:08:10.578
in theory you have

00:08:11.735 --> 00:08:13.453
negative mosquitos, I guess.

00:08:13.453 --> 00:08:14.963
I don't know what that would mean,

00:08:14.963 --> 00:08:16.124
and that might just be a situation

00:08:16.124 --> 00:08:17.633
where the model just breaks down.

00:08:17.633 --> 00:08:18.980
You obviously, I don't know what negative

00:08:18.980 --> 00:08:21.302
million mosquitos would actually mean.

00:08:21.302 --> 00:08:22.324
The relevant,

00:08:22.324 --> 00:08:26.526
the relevant area where
this model should hold

00:08:26.526 --> 00:08:28.802
is between zero centimeters of rainfall

00:08:28.802 --> 00:08:30.543
and four centimeters, because after that

00:08:30.543 --> 00:08:34.792
the model starts describing
negative values for M.

00:08:34.792 --> 00:08:37.066
So anyway, hopefully you
found that interesting.


Simplifying an exponential expression

WEBVTT
Kind: captions
Language: en

00:00:00.430 --> 00:00:01.835
- [Voiceover] What I
hope to do in this video

00:00:01.835 --> 00:00:04.017
is get some practice simplifying some

00:00:04.017 --> 00:00:06.989
fairly hairy exponential expressions.

00:00:06.989 --> 00:00:08.812
So let's get started.

00:00:08.812 --> 00:00:10.890
Let's say that I have the expression

00:00:10.890 --> 00:00:14.999
10 times nine to the

00:00:14.999 --> 00:00:19.108
t over two plus two power,

00:00:19.108 --> 00:00:23.370
times five to the three t.

00:00:23.370 --> 00:00:26.405
And what I wanna do is simplify
this as much as possible,

00:00:26.405 --> 00:00:29.522
and preferably get it in the form of

00:00:29.522 --> 00:00:33.223
A times B to the t.

00:00:33.223 --> 00:00:35.158
And like always, I encourage
you to pause this video

00:00:35.158 --> 00:00:37.591
and see if you can do this on your own

00:00:37.591 --> 00:00:39.947
using exponent properties, your knowledge,

00:00:39.947 --> 00:00:42.687
your deep knowledge of
exponent properties.

00:00:42.687 --> 00:00:44.916
All right, so let's work
through this together,

00:00:44.916 --> 00:00:47.609
and it's really just about
breaking the pieces up.

00:00:47.609 --> 00:00:50.175
So 10, I'll just leave that as 10 for now,

00:00:50.175 --> 00:00:52.508
there doesn't seem to be much to do there.

00:00:52.508 --> 00:00:54.191
But there's all sorts
of interesting things

00:00:54.191 --> 00:00:55.387
going on here.

00:00:55.387 --> 00:00:58.928
So nine to the t over two, plus two,

00:00:58.928 --> 00:01:02.364
so this right over here,
I could break this up,

00:01:02.364 --> 00:01:04.651
using the fact that, and I'll just

00:01:04.651 --> 00:01:06.600
write the properties over here.

00:01:06.600 --> 00:01:10.932
If I have nine to the a plus b power,

00:01:10.932 --> 00:01:13.207
this is the same thing as nine to the a,

00:01:13.207 --> 00:01:16.214
times nine to the b power.

00:01:16.214 --> 00:01:19.695
And over here I have
nine to the t over two,

00:01:19.695 --> 00:01:22.132
plus two, so I could rewrite this as

00:01:22.132 --> 00:01:25.708
nine to the t over two power,

00:01:25.708 --> 00:01:29.086
times nine squared.

00:01:30.789 --> 00:01:32.588
All right, now let's move over to

00:01:32.588 --> 00:01:35.490
five to the three t.

00:01:35.490 --> 00:01:40.143
Well, if I have a to the bc,

00:01:40.143 --> 00:01:43.492
so you could view this as
five to the three times t,

00:01:43.492 --> 00:01:46.093
this is the same thing as a to the b,

00:01:46.093 --> 00:01:48.995
and then that to the c power.

00:01:48.995 --> 00:01:50.957
So I could write this
as, this is going to be

00:01:50.957 --> 00:01:53.534
the same thing as five to the third,

00:01:53.534 --> 00:01:56.413
and then that to the t power.

00:01:56.413 --> 00:01:58.038
And the whole reason I did that is well

00:01:58.038 --> 00:01:59.048
this is just going to be a number,

00:01:59.048 --> 00:02:00.488
then I'm going to have
some number to the t power.

00:02:00.488 --> 00:02:02.310
I want to get as many things just raised

00:02:02.310 --> 00:02:03.936
to the t power as possible,

00:02:03.936 --> 00:02:06.269
just to see if I can simplify this thing.

00:02:06.269 --> 00:02:10.088
So this character right
over here is going to be 81.

00:02:10.088 --> 00:02:11.737
Nine squared is 81.

00:02:11.737 --> 00:02:14.256
Five to the third power, 25 times five,

00:02:14.256 --> 00:02:17.054
that's 125.

00:02:17.054 --> 00:02:19.550
So we're making good progress and so

00:02:19.550 --> 00:02:21.953
the only thing we really have
to simplify at this point

00:02:21.953 --> 00:02:24.774
is nine to the t over two.

00:02:24.774 --> 00:02:26.654
And actually let me do that over here.

00:02:26.654 --> 00:02:30.056
Nine to the t over two.

00:02:30.056 --> 00:02:32.691
Well that's the same thing as

00:02:32.691 --> 00:02:35.675
nine to the one half times t.

00:02:35.675 --> 00:02:37.567
And by this property right over here,

00:02:37.567 --> 00:02:40.190
that's the same thing as nine to the one,

00:02:40.190 --> 00:02:44.114
nine to the one half...

00:02:47.442 --> 00:02:49.381
And then that to the t power.

00:02:49.381 --> 00:02:51.528
So what's nine to the one half?

00:02:51.528 --> 00:02:53.130
Well that's three, so this is going to be

00:02:53.130 --> 00:02:55.232
equal to three to the t power.

00:02:55.232 --> 00:02:57.743
So this right over here is three

00:02:57.743 --> 00:03:01.068
to the t power.

00:03:01.068 --> 00:03:02.766
So now this is getting interesting.

00:03:02.766 --> 00:03:04.814
So I have the 10 out front,

00:03:06.992 --> 00:03:09.952
times three to the t,

00:03:12.495 --> 00:03:14.491
Actually let me write the 81 first.

00:03:14.491 --> 00:03:18.798
10 times 81 times three to the t,

00:03:18.798 --> 00:03:20.296
so all I did is I just

00:03:20.296 --> 00:03:22.245
swapped the order that I'm multiplying.

00:03:22.245 --> 00:03:25.617
Times 125 to the t.

00:03:25.617 --> 00:03:29.353
Times 125 to the t power.

00:03:29.353 --> 00:03:32.975
Now, the 10 times 81, I
can just multiply that out.

00:03:32.975 --> 00:03:35.030
That's going to be 810.

00:03:35.030 --> 00:03:36.890
And then what's three to the t

00:03:36.890 --> 00:03:39.270
times 125 to the t?

00:03:39.270 --> 00:03:41.998
Well this is another exponent
property at play here.

00:03:41.998 --> 00:03:45.353
Because if I have a
times b to the t power,

00:03:45.353 --> 00:03:48.673
that's a to the t times be to the t.

00:03:48.673 --> 00:03:49.811
Or another way to think about it,

00:03:49.811 --> 00:03:52.109
if I have a to the t times b to the t,

00:03:52.109 --> 00:03:54.698
that's the same thing as
a times b to the t power.

00:03:54.698 --> 00:03:57.136
And so over here I have three to the t

00:03:57.136 --> 00:04:00.096
times 125 to the t, so it's
going to be the same thing

00:04:00.096 --> 00:04:03.857
as three times 125 to the t power.

00:04:03.857 --> 00:04:07.479
So this part of it right over here,

00:04:07.479 --> 00:04:09.803
I could rewrite it as

00:04:09.803 --> 00:04:13.639
three times 125,

00:04:13.639 --> 00:04:16.720
and I'm gonna raise that
whole thing to the t power.

00:04:16.720 --> 00:04:18.219
So I'm in the home stretch.

00:04:18.219 --> 00:04:20.680
This is going to be 810,

00:04:20.680 --> 00:04:25.287
times three times 125 is 375.

00:04:26.218 --> 00:04:29.165
Times 375,

00:04:29.165 --> 00:04:32.850
to the t power.

00:04:32.850 --> 00:04:35.060
And that's about as
simplified as we can get,

00:04:35.060 --> 00:04:37.357
and we did indeed write it in the form

00:04:37.357 --> 00:04:38.681
that we hope to write it in.

00:04:38.681 --> 00:04:40.585
That a times be to the t.

00:04:40.585 --> 00:04:43.081
Where if we actually match
this form right over here,

00:04:43.081 --> 00:04:45.344
this right over here would be our A,

00:04:45.344 --> 00:04:50.137
and then our B would be the 375.

00:04:50.757 --> 00:04:52.719
So anyway, I know it looked
a little bit daunting

00:04:52.719 --> 00:04:54.774
when you first saw it, but
if you just keep saying

00:04:54.774 --> 00:04:56.562
okay let's just keep
applying some exponent

00:04:56.562 --> 00:04:58.396
properties here, let's see if I can get

00:04:58.396 --> 00:05:00.300
multiple things to the t power,

00:05:00.300 --> 00:05:02.854
and keep breaking it down
using these properties,

00:05:02.854 --> 00:05:04.735
you see that in not too many steps

00:05:04.735 --> 00:05:07.252
you get to something
that's a lot less hairy.


How to find the area of a triangle on a grid

WEBVTT
Kind: captions
Language: en

00:00:00.366 --> 00:00:01.700
- [Voiceover] What I would like to do

00:00:01.700 --> 00:00:04.574
is find the area of this green triangle.

00:00:04.574 --> 00:00:05.989
And if you get so inspired,

00:00:05.989 --> 00:00:07.700
and I encourage you to get inspired,

00:00:07.700 --> 00:00:08.767
pause the video and see if you can

00:00:08.767 --> 00:00:10.753
figure it out on your own.

00:00:10.753 --> 00:00:12.400
So whenever you start thinking about

00:00:12.400 --> 00:00:14.172
areas of triangle, or least my brain says,

00:00:14.172 --> 00:00:16.900
well look, I can figure
out the area of a triangle

00:00:16.900 --> 00:00:20.200
if I know the base and the
height of the triangle.

00:00:20.200 --> 00:00:21.225
I can just multiply them and then

00:00:21.225 --> 00:00:23.166
multiply that by one half.

00:00:23.166 --> 00:00:24.733
So, for example, if I have a triangle

00:00:24.733 --> 00:00:26.804
that looks like this.

00:00:26.804 --> 00:00:29.033
A triangle that looks like this.

00:00:29.033 --> 00:00:32.523
And if, if this is the base,

00:00:32.523 --> 00:00:34.238
this is b right over here.

00:00:34.238 --> 00:00:36.300
The length of this side is b.

00:00:36.300 --> 00:00:39.033
I'm gonna put the b in magenta.

00:00:39.033 --> 00:00:41.673
And then the height, let
me do that in yellow.

00:00:41.673 --> 00:00:43.866
This is the height.

00:00:43.866 --> 00:00:46.738
This is the height.

00:00:46.738 --> 00:00:48.789
Then I just multiply base
times height times one half

00:00:48.789 --> 00:00:51.066
and I'll get the area of this triangle.

00:00:51.066 --> 00:00:53.366
Or if the triangle looked like this.

00:00:53.366 --> 00:00:54.866
If it looked like,

00:00:54.866 --> 00:00:57.233
if it looked like this.

00:00:57.233 --> 00:01:00.333
If it looked like this, I
could do the same thing.

00:01:00.333 --> 00:01:03.633
Where if this is the base, b.

00:01:03.633 --> 00:01:05.600
So that is the base b.

00:01:05.600 --> 00:01:06.625
And now the height, I guess you could

00:01:06.625 --> 00:01:08.161
say this, if you were to drop

00:01:08.161 --> 00:01:10.666
a penny from here, it's
sitting outside the triangle.

00:01:10.666 --> 00:01:12.433
So it looks different from this one.

00:01:12.433 --> 00:01:15.141
But this would still be the height.

00:01:15.141 --> 00:01:17.433
This would still be the height.

00:01:17.433 --> 00:01:19.533
Right over here, ya do the same thing.

00:01:19.533 --> 00:01:21.867
One half times base times height

00:01:21.867 --> 00:01:24.033
would give you the area of this triangle.

00:01:24.033 --> 00:01:26.066
So how can we apply that over here?

00:01:26.066 --> 00:01:28.433
Well, this triangle is on this grid,

00:01:28.433 --> 00:01:29.600
but it's kind of at an angle.

00:01:29.600 --> 00:01:31.033
With this grid it's hard to pick out

00:01:31.033 --> 00:01:32.566
the base and the height.

00:01:32.566 --> 00:01:35.366
for this triangle as a whole.

00:01:35.366 --> 00:01:37.200
But what we could do,
there's actually several ways

00:01:37.200 --> 00:01:38.500
that we can approach this.

00:01:38.500 --> 00:01:39.852
Is we can break this triangle up

00:01:39.852 --> 00:01:41.933
into two or more triangles where we can

00:01:41.933 --> 00:01:44.466
figure out the base and the
height for each of them.

00:01:44.466 --> 00:01:47.166
So for example, I can break this one.

00:01:47.166 --> 00:01:49.000
Let's see I could, I'm picking this point

00:01:49.000 --> 00:01:52.270
and this point because it breaks it up

00:01:52.270 --> 00:01:54.200
into two triangles where I can figure out

00:01:54.200 --> 00:01:55.402
the base and the height.

00:01:55.402 --> 00:01:56.900
Well what am I talking about?

00:01:56.900 --> 00:01:58.853
Well this triangle over here, that I am

00:01:58.853 --> 00:02:01.130
shading in blue.

00:02:01.130 --> 00:02:04.067
If I, and I've switched the orientation.

00:02:04.067 --> 00:02:05.733
I've rotated 90 degrees.

00:02:05.733 --> 00:02:08.089
But if you view this yellow.

00:02:08.089 --> 00:02:10.266
If you view this yellow as the base

00:02:10.266 --> 00:02:13.933
of this triangle, you see
that the base is three.

00:02:13.933 --> 00:02:18.835
So, let me write the base
is equal to three units.

00:02:18.835 --> 00:02:20.366
And what's the height here?

00:02:20.366 --> 00:02:23.633
Well the height here is going to be

00:02:23.633 --> 00:02:26.900
The height here is going to be this

00:02:26.900 --> 00:02:29.666
distance right over here, which is four.

00:02:29.666 --> 00:02:31.366
Height is equal to four.

00:02:31.366 --> 00:02:35.389
So the area, the area of that triangle

00:02:35.389 --> 00:02:38.033
right over there, is going to be one half

00:02:38.033 --> 00:02:42.400
times three times four,
which is equal to six.

00:02:42.400 --> 00:02:43.933
So this part right over here,

00:02:43.933 --> 00:02:45.600
the area is six.

00:02:45.600 --> 00:02:47.033
And now we can do a similar thing

00:02:47.033 --> 00:02:48.500
with this other triangle.

00:02:48.500 --> 00:02:50.300
'Cause once again, we can view this

00:02:50.300 --> 00:02:52.933
yellow line, or now I have
this yellow and blue line,

00:02:52.933 --> 00:02:54.033
as the base.

00:02:54.033 --> 00:02:55.566
The base is equal to three.

00:02:55.566 --> 00:02:58.233
So I could write that
base is equal to three.

00:02:58.233 --> 00:02:59.405
And once again, I've rotated.

00:02:59.405 --> 00:03:01.466
So now the base is on the side.

00:03:01.466 --> 00:03:03.233
So the base is three.

00:03:03.233 --> 00:03:05.000
And then the height here,

00:03:05.000 --> 00:03:07.900
the height of this triangle

00:03:07.900 --> 00:03:09.300
is two.

00:03:09.300 --> 00:03:11.483
If this is the base, remember
if this is the base here

00:03:11.483 --> 00:03:12.900
we've just rotated it.

00:03:12.900 --> 00:03:14.553
Then this right over here is the height

00:03:14.553 --> 00:03:16.300
Height would be equal to two.

00:03:16.300 --> 00:03:18.433
So what's the area of this one?

00:03:18.433 --> 00:03:21.400
The area of this one is going to be

00:03:21.400 --> 00:03:25.133
one half times the base, three, times

00:03:25.133 --> 00:03:26.666
the height which is two.

00:03:26.666 --> 00:03:28.300
One half times two is one times three.

00:03:28.300 --> 00:03:29.834
This is going to be equal to three.

00:03:29.834 --> 00:03:30.928
So the area of the whole thing is

00:03:30.928 --> 00:03:32.600
gonna be this area of three

00:03:32.600 --> 00:03:33.933
plus this area of six.

00:03:33.933 --> 00:03:36.367
It's gonna be the area of nine.

00:03:36.367 --> 00:03:38.467
Area is equal to nine.

00:03:38.467 --> 00:03:39.866
Now that's one way you could do it.

00:03:39.866 --> 00:03:41.200
Is you could just break
it up into triangles

00:03:41.200 --> 00:03:43.000
where you could figure out
the base and the height.

00:03:43.000 --> 00:03:44.607
Another way, and this is you can kinda

00:03:44.607 --> 00:03:47.533
view it as a maybe a trickier way.

00:03:47.533 --> 00:03:48.700
Or ya kinda have to think a little bit

00:03:48.700 --> 00:03:50.894
outside of the box, or maybe
outside of the triangle

00:03:50.894 --> 00:03:52.367
to do it this way.

00:03:52.367 --> 00:03:54.200
Is to, instead of doing it this way,

00:03:54.200 --> 00:03:55.833
visualize this triangle.

00:03:55.833 --> 00:03:57.868
And actually, let me, let me
clean this up a little bit.

00:03:57.868 --> 00:04:00.450
Let me undo all this work that I just did.

00:04:00.450 --> 00:04:03.733
Let me undo this to show
you the other approach.

00:04:03.733 --> 00:04:06.066
The other way that we can tackle this.

00:04:06.066 --> 00:04:08.292
So, the other way we could tackle it.

00:04:08.292 --> 00:04:09.389
I'm gonna clean up the whole thing,

00:04:09.389 --> 00:04:12.666
so I get more, so I get
more real estate here.

00:04:12.666 --> 00:04:14.515
So the other way that we could tackle it

00:04:14.515 --> 00:04:18.066
is imagine that this triangle is embedded

00:04:18.066 --> 00:04:19.400
inside of our rectangle.

00:04:19.400 --> 00:04:21.774
So let me draw the rectangle.

00:04:21.774 --> 00:04:25.467
Let me draw a, let me draw
the broader rectangle.

00:04:25.467 --> 00:04:28.048
And I think you might
see where this is going.

00:04:28.048 --> 00:04:29.733
'Cause as soon as you draw that,

00:04:29.733 --> 00:04:32.591
that bigger rectangle, then you see

00:04:32.591 --> 00:04:34.600
that that rectangle is made up of

00:04:34.600 --> 00:04:36.900
the triangle that we're
tryin' to find the area of,

00:04:36.900 --> 00:04:41.400
and three other right triangles.

00:04:41.400 --> 00:04:42.800
We have this right triangle

00:04:42.800 --> 00:04:44.633
that I'm shading in in yellow.

00:04:44.633 --> 00:04:47.300
We have this right triangle that I'm

00:04:47.300 --> 00:04:50.100
shading in, in purple.

00:04:50.100 --> 00:04:52.200
And then we have this right triangle,

00:04:52.200 --> 00:04:54.438
that I am shading in, in blue.

00:04:54.438 --> 00:04:57.071
So if we figure out the area
of the entire rectangle,

00:04:57.071 --> 00:04:58.333
and that's pretty straightforward.

00:04:58.333 --> 00:04:59.556
The area of the entire rectangle

00:04:59.556 --> 00:05:02.500
is gonna be four times six,

00:05:02.500 --> 00:05:03.866
those little mentions of the rectangle.

00:05:03.866 --> 00:05:05.066
Four times six.

00:05:05.066 --> 00:05:07.866
So the area of the entire rectangle is 24.

00:05:07.866 --> 00:05:09.400
And then you subtract out the area

00:05:09.400 --> 00:05:11.700
of the purple, the blue
and the yellow rectangles.

00:05:11.700 --> 00:05:15.133
The purple, the blue and
the yellow triangles,

00:05:15.133 --> 00:05:16.067
then you're gonna be left with the

00:05:16.067 --> 00:05:18.066
area of the green triangle.

00:05:18.066 --> 00:05:19.233
So let's do that.

00:05:19.233 --> 00:05:22.000
So what's the area of the purple one?

00:05:22.000 --> 00:05:23.533
Well this is going to be,

00:05:23.533 --> 00:05:25.100
we're gonna subtract it out.

00:05:25.100 --> 00:05:26.433
It's gonna be one half,

00:05:26.433 --> 00:05:29.209
six, the height right over here.

00:05:29.209 --> 00:05:30.966
If we view this as a height of six.

00:05:30.966 --> 00:05:33.200
And it's a base, right over here is three.

00:05:33.200 --> 00:05:36.833
So it's gonna be one half times six

00:05:36.833 --> 00:05:40.766
times three, that's the
area of the purple triangle.

00:05:40.766 --> 00:05:42.766
And then you have the blue one.

00:05:42.766 --> 00:05:45.666
This is going to be minus one half.

00:05:45.666 --> 00:05:48.668
Let's see, you could say height is one.

00:05:48.668 --> 00:05:49.767
So this is one.

00:05:49.767 --> 00:05:53.189
And then this base, you could say is four.

00:05:53.189 --> 00:05:55.833
So that's four times four.

00:05:55.833 --> 00:05:57.833
And then we want to subtract out

00:05:57.833 --> 00:05:59.933
the area of the yellow rectangle.

00:05:59.933 --> 00:06:02.733
So this is minus one half.

00:06:02.733 --> 00:06:05.000
Let's see we could make this base two.

00:06:05.000 --> 00:06:07.967
One half times two times,

00:06:07.967 --> 00:06:10.299
and this height is four.

00:06:10.299 --> 00:06:11.738
Two times four.

00:06:11.738 --> 00:06:13.367
So what's this gonna be?

00:06:13.367 --> 00:06:16.933
Well, let's see, one half
times six times three.

00:06:16.933 --> 00:06:18.392
That's three times three.

00:06:18.392 --> 00:06:20.766
That's going to be nine.

00:06:20.766 --> 00:06:23.273
One half times one times four.

00:06:23.273 --> 00:06:24.966
That's going to be two.

00:06:24.966 --> 00:06:27.900
And then one half times two times four.

00:06:27.900 --> 00:06:30.376
Well that's just going to be four.

00:06:30.376 --> 00:06:34.900
And so we're left with 24 minus
nine minus two minus four.

00:06:35.900 --> 00:06:38.133
So, this we're gonna do the same color.

00:06:38.133 --> 00:06:39.433
Minus four, so what is that?

00:06:39.433 --> 00:06:41.800
24 minus nine is 15.

00:06:41.800 --> 00:06:43.933
15 minus two is 13.

00:06:43.933 --> 00:06:47.266
13 minus four is equal to nine.

00:06:47.266 --> 00:06:48.856
It's equal to nine.

00:06:48.856 --> 00:06:51.500
So that's the other way, or another way

00:06:51.500 --> 00:06:55.500
to get the area of this green triangle.


Translating expressions with parentheses

WEBVTT
Kind: captions
Language: en

00:00:00.501 --> 00:00:01.511
- [Voiceover] What I
hope to do in this video,

00:00:01.511 --> 00:00:03.473
is give ourselves some practice

00:00:03.473 --> 00:00:05.400
interpreting statements

00:00:05.400 --> 00:00:08.244
and writing them as
mathematical expressions,

00:00:08.244 --> 00:00:11.100
possibly using parentheses.

00:00:11.100 --> 00:00:12.308
So let's get started.

00:00:12.308 --> 00:00:13.914
And for any of these statements,

00:00:13.914 --> 00:00:14.952
if you get so inspired,

00:00:14.952 --> 00:00:17.033
and I encourage you to get so inspired,

00:00:17.033 --> 00:00:18.937
pause the video and see
if you can write them

00:00:18.937 --> 00:00:21.294
as mathematical expressions.

00:00:21.294 --> 00:00:22.339
So this first one says

00:00:22.339 --> 00:00:26.867
700 minus 19, divided in half.

00:00:26.867 --> 00:00:28.017
So we could say,

00:00:28.017 --> 00:00:29.293
another way to think about divided in half

00:00:29.293 --> 00:00:31.476
is divided by two,

00:00:31.476 --> 00:00:32.904
so we could write this as

00:00:32.904 --> 00:00:37.362
709 minus 19,

00:00:37.362 --> 00:00:38.941
and we're going to do that first,

00:00:38.941 --> 00:00:41.333
so that's why I put the
parentheses around it,

00:00:41.333 --> 00:00:45.311
divided by two, or divided in half.

00:00:45.311 --> 00:00:48.276
That's one way that we could write this.

00:00:48.276 --> 00:00:49.367
Now the next one,

00:00:49.367 --> 00:00:51.434
and once again, pause
it if you get inspired,

00:00:51.434 --> 00:00:52.769
and I encourage you to.

00:00:52.769 --> 00:00:57.471
Three times the sum of 56 and seven.

00:00:57.471 --> 00:00:59.433
So it's gonna be three times

00:00:59.433 --> 00:01:01.070
the sum of 56 and seven.

00:01:01.070 --> 00:01:02.428
So the sum of 56 and seven,

00:01:02.428 --> 00:01:04.135
we want to take that first,

00:01:04.135 --> 00:01:07.188
so it's going to be 56 plus the seven,

00:01:07.188 --> 00:01:10.695
that's the sum of 56 and seven,

00:01:10.695 --> 00:01:13.432
and then we want to do three times that.

00:01:13.432 --> 00:01:16.198
We want to do three times this sum.

00:01:16.198 --> 00:01:19.727
So we could write it like that.

00:01:19.727 --> 00:01:20.900
Another way we could write it,

00:01:20.900 --> 00:01:22.235
when you're dealing with parentheses,

00:01:22.235 --> 00:01:23.709
and you're going to see this more and more

00:01:23.709 --> 00:01:26.856
as you get into more
and more fancy algebra,

00:01:26.856 --> 00:01:27.506
I guess you could say,

00:01:27.506 --> 00:01:29.683
but what I'm about to
show you isn't so fancy,

00:01:29.683 --> 00:01:32.034
is, you don't have to write
the multiplication sign here.

00:01:32.034 --> 00:01:35.505
You could just write three,
and then open parentheses,

00:01:35.505 --> 00:01:37.542
56 plus seven,

00:01:41.896 --> 00:01:44.368
and this, too, is three times

00:01:44.368 --> 00:01:47.410
the sum of 56 and seven.

00:01:47.410 --> 00:01:48.733
And you want to be very careful,

00:01:48.733 --> 00:01:50.904
because you might be
tempted to maybe do it

00:01:50.904 --> 00:01:51.934
without the parentheses,

00:01:51.934 --> 00:01:53.993
so you might be tempted
to do something like this,

00:01:53.993 --> 00:01:55.853
three times

00:01:56.992 --> 00:02:01.335
56 plus seven,

00:02:01.335 --> 00:02:02.844
but this one isn't, obviously,

00:02:02.844 --> 00:02:05.422
three times the sum of 56 and seven.

00:02:05.422 --> 00:02:07.337
In fact, the standard
way to interpret this

00:02:07.337 --> 00:02:09.334
is that you would do the
multiplication first.

00:02:09.334 --> 00:02:11.134
You would do three times 56,

00:02:11.134 --> 00:02:12.504
and then add seven,

00:02:12.504 --> 00:02:13.804
which is going to give
you a different value,

00:02:13.804 --> 00:02:15.035
and you could try it out,

00:02:15.035 --> 00:02:18.065
than if you were to add
the 56 and the seven first.

00:02:18.065 --> 00:02:21.020
So, to make sure that you do
the 56 and the seven first,

00:02:21.020 --> 00:02:24.102
you want to put this
parentheses around it.

00:02:24.102 --> 00:02:25.867
So let's keep going.

00:02:25.867 --> 00:02:30.789
The sum of three times 56 and seven.

00:02:30.789 --> 00:02:33.517
So we're gonna take the sum of two things.

00:02:35.640 --> 00:02:37.709
The first thing that we're
gonna take the sum of

00:02:37.709 --> 00:02:40.577
is three times 56.

00:02:40.577 --> 00:02:44.303
So, three times 56,

00:02:44.303 --> 00:02:46.168
and seven.

00:02:47.768 --> 00:02:49.342
Let me do that in a different color.

00:02:49.342 --> 00:02:51.392
And seven.

00:02:53.069 --> 00:02:54.779
So this right over here is the sum

00:02:54.779 --> 00:02:57.701
of three times 56, and seven.

00:02:57.701 --> 00:02:59.443
Now it's always good to
write the parentheses.

00:02:59.443 --> 00:03:01.363
It makes it a little bit cleaner,

00:03:01.363 --> 00:03:02.798
a little bit more obvious.

00:03:02.798 --> 00:03:05.373
Look, I'm gonna take the three times 56,

00:03:05.373 --> 00:03:06.571
I'm gonna do that first,

00:03:06.571 --> 00:03:08.011
and then I'm gonna add seven,

00:03:08.011 --> 00:03:09.300
but based on what I just told you,

00:03:09.300 --> 00:03:11.807
the standard way, if
someone were to just write

00:03:11.807 --> 00:03:16.734
three times 56 plus seven,

00:03:26.763 --> 00:03:28.855
this actually can still be interpreted as

00:03:28.855 --> 00:03:32.004
the sum of three times 56, and seven,

00:03:32.004 --> 00:03:33.304
because as I just said,

00:03:33.304 --> 00:03:35.644
the standard, the convention, so to speak,

00:03:35.644 --> 00:03:37.774
is to do your multiplication first.

00:03:37.774 --> 00:03:39.794
Order of operations,
which you may or may not,

00:03:39.794 --> 00:03:42.407
if you're not familiar, you
will be familiar with it soon,

00:03:42.407 --> 00:03:44.322
is to do the multiplication first,

00:03:44.322 --> 00:03:47.271
and then add the seven,
or then do the addition.

00:03:47.271 --> 00:03:48.444
But just to make it clear,

00:03:48.444 --> 00:03:51.068
it doesn't hurt to put
the parentheses there.

00:03:51.068 --> 00:03:54.469
Three times 56, plus seven.

00:03:54.469 --> 00:03:56.701
Now we have 43

00:03:56.701 --> 00:04:00.826
minus the sum of 16 and 11.

00:04:01.103 --> 00:04:03.569
So, 43 minus,

00:04:03.569 --> 00:04:06.100
so we're gonna have 43 minus,

00:04:06.100 --> 00:04:09.200
minus the sum of 16 and 11.

00:04:09.200 --> 00:04:12.636
So, minus the sum of 16 and 11.

00:04:12.636 --> 00:04:14.738
So, from 43, we're gonna take

00:04:14.738 --> 00:04:19.068
the sum of 16 and 11,

00:04:19.068 --> 00:04:20.996
and so, once again, the
parentheses make it clear

00:04:20.996 --> 00:04:23.422
that we're going to take
the sum of 16 and 11,

00:04:23.422 --> 00:04:25.535
and we're gonna take that from 43.

00:04:25.535 --> 00:04:28.043
The parentheses are very,
very, very important here,

00:04:28.043 --> 00:04:31.133
because if we just did 43

00:04:32.303 --> 00:04:35.165
minus 16

00:04:36.704 --> 00:04:38.376
plus 11,

00:04:38.376 --> 00:04:39.966
the standard way of
interpreting this would be

00:04:39.966 --> 00:04:43.310
43 minus 16, and then adding 11,

00:04:43.310 --> 00:04:45.074
which would give you
a different value than

00:04:45.074 --> 00:04:48.225
43 minus the sum of 16 and 11.

00:04:48.225 --> 00:04:49.277
So once again, the parentheses

00:04:49.277 --> 00:04:51.530
are very, very, very important here

00:04:51.530 --> 00:04:55.396
to make it clear that you're
gonna add the 16 and 11 first,

00:04:55.396 --> 00:04:59.808
and then subtract that sum from 43.

00:04:59.808 --> 00:05:02.002
This is fun, let's keep going.

00:05:02.002 --> 00:05:06.448
10 times the quotient of 104 and eight.

00:05:06.448 --> 00:05:09.003
So, we're gonna do 10 times something.

00:05:09.003 --> 00:05:11.498
10 times the quotient

00:05:11.498 --> 00:05:14.301
of 104 and eight,

00:05:14.301 --> 00:05:18.105
and so the quotient of 104 and
8 we could write like this,

00:05:18.105 --> 00:05:20.391
104

00:05:20.391 --> 00:05:24.940
divided by eight,

00:05:24.940 --> 00:05:27.137
or, based on what we told
you a little earlier,

00:05:27.137 --> 00:05:28.101
you could write this as

00:05:28.101 --> 00:05:31.288
10 times the quotient

00:05:31.735 --> 00:05:35.589
of 104 and eight,

00:05:35.589 --> 00:05:40.210
or 104 divided by eight.

00:05:40.210 --> 00:05:41.870
Now let's just do this last one.

00:05:41.870 --> 00:05:44.110
Four times as large

00:05:48.662 --> 00:05:50.629
as the expression

00:05:52.029 --> 00:05:55.416
175 minus 58.

00:05:55.416 --> 00:05:57.602
So I'm gonna do four times
as large as something,

00:05:57.602 --> 00:06:00.364
so I'm gonna multiply
something times four.

00:06:00.934 --> 00:06:02.443
It's four times as large as the expression

00:06:02.443 --> 00:06:05.648
175 minus 58.

00:06:08.295 --> 00:06:10.756
And once again, I could write it as

00:06:10.756 --> 00:06:13.918
four times as large

00:06:13.918 --> 00:06:15.539
as the expression,

00:06:15.539 --> 00:06:17.304
let me do that in that purple color,

00:06:17.304 --> 00:06:20.207
as the expression

00:06:20.207 --> 00:06:22.037
175

00:06:22.037 --> 00:06:26.745
minus 58.

00:06:26.745 --> 00:06:27.962
Either way,

00:06:27.962 --> 00:06:30.238
and once again, if you
were to do it like this,

00:06:30.238 --> 00:06:32.699
if you didn't write the parentheses,

00:06:32.699 --> 00:06:34.835
then, it wouldn't be the same thing,

00:06:34.835 --> 00:06:36.333
'cause if the parentheses weren't here,

00:06:36.333 --> 00:06:38.666
then you would want to do
the four times 175 first,

00:06:38.666 --> 00:06:40.052
and then subtract the 58,

00:06:40.052 --> 00:06:42.695
which isn't what this
statement is telling us.

00:06:42.695 --> 00:06:44.193
And this last one, I think,

00:06:44.193 --> 00:06:46.944
brings up an interesting
thing for us to think about,

00:06:46.944 --> 00:06:50.195
because if someone were to
walk up to you on the street,

00:06:50.195 --> 00:06:51.809
and they were to show you--

00:06:51.809 --> 00:06:54.073
Whoops, what's going on with my computer?

00:06:54.073 --> 00:06:55.443
And they were to show you

00:06:55.443 --> 00:06:57.484
two different expressions.

00:06:57.484 --> 00:07:01.167
Well, the first expression said two--

00:07:01.167 --> 00:07:01.798
Let's write it this way,

00:07:01.798 --> 00:07:03.017
actually, I'm not gonna
even speak 'em out.

00:07:03.017 --> 00:07:05.207
I'm just gonna write it down.

00:07:05.207 --> 00:07:09.895
I'm just gonna write
some crazy number here.

00:07:10.942 --> 00:07:15.737
Some crazy numbers here.

00:07:15.737 --> 00:07:18.001
So that's one expression

00:07:18.001 --> 00:07:19.766
that someone were to write,

00:07:19.766 --> 00:07:22.574
and let's say another one is this one,

00:07:24.236 --> 00:07:25.861
and I'm intentionally--

00:07:25.861 --> 00:07:27.907
What, I put the commas in the wrong place.

00:07:27.907 --> 00:07:30.273
Let me make sure I get this right.

00:07:30.273 --> 00:07:32.502
Alright, that's 183,576.

00:07:32.502 --> 00:07:34.998
This is 37,399.

00:07:34.998 --> 00:07:36.403
So that's one expression,

00:07:36.403 --> 00:07:38.678
and then another expression is this.

00:07:39.770 --> 00:07:43.473
And I'm intentionally not reading it out.

00:07:43.473 --> 00:07:44.402
Well, I'll read it out a little bit,

00:07:44.402 --> 00:07:46.736
37,399.

00:07:46.736 --> 00:07:48.094
And someone said, "Quick!

00:07:48.094 --> 00:07:50.555
"Which expression is larger?"

00:07:50.555 --> 00:07:51.635
And you might be tempted,

00:07:51.635 --> 00:07:52.703
or you might not be tempted,

00:07:52.703 --> 00:07:53.342
but you might be tempted,

00:07:53.342 --> 00:07:54.712
"Oh, let me calculate this thing.

00:07:54.712 --> 00:07:56.268
"Gee, I'm gonna have to
write this thing down,

00:07:56.268 --> 00:07:58.092
"or use a calculator or something,

00:07:58.092 --> 00:07:59.309
"or whatever else to add

00:07:59.309 --> 00:08:03.265
"183,576

00:08:03.265 --> 00:08:05.208
"plus 37,399,

00:08:05.208 --> 00:08:06.690
"and then I'm gonna have
to multiply that by two,

00:08:06.690 --> 00:08:08.934
"and figure out what
that number is equal to,

00:08:08.934 --> 00:08:09.770
"and then I would have to take

00:08:09.770 --> 00:08:13.555
"183,576 plus 37,399

00:08:13.555 --> 00:08:14.530
"and figure out what that is,

00:08:14.530 --> 00:08:15.900
"multiply that by seven,

00:08:15.900 --> 00:08:17.131
"and figure out what that's going to be.

00:08:17.131 --> 00:08:17.967
"That's hard!

00:08:17.967 --> 00:08:18.605
"That's gonna take--"

00:08:18.605 --> 00:08:20.207
Not hard, it's just
gonna take you some time,

00:08:20.207 --> 00:08:22.239
you might make some careless mistakes.

00:08:22.239 --> 00:08:23.470
But the big realization to say,

00:08:23.470 --> 00:08:25.107
"Well, which one is larger?

00:08:25.107 --> 00:08:27.069
"Well I don't have to even
calculate these things!"

00:08:27.069 --> 00:08:31.202
'Cause this is two times this
craziness right over here,

00:08:31.202 --> 00:08:33.570
this thing that's gonna
be 200 something thousand,

00:08:33.570 --> 00:08:35.277
and this is seven times that thing

00:08:35.277 --> 00:08:38.540
that is going to be 200
and something thousand.

00:08:38.540 --> 00:08:39.770
So seven times that thing

00:08:39.770 --> 00:08:42.777
is going to be larger
than two times that thing,

00:08:42.777 --> 00:08:44.003
and so, one way to--

00:08:44.003 --> 00:08:46.504
Before you dive deep, and
start computing things,

00:08:46.504 --> 00:08:48.176
it's always good to take
a step back and say,

00:08:48.176 --> 00:08:51.102
"Hey, look, can I look at how
the expressions are formed,

00:08:51.102 --> 00:08:53.075
"the structure of these expressions?"

00:08:53.075 --> 00:08:56.442
And say, "Look, this is
two times this thing,

00:08:56.442 --> 00:08:58.543
"and this is seven times this thing."

00:08:58.543 --> 00:09:00.413
Well, the seven is going to be,

00:09:00.413 --> 00:09:03.372
this one right over here is going to be

00:09:03.741 --> 00:09:06.126
a larger expression.

00:09:06.126 --> 00:09:07.934
Anyway, hopefully you enjoyed that

00:09:07.934 --> 00:09:09.703
as much as I did.


More ways to think about multiplying

WEBVTT
Kind: captions
Language: en

00:00:00.418 --> 00:00:01.172
- [Voiceover] What I hope to do

00:00:01.197 --> 00:00:03.018
in this video is get
a little more practice

00:00:03.018 --> 00:00:07.430
and intuition when we're
multiplying multi-digit numbers.

00:00:07.430 --> 00:00:09.334
So let's say that we wanted to calculate

00:00:09.334 --> 00:00:14.117
what 7,000 times six is.

00:00:14.117 --> 00:00:16.008
7,000 times six.

00:00:16.008 --> 00:00:18.713
Now, for some of you, it
might just jump out at you.

00:00:18.713 --> 00:00:21.198
That, hey look, if I
have seven of anything,

00:00:21.198 --> 00:00:23.068
and here I have seven thousands,

00:00:23.068 --> 00:00:25.249
and I multiply that by
six, I'm now going to have

00:00:25.249 --> 00:00:29.069
seven times six of that
thing, or 42 of that thing,

00:00:29.069 --> 00:00:32.087
and in this case we have 42 thousands.

00:00:32.087 --> 00:00:34.073
So you might just be able
to cut to the chase and say,

00:00:34.073 --> 00:00:37.288
hey look, six times seven thousands

00:00:37.288 --> 00:00:41.190
is going to be 42 thousands.

00:00:41.190 --> 00:00:44.150
And that's great if you can
just cut to the chase like that,

00:00:44.150 --> 00:00:45.473
and another way to think
about it is like, look,

00:00:45.473 --> 00:00:47.529
six times seven is 42, and then

00:00:47.529 --> 00:00:49.292
since we're talking about thousands,

00:00:49.292 --> 00:00:50.174
we're not just talking about seven,

00:00:50.174 --> 00:00:51.207
we're talking about seven thousands.

00:00:51.207 --> 00:00:53.169
I have three zeros here,
so I'm going to have

00:00:53.169 --> 00:00:56.491
42 thousands, three zeros there.

00:00:56.491 --> 00:00:58.290
But I want to make sure
that we really understand

00:00:58.290 --> 00:00:59.706
what is going on here.

00:00:59.706 --> 00:01:01.226
This will also help us with a little bit

00:01:01.226 --> 00:01:03.789
of practice of our
multiplication properties.

00:01:03.814 --> 00:01:07.612
So 7,000 is the same thing
as 1,000 times seven,

00:01:07.612 --> 00:01:08.936
or seven times 1,000.

00:01:08.936 --> 00:01:10.328
It's seven thousands.

00:01:10.328 --> 00:01:13.067
Or you could view it as a
thousand sevens, either way.

00:01:13.067 --> 00:01:18.067
So this is the same thing as
1,000 times seven times six.

00:01:21.392 --> 00:01:23.145
And so you could view it as, you could do

00:01:23.145 --> 00:01:26.163
1,000 times seven first,
which would be 7,000,

00:01:26.163 --> 00:01:27.603
and then times six.

00:01:27.603 --> 00:01:30.552
Or you could do the seven times six first,

00:01:30.552 --> 00:01:31.852
and this is, this right over here

00:01:31.852 --> 00:01:34.010
is the associative
property of multiplication.

00:01:34.010 --> 00:01:36.402
It sounds very fancy,
but it just says that,

00:01:36.402 --> 00:01:38.387
hey look, we can multiply
the seven times six first,

00:01:38.387 --> 00:01:40.662
before we multiply by the thousand.

00:01:40.662 --> 00:01:45.358
So we could rewrite this as 1,000 times,

00:01:46.712 --> 00:01:48.493
and if we're going to do
the seven times six first

00:01:48.493 --> 00:01:50.028
we can put the parentheses around that,

00:01:50.053 --> 00:01:52.889
times seven,

00:01:55.281 --> 00:01:56.924
times six.

00:01:56.949 --> 00:01:59.232
Seven times six.

00:01:59.257 --> 00:02:02.105
Notice, it's 1,000 times seven times six.

00:02:02.105 --> 00:02:05.181
I could do 1,000 times
seven first to get 7,000,

00:02:05.181 --> 00:02:08.931
or I could do the seven
times six first to get,

00:02:08.931 --> 00:02:10.440
and you know where there is going,

00:02:10.440 --> 00:02:12.507
so if you multiply the
seven times six first,

00:02:12.507 --> 00:02:15.466
you're going to get 42,

00:02:15.466 --> 00:02:18.079
and you're gong to have 1,000 times 42.

00:02:18.079 --> 00:02:20.203
1,000 times 42.

00:02:20.203 --> 00:02:22.165
So you can view this as a thousand 42's

00:02:22.165 --> 00:02:23.883
or maybe a little bit more intuitively

00:02:23.883 --> 00:02:26.100
you could view this as 42 thousands.

00:02:26.100 --> 00:02:31.100
So, once again, we get to 42,000.

00:02:33.658 --> 00:02:34.668
And so the whole reason, some of y'all

00:02:34.668 --> 00:02:36.061
might have just been able
to do this immediately

00:02:36.061 --> 00:02:37.107
in your head and that's all good,

00:02:37.107 --> 00:02:40.020
but it's good to understand
what's actually going on here.

00:02:40.020 --> 00:02:41.843
And the reason why I also
broke it up that way,

00:02:41.843 --> 00:02:44.002
this way, is that the
exercises on Khan Academy

00:02:44.002 --> 00:02:46.464
make you do this to make sure that you

00:02:46.464 --> 00:02:49.482
really are understanding how
to break up these numbers

00:02:49.482 --> 00:02:53.940
and how you could re-associate
when you multiply.

00:02:53.940 --> 00:02:55.298
Let's do another one.

00:02:55.298 --> 00:02:57.101
Let's say that we wanted to figure out,

00:02:57.126 --> 00:02:59.581
let's say that we wanted to figure out,

00:02:59.581 --> 00:03:01.938
let me give ourselves some space,

00:03:01.963 --> 00:03:03.286
let's say that we wanted to figure out

00:03:03.286 --> 00:03:05.209
what 56

00:03:08.479 --> 00:03:10.416
times eight is.

00:03:10.959 --> 00:03:13.881
And there is a bunch of different
ways that you could do it.

00:03:13.881 --> 00:03:16.436
You could say that, look, 56, this is

00:03:16.436 --> 00:03:20.676
the same thing as 50, five tens,

00:03:20.676 --> 00:03:25.394
that's 50 plus six ones,

00:03:25.419 --> 00:03:30.230
so 50 plus six, and all
of that times eight.

00:03:31.600 --> 00:03:32.735
And then you could distribute the eight

00:03:32.735 --> 00:03:37.601
and you could say, look, this
is going to be 50 times eight,

00:03:37.601 --> 00:03:42.118
So it's going to be 50 times eight,

00:03:42.118 --> 00:03:44.637
plus six times eight.

00:03:44.637 --> 00:03:49.372
Plus six times eight.

00:03:49.803 --> 00:03:51.497
And 50 times eight?

00:03:51.497 --> 00:03:55.213
Well, five times eight
is 40, but we're not

00:03:55.213 --> 00:03:57.895
just saying five, we're saying five tens,

00:03:57.895 --> 00:04:01.516
so five tens times eight
is going to be 40 tens,

00:04:01.516 --> 00:04:03.919
or it's going to be 400.

00:04:03.919 --> 00:04:07.297
Another way to think about
it, five times eight is 40,

00:04:07.297 --> 00:04:08.436
but we're not talking about five,

00:04:08.436 --> 00:04:09.875
we're talking about five tens, so it's

00:04:09.875 --> 00:04:11.721
going to be 40 tens.

00:04:11.721 --> 00:04:14.333
So 50 times eight is 400, and then

00:04:14.333 --> 00:04:19.151
six times eight is, of
course, equal to 48.

00:04:19.151 --> 00:04:24.151
So this is going to be equal to 448.

00:04:27.173 --> 00:04:28.973
This is actually how I
do things in my head.

00:04:28.973 --> 00:04:30.227
When I do it in my head I obviously am not

00:04:30.227 --> 00:04:31.830
writing things down like this but I think,

00:04:31.830 --> 00:04:34.753
okay, 56 times eight,
I could break that up

00:04:34.753 --> 00:04:39.513
into 50 and six, and eight
times 50, well that's 400,

00:04:39.513 --> 00:04:41.290
or 40 tens you could say.

00:04:41.290 --> 00:04:44.355
Eight times five tens is
going to be 40 tens, or 400.

00:04:44.355 --> 00:04:46.478
And then eight times
six is going to be 48,

00:04:46.478 --> 00:04:48.730
so it's going to be 400 plus 48.

00:04:48.730 --> 00:04:49.892
Once you get some practice you're going

00:04:49.892 --> 00:04:52.156
to be able to do things
like this in your head.

00:04:52.156 --> 00:04:55.035
And if it helps, we
can also visualize this

00:04:55.035 --> 00:04:57.669
looking at an area of a rectangle.

00:04:57.669 --> 00:05:00.549
So imagine this rectangle right over here,

00:05:00.549 --> 00:05:05.215
and let's say that this dimension

00:05:05.739 --> 00:05:08.978
right over here is eight,
it is eight units tall.

00:05:08.978 --> 00:05:10.184
So that's the eight.

00:05:10.184 --> 00:05:15.184
And this entire dimension,
this entire length here is 56.

00:05:16.453 --> 00:05:19.588
So the area of this rectangle
is going to be 56 times eight,

00:05:19.588 --> 00:05:21.515
which is what we set to figure out,

00:05:21.515 --> 00:05:22.931
and to do that, well we could break it up

00:05:22.931 --> 00:05:26.217
into 50 and six, so this first section

00:05:26.217 --> 00:05:30.512
right over here, this has length,

00:05:30.512 --> 00:05:32.984
we could say this has length 50,

00:05:32.984 --> 00:05:36.166
that has length 50, and
then this second section,

00:05:36.166 --> 00:05:38.175
this has length six.

00:05:38.175 --> 00:05:39.034
This has length six.

00:05:39.034 --> 00:05:40.252
And the reason why we broke it up this way

00:05:40.252 --> 00:05:42.226
is cause we can, maybe in our heads,

00:05:42.226 --> 00:05:43.433
or without too much work, figure out

00:05:43.433 --> 00:05:44.849
what eight times 50 is and then

00:05:44.849 --> 00:05:47.056
separately figure out
what eight times six is.

00:05:47.056 --> 00:05:48.751
So separately figure out the areas

00:05:48.751 --> 00:05:50.724
of these two pieces of the big rectangle

00:05:50.724 --> 00:05:52.453
and then add them together.

00:05:52.453 --> 00:05:55.344
So what's eight times five tens?

00:05:55.344 --> 00:05:58.804
Well it's going to be 40 tens, or 400.

00:05:58.804 --> 00:06:01.068
This is going to be 400 square units,

00:06:01.068 --> 00:06:03.911
is going to be the area
of this yellow part.

00:06:03.911 --> 00:06:05.688
And then what's eight times six?

00:06:05.688 --> 00:06:08.903
Well we know that's going
to be 48 square units.

00:06:08.903 --> 00:06:10.668
So the entire rectangle is going to be

00:06:10.668 --> 00:06:13.072
the eight times the 50,
or the 50 times the eight,

00:06:13.072 --> 00:06:15.649
the 400, this area, the yellow area,

00:06:15.649 --> 00:06:18.424
plus the magenta area, plus
the eight times the six,

00:06:18.424 --> 00:06:21.732
the 48, which is 448.

00:06:21.732 --> 00:06:25.668
448 is going to be the
area of the whole thing.

00:06:25.668 --> 00:06:28.380
Eight times 56.