Khan Academy Computer Science Launch

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Language: en

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SAL: I'm here with John Resig, our resident Javascript

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and all-things-hardcore guru.

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JOHN RESIG: [LAUGHS] Thanks, Sal.

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And also the architect behind

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what we're looking at right now.

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What is this?

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JOHN: Yeah. this is really exciting.

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This is the new Khan Academy

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computer science content.

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We've been working on this now for a little while.

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and I'm just really excited to get this out now.

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SAL: And so, this is a page that we have.

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I see a bunch of links right here.

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There's an introductory video.

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So, what we are doing right now isn't the introductory video.

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It's the launch video.

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JOHN: Yes.

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There is another introductory video

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that walks through all of the stuff.

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but let's talk a little bit about what this computer science is.

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So, fundamentally you've created a kind of a neat environment.

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Here's some of the example programs

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that people have done already on this platform that you've made.

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So, here's some of the getting started ones.

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Let’s play with—

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So this is the most basic one, Winston.

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JOHN: Yeah, yeah.

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You’ve got some ellipses—[INDISTINCT]

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SAL: And it’s an environment where you have, right over here—

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You have your coding pane on the left.

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And then you have a canvas to draw [the results].

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So it’s immediately visual.

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So this is something that’s pretty basic,

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even for an early programmer.

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And, what I find very exciting is some of this action—

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that you can change things in real time.

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JOHN: Yeah.

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SAL: So—And there’s tutorials that go into more depth.

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I can make Winston closer to my complexion.

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JOHN: [LAUGHS]

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SAL: [LAUGHS] There you go

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And we can change things in real time

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So, even for a beginning programmer,

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they can start to play around with things.

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But obviously this is actually—

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What’s surprising is it’s a fairly powerful environment.

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JOHN: Yeah. Being able to have that responsiveness,

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I feel like it allows you to just, kind of, better understand

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how things interact with each other,

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at a pretty fundamental level.

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SAL: Yeah. We have things like this,

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which is a little bit more involved.

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And we can change, we can change in real time,

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some of the core variables right over here.

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So, [in] the initial spiral configuration—

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Well, let’s see what happens if we change it a little bit.

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We can see how it immediately changes everything

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JOHN: Mmm hmm.

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SAL: Maximum radius—

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What is this? How do you know what that does?

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It does something.

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JOHN: It seems to get larger—

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SAL: Oh yeah. It’s the radius of the circle,

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Number of points in spiral itself.

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We can reduce it.

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JOHN: Looks like you're getting less and less—thinner.

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SAL: Yeah, we get thinner—thicker.

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Yeah, so this is fascinating.

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JOHN: Yeah.

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SAL: There’s all of these—

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Leave this page— [Sal reading the prompt on the screen.]

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And so, there is a ton of examples

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of what people have already done

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all the way to this expert level. You have—

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This is a—

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Ben Kamens wrote this [??].

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It’s literally a bird-flocking simulator.

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JOHN: Yeah. Yeah.

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SAL: But you can actually—

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(Maybe this is less realistic.)

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—hit the birds with rocks.

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JOHN: Yes. [LAUGHING] SAL: [INDISTINCT]

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But this was all done—

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I mean, what’s neat is this was done all in this environment

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It’s all Javascript right over here.

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And anyone can go and look at these — and change them.

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JOHN: Yeah.

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And then you can save it as your [own].

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You can “save as,” and it will save it,

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actually, in your profile.

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JOHN: Yeah. The big thing we want to do is

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we want everyone to be able to see the code

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and the thing that’s running—simultaneously.

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SAL: Yeah.

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JOHN: And you should always feel willing and excited

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to change what’s there and make it your own.

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And so, we want to really heavily encourage

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exploration and experimentation.

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And I think this platform is going to do that.

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SAL: Yeah, this is neat.

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And you can collect the stuff.

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And so, if you go to your profile,

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it shows up there,

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and then you can share it with other people, too.

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JOHN: Mmm hmm.

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SAL: So these are the things that I’ve been working on.

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I still have a little work left to do.

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[INDISTINCT] I wanted to do an Ulam spiral.

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I can actually show the current version of it.

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JOHN: Mmm hmm.

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I’[ve] — made a little video on it, too.

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But yes, you just plot the prime numbers, things like that.

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JOHN: Mmm hmm.

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SAL: Let’s see. How do we get here?

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So, regardless—

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So, you go to Watch—

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JOHN: Mmm hmm.

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SAL: —Computer Science

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JOHN: Yep.

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SAL: —and then you just click there.

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And it takes you to that main “Computer Science” page.

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JOHN: Yep.

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SAL: And what if—

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So, we just looked at a bunch of examples.

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And obviously, people can look at

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the code for those examples.

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They can tweak it.

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They can save their own versions,

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and maybe create derivative versions of them.

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What if someone’s a complete beginner?

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JOHN: Mmm hmm.

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SAL: What should they do?

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JOHN: So, I mean, probably the best way to go about it is

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to look at some things

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that you think are kind of are interesting—

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like looking at some of the introductory explorations.

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Some of these programs here,

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where it’s just basic drawing, for example.

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If you never programmed before,

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this is a great way to get started.

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Start to look at what is being done there

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and see if you can’t kind of intuitively know how

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those function calls work.

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But even so, we have a whole bunch of tutorials.

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So if you go up to the drawing tutorials there—

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We have a number of tutorials—

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We have a few tutorials on drawing,

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along with some of these programs

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that you can learn drawing from.

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Then, if you go up to—

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If you want to learn some of the basics of programming,

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as well, you can view the programming basics.

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And we go even into more depth

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about some of these topics

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that you might want to understand better.

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SAL: And they also have an audio component.

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JOHN: Yes.

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SAL: Our now soon-to-be-famous intern,

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Jessica, recorded many of these.

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JOHN: Yes. [LAUGHS]

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SAL: But if you press play here,

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you’ll actually have here talking through what she’s doing here.

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And she’ll change the code in the canvas as she speaks

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JOHN: Yeah.

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SAL: And then, you can interact with that code.

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JOHN: Yeah.

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SAL: Which is pretty neat—

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JOHN: Yeah.

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—almost magical,

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JOHN: [LAUGHS]

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in my opinion

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And one thing that I’ve found really valuable,

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even for myself is to—

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It’s sometimes fun to tweak other people’s code.

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But once you feel comfortable,

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to actually just look at what they did,

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and try to build ityourself—from scratch.

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Because that’s really how you really learn to program,

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is get inspired, see what’s possible,

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and then try to build it yourself.

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And if you get really stuck,

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then you see how other people have implemented things.

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And yeah—so this is really exciting.

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Did we miss anything?

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JOHN: No.

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I mean, yeah, this is the big thing.

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I think the big thing we want to encourage here is,

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we want you to make your own things;

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not only building off of what we have,

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but create something original.

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You can take components from what we’ve written,

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or just write stuff that’s completely new.

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Mercury Subspace, there, is a pretty awesome one.

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That one, I think people will get a big kick out of that one.

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SAL: And so this whole game was written right here.

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JOHN: Yeah. Yep.

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That was done by David, one of our interns.

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SAL: Oh, this is a real game.

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JOHN: Yeah. [LAUGHS]

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SAL: This is—

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Yeah, David kind of overdoes things sometimes.

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JOHN: [LAUGHS]

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This is pretty good.

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JOHN: [LAUGHS]

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SAL: This is—

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Oh, oh. Yeah. Yeah.

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And this is all written there.

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And someone can change—

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How do I pause it?

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JOHN: Just click over right inside of it.

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SAL: Oh, Okay.

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JOHN: So like, for example there,

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you got your max_Health.

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So why don’t you try changing max_Health.

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And look at the hearts in the display up there.

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SAL: So if I change my max_Health to 10—

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JOHN: There you go.

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SAL: then, all of a sudden, we have more hearts here.

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JOHN: Yep. [LAUGHS]

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SAL: And this is fun.

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I mean, even a very beginner

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could start changing a fairly sophisticated program.

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JOHN: Yeah.

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SAL: You can change the images and things like that.

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JOHN: Oh, yeah. Absolutely.

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Yeah. Why don’t you go down—

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Scroll down a little bit.

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So you got your guy there in a little beetle there.

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SAL: Mmm Hmm. Yeah.

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JOHN: There you go.

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So there’s one, then sub—

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The second one down: “Beetle ship.”

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SAL: Beetle ship.

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JOHN: And why don’t you click—

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and then scroll down over there a little bit.

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SAL: Say, Spunky Sam?

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JOHN: Yeah, go ahead.

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JOHN: So, why don’t you click that.

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SAL: Spunky Sam.

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JOHN: And so, you can see that. Look at, now—

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SAL: Oh. So now, the ship is Spunky Sam—

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JOHN: Yeah. Now you’re—

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SAL: shooting. SAL: Very cool!

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JOHN: [LAUGHS]

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SAL: Very cool!

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JOHN: So, I mean, I think this is sort of—

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One of the things that we really wanted to do is

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we really wanted to make it powerful—

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It’s simple enough that anyone can

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really get in and start to play around.

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So, I think one of the big things here is that

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we want you to make your own things.

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And then down below,

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you see that there is this discussion area,

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where you can ask questions and comment on things.

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And this is a big thing.

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I think you’re going to find this to be a really valuable way

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to get feedback, and to be able to talk with other people,

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and get a better understanding about how things work.

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SAL: And actually become a better—

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I mean, one thing I’ve found

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is—just writing a couple of these tutorials—

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getting feedback from you and other people has,

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I think, made me better.

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Because there’s little silly things that I kept doing

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And now, all of a sudden, other people can look at it—

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find other interesting ways of doing it. or even asking [others]—

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And you can ask the author questions

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about how would they have done something.

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So, it really creates a community of learners here.

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JOHN: Yeah. No, absolutely.

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SAL: This is exciting.

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I already have two or three projects that—

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I’m not getting much things done right now.

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JOHN: [LAUGHS] Great. I— Yeah, I love it too.

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I’ve been having a blast playing around with it.


Unit circle manipulative

WEBVTT
Kind: captions
Language: en

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This here is a screenshot of
the Khan Academy unit circle

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module.

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And the whole point
here is to get

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you familiar with the unit
circle definition of trig

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functions, which is
just an extension

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of the traditional
SOHCAHTOA definition.

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But the reason why the
math world did this

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is so that we can define the
sine and cosines of angles that

00:00:20.390 --> 00:00:23.880
are 90 degrees or more, or
that have actual negative angle

00:00:23.880 --> 00:00:24.780
measures.

00:00:24.780 --> 00:00:28.120
So let's figure out what
the sine of 275 degrees is.

00:00:28.120 --> 00:00:30.780
So we have this dot that's
sitting on a unit circle that

00:00:30.780 --> 00:00:32.570
has a circle of radius 1.

00:00:32.570 --> 00:00:35.360
And let's hit this
dot to 275 degrees.

00:00:35.360 --> 00:00:37.260
Let's make an angle
of 275 degrees

00:00:37.260 --> 00:00:39.360
with the positive x-axis.

00:00:39.360 --> 00:00:43.540
So let's see, that's 50
degrees, 75 degrees, 90 degrees.

00:00:43.540 --> 00:00:45.700
You see the angle
measure right over there.

00:00:45.700 --> 00:00:49.960
So 90 degrees, 180
gets us halfway around.

00:00:49.960 --> 00:00:52.970
270 gets us at 3/4
of the way around.

00:00:52.970 --> 00:00:55.470
So 275 is a little
bit more than that.

00:00:55.470 --> 00:00:58.710
And we need to figure out
the sine of 275 degrees.

00:00:58.710 --> 00:01:02.020
The unit circle
definition of sine

00:01:02.020 --> 00:01:05.190
is that it's the y value of
that angle measure on the unit

00:01:05.190 --> 00:01:05.960
circle.

00:01:05.960 --> 00:01:07.293
So we've made the angle measure.

00:01:07.293 --> 00:01:10.230
We've gone 275 degrees
counterclockwise

00:01:10.230 --> 00:01:13.180
from the positive x-axis to get
to this point right over here.

00:01:13.180 --> 00:01:17.330
And the y value is
this negative 0.996.

00:01:17.330 --> 00:01:20.380
That is the sine of 275 degrees.

00:01:20.380 --> 00:01:23.166
Negative 0.996.

00:01:23.166 --> 00:01:24.790
If they'd asked for
the cosine, then we

00:01:24.790 --> 00:01:28.230
would give the x value of
this point of this dot right

00:01:28.230 --> 00:01:29.030
over here.

00:01:29.030 --> 00:01:31.460
So the y value is
negative 0.996.

00:01:31.460 --> 00:01:32.770
Let's check our answer.

00:01:32.770 --> 00:01:33.280
Very good.

00:01:33.280 --> 00:01:35.550
Let's do another one.

00:01:35.550 --> 00:01:39.310
And we get-- let's see--
sine of negative 55 degrees.

00:01:39.310 --> 00:01:41.980
So now we're going to
go below the x-axis.

00:01:41.980 --> 00:01:43.110
It's a negative angle.

00:01:43.110 --> 00:01:45.290
Negative 55 degrees.

00:01:45.290 --> 00:01:46.900
And it's sine once
again, so we want

00:01:46.900 --> 00:01:50.570
to look at the
y-coordinate of this point.

00:01:50.570 --> 00:01:55.110
So the y-coordinate
is negative 0.819.

00:01:55.110 --> 00:01:58.650
Negative 0.819.

00:01:58.650 --> 00:01:59.435
Check our answer.

00:01:59.435 --> 00:01:59.957
Very good.

00:01:59.957 --> 00:02:00.790
Let's keep doing it.

00:02:00.790 --> 00:02:02.730
Maybe we'll see a
cosine soon enough.

00:02:02.730 --> 00:02:05.660
Sine of 285 degrees.

00:02:05.660 --> 00:02:09.340
So 285 degrees, we're
looking right over there.

00:02:09.340 --> 00:02:11.620
The sine is the y value.

00:02:11.620 --> 00:02:14.380
Negative 0.966.

00:02:14.380 --> 00:02:15.510
Let's check our answer.

00:02:15.510 --> 00:02:16.820
Let's do another one.

00:02:16.820 --> 00:02:17.840
Sine again.

00:02:17.840 --> 00:02:18.880
2/3 pi.

00:02:18.880 --> 00:02:19.890
So now it's in radians.

00:02:19.890 --> 00:02:21.680
We don't see that
little degree sign.

00:02:21.680 --> 00:02:23.550
So 2/3 pi.

00:02:23.550 --> 00:02:27.070
Let's see-- 2/3 pi
is right over here.

00:02:27.070 --> 00:02:30.190
That's the angle in radians
relative to the positive x-axis

00:02:30.190 --> 00:02:32.180
going in the
counterclockwise direction.

00:02:32.180 --> 00:02:34.090
We care about the
sine which is the y

00:02:34.090 --> 00:02:36.480
value of our point sitting
on the unit circle.

00:02:36.480 --> 00:02:40.540
0.866.

00:02:40.540 --> 00:02:43.430
And one thing that bears
mention is the exact

00:02:43.430 --> 00:02:44.422
is square root of 3/2.

00:02:44.422 --> 00:02:46.880
And square root of 3/2 if you
were to write it as a decimal

00:02:46.880 --> 00:02:50.490
would just continue on forever
without actually repeating.

00:02:50.490 --> 00:02:51.970
This is just an approximation.

00:02:51.970 --> 00:02:53.970
For the sake of this
module, we really

00:02:53.970 --> 00:02:56.030
just want to make you
comfortable with the unit

00:02:56.030 --> 00:02:57.894
circle, and maybe
using this tool.

00:02:57.894 --> 00:03:00.310
So you really should just type
in this approximate number.

00:03:00.310 --> 00:03:01.880
You don't have to
do the exact number

00:03:01.880 --> 00:03:03.460
of what square root of 3/2 is.

00:03:03.460 --> 00:03:05.630
This is it rounded to
the nearest thousand.

00:03:05.630 --> 00:03:07.750
So 0.866.

00:03:07.750 --> 00:03:09.140
Let's check our answer.

00:03:09.140 --> 00:03:10.720
Let's do another one.

00:03:10.720 --> 00:03:11.480
Sine again.

00:03:11.480 --> 00:03:13.520
Sine of 15 degrees.

00:03:13.520 --> 00:03:15.580
So 15 degrees.

00:03:15.580 --> 00:03:17.500
0.259.

00:03:17.500 --> 00:03:21.000
I'm going to keep doing
this until I get a cosine.

00:03:21.000 --> 00:03:23.180
So let's do it again-- cosine!

00:03:23.180 --> 00:03:23.680
Very good.

00:03:23.680 --> 00:03:24.780
2 pi.

00:03:24.780 --> 00:03:25.750
2 pi radians.

00:03:25.750 --> 00:03:28.800
Well 2 pi is completely going
around the unit circle once.

00:03:28.800 --> 00:03:31.080
So let's go all the
way around, and we

00:03:31.080 --> 00:03:33.760
see that we get back to
where we were before.

00:03:33.760 --> 00:03:35.440
We've gone around
to 2 pi radians.

00:03:35.440 --> 00:03:36.970
We're back to where
we were before.

00:03:36.970 --> 00:03:39.640
The cosine is the x
value of that point.

00:03:39.640 --> 00:03:41.825
So the x value of
that point is 1.

00:03:41.825 --> 00:03:43.170
The x value of the point is 1.

00:03:43.170 --> 00:03:44.910
If we wanted the
sine of 2 pi, then we

00:03:44.910 --> 00:03:47.950
would get 0 because we have
no y value right over here.

00:03:47.950 --> 00:03:49.720
We're at the x-axis.

00:03:49.720 --> 00:03:51.029
Let's check our answer.

00:03:51.029 --> 00:03:51.570
Let's do one.

00:03:51.570 --> 00:03:52.950
Maybe we'll see another cosine.

00:03:52.950 --> 00:03:53.290
There we go.

00:03:53.290 --> 00:03:53.940
Another cosine.

00:03:53.940 --> 00:03:55.140
Cosine of 112 pi.

00:03:55.140 --> 00:03:56.931
So that's just going
to be a little-- let's

00:03:56.931 --> 00:03:58.680
see, 1/35, 1/18, 1/12.

00:03:58.680 --> 00:03:59.180
Cosine.

00:03:59.180 --> 00:04:01.260
We want the x value,
not the y value.

00:04:01.260 --> 00:04:04.690
The x value is 0.966.

00:04:04.690 --> 00:04:07.530
So 0.966.

00:04:07.530 --> 00:04:08.910
Check our answer.

00:04:08.910 --> 00:04:10.994
And we are done.


Exploring the mean and median

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.080
SALMAN KHAN: I'm here with
our exercise guru Ben Eater.

00:00:03.080 --> 00:00:03.830
BEN EATER: Hi Sal.

00:00:03.830 --> 00:00:05.705
SALMAN KHAN: And we are
looking at this model

00:00:05.705 --> 00:00:08.132
you made called Exploring
mean and median.

00:00:08.132 --> 00:00:09.090
Let's see what it says.

00:00:09.090 --> 00:00:11.450
Arrange the five orange points.

00:00:11.450 --> 00:00:13.730
So these are the orange
points right over here.

00:00:13.730 --> 00:00:16.239
On the number line so
the arithmetic mean

00:00:16.239 --> 00:00:17.880
is negative 0.1.

00:00:17.880 --> 00:00:20.610
Arithmetic mean is really
kind of what we traditionally

00:00:20.610 --> 00:00:21.700
view as the average.

00:00:21.700 --> 00:00:22.270
BEN EATER: It's the average.

00:00:22.270 --> 00:00:22.490
Yeah.

00:00:22.490 --> 00:00:23.948
SALMAN KHAN: You
add up the numbers

00:00:23.948 --> 00:00:25.930
and divide by the
numbers that you have.

00:00:25.930 --> 00:00:27.640
And that the median is 2.5.

00:00:27.640 --> 00:00:30.339
And 2.5 is essentially
the middle number,

00:00:30.339 --> 00:00:32.630
if you have an odd number of
numbers, which we do here.

00:00:32.630 --> 00:00:33.120
BEN EATER: Which we do.

00:00:33.120 --> 00:00:35.369
SALMAN KHAN: Or if you have
an even number of numbers,

00:00:35.369 --> 00:00:38.410
it's the arithmetic mean
of the middle two numbers.

00:00:38.410 --> 00:00:39.670
It's really going to be--

00:00:39.670 --> 00:00:40.360
BEN EATER: --halfway between--

00:00:40.360 --> 00:00:42.526
SALMAN KHAN: Halfway between
the middle two numbers.

00:00:42.526 --> 00:00:45.230
And they're both ways of
measuring central tendency.

00:00:45.230 --> 00:00:46.400
BEN EATER: Right.

00:00:46.400 --> 00:00:48.050
SALMAN KHAN: And the distance
between adjacent tick marks

00:00:48.050 --> 00:00:48.540
is 1.

00:00:48.540 --> 00:00:50.480
So this is like, so this
one right over here is 0.

00:00:50.480 --> 00:00:51.020
This is 1.

00:00:51.020 --> 00:00:51.750
This is 2.

00:00:51.750 --> 00:00:52.730
This is negative 1.

00:00:52.730 --> 00:00:54.351
This is negative 2.

00:00:54.351 --> 00:00:55.850
And so the whole
point here-- what's

00:00:55.850 --> 00:00:57.540
the point of this exercise?

00:00:57.540 --> 00:00:58.125
BEN EATER: The point
of it, well, you

00:00:58.125 --> 00:01:00.710
said they're both ways of
measuring central tendency.

00:01:00.710 --> 00:01:02.770
But we can see here is
that they don't always

00:01:02.770 --> 00:01:04.660
give you the same answer.

00:01:04.660 --> 00:01:09.590
So we're being asked to see an
arithmetic mean of negative 0.1

00:01:09.590 --> 00:01:10.780
and a median of 2.5.

00:01:10.780 --> 00:01:12.220
Those are very
different numbers.

00:01:12.220 --> 00:01:14.290
So how can you take
5 points and say

00:01:14.290 --> 00:01:16.069
the central tendency
is negative 0.1.

00:01:16.069 --> 00:01:16.860
SALMAN KHAN: Right.

00:01:16.860 --> 00:01:18.570
BEN EATER: And the
central tendency is 2.5.

00:01:18.570 --> 00:01:18.940
SALMAN KHAN: Right.

00:01:18.940 --> 00:01:19.890
And they're just different
ways of measuring it.

00:01:19.890 --> 00:01:20.960
BEN EATER: There's different
ways of measuring it

00:01:20.960 --> 00:01:22.920
and sometimes one way is
better than the other.

00:01:22.920 --> 00:01:23.390
SALMAN KHAN: Exactly.

00:01:23.390 --> 00:01:24.450
BEN EATER: And you
kind of have to know.

00:01:24.450 --> 00:01:25.310
SALMAN KHAN: I was just
starting to play with it

00:01:25.310 --> 00:01:28.130
and you see, if I take this
the highest number here

00:01:28.130 --> 00:01:29.927
and I change it, I
can change the mean.

00:01:29.927 --> 00:01:32.260
But I didn't change the median
because the middle number

00:01:32.260 --> 00:01:34.700
is still this middle
number, no matter what's

00:01:34.700 --> 00:01:36.320
going on out here.

00:01:36.320 --> 00:01:38.690
And that's why
actually the median

00:01:38.690 --> 00:01:42.720
tends to be viewed as a better
measure when you have outliers.

00:01:42.720 --> 00:01:46.776
When you have really, really,
really large values or really,

00:01:46.776 --> 00:01:48.400
really, really small
values, the median

00:01:48.400 --> 00:01:49.566
is a little bit more robust.

00:01:49.566 --> 00:01:51.634
So if like this thing is
some crazy large number,

00:01:51.634 --> 00:01:52.925
the median doesn't change much.

00:01:52.925 --> 00:01:53.230
BEN EATER: Right.

00:01:53.230 --> 00:01:55.396
SALMAN KHAN: Even though
this one number being large

00:01:55.396 --> 00:01:57.987
took the whole
arithmetic mean up.

00:01:57.987 --> 00:01:58.820
Same thing out here.

00:01:58.820 --> 00:02:01.570
If I have an outlier over here
it doesn't change the median.

00:02:01.570 --> 00:02:03.320
BEN EATER: So in this
case, it's asking us

00:02:03.320 --> 00:02:05.410
a mean of negative 0.1
and a median of 2.5.

00:02:05.410 --> 00:02:06.470
Those are very far apart.

00:02:06.470 --> 00:02:06.830
SALMAN KHAN: Right.

00:02:06.830 --> 00:02:08.410
BEN EATER: So you might
expect a lot of outliers.

00:02:08.410 --> 00:02:08.800
SALMAN KHAN: Right.

00:02:08.800 --> 00:02:10.591
BEN EATER: So when
we're done, you probably

00:02:10.591 --> 00:02:13.290
would think a lot of the points
are going to be closer to 2.5.

00:02:13.290 --> 00:02:14.070
SALMAN KHAN: Let's
think about this.

00:02:14.070 --> 00:02:15.230
Let me get the median first.

00:02:15.230 --> 00:02:16.610
So the median is
just a middle number.

00:02:16.610 --> 00:02:17.240
BEN EATER: Right.

00:02:17.240 --> 00:02:19.198
SALMAN KHAN: And so, and
here there's 5 numbers

00:02:19.198 --> 00:02:21.073
so the middle number
literally has to be 2.5.

00:02:21.073 --> 00:02:23.114
BEN EATER: So one of those
dots has to be on 2.5.

00:02:23.114 --> 00:02:25.140
SALMAN KHAN: Let me make
the middle number 2.5.

00:02:25.140 --> 00:02:28.870
So that gets me to 2.5.

00:02:28.870 --> 00:02:29.620
That's got to be--

00:02:29.620 --> 00:02:30.920
BEN EATER: You are
looking at the mean there.

00:02:30.920 --> 00:02:31.480
SALMAN KHAN: Oh sorry.

00:02:31.480 --> 00:02:31.950
Yeah, yeah, yeah.

00:02:31.950 --> 00:02:32.200
Sorry.

00:02:32.200 --> 00:02:33.199
I'm looking at the mean.

00:02:33.199 --> 00:02:35.000
2.5 is the median.

00:02:35.000 --> 00:02:36.570
Median is 2.5.

00:02:36.570 --> 00:02:40.090
And now I want to get the
mean at negative 0 point

00:02:40.090 --> 00:02:42.960
negative 1, or a negative 0.1.

00:02:42.960 --> 00:02:45.245
So let me start
making this, let me

00:02:45.245 --> 00:02:49.940
see what I can do
to make negative.

00:02:49.940 --> 00:02:53.420
The mean still is
not, so let's see.

00:02:53.420 --> 00:02:54.230
Median, look.

00:02:54.230 --> 00:02:54.830
There you go.

00:02:54.830 --> 00:02:55.430
BEN EATER: There we go.

00:02:55.430 --> 00:02:56.471
SALMAN KHAN: There we go.

00:02:56.471 --> 00:02:57.620
So let me check my answer.

00:02:57.620 --> 00:02:58.120
Yes.

00:02:58.120 --> 00:02:59.454
So this you see that there are--

00:02:59.454 --> 00:03:01.078
BEN EATER: Yeah,
which of those numbers

00:03:01.078 --> 00:03:03.346
do you think better represents
the central tendency?

00:03:03.346 --> 00:03:04.970
SALMAN KHAN: Yeah well.

00:03:04.970 --> 00:03:05.670
You--

00:03:05.670 --> 00:03:06.440
BEN EATER: I don't know.

00:03:06.440 --> 00:03:08.106
SALMAN KHAN: You don't
know in this case

00:03:08.106 --> 00:03:09.411
I guess it's a, either one is.

00:03:09.411 --> 00:03:09.910
Let's see.

00:03:09.910 --> 00:03:11.980
If they were all grouped and
there was a couple outliers,

00:03:11.980 --> 00:03:13.600
then the median probably
would've been better.

00:03:13.600 --> 00:03:14.075
BEN EATER: Yeah.

00:03:14.075 --> 00:03:16.350
And there might be different
ways to arrange those dots too.

00:03:16.350 --> 00:03:17.310
And you can play with them.

00:03:17.310 --> 00:03:18.330
SALMAN KHAN: Yeah, that
would be interesting.

00:03:18.330 --> 00:03:19.390
Let me check.

00:03:19.390 --> 00:03:20.200
I got it right.

00:03:20.200 --> 00:03:21.710
Let's do another one.

00:03:21.710 --> 00:03:23.380
Next question.

00:03:23.380 --> 00:03:23.880
All right.

00:03:23.880 --> 00:03:24.815
Arrange the 10 dots.

00:03:24.815 --> 00:03:25.440
OK, same thing.

00:03:25.440 --> 00:03:26.770
We want the mean 3.6.

00:03:26.770 --> 00:03:28.370
Median at 6.

00:03:28.370 --> 00:03:30.420
So let me think about
the median first.

00:03:30.420 --> 00:03:35.580
So I want to get the middle
number to median at 6.

00:03:42.230 --> 00:03:44.307
The median is six.

00:03:44.307 --> 00:03:46.390
BEN EATER: You need to
drag one more dot over here

00:03:46.390 --> 00:03:48.590
because you're going to be
between the two middle numbers

00:03:48.590 --> 00:03:50.170
because you have an
even number of points.

00:03:50.170 --> 00:03:51.020
SALMAN KHAN: Yes, yes, yes.

00:03:51.020 --> 00:03:51.561
You're right.

00:03:51.561 --> 00:03:54.680
So the median is taking
them, right, right, right.

00:03:54.680 --> 00:03:55.560
So here we go.

00:03:55.560 --> 00:03:57.595
Median is now 6.

00:03:57.595 --> 00:03:58.220
BEN EATER: Yep.

00:03:58.220 --> 00:03:58.750
SALMAN KHAN: Very good.

00:03:58.750 --> 00:04:00.624
Right I had to worry
about middle two numbers

00:04:00.624 --> 00:04:01.840
because I had an even number.

00:04:01.840 --> 00:04:02.760
And now, let me see.

00:04:02.760 --> 00:04:05.780
The mean, I need to get to 3.6.

00:04:05.780 --> 00:04:07.840
The mean, I need to get to.

00:04:07.840 --> 00:04:09.840
And I can't make this one
of the middle numbers,

00:04:09.840 --> 00:04:12.460
then it'll distort my median.

00:04:12.460 --> 00:04:15.930
3 point, I guess it can
be one of, it can't get.

00:04:15.930 --> 00:04:17.897
BEN EATER: Yeah, you
can't go past that number.

00:04:17.897 --> 00:04:18.980
SALMAN KHAN: So let's see.

00:04:18.980 --> 00:04:22.610
I want to go 3.6.

00:04:22.610 --> 00:04:25.860
3.6 gets me right over there.

00:04:25.860 --> 00:04:26.860
BEN EATER: There you go.

00:04:26.860 --> 00:04:27.670
SALMAN KHAN: There you go.

00:04:27.670 --> 00:04:28.390
BEN EATER: So here,
which of those numbers

00:04:28.390 --> 00:04:31.072
do you think have a
better central tendency?

00:04:31.072 --> 00:04:33.030
SALMAN KHAN: I would say
the median in this one

00:04:33.030 --> 00:04:35.202
because this feels a little
bit more outlier like.

00:04:35.202 --> 00:04:35.910
BEN EATER: Right.

00:04:35.910 --> 00:04:37.510
SALMAN KHAN: At least in
this situation, the way

00:04:37.510 --> 00:04:39.468
I have, we have we've
set up this distribution.

00:04:39.468 --> 00:04:40.480
Let's check the answer.

00:04:40.480 --> 00:04:41.240
That's satisfying.

00:04:41.240 --> 00:04:42.240
Get the smiley face.

00:04:42.240 --> 00:04:44.040
And you also have hints here
if someone has trouble with it.

00:04:44.040 --> 00:04:44.440
BEN EATER: That's right.

00:04:44.440 --> 00:04:47.023
And they can actually show you
maybe a different way to do it.

00:04:47.023 --> 00:04:49.760
SALMAN KHAN: So you define the
median is the middle number.

00:04:49.760 --> 00:04:52.499
Then you show dragging points.

00:04:52.499 --> 00:04:53.040
So let's see.

00:04:53.040 --> 00:04:55.070
Show us an example.

00:04:55.070 --> 00:04:57.570
This is an example where the
median and the mean are both 6.

00:04:57.570 --> 00:04:57.810
BEN EATER: Right.

00:04:57.810 --> 00:04:59.190
So that just puts
half the points

00:04:59.190 --> 00:05:01.610
to the left of the median and
half the points to the right.

00:05:01.610 --> 00:05:02.693
SALMAN KHAN: Right, right.

00:05:02.693 --> 00:05:03.420
That makes sense.

00:05:03.420 --> 00:05:04.702
And then you could keep going.

00:05:04.702 --> 00:05:06.160
But that doesn't
solve the problem.

00:05:06.160 --> 00:05:07.320
BEN EATER: No, it's--

00:05:07.320 --> 00:05:08.403
SALMAN KHAN: So let's see.

00:05:08.403 --> 00:05:09.999
And then you explain.

00:05:09.999 --> 00:05:12.040
So you just give a little
more explanation there.

00:05:12.040 --> 00:05:14.248
And then finally, you talk
about how you changed this

00:05:14.248 --> 00:05:16.610
so that you can get
the mean of 3.6.

00:05:16.610 --> 00:05:18.230
So let's see if you
show that example.

00:05:18.230 --> 00:05:19.000
So there you go.

00:05:19.000 --> 00:05:20.450
BEN EATER: So we just moved
a couple of outliers out.

00:05:20.450 --> 00:05:21.780
SALMAN KHAN: Outliers
affect the mean,

00:05:21.780 --> 00:05:23.360
but then the median
doesn't change.

00:05:23.360 --> 00:05:25.190
Very interesting.


Exploring standard deviation 1 module

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.034
SAL: I'm here with our
exercise guru Ben Eater.

00:00:03.034 --> 00:00:03.950
BEN EATER: Thanks Sal.

00:00:03.950 --> 00:00:08.420
SAL: And we're looking at your
exploring standard deviation 1

00:00:08.420 --> 00:00:09.262
exercise.

00:00:09.262 --> 00:00:11.220
And let me think about
what's going on here is.

00:00:11.220 --> 00:00:14.551
It says arrange the 13 orange
sample points in the histogram

00:00:14.551 --> 00:00:15.050
below.

00:00:15.050 --> 00:00:17.560
So these are the orange
points you're talking about.

00:00:17.560 --> 00:00:21.016
So the sample standard
deviation is approximately 1.6.

00:00:21.016 --> 00:00:22.650
So just so I
understand these, these

00:00:22.650 --> 00:00:25.112
are points that are being
sampled from sum distribution?

00:00:25.112 --> 00:00:25.820
BEN EATER: Right.

00:00:25.820 --> 00:00:27.840
SAL: We get these
points right over here.

00:00:27.840 --> 00:00:31.030
And then your module
is automatically

00:00:31.030 --> 00:00:32.232
calculating the sample mean.

00:00:32.232 --> 00:00:32.940
BEN EATER: Right.

00:00:32.940 --> 00:00:34.780
SAL: Which is kind of
the central tendency

00:00:34.780 --> 00:00:36.070
of at least these points.

00:00:36.070 --> 00:00:38.870
It's estimating the
mean of this population

00:00:38.870 --> 00:00:40.130
you're sampling from.

00:00:40.130 --> 00:00:42.520
And then also the sample
standard deviation,

00:00:42.520 --> 00:00:47.940
which is really kind of a way
to measure on average how far

00:00:47.940 --> 00:00:50.350
away they are from
that sample mean.

00:00:50.350 --> 00:00:52.350
BEN EATER: Yeah and what
we're trying to do here

00:00:52.350 --> 00:00:55.280
is to kind of get a sense
of what that actually means.

00:00:55.280 --> 00:00:56.940
So we say the
standard deviation is

00:00:56.940 --> 00:00:59.461
1.9, what does that
sort of feel like?

00:00:59.461 --> 00:00:59.960
SAL: Right.

00:00:59.960 --> 00:01:02.707
It seems clear that the sample
mean it looks like it literally

00:01:02.707 --> 00:01:04.040
is the average of these numbers.

00:01:04.040 --> 00:01:04.590
BEN EATER: Right, it is.

00:01:04.590 --> 00:01:06.604
SAL: And it looks like
if I move this out,

00:01:06.604 --> 00:01:08.270
then it moves to the
right a little bit.

00:01:08.270 --> 00:01:09.354
But that's just one point.

00:01:09.354 --> 00:01:10.520
The mean moves to the right.

00:01:10.520 --> 00:01:13.360
The mean I think, it's easier to
have an intuition for the mean.

00:01:13.360 --> 00:01:15.550
BEN EATER: A lot of people have
a pretty good sense of it here.

00:01:15.550 --> 00:01:16.150
SAL: If stuff
moves to the right,

00:01:16.150 --> 00:01:17.530
then the mean
moves to the right.

00:01:17.530 --> 00:01:20.260
If we move stuff to
the left, then the mean

00:01:20.260 --> 00:01:21.300
moves to the left.

00:01:21.300 --> 00:01:22.210
That makes sense.

00:01:22.210 --> 00:01:24.520
But the standard
deviation in this case,

00:01:24.520 --> 00:01:27.450
we're measuring the
sample standard deviation.

00:01:27.450 --> 00:01:29.802
This is a measure of dispersion.

00:01:29.802 --> 00:01:31.760
And I say on average in
quotes because it's not

00:01:31.760 --> 00:01:33.702
exactly the on average
that we typically--

00:01:33.702 --> 00:01:35.910
BEN EATER: And that's kind
of point of this exercise.

00:01:35.910 --> 00:01:39.170
It's not kind of exactly
on average, so what is it?

00:01:39.170 --> 00:01:39.960
SAL: Right.

00:01:39.960 --> 00:01:41.850
But you see that
the further-- so

00:01:41.850 --> 00:01:44.590
let's see, the further
the points are spread out,

00:01:44.590 --> 00:01:47.430
the wider this sample
standard deviation is.

00:01:47.430 --> 00:01:49.570
If I bring the points
closer together,

00:01:49.570 --> 00:01:52.130
then the standard deviation
gets smaller and smaller.

00:01:52.130 --> 00:01:55.620
If I bring them closer to
the sample mean, as you see

00:01:55.620 --> 00:01:57.270
they get tighter and tighter.

00:01:57.270 --> 00:01:59.360
So let's see, let's do
the actual exercise.

00:01:59.360 --> 00:02:01.400
So arrange the 13
order sample points

00:02:01.400 --> 00:02:03.966
so the sample standard
deviation is approximately 1.6.

00:02:03.966 --> 00:02:06.550
Right now it's 2.1, so I want
to squeeze things together.

00:02:06.550 --> 00:02:07.925
And there's no
right answer here.

00:02:07.925 --> 00:02:10.788
I just have to get them in
a way that I'm close to 1.6.

00:02:10.788 --> 00:02:14.170
So I want to move things
closer to the sample mean.

00:02:14.170 --> 00:02:17.800
And so let's see this one,
so 1 point by itself--

00:02:17.800 --> 00:02:20.660
actually just moving
this 1 point--

00:02:20.660 --> 00:02:21.640
BEN EATER: Not quite.

00:02:21.640 --> 00:02:23.820
SAL: Just that 1 point
didn't quite do the trick.

00:02:23.820 --> 00:02:25.160
But I could move more things in.

00:02:25.160 --> 00:02:27.440
And it really could be I
could pick another point,

00:02:27.440 --> 00:02:28.635
move it closer to the--

00:02:28.635 --> 00:02:30.010
BEN EATER: Lots
of ways to do it.

00:02:30.010 --> 00:02:31.930
SAL: I'll have a bigger impact
if I move something that's

00:02:31.930 --> 00:02:33.780
right now far away and
move that closer in.

00:02:33.780 --> 00:02:34.330
BEN EATER: That's right.

00:02:34.330 --> 00:02:35.810
SAL: Because right
now that's kind

00:02:35.810 --> 00:02:38.840
of taking that-- OK,
that looks right.

00:02:38.840 --> 00:02:42.310
1.6, so let me check
my answer, correct.

00:02:42.310 --> 00:02:45.330
BEN EATER: Now, look
at this show solution.

00:02:45.330 --> 00:02:47.570
SAL: OK, show solution.

00:02:47.570 --> 00:02:48.945
The standard
deviation is smaller

00:02:48.945 --> 00:02:50.699
if the points are
closer to the mean, yes.

00:02:50.699 --> 00:02:52.990
The standard deviation is
larger if the points are more

00:02:52.990 --> 00:02:53.830
spread out.

00:02:53.830 --> 00:02:56.040
Try removing a point
closer to and further away

00:02:56.040 --> 00:02:58.623
from the sample mean to see how
the sample standard deviation.

00:02:58.623 --> 00:03:00.734
BEN EATER: So that's
telling us what we just did.

00:03:00.734 --> 00:03:01.650
Look at the next step.

00:03:01.650 --> 00:03:02.901
SAL: Oh, there's another step.

00:03:02.901 --> 00:03:05.108
BEN EATER: And what this
will do is actually show you

00:03:05.108 --> 00:03:06.820
different examples
of how to get to 1.6.

00:03:06.820 --> 00:03:09.735
So if you click that, it'll
show you maybe a different way.

00:03:09.735 --> 00:03:10.610
SAL: Oh, that's neat.

00:03:10.610 --> 00:03:12.696
BEN EATER: Now, click it again.

00:03:12.696 --> 00:03:15.070
And so what you can see is
there's lots of different ways

00:03:15.070 --> 00:03:18.950
to have a sample standard
deviation of 1.6.

00:03:18.950 --> 00:03:21.840
And they may look different,
but if you keep clicking that,

00:03:21.840 --> 00:03:24.340
you'll kind of see there are
some things in common as far as

00:03:24.340 --> 00:03:25.810
how they are grouped
close to the mean.

00:03:25.810 --> 00:03:27.851
SAL: And one of the things
I just want to clarify

00:03:27.851 --> 00:03:30.670
is you've drawn this
kind of bell curve

00:03:30.670 --> 00:03:33.100
with that area
shaded in the back.

00:03:33.100 --> 00:03:36.900
That is our estimate of what
the real distribution might

00:03:36.900 --> 00:03:38.380
look like based
on the information

00:03:38.380 --> 00:03:39.680
we have of the sample.

00:03:39.680 --> 00:03:42.850
So if our sample
mean is accurate,

00:03:42.850 --> 00:03:46.010
if it really is estimating the
true mean of this population,

00:03:46.010 --> 00:03:47.890
then that's what we're
drawing in the middle.

00:03:47.890 --> 00:03:50.820
And then we're assuming it's a
normal just kind of bell curve

00:03:50.820 --> 00:03:51.640
shape distribution.

00:03:51.640 --> 00:03:53.590
So that's what it would look
like based on information

00:03:53.590 --> 00:03:53.960
we have.

00:03:53.960 --> 00:03:54.668
BEN EATER: Right.

00:03:54.668 --> 00:03:56.690
SAL: This is very cool.


Fill-in-the-blank triangle proofs example 2

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.140
I have another one of these
fill-in-the-blank triangle

00:00:03.140 --> 00:00:04.154
proof exercises.

00:00:04.154 --> 00:00:05.570
But this time,
instead of doing it

00:00:05.570 --> 00:00:08.004
on my own little
digital notebook,

00:00:08.004 --> 00:00:09.670
I'm going to do it
right on the exercise

00:00:09.670 --> 00:00:12.100
because there actually are
tools so you can mark up

00:00:12.100 --> 00:00:13.440
the diagrams.

00:00:13.440 --> 00:00:20.110
So first they tell us that
segment BC is parallel to DF,

00:00:20.110 --> 00:00:22.587
figure not drawn to
scale, fair enough.

00:00:22.587 --> 00:00:24.920
And it's always important,
if we're going to do a proof,

00:00:24.920 --> 00:00:26.336
to know what we're
going to prove.

00:00:26.336 --> 00:00:28.390
We're going to prove the
triangle EBD-- so that's

00:00:28.390 --> 00:00:33.292
this triangle down here-- is
congruent to triangle EBC.

00:00:33.292 --> 00:00:35.250
So this one right over
here, that this triangle

00:00:35.250 --> 00:00:36.500
is congruent to that triangle.

00:00:36.500 --> 00:00:38.330
That's what we're
going to try to prove.

00:00:38.330 --> 00:00:40.384
So let's see what
givens they gave us.

00:00:40.384 --> 00:00:42.550
They say that-- and you see
when you scroll over it,

00:00:42.550 --> 00:00:44.133
they show you which
two angles they're

00:00:44.133 --> 00:00:47.280
talking about-- that
angle EBD, or BDE,

00:00:47.280 --> 00:00:49.910
is congruent to angle ACB.

00:00:49.910 --> 00:00:51.310
So let me mark those.

00:00:51.310 --> 00:00:54.280
So I just click once to show
that those two are congruent.

00:00:54.280 --> 00:00:57.277
Then they say that these
two segments, DE and AC

00:00:57.277 --> 00:00:57.860
are congruent.

00:00:57.860 --> 00:00:58.735
So let me mark those.

00:00:58.735 --> 00:00:59.790
You just click on them.

00:00:59.790 --> 00:01:01.540
Click on them once it
does one slash mark.

00:01:01.540 --> 00:01:02.914
If you click on
it more than once

00:01:02.914 --> 00:01:04.910
it will do more
than one slash mark.

00:01:04.910 --> 00:01:10.930
Then we also know that these
two are congruent, BED and BAC.

00:01:10.930 --> 00:01:13.150
So I'm going to
put two arcs here

00:01:13.150 --> 00:01:15.090
to show that they
are not necessarily

00:01:15.090 --> 00:01:16.950
congruent to these other two.

00:01:16.950 --> 00:01:18.260
So I did those.

00:01:18.260 --> 00:01:22.270
Now we have this one, BED--
which we've already marked,

00:01:22.270 --> 00:01:26.340
that's BED right over
there-- is congruent to CEF.

00:01:26.340 --> 00:01:28.350
So that's just telling
us that CEF, this angle,

00:01:28.350 --> 00:01:29.349
is the same as this one.

00:01:29.349 --> 00:01:31.800
So I can put the double
hash marks there.

00:01:31.800 --> 00:01:34.055
And then they tell us that
BE is congruent to EF.

00:01:38.930 --> 00:01:42.490
And then finally, they
tell us that angle DBE

00:01:42.490 --> 00:01:45.325
is congruent to CFE.

00:01:45.325 --> 00:01:46.670
I'll put three.

00:01:46.670 --> 00:01:48.756
DBE is congruent to CFE.

00:01:51.430 --> 00:01:55.820
So now let's try
to do our proof.

00:01:55.820 --> 00:01:58.210
And they've already kind of
done a scaffold of our proof.

00:01:58.210 --> 00:02:00.630
And we need to essentially
fill in the blanks.

00:02:00.630 --> 00:02:02.831
So this first thing, they're
comparing two triangles

00:02:02.831 --> 00:02:04.330
and they say these
two triangles are

00:02:04.330 --> 00:02:07.390
congruent because
of angle side angle.

00:02:07.390 --> 00:02:10.259
So where can we even
make that statement?

00:02:10.259 --> 00:02:13.310
Well we have this
triangle right over here.

00:02:13.310 --> 00:02:15.330
We can say it's congruent
to this triangle

00:02:15.330 --> 00:02:19.670
up here by angle side angle.

00:02:19.670 --> 00:02:22.284
Or, we could say
that this triangle

00:02:22.284 --> 00:02:24.700
is congruent to this triangle,
because you have this angle

00:02:24.700 --> 00:02:27.660
side angle, this
angle side angle.

00:02:27.660 --> 00:02:30.880
So it's not clear, based
on just this blank,

00:02:30.880 --> 00:02:32.750
which one they want
us to describe.

00:02:32.750 --> 00:02:34.180
Let's go further
down in the proof

00:02:34.180 --> 00:02:36.790
and maybe it'll
become more obvious.

00:02:36.790 --> 00:02:39.150
Well then they have this
blank, right over here.

00:02:39.150 --> 00:02:41.320
Blank equals blank because
corresponding parts

00:02:41.320 --> 00:02:43.370
of congruent triangles
are congruent.

00:02:43.370 --> 00:02:46.570
Once again, it's probably
based on which two triangles

00:02:46.570 --> 00:02:47.970
we picked to be congruent.

00:02:47.970 --> 00:02:49.350
This is going to be one of them.

00:02:49.350 --> 00:02:50.600
But we don't know
whether it's going

00:02:50.600 --> 00:02:52.183
to be that one, or
that one, depending

00:02:52.183 --> 00:02:54.200
on how this proof is going down.

00:02:54.200 --> 00:02:59.810
Then they say that triangle
ABC is congruent to EBD.

00:02:59.810 --> 00:03:01.520
So then they say
that this triangle

00:03:01.520 --> 00:03:04.360
is congruent to this
triangle, right over here.

00:03:04.360 --> 00:03:07.340
Well, if down here, they're
making that statement,

00:03:07.340 --> 00:03:10.510
then up here they must want us
to say that this triangle is

00:03:10.510 --> 00:03:12.900
congruent to this
triangle, right over here.

00:03:12.900 --> 00:03:25.850
So we can say that EBD, the
double arc, double side,

00:03:25.850 --> 00:03:28.870
triple side is going to be
congruent to double arc,

00:03:28.870 --> 00:03:35.650
double slashes, triple side,
so EFC, because of ASA.

00:03:35.650 --> 00:03:36.410
There you go.

00:03:36.410 --> 00:03:38.710
And it filled it in which
is a good indication

00:03:38.710 --> 00:03:41.900
that we did it right.

00:03:41.900 --> 00:03:43.270
Let's see, what else can we say?

00:03:43.270 --> 00:03:46.050
So now, remember, we're trying
to prove that this triangle

00:03:46.050 --> 00:03:49.160
right over here is
congruent to this triangle.

00:03:49.160 --> 00:03:52.420
So based on this, we just said
that this triangle on the left

00:03:52.420 --> 00:03:54.930
is congruent to this
triangle on the right.

00:03:54.930 --> 00:03:58.380
Given that they're congruent,
what is useful information so

00:03:58.380 --> 00:04:01.290
that we can get information
about this middle triangle?

00:04:01.290 --> 00:04:05.640
Well this middle triangle
shares a side-- shares side EC,

00:04:05.640 --> 00:04:07.250
with this triangle over here.

00:04:07.250 --> 00:04:10.670
So we should say that side
EC is congruent to something

00:04:10.670 --> 00:04:13.340
over here, which
will help us set up

00:04:13.340 --> 00:04:15.470
that this middle
triangle is congruent.

00:04:15.470 --> 00:04:20.130
So based on the fact that
EBD and EFC are congruent,

00:04:20.130 --> 00:04:22.920
EC's corresponding
side would be what?

00:04:22.920 --> 00:04:26.050
Well, it's the side that's
opposite the triple angle.

00:04:26.050 --> 00:04:30.326
So it's going to
correspond to side DE.

00:04:30.326 --> 00:04:37.680
So we could say that
DE is congruent to EC

00:04:37.680 --> 00:04:39.700
because they're
corresponding sides.

00:04:39.700 --> 00:04:42.050
They're both the sides
opposite this triple angle.

00:04:45.730 --> 00:04:49.830
Segment DE is congruent
to segment EC.

00:04:49.830 --> 00:04:52.550
There you go, and
that worked out.

00:04:52.550 --> 00:04:56.400
And then they say triangle ABC
and EBD-- so that's this one

00:04:56.400 --> 00:04:58.430
and this one-- are
congruent because--

00:04:58.430 --> 00:04:59.680
and we already went over this.

00:04:59.680 --> 00:05:00.740
There's multiple
ways that you could

00:05:00.740 --> 00:05:03.150
say that could-- well
actually, no, there's only one.

00:05:03.150 --> 00:05:07.090
They have this angle side angle.

00:05:07.090 --> 00:05:09.200
That's why they are congruent.

00:05:09.200 --> 00:05:12.720
So we have angle side
angle congruence.

00:05:12.720 --> 00:05:17.520
And then we can say that
BC is congruent to BD.

00:05:17.520 --> 00:05:20.135
Well those are corresponding
sides of congruent triangles.

00:05:20.135 --> 00:05:22.990
Those are both the sides
opposite the double arc.

00:05:22.990 --> 00:05:25.560
Opposite the double
arc, and then up here,

00:05:25.560 --> 00:05:26.950
opposite the double arc.

00:05:26.950 --> 00:05:30.910
So that's because they
are corresponding parts

00:05:30.910 --> 00:05:32.880
of congruent triangles.

00:05:32.880 --> 00:05:37.620
And then finally, so
let me mark those off.

00:05:37.620 --> 00:05:41.850
So that is congruent to
that, corresponding parts

00:05:41.850 --> 00:05:43.390
of congruent triangles.

00:05:43.390 --> 00:05:45.757
And then finally, we
can prove the statement.

00:05:45.757 --> 00:05:47.590
And actually you could
have probably guessed

00:05:47.590 --> 00:05:50.620
what this last one was because
this was the end of the proof,

00:05:50.620 --> 00:05:55.040
that EBD is congruent to EBC.

00:06:01.740 --> 00:06:04.420
And that's, once again, because
you have side side side.

00:06:04.420 --> 00:06:06.200
We've shown that
these three sides

00:06:06.200 --> 00:06:07.910
are congruent to
these three sides.

00:06:07.910 --> 00:06:09.277
It's congruent to EBC.

00:06:09.277 --> 00:06:11.360
And you want to make sure
you get the order right.

00:06:11.360 --> 00:06:16.850
So E is the angle opposite the
three sides on this triangle.

00:06:16.850 --> 00:06:19.810
And it's also the side opposite
the three sides on that one.

00:06:19.810 --> 00:06:22.960
And then, corresponding to B,
B is opposite the single side.

00:06:22.960 --> 00:06:25.170
And over here B is
opposite the single side.

00:06:25.170 --> 00:06:27.250
So it's going to be EBC.

00:06:27.250 --> 00:06:31.740
It's congruent to EBC
because of side side side.

00:06:31.740 --> 00:06:33.300
And let's check our answer.

00:06:33.300 --> 00:06:36.590
Drum roll, and it's right.

00:06:36.590 --> 00:06:38.390
Very good.


Wrong statements in proofs example 1

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.700
We're asked, what's the
first wrong statement

00:00:03.700 --> 00:00:07.600
in the proof below that
triangle BCE-- so that's

00:00:07.600 --> 00:00:09.560
this middle triangle
right over here--

00:00:09.560 --> 00:00:11.930
is congruent to triangle BCA?

00:00:11.930 --> 00:00:14.230
That's this top triangle
right over there.

00:00:14.230 --> 00:00:18.050
And they tell us that
BC is parallel to DF

00:00:18.050 --> 00:00:20.101
and that the diagram
is not drawn to scale.

00:00:20.101 --> 00:00:20.600
Fair enough.

00:00:20.600 --> 00:00:23.209
So we're essentially
going to check their proof

00:00:23.209 --> 00:00:25.000
to see if it's either
correct or if there's

00:00:25.000 --> 00:00:26.950
a wrong statement in it.

00:00:26.950 --> 00:00:28.960
So let's look at the givens.

00:00:28.960 --> 00:00:32.985
They tell us that angle BAC
is congruent to angle BED.

00:00:37.280 --> 00:00:38.852
AC is congruent to DE.

00:00:42.430 --> 00:00:45.220
ACB congruent to BDE.

00:00:45.220 --> 00:00:47.050
So this is BDE.

00:00:47.050 --> 00:00:49.110
This is ACB.

00:00:49.110 --> 00:00:50.390
BC is congruent to CF.

00:00:54.120 --> 00:00:56.820
Just double-click there
to get the two slashes.

00:00:56.820 --> 00:00:59.140
Angle ABC is congruent
to angle CFE.

00:01:02.158 --> 00:01:03.970
So that's this one.

00:01:03.970 --> 00:01:06.444
Triple-click there
to get three of those

00:01:06.444 --> 00:01:08.360
to show that that's not
necessarily congruent,

00:01:08.360 --> 00:01:09.450
or we don't know whether
they're congruent

00:01:09.450 --> 00:01:10.517
to these other angles.

00:01:10.517 --> 00:01:11.350
So that's those two.

00:01:11.350 --> 00:01:16.250
And then angle BAC, which
we've already marked,

00:01:16.250 --> 00:01:17.530
is congruent to CEF.

00:01:17.530 --> 00:01:19.400
So this angle is
congruent to this one.

00:01:19.400 --> 00:01:21.600
So this one will
be just one slash.

00:01:21.600 --> 00:01:23.300
So now let's go
through their proof

00:01:23.300 --> 00:01:26.250
and see if it
makes actual sense.

00:01:26.250 --> 00:01:27.790
So this first
statement they make

00:01:27.790 --> 00:01:34.475
is, BDE is congruent to BCA
because of angle side angle.

00:01:37.260 --> 00:01:39.050
Let's see.

00:01:39.050 --> 00:01:41.600
It looks like that's
right because you

00:01:41.600 --> 00:01:44.400
have your double arc angle,
then your single side, and then

00:01:44.400 --> 00:01:45.650
your single arc angle.

00:01:50.790 --> 00:01:52.530
So you have angle, side, angle.

00:01:52.530 --> 00:01:53.340
It is legitimate.

00:01:53.340 --> 00:01:55.294
This is a legitimate
statement here.

00:01:55.294 --> 00:01:56.210
So I'm cool with that.

00:01:56.210 --> 00:01:58.070
That's not going
to be incorrect.

00:01:58.070 --> 00:02:04.860
Then they say angle
ACB is congruent to DBE

00:02:04.860 --> 00:02:07.556
because vertical
angles are equal.

00:02:11.350 --> 00:02:14.400
Well, this makes
absolutely no sense.

00:02:14.400 --> 00:02:16.180
Those aren't vertical angles.

00:02:19.690 --> 00:02:23.560
Those just aren't
vertical angles.

00:02:23.560 --> 00:02:25.476
Actually, the way that
this is drawn here,

00:02:25.476 --> 00:02:26.850
there aren't any
vertical angles.

00:02:26.850 --> 00:02:30.682
But ACB and DBE are definitely
not vertical angles.

00:02:30.682 --> 00:02:31.640
So we're actually done.

00:02:31.640 --> 00:02:33.510
We have found the
incorrect statement.

00:02:33.510 --> 00:02:34.670
This is not true.

00:02:34.670 --> 00:02:38.540
ACB is not congruent to DBE
because of vertical angles.

00:02:38.540 --> 00:02:40.210
So the wrong statement
in this proof

00:02:40.210 --> 00:02:41.626
was the statement
that tries to do

00:02:41.626 --> 00:02:44.020
that because of vertical angles.

00:02:44.020 --> 00:02:45.650
So let's check our answer.

00:02:45.650 --> 00:02:47.402
And it is correct.


Fill-in-the-blank triangle proofs example 1

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.520
I've taken a screen shot from
our fill in the blank proof

00:00:03.520 --> 00:00:04.130
exercises.

00:00:04.130 --> 00:00:06.680
And I'm going to work it out,
essentially, on digital paper.

00:00:06.680 --> 00:00:09.470
And then we can fill in the
blanks on the actual exercise.

00:00:09.470 --> 00:00:11.960
So they tell us that
BC is parallel to DF.

00:00:14.870 --> 00:00:17.500
The figure is not drawn
to scale-- fair enough.

00:00:17.500 --> 00:00:22.574
Then they tell us that angle
CEF is congruent to angle BAC.

00:00:22.574 --> 00:00:25.110
CEF, so that's this
right over here.

00:00:29.330 --> 00:00:32.340
CEF is congruent to BAC.

00:00:32.340 --> 00:00:34.560
So it's congruent
to that over there.

00:00:34.560 --> 00:00:39.010
Then they tell us that
CFE is congruent to ABC.

00:00:39.010 --> 00:00:44.065
Angle CFE is congruent
to angle ABC.

00:00:47.720 --> 00:00:51.048
Then they tell us that
segment CF is congruent to BC.

00:00:57.100 --> 00:01:01.120
Then they tell us that
BDE is congruent to ACB.

00:01:07.530 --> 00:01:10.090
So that's that right over there.

00:01:10.090 --> 00:01:13.950
Then they tell us DE
is congruent to AC.

00:01:21.150 --> 00:01:27.080
And then finally, they
tell us that-- let

00:01:27.080 --> 00:01:29.410
me do that in a
different color--

00:01:29.410 --> 00:01:32.997
so BED is congruent to BAC.

00:01:32.997 --> 00:01:34.080
Let me make sure I get it.

00:01:34.080 --> 00:01:38.070
BED-- so it's this
angle right over here--

00:01:38.070 --> 00:01:39.905
is congruent to BAC.

00:01:39.905 --> 00:01:42.060
BAC is this blue one
over there, so we already

00:01:42.060 --> 00:01:43.410
have a marker for that.

00:01:43.410 --> 00:01:45.630
So I'll write the blue
one just like that.

00:01:45.630 --> 00:01:48.160
So that's that piece
of information as well.

00:01:48.160 --> 00:01:49.932
Now we're ready for the proof.

00:01:49.932 --> 00:01:51.390
Now whenever you
do any proof, it's

00:01:51.390 --> 00:01:52.980
good to know what
you're trying to prove.

00:01:52.980 --> 00:01:54.570
So let's just cut
to the chase and see

00:01:54.570 --> 00:01:56.087
where this proof actually goes.

00:01:56.087 --> 00:01:57.670
So in the last
statement of the proof,

00:01:57.670 --> 00:02:01.630
they say triangle BCA is
congruent to triangle BCE

00:02:01.630 --> 00:02:03.150
because-- and we
have to figure out

00:02:03.150 --> 00:02:05.160
why we can say that
based on everything

00:02:05.160 --> 00:02:07.289
that this proof has
proven beforehand.

00:02:07.289 --> 00:02:08.830
But so we're
essentially just proving

00:02:08.830 --> 00:02:16.950
that triangle BCA, which is this
triangle right up here-- this

00:02:16.950 --> 00:02:22.860
is BCA-- that it is
congruent to triangle BCE.

00:02:22.860 --> 00:02:26.310
BCE is this triangle-- this
downward-pointing triangle.

00:02:26.310 --> 00:02:28.326
So this is what we're
trying to prove.

00:02:28.326 --> 00:02:30.707
This is congruent to
that right over here.

00:02:30.707 --> 00:02:32.540
This triangle is congruent
to that triangle.

00:02:32.540 --> 00:02:34.712
So that's important
to keep in mind.

00:02:34.712 --> 00:02:36.420
And all we know about
these two triangles

00:02:36.420 --> 00:02:38.880
is it they share a
side in common, which

00:02:38.880 --> 00:02:41.154
is equal to itself by
the reflexive property.

00:02:41.154 --> 00:02:42.820
But we don't have any
other information.

00:02:42.820 --> 00:02:45.194
We're going to have to figure
out whether other sides are

00:02:45.194 --> 00:02:47.860
congruent or maybe other
angles or something else

00:02:47.860 --> 00:02:50.310
in order to actually
make this statement.

00:02:50.310 --> 00:02:52.690
So let's see what they've
done in the proof.

00:02:52.690 --> 00:02:54.132
So first they make a statement.

00:02:54.132 --> 00:02:55.840
And we don't need to
correct it, but it's

00:02:55.840 --> 00:02:58.350
good to follow the
narrative of their proof.

00:02:58.350 --> 00:03:09.690
BCA is congruent to BDE
because of angle, side, angle.

00:03:09.690 --> 00:03:13.105
So let's see, we have an
angle, side, angle; angle,

00:03:13.105 --> 00:03:15.360
side, angle.

00:03:15.360 --> 00:03:20.200
This triple arc corresponds to
D here, corresponds to C there.

00:03:20.200 --> 00:03:21.930
And you see that even over here.

00:03:21.930 --> 00:03:24.070
The way they wrote it,
D is the middle here,

00:03:24.070 --> 00:03:25.320
and C is the middle there.

00:03:25.320 --> 00:03:27.290
So they do correspond
to each other.

00:03:27.290 --> 00:03:30.380
That makes sense.

00:03:30.380 --> 00:03:33.946
This side, DE, should
correspond to CA.

00:03:33.946 --> 00:03:35.070
And you see that over here.

00:03:35.070 --> 00:03:38.340
DE corresponds to CA based
on the order it's written.

00:03:38.340 --> 00:03:43.000
E comes right after D.
A comes right after C.

00:03:43.000 --> 00:03:46.270
And angle E should correspond
to angle A, and it does.

00:03:46.270 --> 00:03:49.920
Angle E corresponds to
angle A, and it does.

00:03:49.920 --> 00:03:52.440
So this is completely
legitimate-- angle, side,

00:03:52.440 --> 00:03:55.750
angle; angle, side, angle
It's a little bit hard

00:03:55.750 --> 00:03:57.000
based on how they're oriented.

00:03:57.000 --> 00:03:59.583
It would've been easier if they
were oriented in the same way.

00:03:59.583 --> 00:04:00.690
But this is legitimate.

00:04:00.690 --> 00:04:06.590
By angle, side, angle, these
two triangles are congruent.

00:04:06.590 --> 00:04:08.990
Then they say, something
is equal to something

00:04:08.990 --> 00:04:11.790
because corresponding parts
of congruent triangles

00:04:11.790 --> 00:04:13.430
are congruent.

00:04:13.430 --> 00:04:17.089
And this is a little bit
harder because they're probably

00:04:17.089 --> 00:04:21.790
making a statement-- we just
proved that this triangle

00:04:21.790 --> 00:04:24.434
right over here is congruent
to this triangle up here.

00:04:24.434 --> 00:04:26.850
And so if we're going to say
that two corresponding things

00:04:26.850 --> 00:04:29.058
are congruent, it's probably
two corresponding things

00:04:29.058 --> 00:04:30.620
in these two triangles.

00:04:30.620 --> 00:04:34.470
And what is corresponding
in these two triangles that

00:04:34.470 --> 00:04:36.440
will help us get to
our end goal of saying

00:04:36.440 --> 00:04:39.700
that this top triangle is
congruent to this bottom one?

00:04:39.700 --> 00:04:42.590
Well the only information that
this side triangle right over

00:04:42.590 --> 00:04:47.640
here, that BDE shares with the
central triangle is this side.

00:04:47.640 --> 00:04:51.100
So maybe we can say that this
side corresponds to some side

00:04:51.100 --> 00:04:52.260
up here.

00:04:52.260 --> 00:04:54.090
And it does because
they're congruent.

00:04:54.090 --> 00:04:55.650
And then that'll
give us information

00:04:55.650 --> 00:04:56.870
for this middle triangle.

00:04:56.870 --> 00:04:58.430
So let's just think about that.

00:04:58.430 --> 00:05:03.170
And it may or may not be right,
but it's a good working guess.

00:05:03.170 --> 00:05:05.130
And there's different
ways to tackle these.

00:05:05.130 --> 00:05:11.050
So this side over here, if
we look at how this is-- BE

00:05:11.050 --> 00:05:13.972
should correspond to BA.

00:05:13.972 --> 00:05:19.370
BE-- and remember,
it's between the blue

00:05:19.370 --> 00:05:20.945
and the not triple green side.

00:05:24.580 --> 00:05:27.020
So these two should
correspond based

00:05:27.020 --> 00:05:30.420
on the information
right over here.

00:05:30.420 --> 00:05:33.130
Let me do this in
a different color.

00:05:33.130 --> 00:05:38.482
This thing should be congruent
to that thing right over there.

00:05:38.482 --> 00:05:40.190
And that gives us
interesting information

00:05:40.190 --> 00:05:42.370
because now we've built
up two congruent sides

00:05:42.370 --> 00:05:44.780
of this middle one
and this top one.

00:05:44.780 --> 00:05:47.310
So let's write
that as our working

00:05:47.310 --> 00:05:49.310
thesis of what they want here.

00:05:49.310 --> 00:05:58.970
So we can say BE-- well, we
could say the length is equal--

00:05:58.970 --> 00:06:06.600
but we could even say
BE is congruent to BA

00:06:06.600 --> 00:06:08.840
because corresponding parts
of congruent triangles

00:06:08.840 --> 00:06:09.450
are congruent.

00:06:09.450 --> 00:06:10.630
And this is not an
easy thing to do.

00:06:10.630 --> 00:06:12.213
But the real hint
is they just told us

00:06:12.213 --> 00:06:14.287
that these two
things are congruent.

00:06:14.287 --> 00:06:16.620
And we want some information
about this middle triangle.

00:06:16.620 --> 00:06:18.036
And they share a
common side here.

00:06:18.036 --> 00:06:20.100
So it seems reasonable enough.

00:06:20.100 --> 00:06:22.740
Hopefully that gets us
to the right end point.

00:06:22.740 --> 00:06:29.605
Now they tell us that triangle
BCA is congruent to FCE.

00:06:32.680 --> 00:06:34.630
And we started at F.
We started at B. That's

00:06:34.630 --> 00:06:36.110
the double angle side.

00:06:36.110 --> 00:06:38.071
Then you went across
this one side,

00:06:38.071 --> 00:06:40.320
and you went across this
side with the magenta marker.

00:06:40.320 --> 00:06:44.080
So that seems right because
of angle, angle, side.

00:06:44.080 --> 00:06:47.820
Angle, angle, side--
angle, the blue angle,

00:06:47.820 --> 00:06:50.610
double magenta angle, and then
this single, magenta slash

00:06:50.610 --> 00:06:51.110
side.

00:06:51.110 --> 00:06:53.860
Angle, angle, side--
angle, angle, side.

00:06:53.860 --> 00:06:56.280
That seems legitimate,
and it should

00:06:56.280 --> 00:06:58.625
unless there was some type
of error in this code.

00:06:58.625 --> 00:07:04.130
And then they say AC
is congruent to CE

00:07:04.130 --> 00:07:11.600
because-- well we just figured
out that this top triangle is

00:07:11.600 --> 00:07:13.720
congruent to this
one down over here.

00:07:13.720 --> 00:07:16.200
So it's probably going to be
because of corresponding sides

00:07:16.200 --> 00:07:17.840
of congruent triangles.

00:07:17.840 --> 00:07:18.820
And we can verify it.

00:07:18.820 --> 00:07:21.130
If this is congruent to
that, then let's see,

00:07:21.130 --> 00:07:29.590
AC is between the blue angle
and the non-magenta angle.

00:07:29.590 --> 00:07:33.670
And EC is between the blue
angle and the non-magenta angle.

00:07:33.670 --> 00:07:34.880
I haven't marked it here.

00:07:34.880 --> 00:07:42.710
And so we have AC is
indeed congruent to CE.

00:07:47.450 --> 00:07:51.434
And now this is interesting,
and this was useful

00:07:51.434 --> 00:07:53.600
because this shared a side
with this middle triangle

00:07:53.600 --> 00:07:54.780
that we care about.

00:07:54.780 --> 00:07:57.460
And so we now show that this
top triangle, all the sides

00:07:57.460 --> 00:08:00.986
have the same lengths as
all the sides down here.

00:08:00.986 --> 00:08:02.610
So this was actually
useful information

00:08:02.610 --> 00:08:05.370
that we figured out, so I
think that was the right way

00:08:05.370 --> 00:08:06.670
to go about it.

00:08:06.670 --> 00:08:08.100
Oh, I need to finish this.

00:08:08.100 --> 00:08:14.150
AC is congruent to CE
because corresponding parts

00:08:14.150 --> 00:08:15.955
of corresponding angles.

00:08:15.955 --> 00:08:17.437
Let me write it in black.

00:08:17.437 --> 00:08:19.020
And when we actually
fill it in, there

00:08:19.020 --> 00:08:23.265
will be a pull-down--
so corresponding parts.

00:08:27.430 --> 00:08:29.350
And then finally, we're
in the home stretch.

00:08:29.350 --> 00:08:32.429
We're ready to say that
BCA, this top triangle,

00:08:32.429 --> 00:08:34.580
is congruent to triangle BCE.

00:08:34.580 --> 00:08:36.090
All their sides are the same.

00:08:36.090 --> 00:08:40.890
So this is side,
side, side congruency.

00:08:40.890 --> 00:08:42.990
So now let's verify our answer.

00:08:42.990 --> 00:08:47.220
So let's take out
the actual exercise.

00:08:47.220 --> 00:08:48.955
And so let's go down here.

00:08:52.320 --> 00:08:56.610
The first thing I want to type
in is that BE is equal to BA.

00:08:56.610 --> 00:09:00.430
So let me type that in,
BE-- and if this is right,

00:09:00.430 --> 00:09:05.350
it'll actually just process
it-- is equal to BA.

00:09:05.350 --> 00:09:05.990
And it is.

00:09:05.990 --> 00:09:08.050
There you go.

00:09:08.050 --> 00:09:09.369
Segment BE is congruent AB.

00:09:09.369 --> 00:09:11.160
So that's how we know
we've got that right.

00:09:11.160 --> 00:09:13.130
I wrote it as BA, but
that's the same thing.

00:09:13.130 --> 00:09:14.700
And that's why it processed it.

00:09:14.700 --> 00:09:17.945
And then AC is congruent to CE
because of corresponding parts.

00:09:21.910 --> 00:09:23.149
There you go.

00:09:23.149 --> 00:09:24.690
And then finally,
these two triangles

00:09:24.690 --> 00:09:29.140
are congruent because of
side, side, side congruence.

00:09:29.140 --> 00:09:34.600
And now let's verify our answer.

00:09:34.600 --> 00:09:38.940
I feel good about it, and I
am ready to check my answer.

00:09:38.940 --> 00:09:41.344
Yes, it is correct.


Basic Triangle Proofs Module Example 2

WEBVTT
Kind: captions
Language: en

00:00:01.110 --> 00:00:06.030
We're given that angle ADB
is congruent to angle ADC.

00:00:06.030 --> 00:00:07.330
So let me write that down.

00:00:07.330 --> 00:00:12.270
So angle ADB, so that's
this angle right over here.

00:00:12.270 --> 00:00:15.970
They're telling us that
that's congruent to ADC,

00:00:15.970 --> 00:00:18.450
so that's this angle
right over there.

00:00:18.450 --> 00:00:25.350
They also tell us that
angle BAD, angle bad,

00:00:25.350 --> 00:00:26.260
so that's this angle.

00:00:26.260 --> 00:00:28.468
I'll do it with two arcs
because it's not necessarily

00:00:28.468 --> 00:00:29.960
the same measure
as this one here.

00:00:29.960 --> 00:00:34.040
So angle BAD is
congruent to angle CAD.

00:00:34.040 --> 00:00:38.630
That's angle CAD, so this is
this angle right over here.

00:00:38.630 --> 00:00:43.770
Prove that triangle BAD-- this
is the triangle on the left--

00:00:43.770 --> 00:00:45.630
is congruent to triangle CAD.

00:00:45.630 --> 00:00:48.270
That's this triangle
on the right.

00:00:48.270 --> 00:00:50.660
And so they give
the givens here.

00:00:50.660 --> 00:00:54.590
Angle BAD is congruent to CAD.

00:00:54.590 --> 00:00:56.440
Well, they've
already told us that.

00:00:56.440 --> 00:00:59.250
And they also told us that
angle ADB is congruent to ADC.

00:01:02.440 --> 00:01:05.530
So we've already marked that
up, so how do we prove this?

00:01:05.530 --> 00:01:08.600
Well, the one thing we know,
and this is almost intuitively

00:01:08.600 --> 00:01:11.790
obvious, is that the
measure of this segment

00:01:11.790 --> 00:01:13.460
is equal to itself.

00:01:13.460 --> 00:01:15.450
And the reason why
that's a useful thing

00:01:15.450 --> 00:01:19.570
to state explicitly is because
that side, the segment AD,

00:01:19.570 --> 00:01:21.760
is part of both triangles.

00:01:21.760 --> 00:01:23.710
So let's just mark that there.

00:01:23.710 --> 00:01:25.110
It's equal to itself.

00:01:25.110 --> 00:01:26.554
So let me write this here.

00:01:26.554 --> 00:01:27.970
And the reason why
it's useful, is

00:01:27.970 --> 00:01:30.430
you're kind of saying, hey,
look, this is a side on both

00:01:30.430 --> 00:01:32.180
of those triangles
that we care about.

00:01:32.180 --> 00:01:35.321
So we could write AD
is congruent to AD,

00:01:35.321 --> 00:01:37.320
and it would be by the
reflexive property, which

00:01:37.320 --> 00:01:39.480
is the most obvious
property that something

00:01:39.480 --> 00:01:41.530
is going to be equal to itself.

00:01:41.530 --> 00:01:44.760
So reflexive property,
there you go.

00:01:44.760 --> 00:01:46.690
But now, by stating
that we've explicitly

00:01:46.690 --> 00:01:48.910
put up enough
information in our proof,

00:01:48.910 --> 00:01:52.250
to now say that
the left triangle

00:01:52.250 --> 00:01:53.370
is congruent to the right.

00:01:53.370 --> 00:01:55.070
And the reason why
we can say that is we

00:01:55.070 --> 00:01:57.690
have a corresponding angle,
corresponding side, and then

00:01:57.690 --> 00:01:59.550
another corresponding
angle that is

00:01:59.550 --> 00:02:02.604
congruent to this angle,
this side, and this angle.

00:02:02.604 --> 00:02:04.020
And we talk about
in other videos,

00:02:04.020 --> 00:02:06.430
about why
angle-side-angle works.

00:02:06.430 --> 00:02:09.999
If I have one triangle that has
congruent one angle, a side,

00:02:09.999 --> 00:02:12.290
and another angle that's
congruent to another triangle,

00:02:12.290 --> 00:02:16.420
it's impossible for me to draw a
triangle that is not congruent.

00:02:16.420 --> 00:02:18.540
This is a basic
axiom that we assume.

00:02:18.540 --> 00:02:20.200
It's a basic assumption.

00:02:20.200 --> 00:02:22.210
So now we have
enough information.

00:02:22.210 --> 00:02:25.776
We can now write
that triangle BAD,

00:02:25.776 --> 00:02:28.150
and we want to make sure we
have the corresponding sides.

00:02:28.150 --> 00:02:32.630
So what corresponds to angle
B, or vertex B, right over here

00:02:32.630 --> 00:02:36.310
is C. So it's going
to be C, and then

00:02:36.310 --> 00:02:39.510
we'd want to go to
A and then D, CAD.

00:02:39.510 --> 00:02:42.830
And the reason is we have an
angle, a side, and an angle.

00:02:42.830 --> 00:02:47.020
So let me get an
angle-side-angle congruence.

00:02:47.020 --> 00:02:47.760
There you go.

00:02:47.760 --> 00:02:49.000
And I think we are done.

00:02:49.000 --> 00:02:50.520
Let's see how we did.

00:02:50.520 --> 00:02:52.370
There you go.


Basic Triangle Proofs Module Example

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.450
We're asked to prove
that triangle ADC is

00:00:04.450 --> 00:00:08.810
congruent to triangle BEC, so
this left triangle's congruent

00:00:08.810 --> 00:00:11.980
to this right triangle, and the
figure is not drawn to scale.

00:00:11.980 --> 00:00:14.110
And we should never
assume that it is.

00:00:14.110 --> 00:00:16.010
And then we have
some givens here.

00:00:16.010 --> 00:00:19.700
So they tell us that segment
AC is congruent to BC.

00:00:19.700 --> 00:00:23.050
So AC is congruent to BC.

00:00:23.050 --> 00:00:26.080
And they also tell us that
CD is congruent to CE.

00:00:26.080 --> 00:00:28.530
So CD is congruent to CE.

00:00:28.530 --> 00:00:31.160
And I've got to mark it twice,
because CD is not necessarily

00:00:31.160 --> 00:00:32.580
the right length as CA.

00:00:32.580 --> 00:00:34.274
In fact, they don't
look anywhere close,

00:00:34.274 --> 00:00:35.190
so we can't assume it.

00:00:35.190 --> 00:00:39.640
But they tell us that
CD is congruent to CE.

00:00:39.640 --> 00:00:42.892
So what else can we figure out
about this little diagram here

00:00:42.892 --> 00:00:44.850
in order to be able to
say this left triangle's

00:00:44.850 --> 00:00:46.054
congruent to the right?

00:00:46.054 --> 00:00:48.220
Well, the one thing we know
is that this angle right

00:00:48.220 --> 00:00:52.910
over here, angle ACD, is
congruent to angle BCE.

00:00:52.910 --> 00:00:54.430
So let's write that down.

00:00:54.430 --> 00:00:58.455
Angle ACD, and we don't have
to write the angle symbol.

00:00:58.455 --> 00:01:00.580
And it's not easy to write
the angle symbol anyway,

00:01:00.580 --> 00:01:02.630
it figures it out later
that we're actually

00:01:02.630 --> 00:01:03.920
talking about an angle.

00:01:03.920 --> 00:01:08.170
Angle ACD is congruent
to BCE, and we

00:01:08.170 --> 00:01:10.440
can say that because
they're vertical angles.

00:01:10.440 --> 00:01:12.690
Vertical angles are equal.

00:01:12.690 --> 00:01:13.190
Fair enough.

00:01:13.190 --> 00:01:15.720
And there you go, it added
the angle symbols for us.

00:01:15.720 --> 00:01:18.860
And now, we can, I think,
well, that's enough information

00:01:18.860 --> 00:01:20.630
to say that they are congruent.

00:01:20.630 --> 00:01:28.476
We can now say that triangle ADC
is congruent to triangle BEC.

00:01:31.580 --> 00:01:34.885
And we know that because we have
a side, an angle, and a side.

00:01:37.740 --> 00:01:43.114
So by side-angle-side
congruence.

00:01:43.114 --> 00:01:44.780
Sometimes it takes a
little bit of time,

00:01:44.780 --> 00:01:45.910
it hangs up a little bit.

00:01:45.910 --> 00:01:48.570
Luckily for me, this
time it did not do that.

00:01:48.570 --> 00:01:50.100
And so I think I am done.

00:01:50.100 --> 00:01:52.150
So let me check my answer here.

00:01:52.150 --> 00:01:54.400
Drum roll.

00:01:54.400 --> 00:01:56.388
And it worked.


Khan Academy Vision

WEBVTT
Kind: captions
Language: en

00:00:00.960 --> 00:00:02.840
What Khan Academy
is today could best

00:00:02.840 --> 00:00:07.140
be described as a
tool or a resource.

00:00:07.140 --> 00:00:10.120
It's right now if you go
to the site we have videos,

00:00:10.120 --> 00:00:12.360
we have interactive
exercises, we

00:00:12.360 --> 00:00:14.380
have dashboards for teachers.

00:00:14.380 --> 00:00:16.460
And so depending it could
be used as a teacher

00:00:16.460 --> 00:00:17.470
to help their students.

00:00:17.470 --> 00:00:19.240
It could be used by
a homeschooler who

00:00:19.240 --> 00:00:20.920
wants to learn
something independently.

00:00:20.920 --> 00:00:23.390
It could be used
for an adult who

00:00:23.390 --> 00:00:26.180
wants to review things that
might be relevant to whatever

00:00:26.180 --> 00:00:27.884
their exploring at the moment.

00:00:27.884 --> 00:00:29.300
But one thing I'd
like to be clear

00:00:29.300 --> 00:00:34.700
is we don't view this as the
end all of what we think one,

00:00:34.700 --> 00:00:37.010
this could be, and
even how students

00:00:37.010 --> 00:00:38.290
should experience learning.

00:00:38.290 --> 00:00:40.730
We think that this is
really just version one.

00:00:40.730 --> 00:00:42.450
And it's a very
primitive version right

00:00:42.450 --> 00:00:45.010
now of a tool that
will eventually

00:00:45.010 --> 00:00:50.635
empower a true interactive
vision of education.

00:00:50.635 --> 00:00:52.510
And the other thing I
want to make very clear

00:00:52.510 --> 00:00:55.290
is we don't see it is an
either or proposition.

00:00:55.290 --> 00:00:59.246
We don't see it as a virtual
versus physical a amazon.com

00:00:59.246 --> 00:01:00.920
versus Barnes Noble.

00:01:00.920 --> 00:01:03.330
We see it as a
virtual and physical.

00:01:03.330 --> 00:01:06.740
Or even better, how do we
use the tools and resources

00:01:06.740 --> 00:01:07.960
of the virtual?

00:01:07.960 --> 00:01:11.760
How do we use those to
actually empower the physical,

00:01:11.760 --> 00:01:16.550
to make the physical the
best possible experience.

00:01:16.550 --> 00:01:20.720
And in this vision of what the
total education experience can

00:01:20.720 --> 00:01:24.600
be we have a couple
of core philosophies.

00:01:24.600 --> 00:01:28.639
The first is the idea
of mastery learning.

00:01:28.639 --> 00:01:30.430
And I'll write it in
quotes because mastery

00:01:30.430 --> 00:01:32.450
can mean different things
to different people.

00:01:32.450 --> 00:01:35.830
To us this is the general
idea that students

00:01:35.830 --> 00:01:39.820
should have deep conceptual
understanding of core ideas,

00:01:39.820 --> 00:01:43.640
of more fundamental ideas,
before they are pushed ahead

00:01:43.640 --> 00:01:48.130
to the more derivative ideas,
or the more advanced ideas.

00:01:48.130 --> 00:01:50.220
And so if you think of a
traditional progression

00:01:50.220 --> 00:01:52.345
in mathematics, although
we think that this applies

00:01:52.345 --> 00:01:54.640
beyond just mathematics,
traditional progression

00:01:54.640 --> 00:01:57.450
in mathematics you start
off with arithmetic.

00:01:57.450 --> 00:02:01.100
And then you go to algebra.

00:02:01.100 --> 00:02:03.842
And then you go through
trigonometry, and geometry,

00:02:03.842 --> 00:02:04.550
and all the rest.

00:02:04.550 --> 00:02:09.000
But you eventually
end up in calculus.

00:02:09.000 --> 00:02:11.880
And what we believe is happening
for too many students--

00:02:11.880 --> 00:02:14.210
and this is why so many
students end up having trouble

00:02:14.210 --> 00:02:16.840
with algebra and
definitely calculus--

00:02:16.840 --> 00:02:21.380
is they are not getting mastery
of the more fundamental ideas.

00:02:21.380 --> 00:02:23.320
We believe if
students have the time

00:02:23.320 --> 00:02:26.380
to really build a strong
foundation in arithmetic,

00:02:26.380 --> 00:02:28.880
including exponents
and whatever else,

00:02:28.880 --> 00:02:31.072
then by the time algebra
comes around they

00:02:31.072 --> 00:02:32.530
will have a better
intuition there.

00:02:32.530 --> 00:02:34.720
They will have a better
chance of understanding it.

00:02:34.720 --> 00:02:37.010
And if students have a
conceptual deep understanding

00:02:37.010 --> 00:02:40.724
of algebra, then calculus is
going to make a lot more sense.

00:02:40.724 --> 00:02:42.140
If you do it the
other way around,

00:02:42.140 --> 00:02:43.590
if you just push
students through

00:02:43.590 --> 00:02:47.360
so that they get through
with 80% 90% understanding,

00:02:47.360 --> 00:02:50.074
then what they essentially
end up doing just to survive

00:02:50.074 --> 00:02:52.240
is start doing a superficial
level of understanding.

00:02:52.240 --> 00:02:54.690
They start kind of
pattern matching their way

00:02:54.690 --> 00:02:56.230
through algebra or calculus.

00:02:56.230 --> 00:02:58.280
And not only does
that harm you later on

00:02:58.280 --> 00:02:59.920
as you keep building
on it, it also

00:02:59.920 --> 00:03:02.300
makes it almost impossible
to use these things

00:03:02.300 --> 00:03:04.040
in your everyday life.

00:03:04.040 --> 00:03:04.820
In our minds.

00:03:04.820 --> 00:03:07.950
It's much more valuable to have
a deep understanding of algebra

00:03:07.950 --> 00:03:10.720
and being able to see the
world in an algebraic way,

00:03:10.720 --> 00:03:12.750
than just getting
through algebra

00:03:12.750 --> 00:03:14.590
and kind of
understanding calculus,

00:03:14.590 --> 00:03:18.910
but being able to apply or
really fundamentally understand

00:03:18.910 --> 00:03:20.076
neither.

00:03:20.076 --> 00:03:21.450
The other idea--
and this kind of

00:03:21.450 --> 00:03:22.930
gels with
mastery-based learning.

00:03:22.930 --> 00:03:24.430
You kind of can't
have mastery-based

00:03:24.430 --> 00:03:30.040
learning, at least in our minds,
without a kind of self paced

00:03:30.040 --> 00:03:30.855
reality.

00:03:30.855 --> 00:03:32.980
Because if we really want
students to master things

00:03:32.980 --> 00:03:35.380
we really have to give
them the time and the space

00:03:35.380 --> 00:03:37.100
to master those concepts.

00:03:37.100 --> 00:03:40.044
If everyone else has mastered
it and they're ready to move on.

00:03:40.044 --> 00:03:41.960
There's no reason why
we have to move them on.

00:03:41.960 --> 00:03:44.730
We should let them
work at their own pace.

00:03:44.730 --> 00:03:47.160
The other big thing
about self-paced learning

00:03:47.160 --> 00:03:49.090
is that it creates
student ownership.

00:03:49.090 --> 00:03:51.160
And this is a huge,
huge, huge thing.

00:03:53.960 --> 00:03:55.750
Probably the most
important thing

00:03:55.750 --> 00:03:58.770
that you can learn while
you were learning in school

00:03:58.770 --> 00:04:02.050
isn't necessarily any one
of these subject areas,

00:04:02.050 --> 00:04:04.450
figuring out how to solve
a system of equations

00:04:04.450 --> 00:04:05.590
or take an integral.

00:04:05.590 --> 00:04:09.130
The most important thing
to learn is how to learn.

00:04:09.130 --> 00:04:11.270
How do you take
ownership of something?

00:04:11.270 --> 00:04:15.780
How do you, to some degree, how
do you become your own guide?

00:04:15.780 --> 00:04:18.450
How do you set your own goals?

00:04:18.450 --> 00:04:21.310
And this can only happen
in a self-paced environment

00:04:21.310 --> 00:04:24.150
where, with mentorship, with
people who can guide you,

00:04:24.150 --> 00:04:26.150
you can say, look, this
is what I want to learn.

00:04:26.150 --> 00:04:28.204
This is how I want to
go about learning it.

00:04:28.204 --> 00:04:30.120
And this is what I'm
going to do to get there.

00:04:30.120 --> 00:04:31.500
And my goal is mastery.

00:04:31.500 --> 00:04:34.504
My goal isn't just to get by.

00:04:34.504 --> 00:04:36.170
The other thing we
believe strongly in--

00:04:36.170 --> 00:04:38.740
and we're been working
with many, many schools.

00:04:38.740 --> 00:04:40.240
We've been working
with many schools

00:04:40.240 --> 00:04:42.050
across all of this,
including just how we

00:04:42.050 --> 00:04:44.220
define our tools
and our resources.

00:04:44.220 --> 00:04:46.820
--is the idea of students
teaching students,

00:04:46.820 --> 00:04:49.650
or we could call it
may be peer-to-peer

00:04:49.650 --> 00:04:51.420
teaching and learning.

00:04:51.420 --> 00:04:53.570
And there's an
obvious benefit there

00:04:53.570 --> 00:04:55.429
of if you have other
students around,

00:04:55.429 --> 00:04:57.720
maybe some who've gotten a
little bit further ahead who

00:04:57.720 --> 00:04:58.590
can tutor you.

00:04:58.590 --> 00:05:01.360
You have more access to people
who might be able to help you.

00:05:01.360 --> 00:05:05.250
But we think it's at least
as powerful for the student

00:05:05.250 --> 00:05:08.780
getting tutored as it is for
the student doing the tutoring.

00:05:08.780 --> 00:05:11.820
Because it's one thing to
do a lot of problem sets

00:05:11.820 --> 00:05:15.420
and to kind of go
through subjects

00:05:15.420 --> 00:05:16.670
in the traditional way.

00:05:16.670 --> 00:05:18.650
But to truly get
mastery of something,

00:05:18.650 --> 00:05:20.800
I think most teachers
would agree with this,

00:05:20.800 --> 00:05:22.200
you really have to teach it.

00:05:22.200 --> 00:05:24.980
You really have to be able
to distill it, and explain

00:05:24.980 --> 00:05:28.330
it, and help mentor
other people to it.

00:05:28.330 --> 00:05:31.100
And on top of that you'll
build other soft skills, which

00:05:31.100 --> 00:05:34.540
are at least as important like
ownership, as these other kind

00:05:34.540 --> 00:05:37.240
of tangible skills,
the skills of empathy,

00:05:37.240 --> 00:05:39.200
the skills of being able
to listen to someone,

00:05:39.200 --> 00:05:43.150
the skills of being
able to guide someone

00:05:43.150 --> 00:05:47.470
without making them feel
intimidated or insecure.

00:05:47.470 --> 00:05:50.420
And then finally, the fourth
guiding principle for us

00:05:50.420 --> 00:05:52.890
is the idea that
students-- and this

00:05:52.890 --> 00:05:54.930
is very core to the
physical experience--

00:05:54.930 --> 00:05:57.830
that when students are together
they should be interacting.

00:05:57.830 --> 00:05:59.334
So it should be interactive.

00:05:59.334 --> 00:06:00.750
And when we talk
about interactive

00:06:00.750 --> 00:06:02.380
and we're not talking
about computer interactive.

00:06:02.380 --> 00:06:03.890
We're talking about
physical interactive.

00:06:03.890 --> 00:06:05.570
They should interact
with other human beings.

00:06:05.570 --> 00:06:06.611
They should talk to them.

00:06:06.611 --> 00:06:07.420
They should smile.

00:06:07.420 --> 00:06:08.730
They should see them.

00:06:08.730 --> 00:06:12.710
And it should be inquiry based,
or exploration based, inquiry,

00:06:12.710 --> 00:06:14.530
exploration.

00:06:14.530 --> 00:06:16.930
And this is super key to
everything we believe.

00:06:16.930 --> 00:06:19.810
And this is why we are
actively experimenting

00:06:19.810 --> 00:06:21.380
with summer camps
that are driven

00:06:21.380 --> 00:06:25.260
all around this part
of the experience.

00:06:25.260 --> 00:06:27.200
It's because we
believe that in order

00:06:27.200 --> 00:06:29.084
for a student to
truly internalize

00:06:29.084 --> 00:06:31.000
all of these things--
and it all goes together

00:06:31.000 --> 00:06:33.230
with especially
the idea of mastery

00:06:33.230 --> 00:06:35.210
--they need to struggle
with the ideas.

00:06:35.210 --> 00:06:37.740
They need to experience
them tangibly.

00:06:37.740 --> 00:06:39.452
And for example,
we have students

00:06:39.452 --> 00:06:41.410
they do things that are
implicitly probability.

00:06:41.410 --> 00:06:42.330
But they don't know.

00:06:42.330 --> 00:06:44.142
They think they're just
doing a simulation.

00:06:44.142 --> 00:06:46.600
They think that they're just
playing some type of the game.

00:06:46.600 --> 00:06:49.800
But by doing those simulations
and those games, later

00:06:49.800 --> 00:06:52.570
on when they are exposed
to kind of the more

00:06:52.570 --> 00:06:58.040
formal academic ideas around
probability or expected value,

00:06:58.040 --> 00:07:00.460
they now have a tangible
understanding for them.

00:07:00.460 --> 00:07:03.200
They now have an intuitive
understanding for them.

00:07:03.200 --> 00:07:07.190
And they also see the value in
the more formal representation

00:07:07.190 --> 00:07:08.000
of it.

00:07:08.000 --> 00:07:11.030
So hopefully this gives a big
picture of what we are about.

00:07:11.030 --> 00:07:13.404
We are a tool and
a resource today.

00:07:13.404 --> 00:07:15.320
But the whole goal is
to keep [? iterating. ?]

00:07:15.320 --> 00:07:17.960
And we want people to use it
in any way that they see fit.

00:07:17.960 --> 00:07:19.529
It could be part of
a formal program.

00:07:19.529 --> 00:07:20.570
It could be a supplement.

00:07:20.570 --> 00:07:22.670
It could be used as
a review of somehow.

00:07:22.670 --> 00:07:25.330
But our real goal is how
can we optimize this tool.

00:07:25.330 --> 00:07:28.180
How can we improve this
tool so that we can really

00:07:28.180 --> 00:07:31.710
drive these values to make
a truly holistic education

00:07:31.710 --> 00:07:33.077
experience?