1 00:00:00,145 --> 00:00:03,553 I'm here with Jesse Ro, whose a math teacher at Summit San Jose 2 00:00:03,553 --> 00:00:05,160 and a Khan Academy teaching fellow 3 00:00:05,160 --> 00:00:08,403 and you had some interesting ideas or questions. 4 00:00:08,403 --> 00:00:11,599 Yeah, one question that students ask a lot when they start Algebra 5 00:00:11,599 --> 00:00:16,106 is why do we need letters, why can't we just use numbers for everything? 6 00:00:16,106 --> 00:00:18,076 Why letters? So why do we have all these 7 00:00:18,076 --> 00:00:21,811 Xs and Ys and Zs and ABCs when we start dealing with Algebra? 8 00:00:21,811 --> 00:00:22,680 Yeah, exactly. 9 00:00:22,680 --> 00:00:28,286 That's interesting, well why don't we let people think about that for a second. 10 00:00:28,286 --> 00:00:30,232 So Sal, how would you answer this question? 11 00:00:30,232 --> 00:00:32,193 Why do we need letters in Algebra? 12 00:00:32,193 --> 00:00:35,824 So why letters. So there are a couple of ways I'd think about it. 13 00:00:35,824 --> 00:00:38,026 One is if you have an unknown. 14 00:00:38,026 --> 00:00:44,239 So if I were to write X plus three is equal to ten 15 00:00:44,239 --> 00:00:46,577 the reason why we're doing this is that we don't know what X is 16 00:00:46,577 --> 00:00:48,298 It's literally an unknown. 17 00:00:48,298 --> 00:00:50,197 And so we're going to solve for it in some way. 18 00:00:50,197 --> 00:00:51,710 But it did not have to be the letter X. 19 00:00:51,710 --> 00:00:55,703 We could have literally written blank plus three is equal to ten. 20 00:00:55,703 --> 00:00:59,725 Or we could have written Question Mark plus three is equal to ten. 21 00:00:59,725 --> 00:01:03,147 So it didn't have to be letters, but we needed some type of symbol. 22 00:01:03,147 --> 00:01:07,434 It literally could've been Smiley Face plus three is equal to ten. 23 00:01:07,434 --> 00:01:12,181 But until you know it, you need some type of a symbol to represent whatever that number is. 24 00:01:12,181 --> 00:01:15,700 Now we can go and solve this equation and then know what that symbol represents. 25 00:01:15,700 --> 00:01:17,916 But if we knew it ahead of time, it wouldn't be an unknown. 26 00:01:17,916 --> 00:01:20,387 It wouldn't be something that we didn't know. 27 00:01:20,387 --> 00:01:23,576 So that's one reason why I would use letters 28 00:01:23,576 --> 00:01:26,489 and where just numbers by itself wouldn't be helpful. 29 00:01:26,489 --> 00:01:28,942 The other is when you're describing relationships between numbers. 30 00:01:28,942 --> 00:01:32,286 So I could do something like - I could say - that 31 00:01:32,286 --> 00:01:38,021 whenever you give me a three, I'm going to give you a four. 32 00:01:38,021 --> 00:01:43,762 And I could say, if you give me a five, I'm going to give you a six. 33 00:01:43,762 --> 00:01:46,050 And i could keep going on and on forever. 34 00:01:46,050 --> 00:01:51,626 If you give me a 7.1, I'm going to give you an 8.1. 35 00:01:51,626 --> 00:01:54,431 And I could keep listing this on and on forever. 36 00:01:54,431 --> 00:01:57,477 Maybe you could give me any number, and I could tell you what I'm going to give you. 37 00:01:57,477 --> 00:02:00,879 But I would obviously run out of space and time if I were to list all of them. 38 00:02:00,879 --> 00:02:06,259 And we could do that much more elegantly if we used letters to describe the relationship. 39 00:02:06,259 --> 00:02:11,296 Maybe what you give me we call X, and what I give you we call Y. 40 00:02:11,296 --> 00:02:14,678 And so I say, look, whatever you give me, I'm going to add one to it. 41 00:02:14,678 --> 00:02:16,867 And that's what I'm going to give back to you. 42 00:02:16,867 --> 00:02:20,670 And so now, this very simple equation here 43 00:02:20,670 --> 00:02:24,717 can describe an infinite number of relationships between X 44 00:02:24,717 --> 00:02:28,219 or an infinite number of corresponding Ys and Xs. 45 00:02:28,219 --> 00:02:31,497 So now someone knows whatever X you give me 46 00:02:31,497 --> 00:02:34,612 you give me three, I add one to it, and I'm going to give you four. 47 00:02:34,627 --> 00:02:38,200 You give me 7.1, I'm going to add one to it and give you 8.1. 48 00:02:38,200 --> 00:02:41,166 So there is no more elegant way that you could've done it than by using symbols. 49 00:02:41,166 --> 00:02:43,959 With that said, I didn't have to use Xs and Ys. 50 00:02:43,959 --> 00:02:46,648 This is just a convention that kind of comes to use from history. 51 00:02:46,648 --> 00:02:49,725 I could've defined what you give me as Star 52 00:02:49,725 --> 00:02:54,827 and what I give you as Smiley Face 53 00:02:54,827 --> 00:02:57,733 and this also would've been a valid way to express this. 54 00:02:57,733 --> 00:03:01,733 So the letters are really just symbols. Nothing more.