English
{\rtf1 (Ms. Bowell) Euler's number “e”, \par and it's about 2.71828, }{\rtf1 and it doesn’t repeat even \par though it seems like it does }{\rtf1 for what we can get \par on our scientific calculators. }{\rtf1 So, if we want to graph a few different \par irrational numbers on a number line,}{\rtf1 we call this 0. \par We call this one 2. This one 4.}{\rtf1 Then where did you \par end up graphing √2? }{\rtf1 Where does it go? \par (Student) Between 3 & 4.}{\rtf1 (Ms. Bowell) Okay.}{\rtf1 We can put Pi there. 3.14. }{\rtf1 We'll put Pi there. }{\rtf1 √2, however, \par would go at? }{\rtf1 1 point...\par What's √2? 1.41. }{\rtf1 Okay, \par so where does it go? }{\rtf1 About right there. }{\rtf1 A little bit less than... Right.\par Shy of 1/2. So that's where √2 lives. }{\rtf1 "e" is a little bit \par less than 2 3/4. }{\rtf1 So that would be the \par spot where "e" lives. }{\rtf1 “e” is a number.\par It's 2.71828.}{\rtf1 "e" to the 0 power.\par Where does it go? }{\rtf1 (Students) It goes at 1.}{\rtf1 (Ms. Bowell) It goes at 1. }{\rtf1 Anything to the 0 power \par is equivalent to 1. }{\rtf1 e^0 is 1.}{\rtf1 e^-1. Where's it going to go? }{\rtf1 (Student) 0.3.}{\rtf1 (Ms. Bowell) Okay, so here's 1/2, \par so 0.3 is a little less than 1/2. }{\rtf1 So that's where \par we are going to put e^-1, }{\rtf1 and I'm going to have to make my number \par line longer to be able to put in e^2. }{\rtf1 4, 6, 7.}{\rtf1 (Student) What was e^-1?}{\rtf1 (Ms. Bowell) e^-1 is about 0.3. }{\rtf1 Still positive though. }{\rtf1 Having a negative exponent doesn't \par mean you have a negative number.}{\rtf1 Still positive. \par e^2 is about 7 something. }{\rtf1 So 7.29 or so. }{\rtf1 (Student) (inaudible)}{\rtf1 (Ms. Bowell) Oh, you already \par graphed those numbers. Okay. }{\rtf1 (Student) 7.38.}{\rtf1 (Ms. Bowell) Okay, \par so about right there. }{\rtf1 So "e" is a number, \par and you can raise it to exponents }{\rtf1 just like you can with all the \par other kinds of number that you can. }{\rtf1 (Student) So when we're doing this, \par you give us a number line. }{\rtf1 You don't want the actual number. }{\rtf1 You want us to figure it out, \par and you want us to put the symbols?}{\rtf1 (Ms. Bowell) Right. \par (Student) On the number line?}{\rtf1 (Ms. Bowell) Right. \par Because I mean, }{\rtf1 the reason why we want \par you to write by the symbols, }{\rtf1 I mean, the number line scale \par tells you that this is a little bit past 7. }{\rtf1 It's more powerful to say, \par okay, that's where e^2 lives. }{\rtf1 So in other words, don't label them \par with their decimal approximation, }{\rtf1 because that's \par kind of self evident }{\rtf1 because of where \par you plot the point. }{\rtf1 What's more powerful is to know is that, \par okay, e^2 is something a little bit past 7. }{\rtf1 So I would instead of \par putting the actual decimals,}{\rtf1 I would have you put, label it \par with the original expression. Okay.}{\rtf1 (Student) \par How do you know the powers? }{\rtf1 (Ms. Bowell) \par How do we know the powers?}{\rtf1 To do it on your calculator? \par (Student) Yeah.}{\rtf1 (Ms. Bowell) Okay. \par (Student) Like you told me to plug it into -1.}{\rtf1 (Ms. Bowell) Alright. So...\par (Student) So why would I plug it in?}{\rtf1 (Ms. Bowell) Well, because these \par are the numbers that I wanted to graph.}{\rtf1 (Student) Oh.}{\rtf1 (Ms. Bowell) These are the six numbers.}{\rtf1 I wanted to find out where these \par six numbers lived on the number line. }{\rtf1 And then when you \par do your worksheet E, }{\rtf1 you are going to \par need one of those. }{\rtf1 Worksheet E. Okay now, April said \par that it was too hard to read that exponent,}{\rtf1 which it really is, so #74, the \par exponent “e” is supposed to be 0.0137x. }{\rtf1 0.0137x. \par (Student) Where?}{\rtf1 (Ms. Bowell) Right here this \par exponent that's too hard to read. }{\rtf1 (Student) 0 point...\par (Ms. Bowell) 0.0137x. }{\rtf1 (Student) Isn't it...}{\rtf1 (Ms. Bowell) \par There's a decimal point before it. }{\rtf1 (Student) \par Did you say point 0, or it's just 0?}{\rtf1 (Ms. Bowell) \par I always put it in the ones place. }{\rtf1 (Student) \par Now we can graph this, right?}{\rtf1 Yeah, but it is... Yeah. }{\rtf1 Your graphing calculator will graph \par it for you. It's supposed to be this. }{\rtf1 They didn't put the 0 before it. }{\rtf1 I always put it there. \par I wish they would, }{\rtf1 because I think \par it's better to do that. }{\rtf1 (Student) \par But you can just add it.}{\rtf1 (Ms. Bowell) Yeah. \par Yeah. Okay, here's one. }{\rtf1 Here’s another problem \par I want you to do. Okay.}{\rtf1 This is your note card. \par Try this one now. }{\rtf1 Let me go downstairs and \par get a different calculator. }{\rtf1 So. Yeah, I know. Okay.}{\rtf1 So, what we are finding out \par with our example here is that}{\rtf1 this is telling us that \par it's growing exponentially. }{\rtf1 I mean see the graph \par curves up pretty sharply }{\rtf1 when we're looking \par at the graph as (x)…}{\rtf1 In year 5, it's a whole lot \par more than it was in year 2. }{\rtf1 Okay, so that's \par exponential growth. }{\rtf1 Alright. Here's what I want to \par make sure that we're understanding.}{\rtf1 What does the empty \par function look like? }{\rtf1 Well, it's this. \par 0.08e^0.42( ) open parenthesis. }{\rtf1 (Student) Oh, so what my problem \par is I don’t know how to punch this in. }{\rtf1 (Ms.Bowell) I know, \par but punching it in is the least...}{\rtf1 And it just seems like it not now, but that's \par really the least of your problems right now.}{\rtf1 We want to make sure that you \par can put some stuff on paper first. }{\rtf1 (Student) Do we want to change \par that to f(x), because it's a function?}{\rtf1 (Ms. Bowell) Okay.\par (Student) I mean, is that something...}{\rtf1 (Ms. Bowell) You can. Maybe they \par should of put D(x), debit card transactions, }{\rtf1 and that’s in function notation\par and not in equation notation. }{\rtf1 Just for whatever reason \par they decided to do it that way, }{\rtf1 but that way we could \par see it as an empty...}{\rtf1 So, if we want to find out \par how many debit cards transactions }{\rtf1 there were in 1995, \par what value of (x) are we using there? }{\rtf1 (Students) 5.}{\rtf1 (Ms. Bowell) 5. \par We have to evaluate that for 5. }{\rtf1 Okay. So that's what you \par need to have on your paper first, }{\rtf1 and do as much \par as you can on paper. }{\rtf1 You can go ahead \par and do 0.42 * 5,}{\rtf1 and that's going to be 2.1, }{\rtf1 Okay. 2.1. Okay.}{\rtf1 And then, what you have to do, \par if you got a grapher. }{\rtf1 Now are you going to have \par a different calculator, Nora? }{\rtf1 (Nora) Huh?}{\rtf1 (Ms. Bowell) Are you going \par to have a different calcuator?}{\rtf1 (Nora) I'm I going to have one later? \par Maybe. Can I just borrow this one?}{\rtf1 (Ms. Bowell) Okay, cause even today, \par I mean you won’t be able to do }{\rtf1 what we have to do \par with a regular checkbook calculator. }{\rtf1 You’ve got to have \par a scientific at least. }{\rtf1 Okay. So...}{\rtf1 (Student) Where did you \par come up with the 2.1?}{\rtf1 (Ms. Bowell) Well, this rule says that \par you have to take 0.42 and multiply it by (x). }{\rtf1 Well, your (x) value is 5,}{\rtf1 so you have to simplify \par in the exponent first. }{\rtf1 That's why, um, the \par calculator can be misleading. }{\rtf1 I want you to do \par more on paper first, }{\rtf1 and then we'll go punch it into the \par calculator when you absolutely have to, }{\rtf1 but we can multiply, \par and you get 2.1. }{\rtf1 The exponent on “e” is going to be \par the product of those two which is 2.1. }{\rtf1 Alright, and then when we \par put that in our calculator, }{\rtf1 because what you're going to have \par to do is you're going to have to take...}{\rtf1 (Student) Okay.\par (Ms. Bowell) It didn’t do it here. }{\rtf1 (Student) No, it didn't.}{\rtf1 (Ms. Bowell) Well. I don’t know.\par Tell me what you're doing.}{\rtf1 (Student) Because I don't... I'm so used to that.\par I don’t even know how to punch this stuff in.}{\rtf1 What I tried was this. 0.42, um. \par It gives you a parenthesis over here, right?}{\rtf1 A 5 equals.}{\rtf1 (Ms. Bowell) Okay. Well, you need \par to use the times key on this one,}{\rtf1 because you're just \par multiplying those two number. }{\rtf1 Alright. So, you get 0.653, \par and then I made Kimberly write more. }{\rtf1 That's not the answer, }{\rtf1 because if you put that \par as the answer, that's the number...}{\rtf1 (Student) \par We're still working on (A), right?}{\rtf1 (Ms. Bowell) \par It's still (A). Right.}{\rtf1 That is not the answer though, }{\rtf1 because what you have to understand \par from the way they described this (y) value. }{\rtf1 The (y) is in billions. }{\rtf1 So, if you leave it there, then you \par haven't interpreted it in the right way. }{\rtf1 You have to go ahead \par and interpret it as billions. }{\rtf1 So, in 1995, there were...\par (Student) Approximately.}{\rtf1 (Ms. Bowell) Yeah. Approximately \par would be a good word to put there,}{\rtf1 but that may be hard to \par understand what that is, }{\rtf1 so that’s the same \par thing as 653 million. }{\rtf1 There it is in \par an ordinary number. }{\rtf1 (Student) \par I see what you're saying.}{\rtf1 (Ms. Bowell) Yeah, because \par that's a number where there's 0’s, }{\rtf1 and you have a better intuition \par about what that number is. }{\rtf1 (Student) Why is it in billions?}{\rtf1 (Ms. Bowell) Because the\par function says (y) is in billions. }{\rtf1 If we don't know that, }{\rtf1 then we haven't interpreted \par that model correctly. }{\rtf1 Okay, so (B) \par was a problem? No. }{\rtf1 (student talking)}{\rtf1 Okay, we went over a whole lot, \par because that’s like a whole lot more.}