 Again, thank you for your patience with us. So I apologize to those of you who are out in the webinar land that you couldn't hear. We just talked a little bit about kind of the overview for the day, and I was talking about Common Core State Standards, and in particular, which Common Core State Standards, in terms of the content, this webinar will hit. So if you look at the – it's basically talking about functions, but in particular about recursive functions, and we're going to think about writing arithmetic and geometric sequences from real-world problems. So I'm going to keep moving. So as we work through the math today, what I'd like for us all to think about is to think about how your students will approach these problems, to anticipate their questions, to think about possible roadblocks and how you might push them forward in relation to the particular application. Also, if you want to think about how students and the more students centered learning, we give kids an opportunity to talk to each other and come up with their own ideas, and then you as a facilitator, oftentimes, you know, you walk around, you listen to what the kids are saying, you look at the different kinds of solution methods, and ultimately when you're going to summarize the results, you have to think about how they're going to share those results out. So in terms of how we kind of wrap things up after we let kids explore ideas, how can we present the solutions in a way that helps the kids make connections both across the particular topics or those representations that they might choose, but even in a broader spectrum, other topics in the common core or even other applications. So let's start with the problem. So you guys here with me have a handout that has this problem on it. This is a pretty simple problem. It might be kind of a warm-up problem in terms of how we might think about this as a recursive problem. So we have Selena, and she's going to buy a new smartphone, so she's going to borrow some money from her parents. My daughter just did that. I have a teenager who borrowed some money from us. She wanted to borrow $250, and she's going to try to pay them back $20 per week, and we just want to know how long it'll take to pay off the loan. So it's a real simple problem, but what I want to do is just have you guys think about how your kids might approach this problem depending on where they are and also think about the different representations they might use. So I'm going to give you a minute if you want to talk to each other, those of you who are alive here with me. Again, real simple problem, but think about the different ways kids can think about representing this problem. So I'm just going to be quiet for a minute. So I think here everybody, so folks are still chatting, but I just walked around just a little bit and listened to some of the things that you guys were saying. Some of you were talking about how the kids would just pick up their calculator right away. What would your kids do if they picked up their calculator right away? Subtract what? Twenty. Twenty, just one time? No. Lots of times. So they could use the answer key, for example, on their calculator, or they could just take minus 20 and keep hitting enter. So you could just do it numerically right away, right? Did anybody think about writing equations? Anybody write an equation for them? Yeah, so we might think about creating a particular, what kind of equation does you write? A linear equation? Okay, so there are different ways to think about this. What I'd like to do is think about it kind of an in-between stage, not just taking numbers and subtracting 20 from it. We could certainly do that. Now I have an emulator on my computer, and I'll show you how to use the answer key. You can do this just on the home screen with the answer key. No problem. But if you think about it more from a more general aspect, again, this is the way we talk about it as teachers and then you think about how to implement it in your classroom, we might start with an initial balance of 250, and you're going to write an equation that says, my balance this week depends on my balance last week, minus 20, which you might be thinking is pretty much what you're doing on your calculator, but I'm abstracting it a little bit, so I can think about it in terms of an equation. So if I'm going to do this, what we might do is just, if we're just subtracting 20, you might just build a list of numbers like this, and then just keep on going until you hit zero, right? Are you going to hit zero? You don't hit zero, which is another issue, right? So there's some interesting details in here. If you think about, actually, the notation-wise, how we might represent that recursive idea, that recursive idea by recursive idea, I mean, how much I have this week depends on what I had last week. So it's not a big picture idea. It's a very local idea. What I have now depends on what I have before, minus 20. So if you think about notation, we could use function notation. I'm going to show you that. We can use subscript notation, or we can even use this idea of now versus next. Does anybody in here see that now versus next idea? Okay. And then as we think about notation, again, when we're thinking about how to present this in our classes, you need to think about what's going to be obstacles for your students, what's going to make things more easier to understand, and then which of those notations might help us build a deeper understanding of not only this context, but again, the concepts, I'm sorry, but the concepts in a broader notation or idea like in terms of function. So what I'd like to do is think about it in terms of function. So if you write a function B where B stands for balance, and I start thinking about how much Selena owes when she bars that $250 for her parents, we can say B of zero is $250, and then B of one is B of zero minus $20, or like we said before, we could say this week's balance is last week's balance minus 20, but I'm given some kind of mathematical notation here. So this is one way I see recursive equations expressed. And that's the only way. I'll use subscript notation later. And again, depending on where your kids are, you might build this from the ground up using now and next. Okay. So if we keep writing these equations, I can write B of zero is 250. B of one is B of zero minus $20. But now what I'd like to do is step out of that just a minute and think about what would happen if I start kind of generalizing some recursive equations. So before I do this, let me just escape out of here. And let's go to our calculators. I have a calculator here. Hopefully some of you guys have calculators with you. And hopefully those of you who are watching far away also have calculators. So I'm just going to go to the calculator here. And I'm just going to say we could say it's 250 minus 20 if I do it on the home screen. And then I could say, for example, if I wanted to use the answer key, I could say, second answer minus 20. So this is just using the home screen. No fancy notation. And I can just keep hitting enter. This was the idea that was presented at first, maybe not with the answer key. Or I'm going to go ahead and show you how to use sequence mode on the calculator with this simple problem. So let's hit mode. And normally you're in function mode. So I'm going to arrow down here to where it normally says function. You have four different modes. You can be on the 83 or 84 calculator. You can be in parametric mode or polar. And what we're going to do is use sequence mode. So let's hit enter. Well, if you don't have a calculator, there are a couple here in the folks in the room if you want to borrow a calculator. So what I'm going to do is talk a little bit about this notation because it looks similar to what I have on the slide. But it's got some variables. It's got this n min and u of n and u of n min. So if I go back to the slide, I said b of 0 is 250. And b of 1 is b of 0 minus 20. So if I think about these recursive equations, n min is where we start things, the starting value for the n, which is going to be 0. So I'm just going to type in a 0 and hit enter. And then now I'm going to type in, sorry for some reason I'm not in the right place. Hit y equals. That's where I am. Sorry. So if you hit your y equals, type in 0 and hit enter. That's the n min. And then here what I'm going to do is say that b of previous value minus 20. Before I had b of 0, b of 1 minus 20, b of 1 is b 0 minus 20, b 2 is b 1 minus 20. So what I'm going to do is I'm going to use the special u on your calculator, which is second seven, the blue u. And I'm going to open a set of parentheses. And I'm going to say my value now is my previous value. So n minus 1. Now the way I got the n was I typed in. I hit the x, t, theta, n key. That key on your calculator is tied to whatever mode you're in. So those of you who are in the room, don't look at the right hand side. Look at the left hand side. I think it'll be easier if you see what I'm typing. Let me clear this out here so you can just see the screen. There you go. So I'm going to do u of n minus 1. That n, again, is tied to the x, t, theta, n key. And I close my parentheses. And that's like saying previous value minus 20. So I hit Enter. Is that OK with everybody in here? Are we good? Anybody need me to go back? I'm happy to go back. OK. Now u of n, then, is the starting value for the balance. What's the starting value for my balance? 250. So I'm going to type in 250 for that. And what I'm doing is that using that recursive idea that the value now is my previous value minus 20. Is that OK with everybody in here? We're good. So now if you hit Second, Window, which is table set, I can look at a list of numeric values. I'm going to start my table at 0 and have my delta table as 1. And the other two parameters should be auto. And now when I hit Second, Graph, which is table, I should see a list of my balances for my account. So you can scroll down. And those are the numbers that were being generated on the home screen. And I'm going to go back and talk about notation. I introduced the idea of function notation because your calculator makes you do function notation. OK. If you want to look, let's go back then to this idea, that in terms of trying to create a closed form expression, which some of you said you did, you went right to the linear expression. That might not be intuitive for the students to say, OK, I'm going to go straight to the linear expression. But this idea of generating one value based on a previous value, I think a lot of times is more intuitive. So if I think about writing my equation, I have B of 2 is B of 1 minus 20. But what I'm going to do is replace B of 1 with the entire expression B of 0 minus 20, and then subtract 20. So if I do the algebra to simplify this expression, I have B of 2 is B of 0 minus 2 times 20, and then I can do it for B of 3, and then, et cetera. So I can think about writing, turning my recursive function into a closed form function. So I get B of n would be, if I write B of n here, it's B of 0 minus n times 20, which is our linear expression, right? So we could also create a graph over time, and it could convince you that it's linear. And you could see I haven't graphed enough to see how long it'll take me to pay off Selena to pay off the loan. So this is my closed form expression, and it's probably what the expression that you guys jumped to right away. So if you think about it in terms of your linear expression, those of you who wrote linear expressions did you have something that looks like this, 250 minus 28. So we can make sense of the parameters and the problems. What does the 250 mean? Why is it negative 20x? And why is it written this way? A lot of times we write mx plus v. In this case, I wrote it as B minus mx, right? Because I thought about it the starting balance, and now I'm going to be taken off 20 each time. So my point is that we might jump straight to the linear expression, but this idea of recursion helps the kids build the linear expression. A simple example, not too complicated. Questions in here? Any questions? What are we doing out there? If you guys have questions, Carol is monitoring the screen for me. And then we haven't really answered the question, how long is it going to take Selena to pay off the loan? Now we have all these different ways to look at it. We can use a recursive equation, we can use a home screen on our calculators, you can use a closed form expression, or you can also look at a graph. So this example, I've actually graphed all those values there, it's kind of nice because we talked about how you're never going to get to zero. None of these expressions, well, that's not true. If we use the closed form expressions, you could set that equal to zero, right? And solve for x, but the problem is you're not going to get a reasonable answer. Because these are discrete values, and I'm using a continuous function to model a discrete scenario. So again, those are big ideas that kids can think about. When is the domain discrete values and the range discrete values, or when are we going to have some kind of continuous model? We're using a continuous model to represent a discrete situation. So again, I'm not going to go through all these questions because we don't have time, but I put them in there so that if folks are watching the webinar or if you guys are thinking about it later, you can think about these big ideas and how important they are in terms of the mathematics that we're doing at present and how it connects to different kinds of mathematical ideas. Okay, we got ready for another example. Selena, she decided to use her credit card. So she spent $250, but the credit card company charges her 2.1% monthly interest. Okay? And she's not going to make any payments for 12 months. So it's not this idea that she's paying off those $20 a week anymore. She doesn't have that job. She can go wait 12 months before she pays anything on her credit card bill. Okay? So again, we'd like to think about it from a recursive standpoint. So I'm changing notation. And again, I'm doing this for y'all so you can decide which notation makes more sense for your students as you do recursive problems. So instead of using function notation, I let A to then be the amount that Selena owes and month back to the beginning of the loan. A sub zero is a $250 initial balance. And then A1 is the amount she owes after one month. A2 is the amount she owes after two months, et cetera. Okay? How much does she owe in one month? Well, if you think about how much she owes in one month, she owes what she owed before. So that's that now versus next kind of idea. What she owed before plus the interest that they're going to charge her on that $250. So if I write it in subscript notation, I'll have her balance this month is that A naught or A sub zero plus .021 times A sub zero, which will be $255.25. So that's her balance after one month. Okay? So why don't you guys take a minute to write down the recursive equations. I'll let y'all have a minute to do that. Write the recursive equation for A2, for A3, for A4. The recursive notation, you can either use function notation or subscript, I don't care, and then we'll go to the calculator and do it. Okay? So we're not jumping to closed form. We're just thinking recursion. Now based on, or next based on now. I like to think about what I owe now is based on what I owe before. Those of you who are in the webinar land, if you want to either pose some answers or some questions, please feel free to do that. And you are a remote webinar participant. Yes? No, I can just repeat it. So there's a question here about Math 1 and what kind of notation. There was a question about whether or not in Math 1 we should just stick with the next and now versus introducing these other kinds of notation and I think you're asking my opinion on it. So I think in terms of where you start next and now is a good place to start because it doesn't kind of, you know, muck up the waters with issues that kids can have in notation. But I think, again, depending on where your students are and probably if you thought about function notation, if kids understand function notation, I would probably push you to think about at least starting the function notation for the linear expressions and the linear functions because then they really think of it as then you can start talking about domains, ranges and how and really is an integer value for these particular cases depending on what our time is really measured in. Many of these are about how things change over time. So I would say at least introducing a function notation after the now next would be a good idea. That's just my opinion in terms of kind of building the foundation for later. I think if we stick too much with one particular kind of notation, I mean for starters it's good, but then later on sometimes kids get hooked on that notation and they're not as open to other kinds of expressions. Now the subscript I found can be difficult for students. So I would wait and do subscript at a higher level. But you'll see with the power of these problems that this opens up a door for all kinds of further study mathematics. And then in terms of this idea of just doing something over and over and over again that's what we're doing with recursion and there are lots of cool applications. Okay, so I think folks have had a chance to write down some of those equations and hopefully they look something like this. So if I have A sub 2 for example, I have her balance from before, which was last month A1 plus 0.021 times A1 and I've actually got values here. There's some comment about how much the folks will charge her whether they're going to round up. And so we can figure out what Selena owes in any month where n is the month that I'm interested in based on the previous iteration, which is n minus 1. And so let's go ahead and go to sequence mode on our calculator and see if we can type in what you guys have written on your pages. So I'm going to go to sequence mode on my calculator. I'm going to clear out my last equation. You don't have to. You could actually just turn it off. Why don't we do that? I'm going to turn mine off so that if you wanted to keep that in there for later, you can. There's three places you can write recursive equations. So I'm going to go to the second set of recursive equations v of n and I'm going to say v of n is v. Remember it's the special v, so it's the blue one, which is second 8. Open a set of parentheses of n. n is on the x, t, theta, n key minus 1. So that's our previous balance. Plus, she's got to pay interest. And it's 2.1% to 021 times previous balance v of n minus 1. I haven't put the starting balance in there. The starting balance is 250. So I'm guessing that's what many of us wrote on our papers. Maybe use some other letters like v for balance. How many of you in here use subscript notation? Can I just see the rest of us use function notation? Okay. So if I want to look at the iterates over time, then what I can do is, again, let's go to second table set. I'm going to start at 0. And my delta table is 1. Again, that's the idea that this 1 is the increment for months. So when I ask my kids to write recursive equations, I tell them to put three things down on their paper. The actual recursive equation, how do you get one iterate from the previous iterate? Where do you start? Without where to start, you don't. This recursive equation makes no sense whatsoever. I actually got a question about this in Greensboro about how can I write a recursive equation if I don't know where to start? You can write how to get one iterate from the previous one, but if I don't know where to start, it doesn't make any sense. And then the third thing I always have them write down is what is your end measured in? Because if this is a 2.1% monthly interest, then I need to be measuring the end in months. Because without those three pieces of information, I can't make sense of what I'm doing with my recursive equation. And now I hit second graph, and there's my balance on the account. So I can scroll down if I want to. Again, I've just looked at numerical representation for this particular problem. And after 12 months, it looks like Selina's going to owe $320.81 on her credit card bill. Okay, she doesn't make any payments. Let's go ahead and use a, not a scatter plot, but a graph. And to do that, what we have to be very careful with is what we're going to look at in terms of the window. So I'd like for you to hit your window button on your calculator. And your life looks very different in sequence mode. Normally, we only set X windows and Y windows, but now you have a third kind of idea, and that's the value of N. So I start N min at zero, and if we're going to iterate 12 times and see 12 of those iterates, I'm going to scroll down and change this N max to 12. My plot start can be one. My plot step can be one. But now the X axis represents the number of months, since this is all started. I'm going to make that zero. Oops, sorry. I'm going to change that. I'm going to leave that back to one. I'm not quite used to this emulator. So I'm going to make my X min zero, and my X max 12, since along the horizontal axis, I have time. Scroll, keep going down. And now you can hit your Y window. So my Y minimum, I'm going to set to zero just because I like to be oriented in the world. You could start at 250 if you like, because we've already seen the values are going to go up from 250. And I'm going to make this maximum value something that makes sense in the context of the problem. Maybe let's try 400. And my Y scale, something that makes sense here in the context of the problem, zero to 40, it means just every 50, I'll have tick marks. And now if you hit graph, look, you can already see down there. There's my graph. So over time, I can see what her balance is going to be. I can hit trace. It actually works a little bit faster off the graphing window that it does in the table. Mine probably doesn't because the emulator's slow. But on your calculators, if your kids want to know how long will it be until the balance is some amount, a lot of times they can just put it in their graphing screen and just trace right across there. So I can see what my balance is at any given time. And what I'd like to do then is now think about how to write a closed form function for this outside of the idea of recursion. Again, building the ideas based on this idea of recursion. So let's go back to the PowerPoint. And now that we've looked at the balance over 12 months, let's develop the closed form function. Okay. So if you think about it in terms of recursive function, first of all, again, depending on what your notation is and the level of your students, you might be able to collapse the terms that a sub zero plus that point zero to one as one point zero to one. That can actually be confusing for students because they don't see that a sub zero as some parameter plus a number times a parameter. But I think if you take some time to develop the notation and help them see that there's a connection there, I can collapse those terms. So there's a one by collapsing the a zero terms. There's a two by collapsing the a one terms. But what I'd like to do now is replace, if I can use my pointer here, I'd like to replace this a one with this expression above it. So if I replace a one with this expression above it, I'll get one point zero to one times a one, which is one point zero to one times one point zero to one times a sub zero. And that low on the whole back guy right there, I can write it as a square term. And I can do it again. So again, let's do it slowly. One point zero to one times a two, it helps to use maybe different colors if you can do that. So I'm going to replace this a two with the entire expression that I have on the line above it. And I get one point zero to one cubed now times base of zero. Do that a couple times, and hopefully the kids start seeing the pattern. And what you'll do is hopefully be able to see that this turns into an exponential function. So this idea of repeated multiplication gives this exponential function. The idea of repeated addition gives this a linear function. So again, these recursive equations help us build a function and make sense of what we're doing instead of just, you know, memorizing our p, you know, our p equals our principal times, eventually exponential functions times rt or p times one plus r raised to the nth power. A lot of times kids will memorize that, and they don't really know where that comes from, but I think this helps them build some idea of where that comes from. So what we could do is we can go ahead and type this into our calculator in closed form. We're going to have to go back to regular old function mode. So let's go back to regular old function mode and type in our exponential equation. I'm going to go to mode, and I'm going to change my mode to function. I'm going to hit y equals. This is the life we're normally living in. Sorry about all these equations. I was doing some crazy stuff. Okay, so what I'd like to do is type in 250 times 1.021 raised to the x times. I can hit my window button, but if my window is still the same window that I used in recursive mode or the sequence mode should be fine, right? We look at this over time. If I hit graph, there's my graph. Is there anything surprising about the graph? What do you think your kids would say about the graph? Yeah, it looks linear, right? Here's this exponential function. We've been trying to distinguish between linear and exponential growth from this looks linear. Anything else they might say about it because of the packet looks linear? Yeah, it looks continuous. Again, this idea of using a continuous function to represent a discrete situation. Maybe it's not the best model. Maybe I can only look at this for some particular n values. So the domain of my function, in general, if I just hand you an exponential function, the domain of exponential functions are all real numbers. But in this context, when I've used this closed-form function to represent a discrete situation, in this case, I only want n values that are integer values because of the way I set up the function. So what can I do to help kids believe that this really isn't a linear function? What's that? Yeah, I could change the window. Other things? Yeah, exactly. Because, again, this is the idea that the repeated addition built a linear function. This is repeated multiplication. So if I look at the list of values, and I built the function so I know how it was built, right? So later on, if you're thinking about helping kids distinguish them, building this function will help, I think, later on as you give them different kinds of scenarios and different kinds of graphs, and say, this is my actually written function. Can you tell me, or even a list of values, can you tell me is this a linear growth or an exponential growth? That's a really important thing to be able to distinguish. And a lot of times we'll say, oh, when you learn about logarithms, you'll understand exponentials. That's a long time to wait, right? We want to really think about trying to help them understand it now while we're studying it, not wait six months down the road when we know more about logarithms or another year down the road when you study logarithms in a different course. So we could expand the window. Let me go ahead and do that and just look at it over a longer time span. And I might need to increase my Y values as well. If I wanted to take a little more time to do this with my kids, then what I might do is use the table feature on my calculator, excuse me, and look at more iterates over time. I'm not convinced by that one. Maybe I shouldn't have changed my Y window so much. So you can play with a window in terms of seeing that. I'm not crazy about, I mean, I know the calculators are what kids have in their hands, but it would be nice to be able to see all these representations at the same time, which they don't have the ability to do on the calculator. I think I have a graph that I produced in my PowerPoint that I use a different tool for. You can also, so there you might be more convinced that it's linear. The other thing you can do to convince students that the graph isn't linear is to slap a line on it. If I were to put a linear function on this, like, for example, do a linear regression on the data point, you'll see right away that there'll be a pattern in the residuals and you'll be able to see that this line isn't an appropriate model. And even if you haven't ever heard of residuals, if you graph it over a long period of time, you'll see the growth exponential versus linear growth. And in terms of other tools, I just wanted to mention something else too. Depending on what your kids have in their hands or what you have in your classroom, you might want to use Excel or some other spreadsheet because spreadsheets are really easy to do recursion in and then you can produce some nice graphs. Okay. So again, I just read some questions to consider as you plan to think about the different kinds of technology and the appropriate tools and to reflect on the mathematical model. We're thinking about arithmetic and geometric secrecy. Okay. So I have one more example for you and I think we're going to get on time. Here's a problem that's kind of interesting. It's about cadmium. So cadmium is a metal used to manufacture batteries also found in cigarette smoke and the way you can absorb cadmium through cigarette smoke is actually through inhaling it. So over time, what we've seen is a person who smokes one pack of cigarettes a day will absorb about 2.7 milligrams of cadmium each year from smoking and let's assume your body can eliminate about 7% of the cadmium each year. So we know how much you absorb each year if you're a smoker and you smoke as much as one pack per day and we know how much your body eliminates. Again, these are basic numbers that I got from... I actually got these numbers from a book by Jim Sandifer. I've got him as a reference for recursion. He's at Georgetown University and actually teaches a lot of cool high-level mathematics but has realized that some of those ideas are really important to introduce at a different level as early as Algebra 1 and he has a lot of nice problems that are real-world problems. So I got these numbers from one of his books and I want to know how much cadmium with a smoker who smokes one pack of cigarettes per day have inhibited her body after three years of smoking. So what I'd like for us to do is to write a recursive set of equations. Again, I'd say a set. I need to know what I start with, how I get the amount of cadmium for a particular year based on a previous year, and then what my end values are going to be measured in. So I'm going to give you a minute to write those equations and I've actually left a blank slide here so I can write them after you guys have a minute to think about it. Also, he suggests drawing a flow diagram if we can do it as well. So we're just writing recursive equations and then if you want to go ahead and try typing them in your calculator, you can do that too and see how much cadmium we'll have in our body after a year. So we're not trying to write first formula. This recursive formula. Okay. Okay. So it's interesting here in the room, lots of conversation going on. I'd like to go ahead and keep moving because I don't want to totally run out of time here, but I'm hearing when to put the 2.7 in and when to take the 7% out, those kinds of questions. So if I write it in terms of function, the amount of cadmium at any given time is what they had before, right? And I like to draw this flow diagram. This is one of the things that Jim Sandefur's advocate, too, is to think about what's flowing in and what's flowing out. So this is what I had before. What flows in? I'm going to add what flows in. What flows in? 2.7 milligrams of cadmium. And then I lose some amount, right? And I lose 0.7 or 7%, which is 0.07 of what I had before. So this is the amount out and this is the amount in. And I heard some people collapsing these terms. Talk about that just so that we just hushed kind of the chat in the room, which I heard people say 0.93 times A of N minus 1, which is fine. You can write that as 0.93. So if I write it as still recursive equation, I'm not talking in closed form. I say I have one of these and I subtract 0.07 of those. So I should have 0.93 times A of N minus 1 plus the 2.7. Is that okay? Questions? Any suggestions or questions about what I've done here? I wrote it as a function. I went back to function notation. You can use it as a subscript, either one. Okay, so there's a question about where to put the 0.07, I think. You're reducing what I just had before versus what I have, like you're saying I should collapse these two and then multiply by 0.07. Right, so for those of you who are far away, let me try to repeat the question. I'm going to insert a slide here just so we can go ahead and write that other representation down. So the question is where to put the 0.07. So if I say what I have now, and there's also a question about notation, if you like this notation better, this is what I had on the previous slide. I have the 0.93 times A of N minus 1 plus the 2.7. Does that answer your question? Okay. Or if you don't collapse the terms, you have what you had before minus what you lose plus what you gain. Okay, and I like to keep those of those on the slide there. Sorry about having that weird thing in the middle there. I'm just clicking and going. Okay, so the other question was when to take out the value. So for example, if I go back to your representation, I think you're saying it's A N minus 1 plus the 2.7 and then 0.93. And then where do you want another 2.7 in here? No. That's it. Okay, so this right here is really kind of where you start counting, right? I think that's what your question is if you say, let's think about starting time right now. What I have right now is what I had before minus what my body eliminated last year. That's the way I'm thinking about it. It's what I have before minus what my body eliminated last year. Right? Does that make sense? No? Okay. Plus the 2.7. I think what you're saying is it's what I had before and then at the end of the year I'm going to put the 2.7 in there and take 0.93. Okay. So it actually can be interpreted either way. It's perfectly fine to interpret it either way. We can look at both ways and see if it changes the solution of problems, okay? So you can look at it both ways. I can't see anything wrong with either one of those ways. Does anybody in here see anything wrong with either one of those ways? Yeah. There's a comment in here that says a smoker that would prefer the second way because you're losing more than the other way. Okay. So I'm going to go with my way because I'm the one with the chalk in my hand as my colleague. So I'm going to use my way but I just invite you all or whoever's interested in the other way, go ahead and put your calculator and let's see if it makes a big difference on, again, interpreting the problem and how you look at the problem. So let's go back to the calculator. Are there any questions about either one of these representations? Again, I started out with function notation but then I moved over to subscript notation. So let's go to the calculator and type this in our calculator. I'm sorry, I don't know what I'm saying. We're going to go to mode, back. I have to switch back to recursive mode which is sequence mode and hit enter. Hit Y equals. I'm going to go ahead and clear out this one so I can use the first one and then clear out this one. And I need to tell you too that there's something funny about sequence mode sometimes and you won't get the right graph if your calculator is in a funny mode. So I'll help you with that just a second. I haven't come across it. Nobody's hollered about it yet. So what I want to do is go ahead and collapse my terms and type in 0.93, just in the interest of time, u of n minus 1 plus 2.7. And then now this is a question for you. How do you want to start? What starting value do you want to have? Yeah, we could say you could start at 0. I mean, there's probably a chance that there is some cadmium already in your system, right? But we don't know what that is unless they can measure it. Which they can, by the way. So I could just start at 0. Some people might say something different here. What else could you say here if we don't know how much cadmium you already have in your system? What's another alternative because of 0? Yeah, start after the first year in which case you could say the 2.7 is from that accumulated previous year. So again, either way is fine. We just need to think about kind of where we're going to start and how we're going to start counting. So if I hit second table set, that should be at 0. And then we're going to look at the table of values. And if it's just a short period of time, then again, depending on what you start with, are you going to say I'm going to start after the first year. Your answer could be the 7.54 or this 9.7, right? So there's some room for interpretation in the problem. I think that's okay. There's room for interpretation in most problems in our lives. There are very few problems that are well posed that have everything expressed for us down to the itsiest, fittiest detail that we want. And I'm adamant about this because we have to be flexible in the answers that we receive and we take unless you want to prescribe every little bitty piece of information, in which case you can say, okay, now you got to pay attention to that little bitty piece of information. I want you to do this problem based on this information. So I'm about right out of time. Of course I have other fun things for you to look at. There was one question about what tool I used to create that particular graph. That was called Fluid Math. Fluid Math is a program that recognizes your mathematics handwriting. Notice I have a stylus because I teach with a tablet. And so you can actually write mathematics on it and it recognizes your mathematics handwriting. And you can get a trial version if you have anything that takes stylus input. I mean, you can create this graph with anything. So let me go back to regular life and just show you a couple of these resources and hopefully I've just kind of wedded your appetite in terms of recursion because there are lots of different applications. There's another problem here. We go back to Selena and she's got this credit card bill but she is going to make some payments. If you'll notice the last example, this is an idea of mixed recursion where we have both the geometric idea where we're multiplying by something and adding. The closed form for this is actually pretty complicated. You need to be able to write some of those terms out and to see the pattern you'll actually see a geometric series in there. But again, it opens the door to some really powerful problems whether it's paying off a loan or this idea of, for example, if you sprain your ankle and you keep taking ibuprofen, how much ibuprofen would you have in your system over a particular time? And it will reach some equilibrium level. So we have lots of materials for you. Let me share some of those with you. Sorry about kind of rushing through this. NCSSM has developed lots of recursive equation materials and even some videos on how to use them. These are brand new materials that were created over the last year. And NCTM has an illumination site that has some really nice applications. I believe one of these generates the closed form for that mixed recursion, adding and multiplying. And it shows you that you use a geometric series for that. So if you teach a higher level mathematics, maybe in pre-calculus, and you want to use a geometric series in a real-world problem, that's a great place to go. Neal Abel is a teacher at Deerfield Academy, and he has written materials that are free for his function statistics and trig courses. He uses Excel, he uses Fathom. So some of those things, you know, he's got actually little videos on how to do things. All up on his Moodle site, and you can just log in as a guest. And then Jim Sanderfer is a great resource. He's at Georgetown University. He has a lot of great materials through his hands on math. Sometimes you have to ask him permission to use his materials, but he's always happy to share his stuff with high school teachers. So our next webinar is on ratio and proportion leading to slope, and that's November 6th. I'd like to thank all of you guys that are here live and all of you guys that are out in webinar remote land for this opportunity. We're going to have probably six of these, and they'll be on various topics. This next one's on ratio and proportion, because I wanted to do something for the middle school teachers thinking about how to develop the idea of slope. The one after that I think will be on linear data analysis, the nonlinear data analysis, and then building functions for various real-world scenarios. So those are kind of the ideas for the school year. We'll probably have one more in the spring, again for some middle school teachers. But thank you guys all very much for coming. I appreciate you being here live, and I appreciate all the folks out in our remote folks. If you have any questions, send us an email. Carol is going to send you all an email asking you for, well, giving you one thing, and that'll be the site that has all our materials available for you when we get that up and running. And also Carol will send you a link to a survey to ask you how this went, what could we do better. This is all experiment. We're flying by the seat of our pants, and we're hoping that it's useful. So if you can have any ideas about how to make it more useful, please send us that information. Thank you. Thank you.