 My name is Maria Hernandez and I teach here at the North Carolina School of Science and Mathematics and again thank you for joining us. What we'd like to do is give you a kind of an outline for what our goals are for this session. We're going to look at a couple of different tasks involving some real-world applications. And hopefully these will give students a chance to explore different real-world scenarios and some specific data. To help them find ways to distinguish between linear and exponential growth, if you look through the common core state standards, the content standards, I will list those for you later at the end of the webinar but I've picked out specifically how these particular topics kind of fit in with the standards and there is a good bit in terms of just thinking about maybe not being super familiar with everything there is to know about exponential functions but the idea of being able to distinguish between linear and exponential growth is a big deal. Also, we'll share some resources at the end of the session including a handout which you might have already received, a PDF document with a handout. The PowerPoint that you received, you probably want to scratch that one because I've already updated this one, fixed some typos and also put some more information in it and that will be available for you on the Distance Education website here at NCSSM. Also, I have found some other sources on the web, some of which we'll be talking about during the session and I've made a list of those for you. So I hope this webinar will be useful for you guys. As we do, the math what we want to do is consider how our students will approach the problem and think about what prior knowledge they'll need. So as we work through the problems, I encourage you please to type in the chat window or raise your hand if you have a question or just type the question in there. Carol Stern is here with me and she is great about helping me monitor that chat window so that if you have questions, we can answer them. Also, I will ask you questions so hopefully you'll feel comfortable typing your answers and ideas in the chat window. Maybe you'll even be picturing some specific students as we talk about the problem and how they might get started. So please feel free to chat in the chat window. This is, I think, a nice opportunity for us to meet as colleagues and to share our ideas with each other and help each other think a little bit about some of the stumbling blocks in terms of the mathematics that kids might come across. The other thing is I've tried to pick out some problems that are focused in terms of what math one students or algebra one students would be familiar with. But I'm also going to at some time extend those topics to higher level courses. I like to do that because I find that teachers who teach different courses like to think about, you know, can I use the same problem in a pre-calculus course or an algebra two class? So, you know, I try to give you as much as you can so that when you go back to your classroom, you can use it in a variety of ways. But, you know, if I do something you don't understand or you're confused about in terms of what students would be expected to know, we can chat about that. Sometimes I don't know exactly because we haven't, as you all know, haven't seen the assessments yet. So if you have knowledge about something that you've come across in some of the assessments like the Smarter Balance Assessment or even the Park Assessment that is related to the topic, please feel free to let us know what you're thinking about that. Also, it's important for us, I think, to think about how to help kids move forward with their own ideas as they get stuck in particular places and, you know, not scratching their ideas and having to start all over, but instead kind of thinking about, well, if a kid gets stuck here, how can we push him forward without giving him way to answer. And then also to think about how, if we have different solution methods for a problem, how we might organize or sequence the students' sharing of their solutions based on kind of the sophistication level of the solutions. So, sorry, this should say example one. I had an example one that I decided to put at the very end. So example one is at the very end in case we have time, but the one that I really wanted to talk about was an example that I found in the illustrative mathematics materials. I'll give you a link for that later and show you how those are organized because I think you'll find them very useful. Maybe you guys already used some of these materials, but those were created, I think, by Bill McCallum's group in Arizona. Bill McCallum's one of the writers of the Common Core State Standards, and it has, these materials have a lot about distinguishing between linear and exponential growth and more kind of, I would think, skill-based type problems, but there are also some nice applications, so I grabbed this application. And what I've done is I'm going to extend it with some real-world data that a colleague of mine here, Christine Belladine, shared with me that she did a project with her students on. So it'll be a combination of the problem here that is in the illustrative mathematics materials, but then also I'll share some real-world data that Christine shared with me. And I can put together an Excel file if you want, but it's in the handout. The data sets in the handout, and also we'll be using a calculator emulator here to work through the problem. So if you have a calculator handy, it would be great if you could kind of play along with us. So the first problem is about food production versus population growth. And it states the population of a country is initially 2 million people and is increasing at 4% per year. The country's annual food supply is initially adequate to be 4 million people and is increasing at a constant rate, adequate for an additional 0.5 million people per year. So they say, based on these assumptions, in approximately what year will this country's first experience shortage of the food? So this is some fictitious country we don't know what it is, but they've kind of boiled down the information into these specifics. I'm going to go ahead and show you the next slide just so you can see the other types of questions we'll have to answer, and then we'll come back and try to work on this particular question. So the other questions that are in this task in the illustrative mathematics materials say if the country doubled its initial food supply and maintained a constant rate of increase in the supply, adequate for an additional 0.5 million people per year, would shortages still occur, and if so, in approximately which year? And then the last question says, if the country doubled the rate at which its food supply increases, in addition to doubling its initial food supply, would shortages still occur? So you can see there are kind of nice scenarios where the kids have to kind of think about pieces of information in terms of food supply versus population growth. So I'm going to go back to the previous slide and then let you guys think about that a little bit and talk to each other. What I would mean is type in the chat window ideas about how your students might approach this problem. And again, this could be at various levels if your students are very familiar with exponential functions. Hopefully they understand how there's some subtleties in the words here, but they're important in terms of the way we set up the model. So the population of the country is initially 2 million people and it's increasing at a rate of 4% per year. And then it talks about the food supply. It's initially adequate for 4 million people and it's increasing and there's another word in there at a constant rate, adequate for an additional 0.5 million people per year. So if we have given this to our students, if we think about some of the things they might come up with, feel free to type that into the chat window and then I'll go to a slide where we can actually write on the slide. I'm going to write down some specific information because I have to fast forward to the slide. And you should have gotten a handout with us. The handout for this problem will be shared via PDF. So again, the first part of the problem says increasing at 4% per year and the other part of the problem, the food, says at a constant rate. I'm going to forward to an empty slide and I'd like for you guys, I'm going to have to escape out of here so that I can write on the slide. I'd like for you guys to give me suggestions about what your students might do for this particular problem and I'm going to write some initial information down here. The initial population is 2 million. I'm going to use peanut for initial population. And we know that there's a growth of 4% per year. And then the food supply, we have an initial amount of food that we don't have like some units of food and so we have an initial amount measured in what would be adequate for 4 million people. So this is all measured in people. But then they say it grows at a constant rate of 0.5 million people per year. Suggestions about what your kids might do to or what you might just think about writing in terms of comparing these and answering the first question that is. It says based on these assumptions, in approximately what year will this country's first experience short as a food supply? Can I get somebody to help me out and touch some stuff in the chat window to help me think about what you're thinking about? Don't be shy. We have lots of people here asking for audience participation. I'll see how long I can wait. Carol, you can see the chat window right there. I can see how there's... Yeah, it says it would put the equation in and look at the graph. It would put the equation in and look at the table for the population. Okay. So let's think about writing the equation then. So if we're thinking about our calculator, you're saying Heather's saying we put these in there and then look at a table. So if I think of the population, since on what kind of notation you want to use, I'm going to say I'm going to let P of T be a function for the population over time where time is measured in years. So if we think about this population at a growth rate of 4% per year... And I have a question. Is that constant or exponential? That's a great question. In the question, it's not like hitting you over the head, whether it's constant or exponential. But because they wrote constant growth for the food production, I'm going to assume it's exponential. And you can talk about linear growth, and that would be kind of an interesting aspect. But I think what the writers of this question wanted to think about was comparing exponential growth to linear based on the solution. So you might edit the problem. You might put in there that the population is growing exponentially to make it clearer if you want. And Jennifer was mentioning that her students would see the word growth and immediately start thinking about growth and decay formulas they used earlier in the year. Okay. So in that case, the growth and decay formulas I'm assuming would be exponential. So the growth for the population for your students might not have to be so explicit and say is growing exponentially at this particular rate. And students often associate the word constant with slope. Yes. So if you think about constant with slope, then again, that would kind of put you towards using a linear model for food and an exponential model for a population. Okay. So let's do that. I'm going to say P of t is the initial population. I'm going to write it this way. And then you guys can change that if you like. We could say if we think about growth rate of 4% per year, one way to write it is to write 1 plus 0.04 raised to the t power. So if I actually put my constant in there, I can have this 2 million times 1 plus. Or let's just write it as 1.04 raised to the t power. And then the linear function that is for the food production. Our food production, F of t is we start out with this 4 million. Have enough food for 4 million people. Plus that growth rate, like someone said, is a slope times t. Okay. And P of t equals 2 times 1.01. 1.04 raised to the t, yes. Okay. So you're writing n millions. If you want to do that, that's fine. You can write 2 instead of the 2 million. And so one of the suggestions was to put these functions in the calculator and then look at a table of values. So let's do that. I have an emulator here. And if you guys would play along with me, that would be great. If you have a calculator, what I'm going to do then is type in the first equation. Let's find my milk. Should we do it in millions? We could. That might be a little bit easier in terms of setting a window. So let's just do 2 and know that it's in millions times 1.04 raised to the t power. In this case, x, because that's what we're using in the calculator. And then we'll hit Enter. And then the second equation is the 4 plus 0.5 times x. So right off hand, we're comparing this idea of linear functions and an exponential function. And some students might say, well, I'm starting with 2 million people, but I got enough food to feed 4 million people. So it's going to last me a long time. So you could actually ask kids to make some conjectures about how long do they think it would last before they even come up with a function. If we use the table feature on the calculator, I'll hit the second window. And I'll start my table at 0 because that's where time is going to start. You can use a delta table of 1 or 2 or 5. There's a comment, I guess. Would we expect first year students to be able to come up with these equations intuitively or would we have them apply a template? So in terms of, it kind of depends on where you are in the sequence for the course, the sequence of topics that is. For example, I've grabbed this from the illustrative mathematics material. But like I said, those folks have created a good number of kind of exercises and skills where kids have an opportunity to create sequences of numbers by continually adding things, arithmetic sequences, so maybe you could have some experience writing a closed form from that arithmetic sequence. Similarly, creating geometric sequences by continually multiplying to build an exponential function. So I think that it would make sense to do this problem after you'd spent some time helping students kind of understand how you build an arithmetic sequence and then how it creates a linear function. Similarly, how you create a geometric sequence and how you can represent it with an exponential function. In fact, as I was doing this, I think this is really nice because the first webinar we did was on recursion. And it was about building these types of functions with different examples. So if you have time and you want to go back and look at that archived version of that webinar, it talks a little bit about how you use recursive equations to build these types of functions. So it's a great question. In terms of using a template, I would think that, you know, I don't know if you'd want to create something like that where you have kids kind of go back to that template or if you would rather then think about, you know, continually summing versus continually multiplying to create functions. I'll also share some other resources. We've created some recursion materials for actually anywhere you want to teach recursion, math one through math two, maybe even algebra two and precalculus that I'll share with you guys at the end too on our website. So for our purposes, if we have these functions and the kids have this experience creating these functions, what we want to do is look at the table as a suggestion. I'm going to leave my Delta table as two. Maybe we want to come back and change it to one. So I'm going to hit second graph, which is table. And then you can see what's going on in terms of the linear growth. Because I skipped by two, you see the change in wise here. Linearly, these are looking pretty good at why two is my food supply and why one is my population. So if I scroll down in the table, I could maybe think that, sure, we're not going to run out of food anytime soon. Or the other thing I could do is make my Delta table bigger. I'm going to go back. I'm going to go back to table set and change my Delta table and hit enter and then hit second table. You let me know if you need me to go back to any kind of calculator stuff. And so then I can see if I'm going up by five, looks like I'm still doing pretty well. But now they're getting close to here. And there we go. After 86 years here, then our population is going to overcome our food production. How did you change and go up by five? If I go to second window, which is table set, what you do is you change this second parameter here. The first one is ask where to start, table start. And I had started at zero and jumped by two. If you go down here to Delta table and change that to five, then you can make bigger increments in your step. So it looks like at about 78, we can actually look at a graph too. So let's do that because I think some students who are more visual learners might prefer a graphical representation. So now we have two representations on the table. We have the numerical values and we also have the function values, I mean the function. So let's go to a graph. So now that we've seen some values here, you can see it in my window here. I can kind of get an idea of what might be an appropriate window too, which is kind of nice. So instead of using some Zoom keys, I like to ask my students to think about what makes sense in the context of the problem and then also after generating some values, how we could set a window that makes sense based on those values. So we've decided that if we go out here to, I think we've said 81 is where it passes, but remember I'm jumping by five. So when I go to window, X is our time. And I can say, well, let's start our time at zero and go up to say 90 with an X scale of something reasonable like five or 10. And then we also know and we have an idea of what the Y values are going to be because we've already looked at the table. So we might go from zero to say 100. Again, choosing a reasonable scale like 10. And now you can see in the middle of my screen, but you won't be able to see it all together. I hit graph. There's our exponential function. That's our population. And there's our food production. So now we can use the intersect key on our calculator. So let me go back, sorry, let me go a little bit slower in case you haven't used this before. So there's my graph. If you hit trace, you can see the equation up there on the top. I'm on the exponential function and you could just trace to the right. If I hit the up arrow, it'll shoot me up there to the linear function. And again, I can trace on there. Or if I go to second, the trace key, which is count, one of my choices is to find the intersection of these two graphs. So if your students are using technology, they can use the intersection button. And it asks me, is this your first curve? And I can say enter. That means yes, it is my first curve. There's my second curve. It's asking me, is that your second curve? And I can hit enter. And now it asks me for a guess. I can either just go ahead and hit enter here or move a little bit closer to the intersection point. And then hit enter. And there we go. We can see that the intersection is at 7843.16. So you might have students graph this, label the axes with words and units that mean something in the context of the problem, and then label that intersection point and maybe write a sentence that explains how this graph informs their solution. So it looks like that for 78 years, we'll have enough food for this particular population. And then when that intersection happens, it'll be when we hit 43 million people. So are there questions about that first part of the problem? Okay, so let's move to the second one. The second question says, if the country doubled its initial food supply, how long would it be until we have a shortage, if that shortage would occur? So if we double the initial food supply, then we can go to the y equals and say, well, my food supply was the second equation. So I can just go down there and change the four to an eight and then do a similar process. So, you know, I don't have to go through that, but it's a nice problem for students to think about building the two models separately. You could ask some of your own questions if you wanted to do that or change the scenarios just slightly. And again, depending on how much experience your kids have with building these functions from scratch, this could be kind of a culminating experience, or it could be something that you introduce. A lot of times what I like to do is introduce a real-world problem first, and even though students might not have enough mathematical knowledge to solve it, what we hope to do is to kind of wet their appetite and show them why the mathematics is important and why we might care about it. And then we go off and do a couple of days, maybe even of some of the mathematical skills, and then bring this back on as kind of a real-world problem that they might have them write about. So the other pieces I think are pretty clear. What I'd like to do is move on to back to the PowerPoint so that we can talk a little bit about how we can use some real data that, like I said, my colleague Christine Belladin had come across. She actually searched pretty hard for this data based on the Uganda population and food supply. So if we go back to slides, this is the real-world data that has to do with grain production for Uganda in thousands of tons for these particular years. Now, I have a lot more data than I put in here, but the data set was so huge I didn't want to give you the whole data set because I thought it would be overwhelming to try to, first of all, think it's unwieldy to think about typing it in our calculators, but I can make that Excel file available on the Web site if you want to go and choose different years and analyze the data. Also, this idea of food production is pretty complicated because Christine was explaining to me that her kids did a good bit of research to try to figure out what it was that most of the, you got into to actually eat in terms of maybe the folks that are in the lower income bracket. So some of the things they came up with were grains, legumes, things like that, and then they found some data that had to do with production of those particular types of food. So there's grain, what they call pulses, which I learned about pulses. They are different kinds of beans and peas, and so I have some data on that, but I just grabbed the grain production because I thought it would make sense. So I'm looking at grain production in Uganda from 1998 to 2007, and like it says, it's measured in thousands of tons. And what we would like to do is build a scatter plot. I think I actually provided that on the handout. Let's build a scatter plot of this data. Let me also show you a couple of places where you can find some nice information. First of all, this World Hunger map is a really cool map that I've provided a link here for you, and I've got it up on one of my browsers here. I got it in a PDF file. There it is. Okay, so this is a map that you have a link to there, and what it has is it has a, let me scroll down here, it has a legend down here that shows you what the colors mean. So this gives you the proportion of total population that's undernourished in the years 2010 and 2012. Also, Christine also mentioned that it's really hard to get data, accurate data across the world, but with that said, this is one of the resources that she used, and it talks to you here, it shows you here the different colors. So very low is less than 5% of the population undernourished, and very high is 35% and over. So if you look at the map, and you can scroll back this way, so these lighter green areas, yellowish green, are less than 5%, and then as the colors get kind of darker, if you will, the red, and then the really dark red here. So Uganda, the country that we're looking at, it's red, and the really dark red are very high in terms of nourishment. I thought that would be kind of a nice resource for you all to look at. And then the other one is where Christine got her data, which is the site I have pulled up here. I just wanted to show you what that looks like so you can use it if you want. What you do is you can come over here to this left menu bar, and you can choose the country. So I've chosen Uganda, and then you could choose just a single year if you wanted to, like 2009. But if you want to look at a list of years, I clicked on that 2009 at the top, and then I might go all the way down to 1998, and then just hold my Shift key down and click so it highlights that whole chunk of data, those years. And then when you do, you want to choose an element. So for example here, the food supply in tons is what I had chosen. It has other suggestions here. I haven't used this site very much. Christine just shared this with me yesterday to show me where her data came from. And then you can do nested by, let me see if I can remember what this is. Try an item. And then what you can do is over here on the left-hand side where it says item, you can choose wheat or one of these other things. Like I had said, pulses was another thing that you could look at. Those are the peas and the beans. You could look at it by potatoes. So if the kids did some research about what, like I had said, most of the population eats in particular the lower income brackets. That might be kind of interesting. Then on the y-axis, I want the item, but that's not clear there. So let me go back here to an element and let's see if this works. So I think what you can do now is say show data. If you're missing something, then it won't come up with any data. And down here in this box, can y'all see my cursor? Yes. Then something would be read like it would say, sorry, you didn't choose an item or you didn't choose a country or you didn't choose the years. And then if you go scroll down, it actually gives you this data here for Uganda in 1998 through 2000. I think you can copy and paste that like into an Excel file. So I just want to show you that because that's where Christine got her data and that might be useful if you wanted to do like a bigger project. If we go back to the PowerPoint, here's a scatter plot of this particular data set. And I'm going to suggest that we look for a linear model for this data. So I thought this was kind of nice because we'll give kids an opportunity to find a linear model depending on how you want to, you know, where your kids are. Maybe they want to, they have the data set, maybe they want to pick two points that would be good representatives in terms of kind of moving that line around. I was just visiting a classroom recently and somebody, the teacher brought in spaghetti strands. And so he gave the kids a handout with the scatter plots on there and gave everybody a spaghetti strand so that they kind of think of coming up with a model and they use what they knew about linear functions to come up with a model through two of the points. If your kids know about linear regression, then we can do a linear regression. And what's nice about this then is with either of those models, either the student built modeling sense or the linear regression, we can do things like interpret the slope and the intercept in contact. So it's a great problem like that. The other thing is that if you are going to have the kids interpret slope and intercept, you might want to have them think about the time, not as 1997, 1998, but years since the beginning of that time so that then the why intercept doesn't make sense. We had a question, would you need to give credit to AO staff for the data? That's a good question. I haven't read their terms of use. So I can certainly do that and if that's okay, I'll certainly add that to my handout and my PowerPoint. And then I guess if we grab, if you guys grab data too, I haven't read the terms of use. And Christine didn't say anything to me about it so I don't know if it's just an open source data. Provider. That's a great question. Thank you. Okay, so let's go ahead and type this into our calculators and come up with a linear regression model. That's why we could do it pretty quickly so that we can move on to the other piece, which I think is also very interesting. And that is to look at the population growth. Oh, sorry. I grabbed my calculator but I have an emulator. So let's go to the emulator. And again, I'm not sure how familiar you are with using the stat editor on the calculator. So I'm going to try to go slowly. If I go too fast, you can slow me down. But I'm just clearing out my y equals. And now if we hit stat and then hit enter, this will shoot us over to the stat list editor. Well, I call it that. Yeah, I guess it's the stat editor on the calculator. And like I said, we have data from 1998-2007, so we can either put 1998-1999 in there or we can use years since 1998. That way, the y intercepts will make sense. So I'm going to use years since 1998. So I'm just going to type in 0, 1, 2, etc. And go all the way to 7. So we need 9. It's 2007, sorry. So it would be 8, sorry. And then let's just type in the data for the amount of grain. And this is in thousands of tons. So I just arrowed over to the right and then just going to type that data in there. This is why I didn't want to have a huge data set because it can take a good while to type it in. So I can make the data available in an Excel sheet. You can choose what you want if you want to kind of play around with it and see if there's another sequence of years that are particularly interesting. Because if you look at the whole data set, I can show you that it's kind of interesting in particular food supply because it goes up quite a bit, up and down quite a bit. And another thing I think is interesting about this problem is that it is real world data. So the kids get an opportunity to see that real world data isn't all beautiful and clean and fabricated like a lot of their problems are. A little bit slow on the emulator. A lot faster than my calculator in my hand. 2508, 2274. So after we get the data typed in our calculators, what we want to do is turn on a stat plot. So if everybody's got that data in there, the way we turn on a stat plot is to hit the second y equals t. That means I'm just trying to access that stat plot command. And I've typed the data in L1 and L2. So this is where we get to turn plots on and off. Let's just hit enter on the first plot. And then we'll hit enter on the word on. And under type, we'll choose this first one that's highlighted here, that scatter. You can do other things like histograms and box plots. And then under the X list, we'll type L1, which is second one. That's how you get access to the L1. Mine's already there, but I'm going to show you how to get it because yours isn't. And hit enter. And then second L2. And hit enter. And then you choose the mark. We can do one of two things here. We can either look at the list of data and set a good window, or we could also use the zoom button and choose nine. It depends on where your kids are. If you want to think carefully about having them think carefully about the window, then certainly they can look at the list of data and choose a good window. Or you can also just hit the zoom. And then if you arrow down to nine, that's the zoom. Trig command. You could just also hit the digit nine. Zoom stat. What it does is it works off of whatever stat plot you have turned on. But since I have stat plot turned on with L1 and L2 in it, it has access to those values. And the calculator can choose a good window. So there you go. If you don't like the window, you can tweak it a little bit. Like I'm not knocked out about the fact that I have this vertical bar squashed up here. Not sure why I have that. Who's got the scale up? Oh, it's because I'm not sure why I have that all squashed up on the Y window. I can change the scale a little bit if you want to make a big difference. It does have negative values for the X min. A little strange that these aren't integer values. There we go. So now you can see the graph there. If we want to do a linear regression, like we did before on the graphing window, we can use calc buttons later if we want to find a model and actually trace along the model. But here, since we have a scatter plot, you can hit the stat button and arrow over to calc. And then we can choose linear regression, which is the third, fourth one down. And when you choose linear regression, it shoots you back to the home screen. You could just hit enter here. And it assumes that you're doing a linear regression on the data. You have an L1 and L2. I'm going to go ahead and type it in so you can see how it works if your data is somewhere else besides L1 and L2. Another thing that's really nice is once it does the linear regression, it'll certainly show you the equation on the home screen. But we can have it automatically dump it or copy it into one of our Y equals. So the way we do that is we're going to hit second one for L1. Hit the comma key, which is right here in the middle of your keyboard there. And then we're going to hit second two, which is L2. Choose comma again, which is right here in the middle of the keyboard. And then now we can try to find this special place in the calculator where it stores function name. You can do that one of two ways. You can hit the bars button and then arrow over to Y bars and then choose function and Y1. I have a set of explanations for how to put this in your calculator that I'll also put on the website. So if you're new to using the calculator and haven't done this before, you'll have that after the webinar. Not today, but I promise I'll get it up there soon. So a suggestion that instead of setting X's years after 1998, choose a different base year. 1995 for example, allowing for visibility of the data point on the Y axis. But for future classes with log models, so we don't run into domain problems. So yeah, I understand what you're saying in terms of you can see that first point better. And if you're thinking about log models, are you thinking about taking the... Oh yeah, because you want to future... You're talking about the log rhythm. That's for visibility of the data point on the Y axis. Right, but the log model is what I was thinking about. So I agree that visibility of the data on the Y axis would be better in terms of using a date like 1995. In terms of the log rhythm, I'm guessing you're thinking about re-expression, which we're going to talk a little bit about. Yeah, re-expression. Yeah, so we're going to talk a little bit about re-expression later in this particular case. We won't have an issue with taking logarithms of negative or zeroes, but you're right. It's a good idea that we won't necessarily have that zero there. So anyway, if I'm going to try to grab that function Y1, I hit enter on function and I'll hit enter on Y1. You have another question. In my calculator, I use bars, Y bars, enter, enter, enter to paste it into Y. Yes. I'd like to see how your calculator pastes the equation into Y as the newer calculators don't work the same. Okay, so let me go back over this. Let me clear this out and we'll start again so you can see how many times I did that. I hit second calc. No, sorry. I want stat calc, linear regression, and I hit second L1, comma. Oh, I did quadratic regression back. This emulator is sometimes slower than me, sometimes faster than me. Then L1, L1, comma, L2, comma. So you said bars, Y bars, enter, enter, and then enter. So that's the old way. You're right. Let me show you the new way and that equation will be pasted in my Y equals. You can see it showed up right here because it's already in my Y equals. If I do it the new way, let me show you this because I did learn this recently and I thought it was so exciting. If I do stat, same thing, calc, linear regression, the only difference is how to get that Y in there. And there's a hot key for it. So if I do second L1, comma, second L2, comma, then I believe if you choose, let me see if I can remember this, alpha, I think it's this trace button. Yes. If you hit second trace, then you get a list of all the Ys, the Y1, 2, 3, 4, 5, 6, 7, 8, 9, 0. So you can just hit enter. So if you have, I think it's a new operating system is what has that in it. Then you don't have to go through all the bars, Y bars, et cetera. So I think that answers your question. And you can see the linear function in there. And now depending on whether you set years from 1995 or 1998, you can ask questions, in particular about the Y intercept that 2083, what would it mean in the context of the problem? And then certainly the slope 61.2545, that will be something important because that was an important factor in terms of our problem from the materials that we first looked at, that we were kind of thinking about, yeah, the slope for this linear equation is steeper than the exponential function at first, but eventually the exponential function grows faster than the linear function. So this, I think, is a nice problem just in itself in terms of linear functions. But let's talk about the second part of the problem and the time we have left because it's about exponential growth. And I think there's a nice way to think about exponential growth with a real-life data set as opposed to building a model that could still be a real-world model, but we know some piece of information like we know the population grows by 4% each year. Would you push that count for bars, Y bars, enter, enter, enter in your calculator? But I can see what happens on your newer calculator. Push. Push. It's the same thing as the emulator. If I say what you're showing me. Push, stat, calc, stat, four. Yeah, four. How do you show what's going to do? Y bars. Oh, no, L1. Bars, Y bars, enter, enter, enter. Okay, so do you want to see that on the emulator? So you don't want me to put L1 and L2. So if I do stat, calc, enter your regression. And then I do bars, Y bars. So it still gives you the regression line because the default is L1 and L2 in terms of where your data lives. And if you hit Y equals, it should have written it up for there. Does that answer your question? Is it pasted into Y? It is. Pasted into Y1. So if I go back to my home screen, you can see that it worked. Oops, maybe not. Maybe if I hit second. Yeah, this is what I actually typed in there. But I didn't have the L1 and L2, so it worked. So let's go back to the PowerPoint and think about the second equation. So we can do all these things. We can make predictions about future food production. And then later on, like I said, we're going to compare the growth food production population growth. So this is the data set that I wanted to talk a little bit about. Again, this was data that was provided through that website. It didn't have the exact same population growth. No, I guess I can't remember where she said she got this one. I'm not sure where she got this data. But again, it's in the Excel spreadsheet. But I just chose a chunk of the data from 1995 to 2009. So it's not exactly aligned. You could change it and do the specific years. But what I'd like to do is put this in my calculator and have you guys play along with me. Because in the previous problem that came from the illustrative mass materials, they told you that the population was growing 4% per year. Now what we'd like to do is think about with a real live data set and how we're going to calculate that population growth. So let's do that. Let's go to the emulator. And let's go ahead and clear out what we've done so far. So we can start from. If you want to, you can put in L3 and L4. But I think it might be good to just clear these out. And we can just do a subset if you want. Or, well, maybe you can help me. I don't have to move. Is it pretty small? I can print out the handout. Or you can pull it out. Oh, there they are. Perfect. I've got it there. Never mind. I can do it right there. So I'm going to do, again, years since 1995. So I'm going to do that really quickly. I'll show you a trick if you want to learn it. If you go all the way up to L1, since these are just consecutive numbers, I'm going to go from 1995 all the way to 2009. What I can do is use the sequence command. So if you hit second, list, and arrow over to ops, we're going to create a sequence. You don't have to do this. You can type them in one at a time if you want. But this emulator is a little pokey for my taste. So I'm going to choose sequence. And the way a sequence works is you give it a function, a variable, a starting value, a stopping value. And I think that's all I need. So the function in our case is just going to be x. And the variable is just going to be x. But if you put in x squared comma x, and then gave it a list of numbers, it would put in all the squares of the x's, for example. So I'm just going to put in x comma x. Don't worry if you don't care about this. It's fine. Just ignore it. And just type in your numbers. I'm going to go from zero to 2,000. And then 9 should be 14. And hit Enter. And press Do. It types all those numbers in form. Then type them in. But there they are. And then I can just type these in here too. Let me see if my keyboard works. Oh, it doesn't. You're coming? No, I'm just following you. OK. And now I had more decimal places here. I just rounded to the nearest. Now this is in millions to the nearest 10. Are there more? No. What I'd like to do now is I'd like to go to y equals. Let me turn this thing off. I don't really need that linear function anymore. And what I'd like to do is turn this stat plot on with my L1 and L2, which I have it on because I used it already. But I need a better window because it has the window set from the previous linear function. So I can do a Zoom 9. And because my stat plot is on, then I have the window set for me for free in a sense. And again, I could use that comment about maybe not doing it from years since 1995. I can go back and do it in years since 1993 if you want to see that first point a little bit better. But if you look at this, and you do this with your students, I'm kind of running out of time so I want to go ahead and push forward. But one of the things you might want to do and what I have on the handout is for the kids to try a linear fit. So if they try a linear fit using linear regression, what they should see is that it doesn't fit just right. In particular, if you try to make predictions way out, the linear growth is going to be slower. This looks locally linear. Exponential functions, if I zoom in close enough, could look locally linear. And that's a big deal in terms of thinking about local linearity later on in math courses. But for this particular case, even if you put a linear model on it, if you kind of zoom out, you'll see that in the long run the linear model will not look great. So instead, I'd like to think about this idea of repeatedly multiplying numbers to get a geometric sequence that created an exponential function. But the deal is I don't really know what that growth rate is. I need to figure out what the growth rate is. So one of the things that I kind of tried to push kids towards in that handout, I didn't do a great job at that. That was just my first kind of try at that. So if you have comments or you use it and you refine it, please feel free to share it and I'll share it with everybody else. But what I thought about was if we took a new y-value and created a new y-value by taking the previous one and multiplying it by a common multiplier, then that's the way I create an exponential function. That would mean if I have something that I think is exponential in terms of a dataset, I could take the ratio of consecutive terms and see if that is close to some constant or maybe even use it as a modeling technique. So if we take these consecutive y-values, these population values, and take the ratio of those consecutive y-values and then maybe, for example, take the average of those ratios, it actually does a really good job of coming up with a function that fits the data. For more advanced courses, what you might want to do is do what was suggested earlier, and that is think about using inverses of functions to re-express the data. So if you think about a dataset being exponential and you think about what an inverse function for exponential functions would be, that would be a logarithmic function. So one of the things you can do is take the logarithm of the y-values and look at those graphed against the x-values. If the original dataset was exponential, that new dataset should be linear, and you could actually come up with a linear regression model for that. So there are two different ways. One is more kind of statistically based in terms of statisticians, scientists use re-expression a lot for various datasets, but you would need to understand logarithms to do that. And on math one level, if kids had a lot of experience building these exponential functions, then what you could ask them to do is take ratio consecutive terms and then come up with an average of those values and use that as your base for your function. It turns out it's a really nice job here. So I know we're kind of running close on time, so what I'd like to do is go back to the PowerPoint and share with you some of the other information I had here. And then I know I promised you a couple of things. One is a set of instructions for typing the data in and coming up with linear regression. I can add to that a set of instructions for using the calculator in a concise way to create these ratios of consecutive terms so that you can use that with your students and then see that the model is actually a pretty good one. And then the other thing, as I mentioned, calculating a residual plot. If we put a linear function on top of this, if you did residuals, had kids calculate residuals, you would see a definite pattern in the residuals. So I can have all of those steps kind of written out for you so that you can use those with this particular problem. As you can see, it's got a lot of cool stuff and in the short time we had, it can go through all of it. Are there questions and comments before I move on? Where can I find good material for residuals for Math 1? That's a good question. We do have some handouts that have had to calculate residual plots for particular datasets on our website. It was originally created for Algebra 2 and ASM, but now that those topics are in Math 1, you probably could adapt those. I haven't seen a lot out about residuals, but there might be some in that illustrative Math website. I haven't looked through that to look for residuals. Good question. Okay, so this was all working through the Math and thinking about different models. And this was the idea that linear growth is governed by constant differences and exponential growth is governed by constant ratios. So you can use that idea. And then you could also use the idea of semi-library expression to straighten the data. I think there was a comment about some good resources that Heather has. I found some great exercises in the contemporary Math in context for course 2. Oh, good. Okay. And Mary is after to send them. That would be great. If you want to send me a reference to that, Heather, I will also add that to the PowerPoint. Great. So there were some specific things about why these problems are useful in terms of helping students distinguish between linear growth and exponential growth and also to interpret some of the constants and contacts. I think it's a great connection to recursion. And like I said, our first webinar was on recursion. So if you want to think about creating new functions or new values from old values using recursion is a great way to think about it. We've kind of, I hate to just say the word covered, but we have, I guess, hit a lot of the content standards, both in terms of distinguishing between exponential and linear growth and using models, but also in terms of statistical topics. So those are some of the statistical topics that we've talked about. In terms of the mathematical practices, I think they also lend themselves, this problems lend themselves to be able to really engage the students in some of these important mathematical practices. Some of the resources that I've mentioned already are the Algebra 2 and Advanced Function websites, and we have brand new recursion materials up there. So if you want to check those out, if you look under linear data and exponential functions for the first two websites, you should be able to find some nice examples. Also, calculator steps are in there as well. Also, the first webinar, there's a link there to that webinar, so there's a typo there. And then here's the Illustrative Mathematics site. I just grabbed just a couple of things that, just to show you how it's set up, it's really nice in that they have kind of different types of activities for the kids to work on. Here's this equal differences over equal intervals and then equal factors over equal intervals to help kids kind of start building the notion of what's the difference between linear and exponential growth. There's a great book by a guy named Al Bartlett who taught for many years at the University of Colorado in Boulder, sorry. And he, I think, is still around, but he had given a great talk here before, came to talk to our students about how wonderful the world would be if everybody understood exponential growth. So he wrote a book called The Essential exponential that I just put in there because I think it's really cool to have this kind of different perspective. So just to put a plug in for our next webinar, it's going to be on middle grade topics. That was a request from some of our folks. That will be in April, April the 25th, 315, and Dr. Alan Maloney from the Friday Institute over at NC State will be a guest speaker. He's going to be there in Raleigh, but I'm going to work with him to try to hopefully put some useful materials together that will help us think about teaching some of the common core probability topics in middle grade. So I'm kind of going down to the middle grade and hopefully some of you will be interested in that just to kind of see some of the new stuff in the common core and how hopefully one day soon our high school students will come to us with some of those topics under their belts and so it might be interesting to think about how what we do might change based on those probability topics. So I think we're out of time. Carol, did you want to say something? Yeah, I wanted to say that we will be sending out an email to all the participants that has links to where these presentations are situated in our archives and we'll also be asking you to complete a survey and let us know if there are things that you liked or things we can improve so we can make this time very worthwhile for everyone. Thanks for participating. And there's one more slide. That was the problem that was going to be my example one but I decided to push it to the back and that was this marvelous math problem where you get to choose your prize and one is exponential worth and one is linear. So if you like that problem and you want to use it, you can do that too. So thank you all very much for participating. Thanks for those of you who also asked questions and gave me answers. It's really nice to have that going on and so it's not just me talking. So thank you all very much. Have a wonderful afternoon.