 I'm really glad that you all are here. Everything that we do today will be made available on the NCSSM distance education website. So we'll have the PowerPoint, we'll have the student handout and the teacher handout, and we'll also have the archived version of the webinar. So the only thing that you might have to bear with me today is that I don't have the emulator, the calculator emulator working on my computer as some of you heard. So we might have to pick up the camera that's on me and actually put it over the calculator because I didn't know that until I got here this morning. So we don't have our webcam or anything. I mean, I'm not our webcam but our document camera. But I'll kind of work on Excel and then you guys can either play along with me on Excel or you can hopefully use your calculators. And if you have questions, please feel free to just type in the chat window or you can raise your hand. You can send the question just to Carol or me or you can send it to everybody. It's fine with me if you send it to everybody. That way everybody can see the question. And if there's anything issue like you can't hear us at any given time, just let us know. Yeah, please, please do let us know about the audio because we can't tell from here if you're having any problems. So let's do chat. Okay, so let's get started. Today's topic is nonlinear data analysis. The last webinar was about linear data analysis. And I think people have a lot of experience with linear data analysis using things like linear regression. We talked some about the criterion for the least squares line or the linear regressions line. We also talked a little bit about residuals. So I'll be assuming some of that knowledge but we'll work through calculating residuals today and also if you have particular suggestions in terms of how you're thinking about using these types of problems or topics with your students or if you do professional development for teachers and you have ideas that you want to share, please feel free to just chime on in there. We're always happy to hear your suggestions. So what I'd like to do today is the goals for the session are just to work through mathematics in terms of a real-world data analysis problem. We'll probably definitely get through one problem. We might be able to have a chance to talk about an extended problem and to think about as we try to find nonlinear models for a given data set how do we assess our models and then use those models to make predictions. We'd also like to discuss how we can implement the common core state standards mathematical practices through these real-world tasks and to share some web resources with you related to nonlinear data analysis. There's some really nice resources out there that you may or may have not seen and I'd like to talk about some of those. So as we work through the task, again I want you to kind of put on your student hat. Think about that you can even picture them. The students in your classroom and how they might approach the problem and of course we have lots of different kinds of learners in our classroom so we might have different approaches and then also to anticipate the responses and their questions and to consider how we can use the students ideas to contribute to building the idea of the mathematical concepts. So here's our first example. We have this data that has to do with predicting how fast the car was going when they measured the skid links on the road. So it says, when a police investigator gets seen of an automobile accident they look for skid marks and use the length of those marks to estimate the speed at which the car was traveling. That's the type of thing, sorry. The results of experiments with a test car are shown below and then what we want to do is think about the data. Sorry about the typos there. We have skid links here in the left hand column. Can you all see my else yet? And then over here you have the estimated speed. So on the next slide the questions that we want to consider are which of these variables are the independent variable and which or is the independent variable and which one is the dependent variable and have students think about this idea of dependence, independence and explain their answers. And then the next question is to ask them to use a linear function of some sort, either their knowledge of linear functions or linear regression if they know about that to create a linear model for the data. So do you guys want to answer my first question? That is, which of these variables should be the independent variable and which should be the dependent variable? Please feel free to answer. The answer helps me a lot since I can't see your faces if I get some response from my participants. You can just type it in the chat window. Which one do you think is the independent variable? So independent variable in the speed is the speed. Can you see this? Yeah, it just took a minute. So thank you, Lisa. So the car speed is the independent variable. Because the skid marker, are you thinking that the skid mark depends on how fast the car was going? Do you think about Lisa? Yes, okay. So in that case, this is kind of interesting because it is true that the length of the skid mark is dependent on how fast the car was traveling. But if you think about this particular data set and how the police might use it, they're actually measuring the skid length and then based on these previous data trying to think about, well, we're just investigating this accident and we're trying to estimate how fast the driver was driving, then what we have to go on is the information from the skid length. So I think this is kind of interesting because of that. You're right that the skid length depends on how fast the driver was going. But if you're the police person, yeah, the perspective matters exactly. If you're the police person and you're measuring the skid length, then what you're trying to do is assess how fast the driver was going. So I put that in there kind of at the last minute. Originally, I was just going to say, use the skid mark as the independent variable, but I think it's interesting to think of it in terms of that perspective. So for our case, if we really were the police person and we're trying to think about how fast the driver was going, we might use the skid mark as the independent variable and then try to come up with a linear function that will fit this data. So again, I've just pushed the linear function. I didn't say look at the scatter plot and determine it. I'm pushing the linear function to begin with. So because I don't have the emulator, I'm going to escape and go out of the PowerPoint and go to Excel, but you guys can certainly play along on your calculators and you have the instructions. If you have the handout that Carol sent you, you could pull that up. It's a Word document or a PDF file, sorry. And then if you get stuck on the calculator, you can just let me know and I'll try to go back and let you know where I can help you. But for right now, what I did was I just opened up an Excel file. I have a skid mark here, speed here, and I've entered in some of the data. So 5, 17, 37, 65, 105. I'll wait for you guys to put that either in your calculator or an Excel spreadsheet. 205, 265. The skid mark is measured in feet and the car speed is measured in miles per hour. And these just go up by 10. So I'm going to add a graph. And the way I'm going to do that is I'm going to highlight, I'm clicking on the top of this column and then I'm holding down my mouse and then scrolling out to the right and then I'll let go. And I'll use insert and then I'll sign scatter. So there's a scatter plot of the data. And I think if we looked at that data before I read the next question, we might say it doesn't look linear. I want to go ahead and think about a linear model because I want to talk a little bit about residuals. So with this given data set, if you ask the student to come up with a line that represents the data, they might choose a couple of representative points here and try to find the equation, and using what they know about linear functions. They could also think about trying to kind of come up with a line that would be in the middle of this data and use that as a reference point and then, for example, play with a slope. So one of the things that a participant suggested last time when we did the linear webinar, linear data webinar, was that some of his students might use the approximate intercept here, the y-intercept to start with, guess at the intercept and then take two points and come up with a slope and then just tweak those two values. So I don't want to spend too much time on kind of different ways to come up with linear models. There are lots of different ways to come up with linear models. And so I just kind of want to keep moving so that we can talk about the non-linearity of this data set. On your calculator course, you can hit the stat button and then arrow over to calc and then choose linear regression. And if you've typed this data in L1 and L2, then the default, when you push that linear regression button, is to use the data in L1 and L2. But if you've typed it somewhere else, you can actually type in linear regression and then type wherever you stored the list, like L3, comma, L5, for example. So if you want to, you can go ahead and do a linear regression on your calculator. You can do it here on Excel by... I think I can double-click on this. Nope. Double-click on the graph. Let me click off on that. I think I just right-click. I can't find it. Add a trend line there. So I right-clicked on the data set and I want to add a trend line. And then a dialog box comes up. I want a linear model. The default is linear. I can say I would like to display the equation on the chart down here at the bottom. I'll click that. I believe that's all I need to do. And say close. And there you go. So I have this linear model here on top of the data set. So if I go back to the PowerPoint, then what I want to do is we created the scatter plot. We're going to ask the question, how good is our fit? And would you feel confident using your model to make predictions? So you probably have a little bit nicer view on your calculator because you probably have a whole line in there. It's just limited to that particular data set. But if you think about it with your students, what might your students say at this point? How good is their fit? And would they feel confident using their model to make predictions? So this idea of making predictions ties back to the idea of which was the independent variable, right? So we decided that the skid link was the independent variable because that's what we're going to measure as a police officer. And so if you think about it, if you're going to try to make predictions, you might say to the kids, be more specific like, well, what if I found, you know, a skid mark of something that's outside of the data set, like 280 feet long? How could, would you feel confident in using your linear model to make a prediction about how fast the cart was going? So do you guys have suggestions about things that your students might say here in terms of assessing the goodness of fit? You can just type in the chat window if you like. What might your students use to think about discussing this idea of feeling confident? Oh, so Lisa asked the question about R. So when she said, would you go look at the R value, which is the R value measures how close the data is to, how spread the data is, right? In terms of how far it's spread out if the data is linear. So when you think about the R value, what R value did you get? What R value did you get, Lisa? Did you have your diagnostics turned on when you did your linear regression? That R value is called correlation coefficient, right? Oh, okay. So she's not playing along. She suspects that it's high and the students would think that that is good. Okay, so again, I didn't do it with my calculator. Did anybody else do it playing along with my calculator? So Sonya had a 0.97, and Johnson County had 0.975. Okay, so just like you said, Lisa, that R value is high because when you're looking at the R value, it measures that closeness of the data to the line. So thank you Sonya for giving us that because what you're thinking about is if the data is a linear dataset, if the data are linear, it's measuring that spread about that line. The problem with R here is that the data's not linear. So I'm so glad you brought that up because I don't think of using R very much because I have other ways to think about whether or not my model is a good fit because we do a lot of data analysis here. So this would be a good example for our students to not rely too much on that R value because if the data's not linear, R doesn't give you information about the data. It gives you that idea of spread. So if I had more data, you might see that the R value would go down because the spread would be wider about that line. So if we go back to our Excel file, what I'd like to do is think about, well let's just advance the PowerPoint here because what I'd like to do is think about calculating residuals, which is a topic we talked about last time, but we can step through it again because the residuals will tell a story about whether or not a model is an appropriate function, the type of function, is a reasonable model to use for a particular data set. And we talked last time about residuals being the error, a measure of error. The official definition is the actual data value, the actual Y value of the data minus the predicted value. So when we think about predictions, because that's what we were thinking about doing, predicted values will come from using whatever model you have and evaluating that model at the independent variable, so in this case the skid length. So to calculate residuals, if we think about the geometry of it, let me go into my one-note file here. Let's just draw a little picture of some data set. In this case, our data looks something like this. And if I think about laying a line on top of this, it might look something like this. It would be nice to have a couple more data points. Yeah, that's true because of this idea of how far out do you want to think about the line and if you had more data points, there would be a big difference in between. Now the line would be different too, right? Maybe if I had a couple more data points down here, the line would come down a little bit. But in any case, we can still see whether or not we want to make predictions using that line because we'll keep drawing the line further out. So if I change the line slightly to accommodate kind of those new points, if I use a linear regression line, maybe I'll be pulled down towards these points. So I keep doing this, right? I'm going to be missing, in a sense, some of these middle points here. Let me just kind of draw this a little bit darker. And this is the idea of the residual is going to help me get some, again, a story or a particular handle on some of these. So Sonya's caught that. She hopes you're not going over it. Right, good point. Me too. Me too, good point. Okay, so let's think about a residual. So residual is going to measure error. So if I think about the actual y value for this particular data point, let's call that y sub a. That's my actual y value and I can't tell which is which in terms of my line. Let me see if I can erase one of these lines. Yeah, there you go. Okay, so suppose I went with this first line where I didn't really have those other data points. I'll get rid of those too. Then if I have my actual y value of my data point, that's right there on the y axis. And if I think of my predicted value, let me change my colors here. If I think about the predicted value, well, if I take this x value and I substitute it into my model, then this will be my predicted y value. I call it y sub p. So the actual is y sub a. I subtract from it y sub p. And that will be my residual for that point. So that you will have as many residuals as you have data points. So in this particular case, this is one residual. It's a residual associated with that data point. So if you're working on your calculator and you've done a linear regression, suppose you've typed your linear regression into y1. I'm going to go back to it, so I'll just real quick. So I have about 0.255x plus 17.95 0.255x plus 17.95 If you type that in there or just use the save command and you actually have more precision in your calculator, if I think about the actual y values, if I type my data in L1 and L2, these are the skid lengths and these are the speeds. So if I think about calculating one of these, I would take a particular speed. Let's say I want the residual associated with that first data point. So the speed was 10 miles per hour. And I'm going to take my model evaluated at 10 miles per hour. So this is actual minus predicted. A little bit messy because I'm right on my screen picking straight up in the air. Okay. So if we go to our calculators, we can calculate all of those at one time, but if this is new to students, I'd like to do one on the home screen. So on the home screen, we could actually type the number 10 minus y1 of 10. Or maybe you'd want to do y1 of 10 and see what the model predicts and then subtract that value from 10. So you can do one of those on your home screen and then you can actually calculate them all at once in your calculator and we can store them in L3. So I would do that with my students and do one and then do the whole list, but since you guys are advanced learners, we'll do it all together. So Sonya has a question? Oh yes, exactly. Thank you. Sonya says we should be using 5 for the skid mark. Exactly. Sorry. I had my cord on top of this. Yep. So the skid mark, skid length is 5 feet. You're going 10 miles per hour. Good catch. Thanks Sonya. Okay. So if I'm going to calculate them all at once in L3 on my calculator then I can do all of the y-values which are in L2 and if I stored my model in y1 then I can save minus y1 of L1. Now if you're using the calculator and you're not used to grabbing this y1 this lives in a special place on your calculator. You'll go to the vars key and this is described in the calculator step. I go to vars. I have to do it on my calculator. Remember how to do it. The vars key is right next to the left of the clear key and then arrow over to y-vars and then once you arrow over to y-vars you can choose function and then you can choose whatever function you stored your model in. So I'm going to do it on Excel and y'all can see if you get the same things I do. I'm going to go up here and I'm going to type these are my residuals. Remember your residuals depend on the particular model. So in this case these are residuals for the linear model. Let's put all this in the linear or line. I'll stretch that a little bit. So if you think about it from an Excel standpoint you're taking all of the y-values so let's just take the speed and the way I can do that if I can type an equal sign click on the speed here and subtract from that. Notice it replaces it with a cell number B2 and I can say minus but I think I'll do is just type in this linear model so I'm going to say minus the quantity 0.2549 times the x-value which is the 5 that Sonya just grabbed for me. Pay attention that needs to be a 5 but I'm going to click on that cell over there and then I'm going to type in the y-intercept. 17.953 so I'm just using precision that showed up on my screen when I did the linear regression and I had to enter. And again you guys were doing this on your calculator and then on the on Excel I can either when I see this full color crosshair I can just double click right there in the corner and it'll fill down or I can drag it down and say fill down too. These are my residuals associated with each one of these data points and for example the first one is negative 9.2275 and it should make sense that it's negative and you can ask students this what does the negative mean in terms of the residual because your y-value your actual y-value is smaller than the y-value predicted by the model so if we're going to do actual minus predicted it would be negative and in particular it means that in terms of the geometry that your point lies below your model. So one of the things I like to do is to actually graph this residual this residual plot I can look at a graph of all of these I can look down these values but I can also look at a graph of these and I'm going to do that on your calculator if you store those in L3 you just need to turn a new stat plot on and choose L1 your independent variable and L3 is your dependent variable and X in the Y list and then do a zoom 9 which will choose an appropriate window for you if you have your model in Y1 for example you might want to go turn that off so that it doesn't try to graph that along on your screen so I'm just going to highlight the first column and then I let go of my mouse and my click and I'm just going to come over here to the top of C control click and that way it just highlights those two columns and then I can do an insert scatter and there's my scatter plot excuse me of my residual so if you look at that we would say that there's a definite pattern in the residuals and what I mean by that is the idea of pattern can sometimes be subjective and very confusing to choose but in this case it's clear I think because if you think about looking at the residual plot you can almost fit like a really nice curve through those residuals which means that there's a definite pattern another way to think about it is I've had colleagues say if I look at kind of being on one of these dots and I'm walking along the dots and then let's say I'm here or even here cover up the other dots could you predict where the next one would be and like I said there's this nice smooth curve what you want is you want them scattered about the x-axis because that's what we call no pattern in the residual or that the residuals are random so that you have some good an idea of the model being a good representative of your data you can see that if I did have more of these data points that I would be going down here yeah the pattern shows up in the table as well exactly we have this negative then these positive then these negatives exactly true Lisa and in particular it's not just negative positive negative but they're actually these are I guess they're increasing to the positive then we're increasing then decreasing to the negative and then increasing again at magnitude again so this idea of a pattern in the residuals tells a story that says you've got the wrong model the wrong model could be it could be that you have the wrong type of function but it also could mean that there's some other things about the characteristics of the data that don't match the function so let's go back to our power point and think about one on one think about trying to find a better model so if you think about what tools do students have to find models for nonlinear data sets they know about linear regression they know about lines they know how to come up with their own linear model even for a data set do you guys have suggestions about what kinds of things students might have at their disposal in terms of tools for trying to come up with a nonlinear model can be it can be broad like big ideas or it can be something specific ideas for what students have at their disposal in terms of coming up with nonlinear models okay so Sonia said that nonlinear will be their last unit so you'll they'll think about exponential and quadratic functions because those are the particular types of functions that they'll study and so Sonia if I go back to your idea and I say okay maybe a student has a quadratic and exponentials and some experience to those functions would they want to use one of those models here well there you go Sonia she said but isn't this a square root right so as the students gain more experience with various types of functions then they'll have more at their disposal they'll be able to say oh this looks like and maybe it's a particular parent function like the square root or x squared or 2 to the x if those are their types of functions that they have in their toolboxes right and Don said power model remind everybody to send your messages to everyone so yeah so if you think about a power model again power function is a broader category of the quadratics in the square roots right a square root function is x raised to the one half power so that is a power function because it's x raised to some power so the variable is in the base in a sense and the other thing is in terms of power functions they could also be reciprocal functions like a 1 over x is a power function so if kids have more experience with those different kinds of models they might say oh yeah this looks like and then choose one of those models it's a problem that can be revisited over and over as students learn more functions in math 2 and math 3 exactly Sonia in fact what I'm trying to do today is lay the ground work for these for those other courses because you can think about as their experience grows they can come back and they say oh now I have a kind of a bigger library of functions to choose from another thing that I think is very powerful about studying nonlinear data is the idea of using inverses to straighten the data so that's what I'd like to talk about now and even though yeah that's true so Lisa Lisa saying is this an algebra 2 or above math 1 students don't they that's exactly true they don't have the experience with power functions or exponentials or quadratics but I will say this Lisa even though they don't have that experience if you look at the common core standards it does say that students are supposed to be able to differentiate about these different kinds of functions so even though they might not have spent a lot of time studying exponentials or quadratics one of the things that I think data can provide for even math 1 students is a place where you can start introducing those ideas of these other types of functions and then think about helping students discern between those models so certainly we can discern between linear and nonlinear and then Don says natural log will be covered in my class first so natural log the X and the Y to see if it's a linear pattern okay so Don hold on to that it's a great idea we're going to go there I talk about re-expressing in a different way so she's talking about what's known as re-expressing the data that has been a topic that traditionally has been taught in statistics courses but I'd like to just talk a little bit about it here because it uses some more traditional topics kind of the traditional topics in algebra one and two and even higher of using this idea of straightening the data and that we can if we think the data is of a particular type if we think a square root function is appropriate instead of taking just the square root and kind of using transformations of functions which you could certainly do another great application here you could think about how does one undo what the square root function does because as you learn more about functions or even use this as an introductory lesson for motivating the topic of inverses so when I start inverses I start really simply I think about a linear function I say you give me a number I double it and I subtract 3 and then I talk to kids about writing that as a function right so I double it and subtract 3 so I got 2x minus 3 but then I might say well how do I undo that process so students have to think about if they give me a number and I double it and subtract 3 how do I undo that to get the number that they originally gave me so that notion of inverses can be started at a pretty elementary level but then later on if you say well okay you give me a number I square it how do I undo that students will say take the square root or you give me a number and I take the square root how do I undo that process I'd square it so this idea of inverses can be introduced at different levels and then expanded upon later on as you said Sonia you can keep revisiting this maybe you could say you know even later on throughout the year you could say well let's revisit this data problem but I didn't have as many tools in my tool bag now I have more let's think about how that broadens our library of solution methods in a sense okay so Don your your idea is a great one I want you to hold on to it because that's really leading further down the road to the more sophisticated notion that we're really talking about is log log or expression so I'm going to go back to the PowerPoint and just talk a little bit about threatening the data thank you all very much for participating y'all are doing a great job so this is the idea how can we use inverses to re-express or linearize the data so if you think the model is quadratic how can you linearize the data we answer that question you could take the square root of the y value right take the cube root if you think that a square root function which is what Sonia suggested this one was how can we linearize the data pretty obvious that you would square the y value so I'd like to experiment that with y'all with that on your calculators now if you've typed the data in L1 and L2 I really don't need those residuals for the linear model anymore so if you want to you can clear out your L3 and in L3 now what I'd like to do is take the square of the y value that is square all the car speeds I'm going to do that on expo and y'all can play along on your calculators don't really need this one we'll leave it down here because we've decided that certainly in math 1 kids should be able to know that a linear model is not appropriate basically that these residuals are patterned that's in the common core state standards for math 1 so let's go over here and what we're going to do is I'm going to keep up with what we're doing and I always suggest to my students that they write this in their notes keep up with what's in L1 and L2 and L3 I'm going to square the speeds but this is just a column heading for me if I'm going to square the speeds in excel I'll just go equals and then go get the cell and square it for you guys using your calculators on the top of L3 for example and all you got to do is type L2 square I'm going to say equals speed square hit enter go back and click on this cell and again you got to move your cursor until it changes to the right crosshair there's this fat kind of x then there's this x and then finally there's the one I want and I can double click and that fills okay so if I thought the original data set was square root which it certainly looks like it is square root I'm going to square all the values and now what I'd like to do is create a scatter plot of the skid marks along horizontal axis and the square speed along the vertical axis so if you're playing along with your calculator if you just do that in L3 your scatter plot is probably still on from your residuals to scatter plot with L1 in the x-list if these are in L3 and then do a zoom 9 so that it chooses an appropriate window along here I'm just going to highlight this so I'll click on this column and let go then I'm going to hit the control button and click on this column and I'm going to say insert scatter wow that's beautiful if you look at that you'll say that looks linear it looks like a great linear data set and then we know we have a way to find a linear model we could choose two points find the slope and the intercept oh look at that Sonya said she's got an R of 0.999999 so she did a linear regression on that is that what you did Sonya she did a linear regression on your calculator and she gets a beautiful R because if the data is linear then R gives you some indication of the spread of the data around the linear regression line in this case not a lot of spread because the data is so beautiful so what I'm going to do is I'm going to right click on it add a trend line show the equation on my screen oh okay I have a message from Lisa as an AP statistics teacher taught the process to the AP students just trying to see how far we could take students in terms of where they are in common core so you just teach Algebra 1 math 1 students that residuals can help us determine the appropriateness of linear models where beyond that do we take Algebra 2 students okay besides just looking at the you're saying if there's a pattern in the residuals then we know a linear model is not appropriate is that what you're saying Lisa yeah okay so beyond linear models for Algebra 2 we do this in our pre-calculus classes too as you try to come up with models for various data sets again as your repertoire in a sense expands as a student in terms of different types of functions you can you keep going back to this idea of residuals telling you in general if you've got the right type of function so if I had said well for example if we have an increasing function that's kind of up well an increasing function that's kind of up a lot of different functions right it could be a power function like x squared or x cubed it could be an exponential function right so there's things like that where you might say well let me see if I can you can even use this method where I'm going to try to guess at the appropriate power and try to undo it using inverses of functions so this is a great application of inverses of functions again for higher level students yeah so you could keep revisiting it and now if you're an AP stats teacher you of course know about log-logary expression and so if we're guessing that is it quadratic or is it cubic and in fact it's the power function that's in between two and three we know that this log-logary expression can be used to come up with a particular power so those of you who teach math one don't worry about this in terms of this log-logary expression that is a more advanced topic but I do that in my pre-calculus classes for my students because they do it a lot in science to be honest with you if the kids don't take AP stats they haven't taken that yet but they're in a biology course or chemistry course often times scientists will use semi-log and log-logary expression and if students haven't had an AP statistics course it kind of leaves them wondering why the heck does this work when in fact your pre-calculus knowledge or even algebra 2 knowledge as long as you know enough about exponentials and logarithms to know that they're inverses of each other can be very powerful and a great application for what is a more traditional topic it's a great way to, data's a great way to introduce some of these what we consider traditional topics like understanding inverse functions so now if I say to Sonia yeah you have an R value that is one and you understand what R means we can stop there but I can also say how else could we figure out if we have been successful in linearizing the data this was a pretty nice data that I'll have to admit but there are times when you're not quite sure you've linearized the data like I even said wow that looks linear and my students will say that looks good and I'll say stop wait we're mathematicians we need to be able to say what it means from a mathematical standpoint in terms of looking good so you might even use this point in the process to say let's look at the residual plot for the quote-unquote linearized data so let me go back to the PowerPoint to see where we are there I tend to kind of skip ahead yeah there we go so we're using this calculator to re-express the data we've tried some particular process to straighten the data we found the linear model for the re-express data and then assess the process to see if we have been successful so Sonya has said looks to our values I might even say let's find the residuals for the quote-unquote linearized data and see if we really have been successful so if we do that does anybody want me to go ahead and do that do you want to find the residuals for the linearized data we can do that easily in the calculator and I can do that on Excel I'll let you guys let me know again this is a very special data set it looks really beautiful I got it from the common core math tools from NCPM but I've seen it in other places too so okay let's do this so let's go back I'm going to do an Excel again if you need help with the calculator holler I'm going to say well let me see if I really have been successful in linearizing this data I'm going to go over here and in this cell I'm going to put the residuals for the re-express data I won't call it linearized because maybe we didn't really linearize it we'll re-express this so if you think about it what I've got here is this beautiful linear data set I think it's linear and this linear model well just like we could calculate residuals for any model in any data set I can do that here what I want to do is say the actual y-values which if you put the squared speeds in your L3 would live in L3 could say L3 minus and now if you stored the linear model is 24.076x minus 5.0322 and say y2 then you could say L3 minus y2 of L1 so Sonya says it will be a shift for our teachers and students to put less stock in R and look at the residuals I think they're used to using R to judge the model because before Common Core they did not make residual plots as a former AP staff teacher I prefer the way that you're developing well thank you Sonya this is powerful because you just saw that that R value was really great for this model I left this picture on the screen because this hits you over the head a linear model is not what I want to use here I don't want to make predictions with this linear model you can always see the obvious pattern in the residual here when you think about these guys being negative these guys being positive etc so that R value I can give you some other examples of some data sets there is going to be good but there is an obvious pattern in the residual or there's even an obvious pattern in the y value you can see that it's cubic data and here you are using a linear model it doesn't make sense so I think it's a great way again to use some of these very powerful important topics in mathematics to help students make sense of what they're doing instead of just hitting magic buttons so let's go back to the residuals for the re-expressed data let's take the squared values here those are my actual because I squared them right so let's say it equals this guy sorry let me go to the cell equals this cell minus the predicted value the predicted value is 24.076 times my skid mark minus 5.0 so that's actual minus predicted but I'm using actual of the re-expressed data there is residual if I wait until I have the right cross here and hit double click maybe not I might have to fill this one down there we go so if I want to make a scatterplot of these residuals it seems awfully good oh I see why they seem really big to me because they were in the 1522 but look at the y-values here I squared all of the speeds so the magnitude of the speed squared is huge so these residuals they're very big but relative to the y-values they're not that big so let's do a scatterplot of the new residuals for the quote-unquote linearized data I'm going to click on this column you guys could just turn a scatterplot on for say L1 and L4 if that's where you store them I'm going to hit control click here I'm going to insert a scatterplot and look at that residual plot that's enough to make your day so look at the scattered residuals they're scattered across the x-axis not a definite pattern in the residuals some of you again might be concerned like I was that they were large but relative to the y-values they're not and if you want to calculate the percent error you could just take these and divide them by their respective squared y-values multiply it by 100 so this is great so Lisa says this really connects traditional topics to the data and stats world and again I think it's the data and stats world that the statistics is coming into the regular kind of the regular courses of mathematics and there's a reason for that it's because it's really helping students become more just educated in terms of mathematical concepts that can help them in the real world so these topics are a wonderful way to show students the relevance of some of these traditional topics so I think we're kind of running down in time here but so what I want to do is just now that we've kind of used this process and said wow we've been a great job we really did guess right in terms of thinking that it was a square root function let's go out of this process back out of it to revisit the original data today so we'll go back to the PowerPoint and we'll think about the mathematics and how the mathematics that we've done has brought us to this spot so Sonya says she wants more data I can share that with you, I have a great resource for you Sonya the NCTM core tool it's a new tool they've just developed has a ton of data sets and I'll share that with you at the end on my last slide as a resource and I'll show you how to grab some of that data in fact that's where this data came from I've seen it other places too so let's think about the mathematics we squared the y values and then we came up with a linear function we used our calculators or our computers to do that you don't have to, if you don't want to hit the linear regression button that data was beautifully linear so you could have just let the kids practice coming up with their own lines and then when we go back to think about how we're going to go back to the original data set if you think about s being the speed of the car depending on the length of the skid again I'm trying to figure out what the speed is based on a length a skid length that I've measured and if you want to solve for the dependent variable now all you've got to do is do the algebra, right? take the square root of both sides and you get s equals the square root of in my particular case for our case we have the m and the b our computers or calculators gave us those numbers and when you're solving this equation this totally depended on what we did with a re-expression so if you can see here if instead of squaring the y values I had taken the square root of the y values that's what would be here so I have the square root of the s is equals mL plus b then you solve for s or maybe I took the cube root so this process depends on the re-expression that I did and then the algebra to solve this equation depends on what the students know in terms of solving equations so if I want to come up with my final model that is the model for the non-linear data set all I got to do is write the square root of mL plus b where m is the slope given by my computer calculator and b is the intercept if I think about looking at that model let's do that let's graph the non-linear model with non-linear data and then we can go back to the idea of residuals here we can use residuals to think about did I get the right model because of all the work I've already done I have a pretty good indication that my residual plot is going to look pretty good because I really did linearize the data and then we can use our non-linear model to make some prediction so I'm going to go to go ahead and do that and y'all can play along with the calculator I'm going to move all this stuff away I don't need this one I think I'm going to delete that much so this was all a process something that you might want to be careful of if you're going to use this in your classes is that you might want to make sure that the kids have their work organized pretty well in their papers because what happens sometimes is they do a bunch of stuff on their calculator and they get stuck and they know exactly what they did and they don't remember exactly what they did so it gets a little bit hard to troubleshoot if they don't keep an account of what they've done so I have my kids write L1, L2, L3, L4, L5 on their papers and write down what each of those holds in terms of this process so let's go to our final model I'm going to put predicted values from the final model in here I actually need this model here so I get that data set back so this is going to be my non-linear y-value if you think about this in terms of mathematics you could put this say in L5 now this is going to be my model my non-linear model evaluated at all my axes so my non-linear model we've established the fact is the square root of all these should they only look at the residual plots the linearized data or is it legitimate I like to look at a residual plot for the final model as well now they should match like I said because of the process that we use but the reason I want to look at the non-linear residual is because the non-linear residuals will help me calculate percent error in my final model so I'll take those residuals and divide them by the y-values of the original data set again not only do I don't have a pattern so that's good news but I can tell you specifically that my residuals are quote-unquote small small by what measure small by percent error measure so that's a great question Sonya, thank you so I'm going to just take my model which is going to be the square root excuse RT of 24.0 76 my linear model but I'm taking the square root of it times my skid marks minus 5.0 3, 2, 2 close the parent and that should be my new model evaluated at 5 feet so it makes sense so it's about 10.7 right look at the data set if anybody needs help with that let me know on your calculator it would be easier just type this linear model or if you stored it say in Y4 if you got your linear model there in Y4 say in Y5 just type the square root of 24.076x minus 5.0322 and then you could just do L2 minus say Y4 of L1 I apologize for not having the emulator working I'm going to go here and double click and again those seem reasonable look at these Y values compared to the data set look at that last one they're all great look at that 79.844 and that Y value is 80 so if you want to go back to this graph I think I can get rid of this trend line now and I can put these guys on that data set maybe I'll just do a whole new graph I'm going to graph these and control click and go over here to the last one control click that's just to highlight all of them and then I'm going to insert a graph look at that you can't even see them because they're on top of each other I'm going to right click on can I right click on this I think I just need to make them smaller so it's on your life that you're doing it itself yeah because you can keep track of what you calculated yeah and then you can always go to a cell if you've forgotten what you've done and you can see lots of different graphs yeah it is nice I mean the calculator is limited in terms of the representations you can see all of your graphs at the same time so yeah I mean I might use this more often I do use it in my classrooms in my classrooms a lot of times just to demonstrate but also I think it's a nice tool to be able to help students use to get literate with Excel or other spreadsheets and I imagine some of you might use Inspires and I think the Inspire I'm not an Inspire person but if you use Inspires the Inspire has ways to do this too so I think we can change the style and then and then see that they're right on top of each other okay so you might change that so that it's a line instead of these dots on top of each other pretty great fit we can calculate residuals again there shouldn't be a pattern and the residuals the pattern should match that residuals for the re-expressed data and then you can find a percent error by taking the y-value of the data set divided by I mean I'm sorry the residuals divided by the y-value of the data set sums 100 so I'm going to keep going because we're almost out of time and go back to the PowerPoint to just share some of those resources with you so you can make predictions I said predict a skid length for a car traveling at a speed of 90 miles per hour that's actually backwards right because in that case what I'd have to do is say what if my speed is 90 miles per hour what I expect my skid length to be or you could say suppose I'm a police person and I'm out there and I've measured a skid mark of say 70 feet what would the predicted speed of that car be so lots of nice predictions you can make you can also make a sense maybe out of the constant in your model too but I'm going to keep moving so we can get to the resources so in terms of the content standards this is a lot of content standards in particular for statistics if you'll notice again in math 1 it says differentiating between linear and quadratic and exponential functions now this is a square root function but if we've gone back to the problem that said let's swap the x's and the y's suppose I want to think about if I have the car speed I expect skid length to be then you would have a quadratic data set and you can do the same process but in reverse of this now functions a lot of stuff with functions understanding concepts with functions and then I just put inverse functions at the bottom because it's a beautiful application of inverses of functions and it's in the common core as well not until later on though not in math 1 and then of course the practices if we have more time we talk a little bit about these this will be archived so if you're watching this later or if folks are watching this in their PLC then maybe you can take this particular task and think about these particular practices but in general it's a great way to get to talking about the mathematics this is the nctm core math tools website that I just grabbed some of these this is the car skid length you can put a linear regression line on it you can graph the residuals and this is a great tool to be able to grab some of these pretty graphs and you could put this on an assessment you could give kids that middle graph there and say sketch a little picture of what you think the residual plot is going to look like and it's not that you want that precise picture on the right but it could be just the notion of do they understand what a residual represents Lisa so she has an inspire but it's not as easy to navigate as excel that's been my experience when I've tried to use the inspire but I have not spent a lot of time doing it so some people are really they love their inspires and I think this is accessible to those but I just have not been very successful on that in terms of resources we've got this nctm core math tools go check them out they're really cool I might pull them up real quick at the end here ncssm has developed materials for AFM and algebra 2 and it's got a lot of data there quadratic and exponential in particular and then this was just a little plug for our conference we're having a conference here at the end of next week on Friday and Saturday and even though it's right up against the deadline if we might still have some room for you we'd love to see you if you're interested you can go to that website or you can also go to the second website you just do a Google search on tcm ncssm and you can see that I gave a talk here last year about nonlinear data in common core and I used a data set that had to do with water flowing out of a jug I made a video of water flowing out of a jug and then I graphed the height over time great nonlinear data set Neal's Ible is an awesome colleague who works in Massachusetts he teaches at Deerfield Academy and has created his own course materials for a course called alternative to pre-calculus or function statistics and trig all of this stuff is free and available on his Moodle site he loves for you to use it and has some great great probability and data actually mostly data analysis on his website now the water jug problem is up on the website for the tcm and so I won't have a chance to talk about that our next session will be in February we haven't decided on the date yet because we haven't decided on whether or not we'll have kind of a separate webinar where it will be just me talking about the PowerPoint or if it'll be some local people here because it'll be for Durham County teachers they're going to come here to NCSS now and Sonia yes we will be sending out the PowerPoint after the session is over it'll be archived and we will send you a link to the archive and also ask you to do a survey complete a survey on this session so thank you all very much for being here as usual comments and suggestions are welcome we're trying to tweak it as we go I think we're getting better and better as we go along and this will be archived for everybody and if you have any questions or any comments you get something and one of those links isn't working just shoot me an email and we'll get it fixed hopefully very quickly and check out those Common Core math tools at NCTM thank you all very much all have been wonderful it's been so nice to have some back and forth and I think this has been a nice vocal group