 Welcome. I'm Carol Stern. I'm here at the School of Science and Math with Maria Hernandez and thank you for joining us this afternoon. This is the fifth session that we're doing on Common Core State Standards and I'm just going to turn it over to Maria. If you have any questions or comments, I have a chat window open and I'm sharing those comments with Maria. So again, welcome. I'm Maria Hernandez. It's great to have you guys with us here this afternoon. In this particular session, what I'd like to do is talk a little bit about a particular hands-on problem that I found that folks like because it's adaptable in terms of using it across grade levels and courses. And so we'll consider a specific problem that I think and we can tackle even in the eighth grade math classroom or maybe even seventh grade in terms of different kinds of approaches. And then we'll kind of also talk about it in terms of higher-level math courses in out of two, pre-calculus, and it can even be extended to a calculus application. So I think I'm hoping that it'll be a problem that you find useful and something that you can do in your classrooms to help students get their heads around the idea of building a function or building a model using a function. So we're going to share lots of materials with you. I created a pretty in-depth lesson plan that has the lesson according to these different levels that kind of split it apart. So the handout that you got from Carol yesterday is a student handout that will certainly be available. I'll give you the website at the end of this session. It will be available on that website. And take the shared content. So I'll give you the handout that the student handout that again is based on different levels. We'll have this PowerPoint on it. So you don't have to worry about taking notes from the PowerPoint. It will have the in-depth lesson plan so that it has possible solutions in there. And then also I have a geometry sketch profile that I didn't create. But when I did this problem in a hands-on session, one of the participants in there, a woman named Greta Mills from New Hampshire, created a geometry sketch path file that goes along with the particular problem that I think you will find useful in terms of visualizing the problem. So let's get started with the problem. So here's the problem. If you're going to do this with your students, I recommend that you give each group a wire. So I've just got a wire here and let me kind of hold this up against the background because I don't know if you can see it very well. I have just a wire here that I went over to like Home Depot or Lowe's or someplace and just asked them to cut a long piece of wire for me and then I took it home and cut it into different sized lengths. So the problem is set up in this way that's kind of more open in it. It says you have a piece of plastic covered electrical wire of the length L. So L is specified. And what we'd like for you to do is to cut the wire into two pieces, one of which you'll form into a square and the other into a circle. And our objective for the first part of the problem is to think about doing this in such a way as the sum of the two figures, areas, I'm sorry, the sum of the areas of the two figures is as small as possible. So again, what I'd like to do is think about this problem in different ways. If you take the wire and you give each group a different length, this becomes kind of a more, I think, a more advanced problem. If you decide instead that we're all going to get, each group is going to get a wire of length 24 inches then we can be more specific and approach the problem from a more elementary level, if you will, in terms of the numerical approach. So I'm going to suppose that we all have the same length of wiring that it's 24 inches long. So like I said, I just go over to, you know, a Home Depot or a Lowe's or some place like that and ask folks to cut me a long piece of wiring. This is about 20 cents a foot, I think. And sometimes they even have a scrap table where you can get scraps of wire from very inexpensive. And if you tell them you're a math teacher, a lot of times they'll cut you a deal because they're like math teachers and they think it's cool that you're using wire in your math class. So anyway, we're going to do the problem from that approach that we all have the same length of wire. And what we'd like to do as we work through the problem is to think about how your students will approach the problem, again, based on maybe different classes, different levels, what kind of prior knowledge they'll need. And what I found for this particular problem when I kind of throw the problem out, it's pretty open-ended problem, but the students sometimes don't know where to start. So we'll think about kind of getting our hands on the problem in a sense and helping students figure out how to push forward and then if they get stuck along the way, how to push forward. And to, I think, really encourage them to share their own ideas that this is a nice problem because of that. They just start writing down some of the things that they know that prior knowledge. They write down some formulas for, say, the area of the square, the area of the circle or circumference and perimeter. Those ideas, at least there's some place they can start. And then if you are doing this where you're going to let students work with their group members and use different approaches to the problem, then how might you think about organizing or sequencing what students share and that process in terms of their approach to the problem. So again, I'm trying to think about it with lots of different hats on, but if we think about everybody having the same length of wire, then I think it'll simplify our approach for this webinar. And then we'll talk a little bit about extensions of the problem for other classes. Okay, so let's think about the various solution methods. This is where, as audience participation, I encourage you to think about how your students would approach this problem, how they might get their hands on it, and just type in the chat box. Carol's watching the chat box for me. And let me know what you're thinking about in terms of, if you gave this problem to your students and they knew that the wire was 24 inches long, what would they do to try to get started on this problem? So I'm going to wait a minute. I'm going to do that waiting time thing so I can get you guys to chat in the chat window there, just type a suggestion of what your students might do to approach this problem. They would cut it in half. So Terry says that some of them would be brave. They take their wire cutters and just cut it in half. So why would you say that, Terry? That's interesting. I've gotten that answer before. They always do. So they're kind of taking middle of the road then. So they just say, well, let's just cut it in half and see what happens. It's an okay approach. We're not actually going to cut it. What we might do is we might encourage them to let's suppose that we did cut it in half. Let's calculate the area of the two figures and see what happens. Does somebody else have another suggestion besides this going straight for the half? Start with the area just a circle and just a square from the 24-inch piece. Great. So I thank you for that comment. So what we've done then is consider the two extremes. If I only make a circle out of my wire, what will the area of that circle be? If I only make a square, what will the area of that square be? And now if we put it together with Terry's suggestion, we'll have actually three pieces of information because what if we cut it in half and made the circle and the square, what would the combined areas be? So in a sense, what we're trying to do here with your suggestions is we're building some actually some intuition in a sense. What's going on with these different kinds of possibilities? So I'm going to take that. I'm going to say if we have a level of students that maybe isn't familiar with functions, they might not go to the functions. They might do exactly what you guys are talking about. And that is actually to think about some specific cases. And one way you could kind of organize those specific cases is to say, well, what I'm going to do is I'm going to divide the students up in pairs and assign each pair of students a specific length to use for, say, the perimeter of the square. That's arbitrary. I just decided to do it that way. And then have them calculate the combined areas of the figures. So if you do that and you have, say, 24 kids, then you should get 12 data points in a sense. We're going to build a table, a data set. And with that, because of the way I've set up the measurement, I could have said, for example, let's suppose we only make a square like someone suggested. And then what if we made a square with only two inches of the wire? And then what if we made a square with four inches of the wire, six, et cetera? So if we do that, then we can get a nice data set. So let me show you the table that I have on the next slide where I actually calculated that. And this, to be honest with you, even calculating these actual values can be powerful for the students because they have to think about, if I only use two inches for the square, then how big is the side? I'm going to use a perimeter as two inches. Then each side will be half an inch, right? And they can calculate the area of their square. And then how big is the other piece with the rest of it is, if this is two inches and the whole thing's 24, I've got 22 inches left, but that's going to be the circumference of my circle. So if I've got the circumference of the circle and I'm looking for the area of the circle, what I'll need to do is take the circumference, which is 22 inches, and figure out what the radius of my circle is. So they're going to use that circumference formula to figure out the radius. And then once they have the radius of the circle, they can calculate the area of their circle. So just kind of collecting that data as a class and have them write those points on the board can be just a nice quick way, maybe not so quick, but a nice kind of hands-on way where they actually get to calculate some values. Now, actually the extreme pieces in terms of only making a circle and only making a square, we'd like to tell the kids today the possibility that you don't have to cut the wire, because that is one of the questions that comes up. Okay, so here's my table, and I have a handout with, like I said, with this numerical approach. So one of the things we could do is we could take this data set and put it in our calculators. I'm not going to do that right now, because I'd like to be able to talk about some of the other solutions. But if we put that in our calculators, then we should get a nice graph. There you go. There's a graph of that data set. So once the kids have this data set, which will take a little while to put it all together, we get the kids to make the calculations, put it up on the board. If the kids are familiar with putting in the data into the stat list editor in their calculators, they could type in the independent variable, that is L1, the way I've got it graphed here, is going to be that x, which is, remember, the piece of wire that I'll use is the perimeter of the square. And then along the vertical axis, you can ask them, what does that y value mean? And hopefully they'll understand, because we built the table, that the y value is actually the area, or the combined areas of the two figures. So what would your students do at this point if we had this on the board? And we said, now remember what we're trying to do? We're trying to create these figures so that the combined area is minimum. The minimum value. Suggestions about what your kids might do at this point. And again, it might depend on what courses you teach. Somebody said quadreg. Yeah, that's one of the things I've had on the table too. So they might just push the quadratic regression button. Other suggestions? So if you think about this, if they've just got the data set, there's a suggestion about using the second calc minimum, which is certainly an option on the calculator. But I'd like for you to think about, if you only have a scatter plot on, you don't actually have a function. I'm just going to look real quick and make sure this is what is true. I'm going to turn a scatter plot on just on my own which is calculated here because I happen to have it here in my hand. And I'm going to see if that second calc feature is something that actually works if you only have a stat plot on minimum. Yeah, if I only have a stat plot on and I try to choose that, nothing happens. So that second calc minimum won't work if you've only got a data set. If you're going to hit the quadratic button like the suggestion that we got, we'll actually have some type of a model. We'll have this quadratic model because it looks like a quadratic function would make sense. We can also talk to the kids about why that might make sense if the independent variable is a linear dimension and if they think about what's on the vertical axis, that's the area. So it would make sense that this would be quadratic in that linear dimension because it's area's quadratic, right? Multiplied. Here's another suggestion. My students might choose the power regression. So a different power if they're not convinced that it's quadratic. I have another chat comment that they would choose the lowest point. Yeah, there you go. So we have this data set. We can just look across that data set even just look across these numbers and say, oh yeah, here we go. We're minimized here. Let me move my chat window out of the way so I can see. The minimum value here is 20.29 and what we're hoping is that they might say, well, that's what my teacher gave. My teacher gave me these specific x values. How do I know that there's not a value that's in between these, maybe even a portion of an inch that might actually minimize the area? And then I've got a comment here they might just guess from the visual. So these are all good comments and what we'd like to do is to kind of lead them to, again, depending on the level of the student to think about things that go beyond just the particular data set because I just chose these increments of two because I thought it would make sense to have 12 data points depending on the number of students I have. So that's those suggestions that you guys gave me. I actually have here. What we have here is a possible transformation of the function. Nobody suggested that, but you could say if you think the function is quadratic could you take our basic parent function x squared if you're doing this with the Algebra 2 students you could say, what if I take my parent function x squared and try to use transformations of function to push it over to the right and up and then also try to think about expanding it in the right way so that we have either a horizontal stretch or vertical compression. So I think that in itself could be kind of an interesting exercise for the students if that's what your goal was for this particular lesson. I'm not going to go from that standpoint because I want to think about it in terms of other kinds of approaches but that certainly is a reasonable thing to do if you were doing this with transformations of function. And if you do that you might think about well how will I know how far right to shift it and how far up to shift it if I don't really know what the vertex is. That's why I'm not sad about that approach because our objective is to find the vertex if you believe that this is quadratic. So what I'd like to do is take the other suggestion that we got and that is use a quadratic regression. So if you hit the quadratic key and your students are familiar with that then this is what the calculator produced for me when I hit the quadratic regression. So again depending on your goal for the lesson this could be enough. You could say well I just wanted to understand if you can use a quadratic use the machine to get a quadratic regression but you can still ask some interesting questions here. You can type this in your calculator and then you can use the minimum key. You could also say well let's just try to make sense of some of these values in here. For example this 45.85 you could ask the students what does that 45.85 mean in the context of the problem. So again I think there are different approaches in your goals for the lesson. You could let your kids go in those directions. So I'm going to keep moving because what I'd really like to do is think about it from a higher level and say what if we actually want to build a function. But I will say that if you have students at these other levels that we've been talking about you could take it from here and still finish the problem, right? You could still minimize the function hit the minimum key and figure out what the x value is and then again there's still lots of interpretation to be done because you have to say what does the x represent? Remember for us, you have to keep reminding the students that x, it's important to define our variables, x is the length of the wire that I'm going to use for the square. So once they know that perimeter of the square they can go ahead and cut the wire so you bring your wire cutters in have them cut it and then go ahead and make the figures the two figures. When they do that they're going to see a beautiful geometric relationship that I'm not going to talk about just yet but even at this particular point this geometric relationship comes out and it's true that the geometric relationship will hold no matter what the length of your wire is. So this is one approach that we chose to have everybody have the same length of wire because it lends itself to this approach where we create a data set. If instead you gave each group a different length of wire and say you were going to build your function in a different way then we wouldn't build the data set it wouldn't make sense to build a data set in that way but instead we would need a function so it actually kind of gives the students some the motivation to seek a more complicated or maybe not complicated but a more sophisticated solution of it. So if you give everybody a different length of wire and we can't really create this data set per se but instead let's think about it from a function modeling approach. So if I go back to thinking about building a function I've drawn a picture here and again this is arbitrary the way I've decided the state is specifically you might not do this depending on the level of students you might just let kids start writing down formulas for the area of a circle and the area of a square and then relate it to the perimeters here that we're talking about. So this is just one approach. If I let x be the piece of wire that I'm going to use for the square just kind of going back to the other problem then the rest of the wire is going to be used for the circumference of the circle. So I can find the area of the square because the perimeter is x so I'm going to take x and divide it by 4 and then square that amount that'll be the area that is the contribution from the square and then the circumference of the circle is the rest of it so it's 24 minus x and the area of the circle well we know the area of the circle is pi r squared but I need to be able to get r in terms of x so if I go back to the idea that I know what the circumference is the circumference is 2 pi r in general I know that's equal to 24 minus x and I can solve that for r in terms of x so we've got the makings of a function now a function that will be representing the combined area of the two figures and that function is our model so we're going to say area is x over 4 quantity squared plus pi times the quantity 24 minus x over 2 pi squared now again that's not the only way to do this if you did this in a more open-ended way then you would let kids get to this point via their route so if instead they let x be the piece that they use for the circumference of the circle this wouldn't make sense, it's a different function or if I've seen kids for example write down something like the area of the square is s squared and the area of the circle is pi r squared and then go from there thinking about which of those pieces would be useful in terms of perimeter and the circumference of these two figures so this is just one particular idea if you give kids wires that are different lengths then this 24 appear will change to whatever the length of their wire is and this will be more powerful I think if you do give kids different lengths of wire because they can see that this function is something that's uniform and what changes about it is this parameter this 24 so if you look at that depending on what your students are used to doing there could be some they could feel like they really want to simplify this expression I use the word simplify almost in quotes because I think the expression or the function is fine just the way it is they can type it in their calculator they gotta be careful with parentheses granted but in terms of expanding this binomial out you don't really need to expand that binomial out if what you're going to do is take this function and use the minimum key on your calculator so depending on again where your students are and how comfortable they are with typing this particular expression in then you could maybe you could clean up the denominator here this would be 4 pi squared and I could divide out the pi but you know there's no reason to expand that binomial right now there's no reason to do that if we're going to just type it in our calculator and find the minimum so since I asked you guys to bring your calculator along let's do that I'm going to go to my emulator and I'm just going to type this function in I don't have the data that's typed in here but we'll just type that function in and then find the minimum I've got x divided by 4 and if you're playing along with me I invite you to type it in as well and this is where we're going to be careful with parentheses I'm going to go ahead and suppose that I've cleaned it up I'm going to just go right here on here and make sure I don't do it incorrectly I have 24 minus x quantity squared in the numerator and I'd have 4 pi in the denominator and that's really just to avoid trouble with parentheses I'm going to have 24 minus x divided by 4 pi, I need to square it and these parentheses I do need and I'll go back and square it so I'm going to scoot you back here for this quantity by inserting the square and if you'll notice it's trying to grab for me let's see if I can make sure I've got this right here okay I hit window I want to think about a good window because I want to think about a good window because it's important for the student to think about the domain of this function and we're going to talk a little bit more about that later but in general remember that x is the length of the wire that I'm going to use to make the square so how small can x be and how big can x be well normally the domain of a quadratic function is all real numbers but because we have this context the domain becomes very specific and it can be a very interesting conversation for students a lot of times the kids want to hit the magic zoom button they'll hit the zoom fit or whatever and I really try to encourage them not to do that to instead make sense of the context so let's go ahead and change the window here and I'm going to change the x min to zero and if I used a wire of length 24 inches the maximum x value would be 24 and I'll make the scale too so if y'all have comments or questions as we're working along please holler what I'd like to do now is to use the table feature on my calculator to think about a good y window because it's not always obvious to students what the y window will be now because if you did it with a 24 inches because we generated a list of values you could say well we can look at our data set if you're going to use this model on top of your data and say I can certainly use the data set to come up with a good y window but I also like to use table feature on the calculator so I'm going to hit second window and I'm going to say suppose I start at zero and I jump by two so if I go too fast on the calculator let me know I can slow down and then if I hit second graph which is table then I can see those values actually that we collected there you go but if you didn't do it from a data perspective or each of us had our own lengths of wire then this will help students find a good window so somehow I've gotten crazy and gone all the way 60-60 doesn't make sense in the context of the problem but I can kind of use this just set my window for the y value so this zero of 130 is what was in there already that seems like a reasonable window the y max is bigger than we need but now if I hit graph you can see the graph that's already on my screen kind of came away so if you think about this this is really the window in context if I hit trace now and I say well let me see what the value of this function is it will hit second trace and let's just plug in a value at zero then you can say the combined areas will be 45.83 that area that total area is 45.83 inches squared you can ask the kids what the units are there and then we could use the second calc feature and choose minimum and when you do that it's asking for a left bound I'm just making sure that I'm to the left of the minimum I'll hit enter and then it's asking me for a right bounce I'm going to scroll to the right until I'm to the right of the minimum emulator is kind of slow so until I get to the right of the minimum I'll hit enter there and then trying to catch up with my keystrokes there we go so I have a comment here if you set the domain in window then press zoom zero it will set the range and the kids can go back and fix the Y min and Y max as needed so it makes sense for the problem yep that's right so the zoom fit again I'm not crazy about using the zoom but they really like zoom fit so the advantage of that is that at least they have to think about the domain of the function and then they can tweak it after they get some kind of a sense for those values but also I like to have them look at the table of values as well because the table is always there and you know what's happening when you look at the table so my kids do like zoom fit so I'm going to go ahead and hit enter here I guess and there you go it calculated that the length that I should use for my perimeter of my square is 13 about 13.4 and that the total area combined is 20.16 so if you were going to stop here I would think that it would be very important for the kids to sketch a graph of the function to label the axes with words and units and numbers and then to label this minimum value and write a sentence that explains what's going to happen in terms of this point that I see on the graph they should be able to say okay now I'm ready to cut my wire I'm going to measure about 13.5 or so inches for the perimeter of the square the length of it is going to be my circle and then they can go ahead and make their two figures and again if you have different groups with different lengths of wires of course their answers are going to be different and actually on the lesson plan the other thing I just thought about is if you do want to use this as a writing assignment where they actually build a function I've provided a list of expectations if you wanted to do kind of just a short project or writing assignment for the kids a list of expectations that you might want them to include in a report and then also a sample rubric that I think it's important for students to have the opportunity to write about mathematics and not just think about math as a list of formulas and facts but instead of a tool for solving a real world problem so when they have to think about standing back and looking at their process being able to put all that stuff together and write you a short report maybe together with a partner so that they have somebody who wants their ideas off of can be a very I think a powerful thing so at this point then you be ready to let the kids cut their wires so when they cut their wires and again if everybody has a different length this can be very powerful because if you make the two figures one thing that's very interesting that happens in this problem is that when you're minimizing the sum of the areas of the two figures no matter what the length of your wire is you will always get this beautiful geometric relationship I need to ruin it for you if I'm going to show you up here and it will be such that the circle will be inscribed in the square so some folks might think well I know that the circle will max if I just make a circle it will maximize the area some people might say well that might mean that if I just make a square it will minimize the area but that's not the case instead you get this beautiful geometric relationship so I'm showing you a pretty big one here I even have a really tiny one here because I gave some other kids a smaller wire and if you get all of the kids to hold up their circle and their square and show that the circle is inscribed in the square it's a pretty powerful notion I just want to show you that because I think it's great for the kids so let's get back to the power point are there questions or comments here for what we've done so far so we have our model and then I want to think about minimizing the area so we've already done that we've minimized it and also I also want to notice that remember when we had just the data set we kind of made some just some kind of prediction that it would be a quadratic it kind of looks like a parabola then somebody said well they might hit some other regression button but if they build the model themselves there's no doubt this is a quadratic and I got a comment that your kids would love this that's great thank you for that comment I think it's very powerful I found it both with teachers and with students to be kind of a classic optimization problem you'll see it a lot of times in calculus books it's been around forever but until I actually built actually cut the wire I didn't see this beautiful geometric relationship so I think it's great because of that too it's also got a lot of nice extensions that we'll talk about so the kids can see that this is a quadratic function we'll talk about that you can talk about the fact that you know that this opens up it's a parabola that opens up because the leading coefficient now remember I said you wouldn't really want to multiple all this out but hopefully they would be able to see that this coefficient of course is positive because it's a quarter squared and the coefficient of the quadratic term here as well should be positive because I'm squaring it out it should be a parabola that opens up and you're convinced that it really is a parabola now if you think about it from an algebraic standpoint if your students are familiar with quadratics they probably know how to find the vertex of the parabola maybe they've figured out the formula maybe they've um maybe you talked about it another way but if you were brave enough to let your students um quote it would quote simplify this algebraically that is multiply out all the terms so that you can identify the coefficient of the quadratic term the linear term and the constant term actually you don't need the constant term just the quadratic and the linear term you could have them identify the vertex of the parabola without hitting the magic button and and then again it gives them some power because this 24 here will be replaced by whatever the length of their wire is and so everybody will be able to find the vertex of the parabola no matter what this parameter in a sense is okay oops sorry let's talk a little bit about the um sorry the domain so we talked a little bit about what does x represent so we have a picture there and again that was arbitrary the way I said it up if you let your kids just dive in that won't be consistent throughout the entire class but we talked about how small x could be and how big x could be and that idea of the domain having some kind of physical property that is that the kids are holding the domain in their hand when they when you give them the wire because what they're thinking about is if x is as small as possible and that is you only make a circle that is zero then I've got that particular instance if x is as large as possible then I've only made a square and they can see that x is whatever the length of the wire is so that's very I think very powerful and the other thing is that when you think about it from a higher level mathematics what we're doing here is we're minimizing a function on a closed interval so there is there's a guarantee that the function because it's a continuous function will have a minimum and it will have a maximum on that closed interval so the idea of minimizing the function of course we've done that because we have parabola we're trying to find the vertex but the idea of maximizing the function becomes a little bit I guess kind of trickier because if I just gave you a parabola and I said what's the maximum parabola well if you're not considering it over a closed interval then you don't really there is no maximum parabola opens up it goes on forever and ever so the maximum is infinity but because I'm concerned about it over a closed interval then I really do have a maximum and if I look at the graph so that is if you think about those extremes what do those extreme values yield and how they relate to the graph let's look at the graph there's my function and there's my graph and if I look at the graph if I ask my kids what if x is zero well this part is zero and then I only get the circle well that's what happens over here that's this point here and you can see that on this closed and bounded interval this function does have a maximum and it occurs at this leftmost end point so it's a kind of a precursor to the extreme value theorem which is an important theorem in calculus so if you think about maximizing the area we would only make a circle and then if we make x be the largest it can be that is over here on this right hand end point that would mean that x is 24 and this piece would go away and we'd only have a square so I think it's got a lot of nice representations here I've got the geometric representation here I've got the function we have a numerical approach to so we could put that up there too and then we have this graphical approach so all those different representations are nice to have together so hopefully the kids can make connections across those the Geometer Schetspad file that I've made available to you was created by Greta Mills she was a participant in one of my workshops a couple years ago and she created this great Geometer Schetspad file and she said it was fine if I shared it so I'm going to click on this and open it so if you have Geosketchpad you should be able to open this file let us know if something goes wrong and you can't open it but what she's got here is across the top here she's got the wire and she's got something here where I can change the length that I use she looks like she's set it up so that this is a square so you see that if I go all the way to the left hand end point square is nothing in the circle the only thing I make and then she's also got this green function across here that's the combined area function and if I go all the way to the right you can see that I use the entire wire to make the square I don't know what she used for her entire length of the wire here and if you just want to ask the kids to kind of scroll until they're trying to minimize this value right here this 10.52 we're keeping our eye on that and you ask the kids to holler when we get there looks like we're somewhere in here 10.48 so maybe for this one I've got some of the areas 10.48 and if you do this she has minimized the sum so actually we've already done that by hand but if you put this button where she has watched with an exclamation point on it she shows you that the circle is inscribed in the square so this is a really nice I think visual tool for the kids and then you're welcome to use that so that's available for you are there any questions or comments so far so again I invite you just to type in a in the chat box if you have any comments or questions so again some of the things that I've already mentioned that I think this problem is so powerful is that the students get to create their own mathematical models and again I've been prescriptive a little bit about which way we created the model but I don't always do that with my students depending on what level they're at I just let them kind of dive in and try to help them with their own ideas and then kind of get to the idea of thinking about different ways to approach the problem if you let kids approach it in their own way then they can compare their models and think about well what do I want to do with this model and which is the easier one to work with so the question could you show the equation again but I wanted to comment you're going to give them the presentation yes you will have the power point and all the other files I'll go back to this equation here is that the equation you wanted okay there was a comment about not expanding the polynomial because yeah because it depends on what you want to do with the mathematics you know if you're at a point where we're building the function trying to make sense of the function there's no reason to expand it in fact if you expand it you might lose some information that kind of hits you over the head here you're looking at this and you tell the kids what if x is 24 they should be able to say wow if x is 24 that numerator is 0 which means I don't have a circle if you write over this this is the part that's contributed by the circle and this is the part that's contributed by the square similarly if x is 0 what's going to happen if I square all of this out and combine like terms you lose some information and I think that's kind of a powerful notion too because you know we do spend a good bit of time getting students to be comfortable with those algebraic processes but we also want to be able to say if you're using it to do a real world problem you don't necessarily have to do that as an exercise. If you think about this from a calculus perspective if you want to extend it to calculus this 24 here would be a parameter we could call it L that's why I introduced the problem not specifically in terms of the number there and if you do this in terms of the parameter then of course we still get this function quadratic function with a parameter in it and that kids have to make sense of the fact that x is the variable and L is fixed L is governed by the piece of wire that I hand you so when you try to think about minimizing this function on the closed and bounded interval if you're a calculus student you might take the first derivative and set it equal to zero and then you'll get some x value that will be in this particular case the perimeter of the square and then when they figure out what the rest of it's going to be in terms of the circle it might not be obvious that the geometric relationship that I showed you is there unless they actually solve for the length of the side of the square and the length or the radius of the circle because then what they'll see is that powerful idea that the length of the side of the square should be two times the radius of the circle so that's kind of nice also even if you didn't do it in terms of calculus but in terms of pre-cal if you wanted to find the vertex of that parabola you can still do it in terms of either thinking about minus v over 2a or using the calculator and they can still see algebraically that if they figure out the radius of the circle and the length of the side of the square the radius should be half that that length. I have a comment. Is this a performance task that you would recommend using in a remediation type playoffs? I don't think so. I think that if you were going to do this you would probably want to think about it as some kind of a collective group effort especially if you were thinking about in terms of remediation what you might be thinking about is can my kids actually approach this open-ended question and get their hands on it in some way so that idea of being more specific and saying each pair of students you do this task just to be able to find the combined areas and then let's put all that information together so it would be more kind of a culminating experience instead of a specific can you do this one particular task, another particular task because I think that again if you're trying to leave kids in one particular direction then it would be kind of I'm doing a piece of an important big project as opposed to I'm just doing one specific thing and that's all you're testing is that one specific thing. Can I move forward and get my hands on this and then after we get it together what do I do with that so I think of it as more kind of an open-ended kind of a almost a lab activity. So let's keep on here just to think again about if we're going to reflect on the problem we've talked a lot about the domain and making sense of the pieces of information in the context of the problem and then of course the extreme cases such as the physical constraints when I talk about which form might be easier to use because I was specific in my approach that might not make a lot of sense here but if you did leave it as a more open-ended task where you let kids build their own model then that might, that will make more sense because if you have a student that set up the problem in a different way then they can compare their final model to somebody else's and say I know I like your model better let's go with that one. More things that are powerful about the problem, of course that beautiful geometric relationship that emerged from the specific problem that I had no idea was I did this problem for 20 years before I solved that relationship and that it can, like I've mentioned it can be extended to calculus and show kids the power of parameters. Another thing I think that is interesting in terms of an extension is that you could ask the kids what if instead of making a circle on a square if I made a circle in an equilateral triangle so if you think about it from that perspective let me go to get that in here I think I'll probably have to comment here about having some students work it numerically graphically while others work it algebraically. Yeah that would be great so again depending on kind of differentiated instruction where your kids are you might say yes you really get your hands on it you guys do the numerical way and then we can compare our model to our data that'd be very powerful I think for students. And then that goes back to the idea of if you're going to have kids do it in those different types of approaches if you're going to share out you can actually have kids come up to the board and share what they've done or if you have a document camera show what they've done then you as the leader of the conversation you can think about how you would kind of layer those approaches so you might have the numerical folks come up first and share their data set and then you'd have the function folks come up next and put it all together. There's also a comment here that says that you like sketchpad yeah you can do this with sketchpad and I have not I don't have a lot of experience with geodrubra but I've been told by folks that use geodrubra which is kind of a free it is free but it's comfortable I guess the geodrubra sketchpad is that it wouldn't be too difficult to create it in geodrubra so if somebody out there feels courageous and wants to try creating something like that with geodrubra please please send me an email and share it with the rest of us and I'll put it up there on the website and give you a byline and everything because that would be great because then you don't have to buy a program geodrubra so I just wanted to sketch out here just real quick we have a little bit of time left I just want to think about it in terms of let's create just a real quick page here if we think about it in terms of making a square now we're going to do a triangle let's do an equilateral triangle and a circle here then if I let X be the perimeter of the triangle and then say 24 minus X the circle then what I need for kids to understand is how to find the total area here we need the area of the triangle plus the area of the circle and again if you were doing it from a numerical perspective depending on where your students are they would need to be able to think about finding the area of the triangle knowing that this each of these side lengths is X over 3 so they need to be able to find the area of that triangle using one half the base times the height where the height of course has to have to be calculated in terms of the side length so this would be a nice kind of twist for kids that are pushing forward and you need something that stretches them a little bit differently and you could certainly do the problem in terms of the circle and the triangle the other beautiful thing that is so powerful about this problem is that if you do it with an equilateral triangle and a circle I have a model here I'll put it in front of my paper here so you can see it is that the circle will be inscribed in the triangle as well and in fact if you make any other polygon and a circle this relationship will hold so I'm hoping to went your appetite and have you explore it in lots of different levels I haven't looked at say for example looking at a square and a triangle what kinds of relationships happen but if you do a circle and any other polygon then you're going to have the circle inscribed in the polygon so let's go back to the power point so in terms of the state standards content standards we're certainly using modeling I didn't put modeling on here because it's really all under the umbrella of modeling but if you're thinking about writing the function that describes the relationship between two quantities and combining standard functions if you use transformations of functions like I mentioned before then of course you'd be thinking about taking a data set and trying to manipulate x squared in terms of the transformation of horizontal shifts, vertical shifts compressions, etc just fit that data set and then for geometry there's lots of nice geometry properties in here too and then like I said of course for modeling hits a lot of content standards if you think about the sorry these are not the content standards these are practices these are the mathematical practices the kids have to make sense of the problems and often times persevere it's really hard they see something that they don't know where to start and so it's kind of our responsibility to kind of get them to start somewhere write down what formulas you know let's do a specific example some of you said well let's suppose we just cut it in half what happens some of you said let's look at the extreme extreme values because it gives an opportunity to kind of jump in wherever they are and then you can hopefully push them forward if you leave it more open ended they get an opportunity to construct their own viable arguments and critique the reasoning of others depending on how they're comparing their models using mathematical models and I really like to think about this precision too because this attending to precision often times often times is thought of in terms of numerical precision but if you look at the standard it really talks about precision in terms of mathematical language too so we can use the mathematical language of domain and range in context we can talk specifically about how you're going to define your variables and how you're going to write the area functions in terms of that variable you know the more the kids have an opportunity to express themselves and communicate their ideas with that precision the better off I think they are so I think it's a nice problem in terms of mathematical practices here's some resources for teachers I actually gave a talk at the teaching contemporary mathematics conference in 2009 which is a conference we have here at North Carolina school sites math it happens end of January usually and the handout for the hands on optimization talk that I gave is available at that TCM website if you just do a Google search on NCSSM TCM you can see my materials from that because you can go to the 2009 year you can see my materials from that hands on optimization talk and I did this problem and I had three other problems that you might want to go look at especially if you are doing optimization problems and pre-calculus or calculus the other problems I think are a little bit more challenging because one of them is one of them is a three-dimensional problem that you have to think about a vertical cross-section but you can go look at them and see if there are things that you might want to use also we have created lessons, complete lessons with classroom ready handout and a handout that is for the teachers for various topics in Algebra 2 and advanced functions and modeling so if you go to either one of those websites then you can see those materials there and they are divided up by topics so if you are studying exponential functions and you are looking for a data problem with exponential functions or linear functions across those different courses you might be able to find something that is very useful. One resource that I did not list here that I have just been exploring recently is are the NCTM core math tools so if you go to the NCTM website and you pull down the core math tools, those are the Java applet I believe that you can pull down onto your machine and it has lots of data sets in it and some different kinds of tools and I have just started exploring those but I found them to be very useful. I have also put my email address down there at the bottom so like I said if you create a geotuber file or you use this and have questions or comments or if you have some other problems that you want to share please feel free to send me those via email and we will post them and I will make sure and give you credit for those. In terms of these sessions we have not set the dates for these yet they are still up in the air so we are thinking about late March, March 25th and then the late April session, April 22nd I would be after the NCTM meeting and this week's materials will be posted at that website. I have found a couple of typos that I have been talking in the PowerPoint so I will fix those and send that right to Carol and she will post that PowerPoint along with all the other supporting materials and the archived version of the webinar because she has been recording the webinar.