Exploring Conic Sections with Sketchpad: Sketchpad Webinar

In doing this, there are several pedagogical advantages. One is that we're not rushing immediately into the algebra here, but we're really going to focus on geometry mostly. Secondly, we'll see that each of the devices that I'll show you tonight gives you a really unique insight into the mathematics of ellipses. We'll see that each device gives us something a little bit different to learn about them. And this is in contrast to the usual approach where it's highly algebraic, and I suspect many students come away just remembering the formula for the ellipse, as opposed to anything very concrete about its construction methods. One of the benefits we'll see of using Sketchpad with von Schutten's models is that if you build models out of things like wood, it's not so easy to vary them. You make one wooden model and it might construct one ellipse. But with Sketchpad, we can build a model and then very easily tinker with its parts and draw lots of different ellipses. Another reason for doing this is that building these devices is really satisfying. It was my first exposure to Sketchpad, and it really just is a beautiful application of dynamic geometry, and when you start watching these models in motion, they really are quite mesmerizing. And finally, the actual proofs of why these devices draw ellipses, they're well within students who are pretty much in Algebra 2, I would say. So they serve. Welcome to GoToTraining, online training made easy. There are 67 other callers on the call. Maybe two polls, just to get some basic information here. So let's see. My polls are, okay, here we go. So here's the first poll. Simple one. Do you teach conic sections? Okay. So it looks like 59% of you say yes, 41% no. Maybe we can bump up those number of no's. Okay. And let's see. Actually, I'll wait for the second question until we move a little further along. So here we have the definition of an ellipse. The set of points, P, such that PA plus PV is constant for two fixed points, A and B, that are called the foci of the ellipse. And what follows below this on my screen is the algebraic derivation of the equation of an ellipse. And this is the one that I think many of us learned in school. I know I did when I was in high school. And it's kind of a hairy derivation in that you've got two square roots and you've got to work with those. And if you like algebra, it's fun. But you come to the end of this and you get this really nice result, actually, here. But I remember feeling kind of lost by all this. Like, I didn't really have a tangible sense of what an ellipse was. Whereby comparison, something like a circle, which I drew all the time, and just had a much better sense in my own, just from a physical sense of what a circle was, whereas an ellipse, not so much so. So this is where we get back now to our star of the evening, Franz von Schutten. Who was this man? Well, he was a friend of Descartes, and he lived from 1615 to 1660 in the Netherlands. And interestingly, he was an assistant professor of engineering, which probably led to his interest in actual building of devices that could draw things like ellipses. He had particular interest, from what I've read, in things like being able to draw large-scale ellipses in gardens. So he really wanted to come up with techniques that would make it easy to draw these different conic sections. He also translated Descartes' geometry from French to Latin, because at the time most scholars were fluent in Latin but not French. And his commentaries on the geometry were actually more popular than the original version that Descartes released. So he published four books. One, I think, was on trigonometry that was just a table of trig values. And the one we're going to be looking at tonight, one of its five sections relates to conic sections. And so when David Dennis, my colleague at Cornell, first told me about this book, it wasn't a book you could go and just see in the bookstore. This was one which was only in rare book rooms of libraries. And we were lucky, because at Cornell they actually had this book in their rare book collection. And David went to see it, and when he came back from having seen it, he was just overjoyed with the models that he had seen in the book. He didn't have copies of the pages of the book, so he just said, go see it. So I remember going there to the library and putting on gloves, and they took out this book and started paging through it. And nowadays we're actually very lucky, because you can see the entire book online in digital form. So I wanted to show you a few pages of it. And when I send you this, when you receive the link for the sketch at the end of the webinar, you'll be able to go to this page too. And so here's the book, and you can see it has on this title page a beautiful little illustration, one you would not see in today's textbooks. And I'm just going to move ahead a little bit here to the section which talks about the conics. And here is one of the devices we'll look at tonight. It's one that we call the congruent triangle construction. And you can see how beautifully Von Schupen has drawn this hand here and the little linkage that draws the device. And if I move forward a little bit, just take a look at, say, this one here, this little flourish on the side, or this one here that makes hyperbolas with these little flowery images on the side. It was just beautiful, beautiful books, pages to go through in this book. And when we looked at this, we realized, like, for all the talk about manipulatives that we hear today about the NCTM and the students need to use concrete manipulatives, Von Schupen had us beat on that years and years ago, centuries ago. I mean, this book, when it comes to things like methods to draw conics, there is no equal to this from what I've seen. There are other books that will show you ways to draw conic sections, but this is just a gorgeous, gorgeous book. So, having seen this book, what I set out to do, and it was around the time that I began to learn about Sketchpad, was to work with Key to write a book about conic sections that focused on a lot of different methods for drawing the conics, but in particular focused on the methods and illustrations from Franz Von Schupen. So, tonight, I'm going to focus on his methods primarily, and at the end, show you a few of the other pictures in the book. I guess I should just stop here for a second to see if there are any questions. The only real question I've seen is how you created the little button, hide show button things with the dot that you're doing on your agendas. People are fascinated by that. Well, I'll just say about that, that that is using a brand new feature of Sketchpad 5 called hot text, where you can assign a action button to, in this case, this symbol of the dot. So, by pressing it, you're actually pressing a hidden action button, which you can see at the bottom of the screen. So, it's a good plug for why to upgrade to Sketchpad 5 if you haven't already. That was it. Okay, let's move on. So, here's an illustration which has a technique for constructing an ellipse that you may very well have seen, the pins and string construction. And I want to ask now, as my second poll question of the night, do you teach the pins and string construction? Okay. So, I will close that poll and share the results. It looks like pretty, well, most, more of you don't than do, but a bunch of you do. So, let's see if we can talk a little bit about that and maybe give you some insights on how to use Sketchpad to show this particular physical model. So, here's the picture from Franz von Schutten. And we've attached a string, or we've looped it around two points, which I will label as F1 and F2 for the foci of the ellipse. And we've pulled the string taut at point E, and we have a pencil at point E. And as we move our hand, keeping that string taut, our pencil draws out a curve. Now, remember that the ellipse was the set of points such that the sum of the distances to two fixed points, in this case, F1 and F2, is constant. So, this string has a constant length from E to F1 and E to F2. We're just changing where that point E is along this piece of the string. So, as we trace our pencil point here at E, we're using the definition of an ellipse to draw one. So, I'd like to show with Sketchpad how we can model the pins and string construction. So, the first thing we need is something to represent our string. So, I'm going to start with a segment. And I'll put that segment here at the bottom of the screen. And that's going to represent our entire string length. And, of course, I can change that as we go through this to make the string shorter or longer. So, now, I'd like a way to be able to construct the two lengths of string that are here, one going from the focal point to E, and the other going from E to the other focal point. So, I'm going to take my segment tool again, and I'm going to draw one segment here on top of my original segment, and then another segment. And let's color the second segment, say, red, just so we have an easy way to see the two segments. So, as I'm dragging this point, the total length of my string, which I'll change those labels, and I use brackets so I can get a subscript, and this is point E. So, these two lengths, F1E and EF2, change, but the total length of the string stays the same. So, now, I'll put on my sketch the two points, F1 and F2, and let's label these also. F1 and F2. And now, I want a way to actually build this model using these lengths. So, in Sketchpad, whenever I want to be able to copy a length, like in this case, the length of F1E, so that it sits here, starting at this point, F1, what I do is I select the segment whose length I want to copy, I select a point, and I choose construct circle by center plus radius. And let's color that circle blue so it matches my string. So, now, when I drag point E, the size of the circle is dependent upon this length here. Its radius is equal to F1E. So, now, I want a second circle, this one centered at F2, and I want its length to be this red segment. So, I'll select the segment and the point, and go to construct, circle by center plus radius, and let's make this circle red. So, now, when I drag point E, I have these two circles that are changing their size. And what I'm interested in is that point E, where I'm pulling the string up, and that point is at the intersection of these two circles. So, I'll draw a segment from F1 to the intersection of one of the intersections, and this other segment, and let's color this one blue to match its circle, and let's label this as E. So, now, as I drag my point E down here, I can see this model of the string now more clearly. It becomes clear why I was making those circles. I can't drag this point directly, but I can control the length of the string through this point E. So, I don't want to have to imagine what that looks like. I want to actually be able to see the trace of point E. And, in fact, really, I want to be able to see the trace of this point here, too, because that also satisfies the definition of being a fixed constant distance to my two focal points. So, I'll select these two points, and I'll go to display, and I'll choose trace intersections. So, now, when I drag point E, I can see their trace, and, indeed, it looks like an ellipse. And I talked about being able to vary things easily when we use Sketchpad. So, one thing we could do is change the distance of our two focal points, and while I'm at it, let's go to preferences, and in the color preference, let's make these traces fade over time. I don't want them to stay there so that I can draw many different ellipses and not have them all appear on screen. So, now, if I drag point E, my ellipse looks fatter, and I can do things like see what happens when these points get pretty much right on top of each other to see how the ellipse looks more and more like a circle. So, using traces is good, but they go away, and we'd like to see if there's a more powerful way to be able to see the locus of all the points E on this point down here. So, what I'll do is I'll select these two points, and let's turn off their traces. And what I'm going to do instead is I'm going to construct the locus of point E, this point E, as this point E down here moves along the segment. So, to do that, I'll select this point, and I'll select what I'd like to actually construct the locus of, and with both of those points selected, I'll go to construct and choose locus, and there I see that entire path rather than having to see it traced out point by point. And for this point down here, I do the same thing. I select point E because that's the point that's confined to this path along F1 and F2, and I select this point, and I choose construct locus. And here now is my ellipse, and as I drag either one of these focal points, I can dynamically see my ellipse changing. I can also drag either endpoint and watch the effect that has by changing the length of my string. So, let me pause here for a minute and see if there are any questions. I'm not sure if the question was answered. I have a feeling the construction you just did answered it, but there was a question from Brian that said, would the compass marks on the foci have been from two points on the ellipse? Because he didn't see their use in the construction. Oh, in the original? Yes. This was asked before you did your whole construction. Yeah, I suspect so. I suspect these were actually, often you can use these to actually find the foci by putting the compass at either endpoint of the minor axis, and then sweeping out a length of half the major axis. Okay, and we have a question from Robert. Does the order of selection of the points matter for the locus? No, it doesn't. Sketchpad will know. Okay, so Sketchpad will know, okay. Wait, there's another question from Cora. I'm not quite sure I understand it. Can you hide the circles to make it clearer? Yes, I could. In fact, I debated about doing that, but I kept them around just a little bit longer for my next piece of this, but I could select them both and choose hide circles. Very much like it, yeah. Yeah, yeah, so that's a good idea. I'm going to bring them back just for a second, because I wanted to do, even though I said I was just going to do ellipses tonight, I wanted to show you one other thing to show you how easy it is to mess around a little with the constructions. So we said that the ellipse was the set of points such that the sum of two distances, these ones here, is a constant. So we're looking at a constant sum. What if we were to look at the set of points where the difference was constant, the absolute value of the difference? What would that look like? So let's see if I can build that by using what I have here. So I'm going to go to the line tool and construct a line through points F1 and F2. And let's make that a little thinner. And what I'd like to do is to break E off of this path between F1 and F2, and instead have it give it more free reign to be able to travel anywhere along this line. So with Sketchpad, I can actually do that without having to rebuild everything from scratch. So what I do is I select point E, and I go to edit, and I'm going to choose split point from segment to rip it free. So there, now E is free to wander wherever it wants, but not for long because I want to attach it to this line. So I'll select E and the line, and I will choose merge point to line. And now when I drag point E, we see that it extends in either direction. Now, if we look at these two distances, two lengths, one of them is EF2, and the other is EF1. And if we subtract EF2 from EF1, we're left with a constant difference, and that difference is this length, F1, F2. So let's see what that actually means in terms of our model. So if I drag this focal point out here, now I no longer see my locus because E is not sitting on this path between F1 and F2. So I'm going to have to construct that locus again. So I'll select, as I did before, this point E and this one, and I'll choose construct locus. And I see that this piece of the locus I can drag and extend, and then I'll do the same for the other intersection point of my two circles. I'll select E and this intersection point and choose locus, and let's drag this one out. And now I see that I have what actually is a hyperbola. And if I drag F1 and F2, I can switch from my hyperbola to my ellipse. And it helps to create a nice connection between the two. So I particularly like this because it allows me to really use Sketchpad to its fullest advantage to ask what-if questions and be able to explore them without having to rebuild models from scratch. Any questions? If anyone has a question right now, it would be a great time to enter it in the chat panel. We have a question from, I think it's Kali, and, oh, just a comment, really. The foci on top of each other gives a circle. So maybe you could actually demonstrate that. That's very cool. I could also increase the, it might help if I, let's see, I go to properties and plot, I can increase the number of samples I have and see if that makes a difference. In fact, the very first activity in my book, students start with a circle and then drag what looks like one center point apart so it becomes two points. So they see their, oh, that's a little better. It's a little better, yeah. A little better. Still looking a little rough around the edges. So I'm not seeing any other questions, Daniel. So again, if anyone has a question, if not, Daniel can move on. Okay. So let's turn to a model that I've never seen anywhere else except in Von Schutten's work. And this is one that my grad school colleague, David Dennis, coined the congruent triangles construction. So what we have here are some, actually, what I'll do is show you this as a physical model first. So when I taught this section on ellipses to students at NYU, one of my students at the time was friends with an architect. And he showed this to his architect friend who built this model of the congruent triangles for me. And I think I still have it somewhere in my closet. I should really fish it out. But it's really cool to actually operate this model because even though it only really draws one ellipse, the actual sensation of putting your pen or pencil right here and operating this device is a really nice counterpart to actually building a sketchpad. So here, with Von Schutten's typical flourishes, is the congruent triangles construction. And we have AB is equal to FC. And BF is equal to CA. And in this activity in my book, the first thing that students do is prove that these two triangles, AEB and FEC, are congruent to each other. So now we're actually going to build the model. And I should also say that for teachers who want to use this material and not build everything from scratch, my book comes with pre-built models of all the different construction techniques. So you can just jump straight into the investigation and proof section. So I'm going to need two segments to represent AB and BF. And I'm going to put those on the bottom of my screen over here. And first, I'm going to put a line, two lines here. And I'm putting those down so that when I draw my two segments to represent AB and BF, and when I drag either endpoint of those, the segments will stay horizontal. Not necessary, but it's nice. I should also point out that notice when I put these segments on top of the lines, the lines were very accommodating. They became dashed. So it became easy to see these segments. And that's a Sketchpad 5 feature that's new. You won't find it in Sketchpad 4. So I'm going to select these two lines and the points that come with them and hide them. We're not going to need those. And let's color code these. And I'll make these a little thicker. I'll make that medium. And I'll make this medium and blue, red. And let's say this is AB. And let's say this segment represents BF. So similar to what we did before in the previous model where we used this length here to control what was happening in the sketch itself. So, okay, I want to actually start building this. So I'll put a point on my sketch. And let's call that point A. And now I'd like to construct point B. And I could use a circle. But it would be nice if AB stayed horizontal just like my segment AB did down here. So what I'm going to do is I'm going to select point A, select point B, and choose Transform Mark Vector. That will create a vector from A to B. And then I'll select my point A here. And I'll translate that by that marked vector. And we'll see that I now have my point B, which is actually A prime. So that when I drag point B, my model of the congruent triangles will react appropriately. And let's move those down just a little bit. Okay, and let's connect those while we're at it. So now I'd like to construct the remainder of my model. And let's make BF a little bit longer so it stands out. So here I have BF. And what I'm going to do is I'm going to select point B. And I'll select my red segment that represents BF. And I'll use the technique I did in the previous model. I'll construct a circle. And I'll color it red just so I remember that it belongs to BF. And it doesn't really matter where along this circle I place point F. So I'll make this segment red to help me with the color coding. And I'll label this point F. So right now I can drag point F anywhere along this circle. And let's see. I can probably hide this circle now. And now I need this segment here which has length of FC. And FC is the same as AB. So I'll select my AB segment there and point F. And I'll choose circle by center and radius. And let's make that blue. And I don't know where point C is. I shouldn't be so quick to just place it somewhere. But I do have some more information about C. I know that C is a distance of AC away from point A. So if I select point A and again my red segment. And choose construct by circle and radius. I now have the set of points that are a distance of BF away from point A. So where these two circles intersect right there. That satisfies the condition that this segment has a blue length. The length I like of AB. And this segment here which I'll color red has the length of AC. So I don't think I'll need these circles anymore. I'll hide those. And I'll label this point C. And my intersection point, the one I'll be tracing, that's point B. So now if I drag point F. I can see a working model of Bronspon-Schutten's construction. Which is really cool to be able to take this illustration. And bring it to life with Sketchpad. Using techniques that are very common for a lot of Sketchpad constructions. And ones that, you know, regardless of how much time you spend doing ellipses. The process of actually building this. And thinking about why these triangles are congruent. Why or how you can use this technique of basically moving the compass up here. Onto the sketch to copy lengths. I think that's all great stuff. Regardless of the ellipse portion of this. The ellipse is kind of just the icing on the cake. So I'd like to trace point E. So let's trace that intersection. And now I'll drag point F. And notice that this works on the upper portion here. But when I come down here my, what is essentially a crossed parallelogram. Gets uncrossed and doesn't draw what I'd like. But I can take care of that. I can mark AB as a mirror line. And then take point E. And reflect that. And I can trace it as well. And I can also vary things like my lengths. So I can see what happens when I drag now in this case. And finally, why does this actually draw an ellipse? Well, if you look at this portion here. From E to A and E to D. That looks a lot like our spring construction. And since we know the focal points seem to be at A and D. Maybe we should try to show that EA plus EB is a constant. Well, if these two triangles are congruent. FEC and AEB, which we could prove. Then we know that EC is equal to EB. So if we want to add AE plus EB. That's the same as AE plus EC. And that length we know is constant. AE plus EC is this segment here. Which is BF. And that's not changing unless I choose to change it. In which case we've drawn a different ellipse. So the proof of this is really quite simple and beautiful. I think this is of all my models that Franz von Schutten did. Probably my favorite. Both in terms of the actual construction involved. And the proof method. So let me pause here and see if there are any questions. No questions that I can see. So if anyone has something they'd like Daniel to demonstrate again. There was one question a while back. Just about how you constructed the parallel lines. Because you did it slightly different than using the construction menu. Right, I kind of cheated on that. When I drew my lines. I held down the shift key on my keyboard. So that the lines would be horizontal. Okay, and then there is a question from Robert. Can you show how you got the fixed lengths to connect again, please? The fixed lengths to connect. Maybe Robert, if you explain exactly what you mean by that again. That might be helpful. In terms of getting all of this to connect. In terms of the actual model. That's when we use these circles to find, for instance, the intersection point C here. And there's a question. Why was there a need for a vector? Well, the reason I used the vector. I could have actually used a circle or done a different method. Is that I wanted to be certain that when I dragged point B. I wanted this length to stay. This segment to be horizontal. I could have done another technique like use a circle. I just chose to do that to show you something different. Okay, in fact, you could also instead of putting AB down here. You could use this AB here as your primary method of changing the length of AB. I'm sorry to interrupt. There was a question about the directions for this. Because we're looking at the picture and you're doing it. Are you going to provide them with the directions for how to actually construct these? That did come up. I can provide the construction for the congruent triangles. I will also talk a little bit at the end about my book. Which contains all of these constructions. Okay, perfect. So, one thing I wanted to show you is that when making the new edition of this book. Using Sketchpad 5. I basically redid all of the models. Because the previous version of this book came out 10 years ago. And Sketchpad has come a long way in that time. And so has my own thinking about how to make things really stand out and pop with Sketchpad. So, what I wanted to do with my models. Was to make them look a little more than just these thick models here. I wanted them to look closer to what Von Schutenz did. So, what I include with the book. And I'll include this particular set of tools with the webinar materials. Is I made some custom tools of different linkages. So that they would look closer to what Von Schutenz has in his models. And using them is really simple. So, for this particular set of tools. I'm going to copy these two parameters into my sketch. Because I'll need them. And I can hide those. And what I'd like to do now is let's hide these segments. Actually, I don't want to hide point B. I'll hide that one and this one. This one and this one. And I'm using my keyboard shortcut to do the hiding. And now let's come back to my tools. And I'm going to use linkage two between A and B. And I'll go into my custom tools and choose linkage two. And I'll just click A and then B. And there we have a nice little broad between them. I'll use that same tool between F and C. And now I'm going to use a slotted linkage between my remaining points using linkage three. So let's come back here and go to linkage three. And we'll go between A and C. And this also creates this point along here that slides. But I don't need that in this case. So I'm just going to delete it. And then I'll use that same linkage to connect F to B. And again, we don't need this point. And point B, the intersection point, looks a little bit lost here. So what I'm going to do is select it. And in Sketchpad 5, we can make our points much bigger and much smaller than they were in Sketchpad 4. Let's make it larger. Let's go there. That fits nicely. So now when I drag point F, my model is the same. But I've added this nice little ability to make the model look more like Bond Students. And this is something that I think students in particular will really enjoy of making their model look very, very fancy, just through the use of those tools. OK, it looks like I have just a little more time. So let's look at the final model, which is the falling ladder construction. I doubt that's what Von Schutten called it. But it's the one we think about, where you have a pail on a ladder, or you're standing on the ladder. And the ladder starts to slip down the side of the wall. While you're slipping down, of course, what you're thinking about is what path is my foot tracing? So let's take a look. So I will make a line here. And I'll select this point. And let's make it perpendicular through it. So here's my wall. And here's the ground. And what I'd like to do is build this ladder. So what I'll do is I'll draw a segment down here, something we've done in all of our models. And let's color it red. And now this will be the length of my ladder. So I'll mark a point on the ground that represents the tip of my ladder. And I'll select that. And I'll select my ladder length. And I'll choose construct, circle by center and radius. Color that red. I will mark this intersection point. And I will connect it. We don't need to see this circle anymore, so I will hide it. And now if I drag this point, we see that this length, which represents my ladder, is staying constant. And it's slipping down the wall. And I can put a point on it to represent either me or the bucket. And I can start by tracing that. And I see it gives me the upper half of my ellipse. And I could also select it, turn off the trace. And let's erase those traces. And I'll use the technique I did before. I'll construct the locus of this point as my point along the ground moves. Because this is the point, as I drag it, that controls the locus of this point. So I'll select them both. And go to construct and choose locus. And I see immediately I have the upper half of my ellipse. I can select this horizontal line and mark it as a mirror. I can then, oh no, I can't reflect my locus itself. I can select this point and reflect it. And then select that point and my point here. And construct this locus to get my entire ellipse. And this is really nice because I can now drag the point along the ladder and see how my ellipse changes for different locations of the point. Interesting, in fact, to think about here when my point is midway along the ladder, why this is a circle. And one other interesting thing I can do with this. Von Schutten has another model that we've dubbed the bent straw. And if I construct the midpoint of my ladder and I connect that with the origin. And I now hide my ladder and just replace it with a segment. Now when I drag this point, I see what we call the bent straw construction. And this is a really nice proof with the ladder or the bent straw proof that we actually have an ellipse. We could use trigonometry, we could use the Baccarin theorem and similar triangles. All of these give proofs that I still think are simpler than that one we saw at the beginning, which had the two square roots. So let me pause again for any questions. The only, there's been some questions about is this appropriate for high school level or as a high school teacher, you know, what resources are provided type of thing. So I think this book is totally appropriate for high school. Each of the activities in this book comes, as I said, with both completed sketchpad constructions. And let me actually show you some more of those here. This is all of the ellipse constructions that come in the book. So you'll forgive me if I do a little advertising. So this is the concentric circles we looked at before, using little pins here to make it look a little more like we're actually doing this with string. Here's one called the folded circle construction, which I'm not sure, but there was actually a comics webinar a few weeks ago that may have included. There was, there was. This is one called the tangent circles construction. This is provided all finished. We don't actually build this one from scratch. And this one has another very beautiful proof. Here's another one of Franz von Schutten models. Congruent triangles, carpenters, construction, sliding ladder. And this is one called Danny's ellipse. And that's Danny is not me. This is a student who actually came up with this construction about 20 years ago and sent it off to So the book comes with all of these different constructions. And in the steps themselves, you can walk through the constructions. You can focus just on the proof. You can focus on the historical aspect of it. And to me, the one thing I resisted doing in this book, even though I was asked to at some points, was that I did not go to the graph menu and start bonding the equations for instance of parabolas. I kept everything very geometric about how these models are built and the kinds of proofs you use to show that they actually do construct the various comics. I think the ways that it meshes with the high school curriculum are very strong and give a good reason to spend a little more time on comics than we typically do. So in summary, we will be providing you a zip file in an email you'll receive, which will have the materials that you've seen today. There will also be a webinar recording of tonight so you can share it with other teachers and educators. The book itself, Exploring Comic Sections, the new edition, will be available by the end of April. For any of you who will be at NCTM in Philadelphia near the end of April, we're expecting the book to be available then, as well as the remaining books that we have like Pythagoras plugged in that are being converted over to Sketchpad 5. Any of you who have iPads, if you don't know about it, you should download Sketchpad Explorer for the iPad. It's free and it allows you to open any Sketchpad model on the iPad and interact with it. And in particular, you'll see that one of the documents that comes with Sketchpad Explorer focuses specifically on the ellipse constructions you've seen tonight. And final plug is for Sketch Exchange. This is Key's reasonably new site where we share Sketchpad models, and any of you can share and exchange Sketchpad models. It's a great place to come and see what other people are doing with Sketchpad and put your models out there for others to share and enjoy. And I think I'm trying to unmute myself here. There's just a couple questions. I may go back in my little list here where I had some questions. One of them was related back to your sketch where you were moving the triangle. Do you ever use animation to move the points? And I think, you know, in any of those when you're moving the model around. Yes. In fact, I could have done that tonight by showing you these models with animation. One that I'll just show you here, if you press animate points, one of the cool things about this model, the way we built it, which was more advanced, is that this congruent triangle construction stays together for the entire way around. So yes, each of these has animate buttons, and you can easily build those into the models yourself, just by selecting the point and going to edit and choosing animation. And there you have the animate button. And that leads to my next question. How long does it take to create one of these models? I think that's a tricky question. I found this particular congruent triangles construction, the first time I built it, when I was new to Sketchpad, to be probably the most difficult thing I had done at that time. I remember just sweating this one, trying to figure out which circles were doing what. You know, it depends. Do you provide students with step-by-step directions? Do you let them do some of the figuring out? You know, in a rush, we could do this one very quickly with students. Maybe it takes 15 minutes. If that's too long, maybe you just provide the completed construction. Right. Okay, I don't see any other questions. If any of you have any more questions before we end this lovely presentation, type them in the chat panel now. I think we've answered all the ones that were up there that I could see. Thank you very much, Daniel. This was great. I always learn something. I never taught this level mathematics. So I always feel like I learn something new every time, especially with Sketchpad as well. So thank you very much. I'm not seeing any questions. Well, thank you, everybody, for coming. And just a reminder, we will post the recording and the documents that Daniel's going to share. It's usually up by Friday. So give us a little time to get that up there. And it's at our website, keypress.com slash webinars. If you look on the left-hand side, you can look at all of our software. And all of our archived webinars are there, along with documents. You'll also receive an email from me with a certificate of attendance, if you can use that. It does depend on your district. But it's just a certificate that says you participated in an hour worth of professional development. And we appreciate all of you joining us this evening. Thanks again, Daniel. And thank you, Andres, for helping with questions. And if you have any questions after the fact, feel free to email me. I'll pass them along to Daniel if I can't answer them. And thank you very much for joining us. Have a lovely evening.