WEBVTT 00:00:11.046 --> 00:00:17.016 Spatial skills are those parts of your brain that enable you to visualize what something looks 00:00:17.116 --> 00:00:20.696 like if you rotate it in space or, if you're standing someplace, 00:00:21.116 --> 00:00:26.136 what something looks like over there, or imagine the path you're going to go 00:00:26.136 --> 00:00:28.316 down as you're traveling somewhere. 00:00:29.036 --> 00:00:35.686 Spatial skills are part of your-what Gardner calls, your "intelligences," 00:00:35.686 --> 00:00:42.526 are your spatial skills, and they enable you to maneuver within the world around you. 00:00:42.776 --> 00:00:44.646 In order to develop 3-D spatial skills, 00:00:44.646 --> 00:00:48.566 one of the things that we found to be very effective is helping students to learn how 00:00:48.566 --> 00:00:51.836 to sketch 3-D objects on a 2-D piece of paper. 00:00:52.206 --> 00:00:55.306 One of the techniques that is used is an isometric drawing. 00:00:55.706 --> 00:00:57.996 If you look at this picture here, 00:00:57.996 --> 00:01:02.826 you can see that this is just a 3-D object in any kind of orientation. 00:01:03.336 --> 00:01:09.086 An isometric drawing though, is a special type of drawing that shows an object 00:01:09.536 --> 00:01:12.826 as if you are looking down a diagonal of a cube. 00:01:13.306 --> 00:01:17.916 So this picture here, then, shows what that isometric view 00:01:17.916 --> 00:01:19.836 of this particular objects looks like. 00:01:19.886 --> 00:01:23.656 Because now, you're looking straight down the diagonal of that cube. 00:01:24.396 --> 00:01:29.806 So, one of the things we work with in developing spatial skills is helping students to be able 00:01:29.806 --> 00:01:32.226 to draw an isometric view of an object. 00:01:33.006 --> 00:01:38.336 Now to draw an isometric view, we kind of cheat a little bit. 00:01:38.706 --> 00:01:41.866 We use isometric dot paper or isometric grid paper. 00:01:42.276 --> 00:01:45.956 And the isometric dot paper and grid paper are related to each other. 00:01:45.956 --> 00:01:50.206 I've found that the dot paper works a little bit better for my purposes. 00:01:50.676 --> 00:01:58.866 But basically the dots are in straight lines and they go that way, that way and then vertically. 00:01:59.396 --> 00:02:05.126 And if you were to draw lines from these, all these angles meet at 120 degrees, 00:02:05.426 --> 00:02:10.806 or the lines meet at 120 degrees, which is where it comes, the word "iso" comes from, 00:02:10.806 --> 00:02:12.676 which is the Greek meaning the same. 00:02:12.676 --> 00:02:16.166 So all the angles are the same, they're all 120 degrees. 00:02:16.806 --> 00:02:20.596 So how do we draw, then, an isometric drawing using dot paper? 00:02:20.716 --> 00:02:30.286 Well, one of the other things that we do is we use blocks like this that I have in my hand. 00:02:31.126 --> 00:02:36.526 These are snap cubes, and they are easily available through lots 00:02:36.526 --> 00:02:39.156 of different places in math manipulatives. 00:02:39.676 --> 00:02:42.566 And we use these to help us build our buildings. 00:02:42.626 --> 00:02:46.236 But the one thing that we tell the students to make sure about is 00:02:46.656 --> 00:02:51.676 that they don't sketch all of the individual blocks. 00:02:51.676 --> 00:02:53.896 They just sketch the outlines. 00:02:53.896 --> 00:02:57.476 So where you have two surfaces intersecting is where you draw a line. 00:02:57.786 --> 00:02:59.606 You don't outline all the blocks. 00:03:00.176 --> 00:03:05.166 Now, we use a coded plan to define what a building will look like. 00:03:05.456 --> 00:03:10.306 So here is a coded plan: it has a two, a three, a one, a one and a one. 00:03:10.876 --> 00:03:17.106 So the numbers on the coded plan are telling you how high the building is 00:03:17.196 --> 00:03:18.936 at that particular location. 00:03:19.276 --> 00:03:24.296 So right here, the building is two squares high, here is three squares high, 00:03:24.546 --> 00:03:26.686 and all of these are just a single block. 00:03:26.686 --> 00:03:32.056 So the first thing you would have students do is build this particular object. 00:03:32.056 --> 00:03:35.336 So if you start, I've got two blocks here. 00:03:35.886 --> 00:03:38.836 So that corresponds to that area. 00:03:39.906 --> 00:03:42.866 Now I'm going to put in three blocks, 00:03:49.596 --> 00:03:53.396 so now I've got the two blocks here and the three blocks here. 00:03:53.396 --> 00:03:57.236 I need three single blocks in those three locations. 00:03:57.796 --> 00:04:12.296 And so this is my completed building, then. 00:04:12.296 --> 00:04:16.245 And you'll see that it's got two blocks high here. 00:04:16.245 --> 00:04:20.726 It's three here; it's one here; it's one here; it's one here, and it's one here. 00:04:21.026 --> 00:04:24.826 So this building then matches this coded plan. 00:04:25.416 --> 00:04:29.486 Now when you have a coded plan, you need to define 00:04:29.626 --> 00:04:32.536 which angle you're looking at the building from. 00:04:33.066 --> 00:04:40.846 In this particular example, we're going to look at this from this corner here labeled X. 00:04:41.176 --> 00:04:44.186 So if this block, if this building is located here, 00:04:44.476 --> 00:04:47.016 we're going to imagine looking at it down like this, 00:04:47.016 --> 00:04:50.346 which means that the building actually looks like this 00:04:50.346 --> 00:04:52.586 because we're looking at it from that X corner. 00:04:53.516 --> 00:05:00.576 And this, then, is the isometric sketch of what that building looks like from that corner. 00:05:01.016 --> 00:05:05.146 So this, we call the X corner view and this is the isometric sketch 00:05:05.146 --> 00:05:07.586 of this particular building made out of blocks. 00:05:08.436 --> 00:05:13.686 Now a building can have more than one-has more than one corner view. 00:05:13.686 --> 00:05:15.366 It actually has four corner views. 00:05:15.756 --> 00:05:20.446 So here we've got a new coded plan. 00:05:20.676 --> 00:05:24.436 This is the particular building for that, two, three, one and one, 00:05:24.876 --> 00:05:28.246 and you'll see that the corner views are W, X, 00:05:28.486 --> 00:05:33.606 Z and Y. And if you look at it from the W view, it looks like this. 00:05:34.046 --> 00:05:37.376 If you look at it from the X view it looks like this. 00:05:37.666 --> 00:05:40.556 If you look at it from the Y view, it looks like this. 00:05:40.596 --> 00:05:43.886 And if you look at it from the Z corner, it looks like this. 00:05:44.256 --> 00:05:50.026 So we have the students then, we give them a coded plan and we say we want a sketch 00:05:50.026 --> 00:05:55.786 of the W view or the Z view, or you know, whatever the problem is. 00:05:55.846 --> 00:06:00.626 But they have to then typically they would build this building out of the blocks 00:06:00.666 --> 00:06:04.336 and sketch what that looks like from the corner that you specify. 00:06:05.186 --> 00:06:12.956 Now when sketching the object, the easiest way to do this is to think about sketching 00:06:12.956 --> 00:06:15.596 in one surface at a time as shown here. 00:06:15.596 --> 00:06:20.016 So you sketch in this first surface, then the second surface, then you keep going. 00:06:20.636 --> 00:06:25.326 Now what I'd like to do is demonstrate what this looks like-what some 00:06:25.326 --> 00:06:30.616 of the problems we've used look like for this particular type of activity. 00:06:31.396 --> 00:06:32.986 So here we have an object. 00:06:34.016 --> 00:06:38.776 This is a multiple choice type of problem, and here we have a coded plan. 00:06:39.156 --> 00:06:42.066 It's two, one, two, two, one, one. 00:06:42.886 --> 00:06:49.406 If you build that particular building, it looks like that. 00:06:52.096 --> 00:06:57.396 Now a lot of times we don't let the student build the object 00:06:57.426 --> 00:06:59.216 with the blocks, but sometimes we do. 00:06:59.806 --> 00:07:07.246 And then the question is, "Of all of these objects shown here, which one is a view - 00:07:07.246 --> 00:07:10.036 one of the corner views of this particular object?" 00:07:10.476 --> 00:07:16.786 Well, if you look at you can see that the correct answer to this problem is A 00:07:16.836 --> 00:07:20.806 and you're actually looking at it from this corner, right? 00:07:21.376 --> 00:07:28.266 Now, to sketch the object is a little more difficult for the students, 00:07:28.266 --> 00:07:32.626 and it's also actually probably one of the more beneficial parts 00:07:32.926 --> 00:07:36.006 of the spatial skills development. 00:07:36.346 --> 00:07:41.086 So, what you would normally do is have the student build this coded plan. 00:07:41.086 --> 00:07:45.266 So is the three, two, one and then all those one spaces there. 00:07:46.156 --> 00:07:57.266 So if I build this object, it looks like this. 00:07:57.966 --> 00:08:05.246 I've got a height of three, a height of one, a height of two, a height of one, 00:08:05.336 --> 00:08:06.866 a height of one and a height of one. 00:08:07.396 --> 00:08:11.326 This is my object that I have defined in my coded plan right here. 00:08:11.816 --> 00:08:15.116 Now, it says here to, "Sketch the indicated corner view." 00:08:15.376 --> 00:08:20.316 So this particular view, I'm looking at from X, which means I'm going to look at the object 00:08:20.716 --> 00:08:24.636 from this direction and sketch what it looks like from that direction. 00:08:25.426 --> 00:08:31.836 So the way that you do this normally would be to start with the edge that's closest to you. 00:08:31.836 --> 00:08:34.515 In this case that's the edge right here on the object. 00:08:35.116 --> 00:08:36.306 So I would draw that. 00:08:36.306 --> 00:08:41.525 That edge has a height of the vertical edge and it's got a height of one. 00:08:42.366 --> 00:08:44.866 So I sketch that line in there. 00:08:44.866 --> 00:08:50.676 Now, I want to pick one of the receding surfaces from that edge. 00:08:50.676 --> 00:08:54.056 So here's a surface that's going back towards the left, 00:08:54.676 --> 00:08:57.786 it's one unit high and it's two units deep. 00:08:58.446 --> 00:09:06.076 So if I start at my edge I've drawn, go back two units, and one unit high, 00:09:06.466 --> 00:09:10.676 and that is what that surface looks like on isometric sketch paper. 00:09:11.716 --> 00:09:17.016 Now I've got this surface right here, kind of an upside down, T-shaped surface, 00:09:17.316 --> 00:09:21.446 and that is going back from my original edge to the right. 00:09:22.256 --> 00:09:28.456 So if I want to sketch this, it goes back three, it goes up one, in one, 00:09:28.456 --> 00:09:31.246 back up one, and then making the T shape. 00:09:31.286 --> 00:09:38.816 So I'm going to go to the bottom, I'm going to go over three units, go up one, 00:09:39.026 --> 00:09:41.966 in one, and then make the T shape. 00:09:42.476 --> 00:09:50.966 So that is what that particular surface looks like in the isometric sketch. 00:09:51.266 --> 00:09:54.046 Now I can add in all the other surfaces. 00:09:54.496 --> 00:09:57.856 I've got this small vertical surface. 00:09:58.186 --> 00:10:00.836 I've got this small horizontal surface. 00:10:01.426 --> 00:10:05.246 Now I've got this part here sticking up one from there. 00:10:05.756 --> 00:10:14.096 So if I got up one, and then I've got a top surface, and I ran out of dots - but that's okay. 00:10:14.796 --> 00:10:17.516 That goes down by two and across, 00:10:18.196 --> 00:10:25.336 and this goes down by two again and across that front like that. 00:10:25.826 --> 00:10:31.286 So that, then, is what the isometric view of this particular object looks like. 00:10:31.976 --> 00:10:38.016 You can see that we normally have students practice several of these problems 00:10:38.056 --> 00:10:41.856 because it's practice, practice, practice-the more you do the better you get 00:10:42.136 --> 00:10:43.346 at doing the sketches. 00:10:43.766 --> 00:10:47.306 But this seems to help students develop their 3-D spatial skills - 00:10:47.306 --> 00:10:54.936 being able to take a 3-D object and translate that into a 2-D image on a sheet of paper.