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- [Voiceover] So here again with Bill McCallum, and what I thought we would talk about today is how the Common Core looks at geometry, especially high level geometry: 8th grade and high school, and what I find especially interesting, at least in my own read of the Common Core, is that historically, arguably for thousands of years, Geometry has really been based on Euclid, and Euclid's elements, and this idea of kind of axiom based proofs, and it was the first math class for most of us where we really got kind of exposed to the idea of a rigorous proof. Well, at least in my reading, it looks like Common Core is much more about kind of the conceptual understanding of manipulating and transforming geometric shapes. - Well, I would, that's not quite true. It's still about rigorous proof in high school. The question is, where do you start? And traditionally Euclidean Geometry starts with askings about lines and points and there's a very very long series of theorems you have to prove about those, which I think for most people, they're thinking, "Why are we proving this?" And so, what Common Core does, is it starts at a higher level, it starts taking for granted the properties of what we call rotations or, well, congruent transformations: rotations, reflections and translations. And then using that as an Axiom addict basis, from there on you're doing pretty much straight deductive Geometry. At least in High School. The Middle School standards we're looking at now are a preparation for that. So really what it does, is it jump starts you into the interesting part of Euclidean Geometry. - [Voiceover] Right, ok, I guess I see what you're saying. because I remember when I first learned proving something was congruent, or proving that they were similar, in my brain I did kind of develop a, "Hey, OK, well similarity? "Ok, I can understand I can do these transformations "that it will get there." It wasn't formerly part of the curriculum. This is kind of doing that, because I think anyone who really does get Geometry to a conceptual understanding does understand the notion of transformations. - That's right and if you look at Euclid's definition of congruents, he says, "well, two figures are congruent "if you can move one on top of another." Now, he never really explains what that means. But, in fact, defining rotations, reflections, and translations as the movements you can make formalize that. And then there are things we assume about those transformations, for example, if I rotate a figure, all the distances stay the same. Now, in Euclidean Geometry, you have to prove that. In what we're doing, is we're saying, "Let's take that as an assumption, "and then let's move on from there." Because a lot of those early proofs, our angles stay the same when you do a similarity transformation. They still have to prove. And so we're sort of jump starting into a level of Euclidean Geometry, where you start proving interesting theorems about intersecting, bisectors and so on. - [Voiceover] Hm, and the other thing that's clearly different is typically Geometry, especially these notions of similarity congruents get introduced in 9th or 10th grade, while in the Common Core, they get introduced in 8th grade. - Yeah, but they're introduced in 8th grade in a pretty hands-on informal way. They're pretty intuitive notions, you can get quite nice things with transparencies where you just show these movements. and you get an idea of that's the way to think of congruents. In fact, in some ways it's more intuitive than the traditional way of introducing congruents in terms of distances and angles. Those end up being things you deduce from playing around with these congruent transformations. - [Voiceover] And if I'm a teacher, or I guess even a content creator, I guess we are kinda a muse and content creator. What, I mean, a lot of this is about using manipulatives. You just mentioned transparencies, how would a teacher go about, I guess, assessing or giving students practice in some of these types of things? Is it literally like, cut things out and move them around? - Well, I mean there's some pretty cool online tools. In fact, Khan Academy has some pretty cool online t-- - [Voiceover] I did not ask you to make that plug, but yes we have been (Bill laughs) developing specifically for rotations and transformations, a lot of fairly unique tools. - And the nice thing about the online tools, and there are plenty available, is you can track what kids are doing, and you can have them manipulate a figure in a certain way and then answer a question that they have answered as a result of that manipulation. - [Voiceover] Right, so your answer is, Khan Academy is the best solution. (Laughs) Or at least it's one of them. But I guess it's both, and some of the standards literally call out, actually, I'm having trouble finding it right now, literally do call out potentially using software tools - Absolutely, mhm. - [Voiceover] And obviously, - we've been investing here at Khan Academy in a lot of these tools specifically for these standards, here. - Right, the other thing I want to emphasize is that a lot of people, there's a subject called Transformational Geometry that some students study in college. This is not that. There's a whole sort of bunch of stuff about the Algebra of transformations that is not what the Common Core is about. And people make that mistake sometimes. - [Voiceover] Right, the Algebra, I mean, we're talking about kind of, transformation matrices and stuff like that. - Transformation matrices, but also, the product of every transformation will be expressed as the product of three reflections, for example. It's beautiful stuff, but it goes beyond the Common Core. We're really just, with the Common Core you can quite quickly derive the standard criteria for triangle congruents: angle, side angle and so on. And once you've gotten there, you could just teach a fairly traditional Euclidean Geometry course. Or, you could continue to use the transformations. There are choices there. And so, it's not as much a departure from Euclidean Geometry, I think, as sometimes people get the impression that it is. - [Voiceover] Right, that's interesting, Because I remember, when I first learned, You know, I've never been much of a memorizer. Whenever I'm trying to figure out which are the valid Geometric, you know, the ASA, SSA, I actually probably am doing these transformations in my head to re-evaluate them - Exactly! exactly, and you can prove these using transformations. There's quite beautiful, some of the proofs are harder than others ASA for example, you might move one of the angles to a corresponding angle then rotate so they match up and notice that the lengths match up, and then notice the other angles match up so you've got all the sides matching up, so the triangle matches up. Well, that was a very, very quick proof of ASA, using rotations translations and reflections. - [Voiceover] Right, right. And so the progression, once again, we get first exposure in 8th grade, really getting this high level understanding of it, and then as we go into the High School standards we just continue that. And all the way from things like, I mean we do start with, there is this notion of precise definitions which is no different than what's always been the case. But then we start representing transformations of plane And here is actually where you talk about transparencies and Geometry software, And then we keep building on that, Geometric descriptions of rigid motions to transform figures. Definition of congruents in terms of rigid motions So these are all of the things that we've just been talking about. - Yeah, and this is where you start proving the classical theorems of Geometry. - [Voiceover] Right, right. and so once again, Your advice to a, I guess there's two teachers that would have to think a lot about this, which is one, the 8th grade teacher now, which this might be a new area for them. Then the traditional 9th grade Geometry teacher. It sounds like the 9th grade Geometry Teacher, it might be, if they still keep the classes at 9th grade Geometry or 10th grade Geometry class, is to kind of introduce these theorems through the transformational mindset but then they can kind of continue from there in a fairly traditional manner. - Yeah, you can take two routes, really. If you wanted to stick as close to traditional development, you would use the transformations to prove the criteria for triangle congruents and triangle similarity which are classically learned. And then from there, you could if you wanted to, follow a completely traditional course. There are some things where it's really nice to use the transformations, however. For example, the theorem about the base angles on an isosceles triangle being congruent, that is a very nice transformational proof using treflections of that theorem. You could also prove it using triangle congruents criteria. - [Voiceover] Yup, awesome. Well thank you, that was very helpful. - Sure! - [Voiceover] Alright.