This is Common Core State Standards Support Video for Mathematics. This is standard 8.G.2. This standard reads: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
In the future, triangles and congruence are going to be something that students do a lot with, so let s start with a triangle. Now the wonders of technology enable us to take a figure such as this and create a carbon copy. So we already know that they’re congruent. A translation is the simplest of the different transformations. The figures are already in the same orientation, so that’s the key. They’re already situated in the same orientation. All that’s needed is a slide to change the location from one over to the other. So all we have to do is do a slide to change the location from one over to the other; and by doing that everything matches up so we know that one is congruent to the other, because they basically form the same figure.
Now let’s look at this scenario. The problem is that they’re not in the same orientation, so we’re going to need to do something else. Now a rotation is a transformation that involves moving a figure in a circular motion, either clockwise or counter-clockwise, depending on which is best. So this is necessary again when the figures are positioned in different orientations, but what students should do first is label the different vertices and so forth so that they’ll know what is it that matches up with what, which is very important when you start getting into more complicated figures.
So in this situation, label the first one A sub 1, B sub 1, and C sub 1, and the corresponding vertices on the other triangle as A sub 2, B sub 2, and C sub 2. Now what I need to do is rotate this figure around to where it’s basically in the same orientation as my other triangle. So I’ll do that, and once that is done, then I can take one and slide it over. I see that, okay, one fits exactly over the other, so I know that they are congruent. Now if you notice, this involved a combination of a rotation to position the figure in the same orientation. Then I did a slide, a translation, to move one figure over to the other; so again, this is going to be a real common combination for showing congruence. Again do a rotation to put them in the same orientation and then a translation to move one figure over to fit exactly over the other.
Let’s try another example of that combination. Let’s say a pentagon, and we use technology to create a carbon copy. Again what we need to do is label all the different vertices to know what matches with what. Then we simply do a translation, slide it over and see that, yes, they are congruent. I can get one to fit exactly over the other. Now this is a little bit more difficult a situation where I’ve got two figures, and what’s interesting, is that they don’t look congruent when they’re in different orientations sometimes. So what I need to do first is again label everything that I think matches up, then rotate it in such a way where they’re in the same orientation. Then I can take one and slide it over, do a translation, and, viola, yes they do, even though initially it didn’t look like they were congruent; they looked a little bit different.
Now one of the standards in this cluster is related to a transversal intersecting parallel lines, so this might be a good way to connect the two as far as congruence of the figures, which then leads to the idea of the congruence of the matching angles at which they can be applied to this context with the parallel lines and the transversal. So notice that I’ve got two different triangles formed here. So label these according to what they look like, as far as them matching, and then take this triangle here, and what I need to do is rotate it first to get it in the same orientation as the other. Then what I need to do is do a translation, slide it over, and bingo. They do fit; one fits over the other, so they are congruent. Then students can make some connections. Again this connects to that other standard. I believe it’s 8.G.5, where again you can do some things with the transversal cutting parallel lines.
Now there will be situations where the combination of a rotation and a translation isn’t going to get the desired result. So, for example here, if I rotate this around, and then I slide it, whoa I can’t get what I want. I can’t get one to fit exactly over the other. So I end up with something like this, and again, I cannot get one over the other. So this situation requires a reflection, which is the more complex of the basic transformations. Now what’s needed is the creation of a mirror image, but you have to use what’s called a line of reflection. One of the problems though, it’s hard for students to really visualize what’s happening with two dimensions. So this would be a good point to do some reflections using solid concrete figures so that students get an idea of what’s really happening. If I borrow from that same idea where I had the letter F, and we start off with a situation like this, and then here’s my line of reflection that I want to use. And basically what happens with the reflection is again you end up with a mirror image where I do what some people would call a flip.
What’s important is that with the line of reflection that all of your matching points and segments and so forth are the same distance from this line of reflection. So notice that this point right here is one, two, three away from the line of reflection as this point is here; one, two, three away. Of course, I don’t have horizontal lines, but assuming that they’re also lined up nice and perfect vertically and you can do the same thing if your line of reflection is horizontal. It doesn’t matter if it is vertical or horizontal. I might start off like this, and let’s say, we’re 4 away from the line of reflection. Well again, I would do something like this where basically I flip it, and I’m 1, 2, 3, 4 away. So then my reflection would look like this assuming that I have these things lined up perfectly this way also.
Let’s try one more figure. Let’s say I start off with something like this and notice that I am two away from my line of reflection. So basically, what I have to do is flip it like this, where again it’s going to look like this. Notice again, my matching vertices my 2 Cs, my two Bs, my 2 As, my D1, and my D2. Again they have to be the same distance, all of the corresponding segments and points have to be the same distance from that line of reflection. The same would have applied if we would have done this like so using a line of reflection that was vertical. So again, let’s say we started off like this, then my reflection would have been okay, I would have done a flip, and I would have ended up 3 away, like so, again matching Bs, As, Ds, and Cs. So again, this way the students will get a much better handle on what really happens physically when you do a reflection like so.
Now that students have a better idea and a handle on what really happens with a reflection physically, we’re able to now take it a step further and really look at this from more of a mathematical standpoint to fulfill all the necessary conditions for a reflection. So let’s say we have this triangle JHK. We want to reflect it across this vertical line here. Again notice that this point H1 is one, two, three away from this vertical line, my line of reflection. So the best thing is probably to do this like one segment at a time. So let’s say we pick segment JH to do the reflection on first. Again just match your points. So here I need to be a horizontal distance of three away from the line of reflection. Over here, let’s see—one, two, three. Let’s see over here—1, 2, 3, 4. So I need to be a distance of four. So I go 4 this way, and then just connect your points. So there is segment JH.
And let’s say we decide to do KH next. Again same kind of thing; we already have the Hs figured out, and then, so over here, horizontally. Let’s see 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. So I simply start here and go 11 this way horizontally, so there’s my point K. Connect the two, and now we have segment HK, as far as the reflection. Now really there is no more work. All I have to do is just connect J and K, and that should be fine. So we’re set. So here’s my reflection of the triangle across that vertical line of reflection where I have my new triangle; we can call it J sub 2, H sub 2, K sub 2.
Let’s do something similar. But this time, let’s use a horizontal line of reflection. Let’s see, point H1 is only one away from my line of reflection. So that might be the easiest thing to start off with, and so let’s work with HJ. Again it’s just a matter now of counting vertically. How far am I from the line of reflection, and then count off the same distance here. So we have that segment taken care of, and let’s say we decide to go with JK next. Again the same thing; J is already taken care of, so we worry about K. Again we just count the vertical distance here—1, 2, 3, 4, 5. We just go a distance straight down vertically of 5, and there’s point K. There’s my segment JK for my second triangle. Then of course, just connect those two, and so now I have a new triangle formed using a reflection across a horizontal line of reflection.
Let’s try one more. Let’s say again we needed to show that this figure and this figure are congruent. So what sequence do we need here? Well first we need probably to do a reflection. So we need to do it across this line of reflection— one, two, three—so this line segment needs to be to 3 away from the line of reflection. So when we go through all that work, the resulting figure should look like this. Now we need to get it in the same orientation, so let’s do a rotation to get them oriented the same way. Now what we need to do is move this over, do a translation, and bingo, there it is. They are congruent.
So again, this last one is a good example. We had to use all three of the different transformations. We had to do a reflection here. Then we did a rotation to get them into the same orientation, and then we did a translation to get one to fit exactly over the other. So again, use concrete figures, especially with a reflection, so that your students will understand what really happens with the reflection, because that is the more complicated of the three. Again just create situations like this where students can actually have a little bit of fun trying to figure out all the different transformations that would be necessary to get one figure to fit exactly over the other to show that they are congruent. One last point of emphasis is that you should use some type of grid paper, such as we did with the last few examples, because again in particular, with the reflections, it is important that students understand all the necessary conditions for some of these transformations to happen.