This is Common Core State Standards Support Video in mathematics. The standard is 8.G.A.4. This standard states: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
Let’s look to see if there’s any other standards that connect to this. In the same grade level, there’s standard 8.G.A.2, and if you look closely, there’s a lot of similarity between the two. The main difference is that 8.G.A.2 deals with congruent figures, whereas 8.G.A.4 deals with similar figures. And it involves the same transformations, except that 8.G.A.4 also includes dilations, whereas 8.G.A.2 does not. Back in seventh grade, we had standard 7.G.A.1 that talks about reproducing a scale drawing at a different scale. Now, even though the term similar isn’t specifically used, it still deals with similar figures.
You have another standard in the eighth grade that deals with using similar triangles. So, it’s important that students have a good handle on 8.G.A.4 before they attempt 8.EE.B.6. Similarity really isn’t addressed again after eighth grade until high school. Specifically, this high school standard HSG.SRT.A.2 gets a lot more deeply into similarity, and it also focuses on the idea of the proportionality of all corresponding pairs of sides.
Let’s do a quick review of our different transformations. Let’s start with a translation, which is the simplest of the different transformations. These figures are already in the same orientation, so all that’s needed is a slide to change the location of them. So, if we do a slide, a translation, we can get one to slide over and fit exactly over the other. A rotation—that’s a transformation that involves moving a figure in a circular motion either clockwise or counterclockwise. That’s necessary when figures are positioned in different orientations. So, like in this case, they’re not in the same orientation, so I’ll take this one and rotate it around. Now they are in the same orientation, and then I could do some other things with this.
A common combination is a rotation to position the figures in the same orientation and then follow this with a translation to move one figure over the other. So, like here, I would start with a rotation. Get them in the same orientation. Now I can do a translation and slide this one over, and now I’ve got one figure exactly over the other. So, in this case, these two would be congruent.
Now, there are situations when the combination of a rotation and a translation won’t have the desired result. So here, this situation requires a reflection, which is the more complex of the basic transformations. It requires the creation of a mirror image using a line of reflection. So here’s our line of reflection. We have to imagine that that is the center axis of a cylinder, and then we’d rotate that figure around to the other side. So that would be in this orientation. So, now I could do something like a rotation to get it into the same orientation, and then I would do a translation and slide it over, and again, in this case I would have shown that these two figures are congruent.
Looking at the related standard 8.G.A.2 that dealt with congruency, there has been a video already produced for that standard, and that video does contain a three-dimensional illustration of a reflection. So, if you need a little bit more illustration on what a reflection involves, please go to that video. That way we don’t have to reinvent the wheel and do it again here.
All right, a dilation: that involves a change in size rather than a change in location or orientation. Now, when you do a dilation, each matching pair of angles remains the same size and all the matching pairs of sides maintain the same ratio. If this figure was slid over here, and then I did a dilation, notice again the matching pairs of angles stay congruent, and the matching pairs of sides all stay in the same ratio to where I would eventually be able to do this and show that these two are now congruent.
This standard talks about understanding a two-dimensional figure to be similar to another, but just by using transformations, rotations, and so forth, along with probably a dilation. But that still won’t give students an understanding of what it really means for two figures to be similar. So, to really have students understand the idea of similarity we need to take this a step further and really look at the concept. What does it mean for two figures to be similar? Well, two conditions have to be met. First, each matching pair of angles must be congruent. So, like in this case with these two pentagons, each of the two angles in each of the matching pairs have to be congruent to each other. The second condition is that all matching pairs of sides have got to be in the same ratio. So, for example, keeping it simple, if these were in a 2 to 1 ratio, then each matching pair of sides has to be in that 2 to 1 ratio for these two to be similar.
Students have to realize that meeting just one set of these conditions doesn’t ensure that the two figures are similar. For example, let’s take these two. They’re all in a 2 to 1 ratio, so the matching pairs of sides are okay. They’re in the same ratio, but the matching pairs of angles are not congruent. That blows the idea of these two being similar. Let’s look at a second example. If we were to take one of these parallelograms and move it over, I can show that that pair of angles is congruent this matching pair’s congruent, that matching pair’s congruent, and this fourth pair is congruent. So the matching pairs of angles are congruent, but the matching pairs of sides are not in the same ratio. Notice that, for example, one matching pair of sides, 5 and 10, that’s in a in a 1 to 2 ratio, but then, let’s look at the top ones. Oh, that’s a 12 to 11 ratio. That’s not equivalent. So, again, that ruins these two being similar to each other. Pairs of angles were congruent, but the matching pairs of sides were not all in the same ratio.
Let’s look at the last part of the standard: given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. So, let’s say we have these two pentagons. If I were to do a translation to get this over here in a proper position, at this point, I can use a translation to show or informally prove that each of the matching pairs of angles are congruent. So, what I could do here, and again, this is going to be one of the key ideas, is that you can use translations to show equality of the angles. So, like here, okay, that pair of angles is congruent, that pair’s congruent, this pair, that pair, and that pair. So that’s going to be a big use of translations, again to slide one figure around to show that the matching pairs of angles are equivalent.
Remember that for two figures to be similar, two conditions have got to be met. So, here I’ve also got to have some idea either with numbers or something to know that the corresponding pairs of sides are all in the same ratio. So, again, they need to verify that the matching pairs are equivalent. So, in this case, all of these would simplify to a 4 to 3 ratio. So again, it meets that second condition.
Let’s take this pair. Well, first of all, they’re not in the same orientation, so I need to do a rotation to accomplish that. So I’ll do a rotation. Now they’re in the same orientation. Now I can do a translation. Move this over. Now we are in a position to use a translation to check for congruency. So we move this over—that pair of angles congruent, this pair congruent, this pair congruent, and the fourth pair congruent. But again, we have to make sure that all matching pairs of sides are in the same ratio, and in this case they are. They’re all in a 2 to 1 ratio.
One more example; let’s take these two triangles. Well, got to slide it over. I can show, okay, that first pair, those angles are congruent. Second pair—those angles are congruent; third pair likewise. We’ve taken care of the angle part of it. Now we need to make sure that the matching pairs of sides are all in the same ratio. But they didn’t give me any numbers here. So, what you might have students do is actually do the measurements to ensure it. So, let’s see, that’s 3. Measure that, that’s 4 units, and that’s 5 units. So, on the other triangle, measure the hypotenuse. That’s 10, that’s 8, and that is 6. So each matching pair is in a 1 to 2 ratio. So that takes care of the second condition dealing with the corresponding pairs of sides being in the same ratio. As an option, what you could also do is use a grid, use a graph, and you can do something like this to create some triangles, and in this case, right triangles. And again, you can create several different similar triangles just using your X axis and that line.
Let’s look at our standards for mathematical practice. Looking at the first four, doing some of the things that are involved with these activities, students would reason abstractly and quantitatively. Especially if they work in groups, they’ll be constructing viable arguments and critiquing the reasoning of others. And then standards 5 through 8, students would be doing three of those. They would use appropriate tools strategically. They would need to attend to precision, and they would look for and express regularity in repeated reasoning.