Mathematics

8.G.A.3 Transcript Parts 1 and 2

This is Common Core State Standards support video in mathematics. The standard is 8.G.A.3. This standard reads: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

First, let’s look at how this standard relates or connects to other standards that deal with this area of translations, rotations, and so forth. In the same grade level, a precursor to this one is standard 8.G.A.1, and it basically talks about the properties of these different transformations. For example, in part A, they’re talking about lines being taken to lines, and segments being taken to segments of the same length, angles being taken to angles of the same measure and parallel lines being taken also to parallel lines. This eighth-grade standard sets the foundation for something a little bit more complex at the high school level. Standard HSG.CO.A.4 is about developing definitions of rotations and reflections and translations in terms of angles, circles, perpendicular lines, parallel lies, and line segments.

In the same grade level, we have standard 8.G.A.2, which is just prior to the one that we’re addressing. It deals with congruency, but it connects it to the idea of rotations and reflections and translations. Following 8.G.A.3 is standard 8.G.A.4, which again, is very similar except this one deals with similarity, and it ties that idea to these different transformations. Notice though that 8.G.A.3 involves dilations, which is, of course, a little bit different than congruency because of the idea of a scale factor.

So, let’s look at dilations. What else in the standards is connected to this idea? Well, back in fifth grade we had standard 5.NF.B.5, and it talked about interpreting multiplication as scaling. Now, the important pieces here are this idea about multiplying a given number by a fraction greater than one results in a product greater than the given number, and also if you multiply a given by a fraction less than one, that results in a product smaller than the given number. So, that kind of parallels what happens with dilations.

With respect to dilations, this standard, 8.G.A.3, sets the foundation for something more complicated in high school. This is addressed in standard HSG.SRT.A.1 that talks about verifying experimentally the properties of dilations given by a center and a scale factor. One of the things that is addressed at the high school level is this idea of the center of the dilation as opposed to no mention of that in eighth grade. However, you do have this idea of a scale factor, and that’s really important.

That idea of a scale factor connects to a ratio. So, back in sixth grade, we have standard 6.R.P that talks about understanding ratio concepts and using ratio reasoning. Then we have a seventh grade standard, 7.R.P that deals with analyzing proportional relationships and using them to solve real-world problems. Although the exact phrase scale factor is not used specifically, teachers need to connect dilations to these clusters of standards that focus on proportional reasoning because that is what happens with dilations. You do have a scale factor that has a huge impact on how large or small the dilation will be in comparison to the original figure.

Let’s see what other standards are connected to the idea of reflections. Back in fourth grade, we had standard 4.G.A.3 that talked about a line of symmetry, which is an integral part of the idea of a reflection. Now, this eighth-grade standard talks about these different transformations but with respect to coordinates. So, connected to that idea is standard 6.NS.C.6b that talks about understanding the signs of numbers in ordered pairs. But, what is especially important is the latter part of this standard that talks about recognizing that when two ordered pairs differ only by their signs, the locations of the points are related by reflections across either one or both axes.

So, let’s start off with translations and see how two-dimensional figures are impacted when it comes to the coordinate plane. Let’s start off with a basic triangle, and there are our coordinates. Let’s say we’re going to shift it to the right and up. So, let’s shift it, and this is where our new location is. Here are our new coordinates for each of the three vertices. Well, let’s see what happens. If we look at all of the x coordinates; if we pair them up, we have 3 compared to 10, 6 compared to 13, and 3 compared to 10. Let’s see. It looks like there’s a difference of 7. So, for all of the x coordinates, the matching pairs, the difference is 7 in the positive direction. If we look at the y coordinates, let’s see, 8 compared to 12, 8 compared to 12, and 13 compared to 17. That’s a difference of 4, and that reflects the idea that we shifted upwards by 4.

Let’s look at another translation. We start off with the same original, but this time let’s shift to the left and down. Here is the new triangle in its new location with the new coordinates. If we investigate the x coordinates, 3 compared to -5, 3 and -5, and 6 and -2, that looks like a difference of 8, and they’re 8 smaller, which reflects a shift of 8 to the left. If we examine the y-coordinates, 13 compared to 10, 8 compared with 5, that looks like it’s a difference of 3, and it’s 3 less. And that reflects our shift of 3 down. So, the translation here involved a shift of 8 to the left and 3 downward. Each of the corresponding pairs of x and y coordinates reflected those.

If we generalize this, if we start off with our original coordinates, we’ll call them x and y, our new coordinates would be...well, let’s see. If we let h be the horizontal shift, then the new coordinates would be, for the x, we simply add whatever the horizontal shift was, and by add we mean we could be adding a positive or a negative number depending on the direction of the shift. If we let v be the vertical shift, then for the y coordinate, it’s the same deal. You’re going to add the vertical shift to the y coordinate, and of course, by add we mean adding a positive number if it’s an upward shift and adding a negative value if it’s a downward shift.

We need to lay the foundation as to how some of these figures are labeled. We start off with our original figure being triangle ABC. There are our coordinates for this particular example. Then we did a translation where we, in this case, shifted to the right and up. Notice that in order to better try to match up each of the vertices, we’re labeling them using the same letters, but to distinguish one from the other, for the new position, A would be A', B would be B', and C would be C'. So, our new triangle after being shifted, is triangle A'B'C', and there are our new coordinates. Again, in the future, this is what students can expect as far as how vertices are labeled after some type of translation.

Now, let us look at reflections. Let’s start off with a triangle, and we’re going to reflect it across the x-axis. Here are our coordinates, and let’s color code the lengths of the sides so that we can identify each one pretty quickly. If we start off with the point (5,3), if we reflect it across the x-axis, there’s our new location. If we reflect the point (8,3), there’s its new location, and then we reflect the point (2,9). Here are our new coordinates. Now we connect all the vertices and color code them to make sure that we can again, distinguish which side is which.

If we examine what happens with the x coordinates, well, let’s see, both of these x coordinates are still 2. Both of these are still 5, and the new x coordinate here is still an 8 like the original. So, if we let x and y be the original coordinates, so far the x coordinates stayed the same, and now we examine the y coordinates. Let’s see. It was 9 but now it’s -9, 3 versus -3, and another 3 versus a -3. So, it looks like our y coordinates, when we reflect across the x-axis, will be the opposite value. Now, notice that this should be described as the opposite of y, not negative y because that can be very confusing. So, it’s very important here that we not confuse things by saying negative because it may not, in fact, be a negative number. We’re just talking about the opposite number, the inverse of it, the additive inverse to be more exact.

What about reflections across the y-axis? Let’s reflect this point across the y-axis then this vertex and the third vertex. Let’s figure out our new coordinates. Let’s connect the vertices and color code the segments accordingly. Let’s examine what happens with the coordinates. Well, let’s see. Here we started off with a coordinate of 2 for the x, and now it’s a negative 2. For this point, originally we had a positive 5 for the x coordinate. Now we have a negative 5, and then here, we started off at 8, and now it’s negative 8.

To generalize, if the original coordinates here reflected across the y axis were (x, y), well, let’s see, it looked like on the x coordinates the values for x are now the opposite, they’re the opposite sign. Again, we need to describe this as the opposite of x, not negative x. For the y coordinates, let’s see, still both 9’s, the 3 remained a 3, and that 3 remained a 3. So, here’s our generalization. If we start off with original coordinates (x, y), if we reflect across the y-axis, the x coordinate will be the opposite sign, and the y coordinate will stay exactly the same.

What if we take a figure and reflect it across both the x and the y-axes? If we reflected the original triangle across the x-axis and the y-axis these would be the results, and now, regardless if you take the triangle that was reflected across the x-axis or the y-axis, when we take this vertex and reflect it, here’s where we are. If we take that vertex and reflect it, there’s our new location, and then reflect this point. Let’s connect those points. Color code our segments. Let’s figure out what our new coordinates are. Now, let’s compare to see what happened with the x coordinates and the y coordinates.

Well, it looks like, let’s see, if our original coordinates are (x, y), our new coordinates—well, on the x’s, all of the values are the opposite signs. So, whatever the original x coordinate is, the new coordinate for the x when we reflect it across both axes will be the opposite of x. When we look at the y coordinates, same thing. All of the values for the new coordinates on the y’s are the opposite sign. So, it looks like if our original coordinate vertically is y then the new coordinate would be the opposite of y.

Let’s try another example just to be on the safe side. Let’s say we start off over here in the second quadrant. We reflect across both axes. So, here’s our new triangle, our new location. Let’s see what happened. Here are our original coordinates. Here are our new coordinates, and we’re labeling these like we mentioned a while ago. Triangle ABC becomes triangle A'B'C'. Now, let’s compare the coordinates. If we look at the x’s, yes, they’re all the opposite signs, and when we look at the y’s, the same thing. All of our y coordinates are the opposite sign of what they started out to be. Again, note that our negative sign, subtraction sign, whatever you want to call it, is being interpreted as the opposite of, not negative.

Now, here’s something interesting. This is what can lead to confusion if we’re not careful. Okay, notice here the way that the original coordinates are labeled and the way the new coordinates are labeled. Notice here that the sign is being interpreted as negative rather than the opposite of. So, in other words, the way that it’s labeled here, being that it’s in the second quadrant, this is saying that our x coordinate will be negative, and our y coordinate is positive, as opposed to being in the fourth quadrant, we would have positive x coordinates and negative y coordinates. But, again, it’s not a good idea to do this with variables because the meaning of that sign can really confuse students. So, labeling it like this is not a good idea. You should always start off labeling your original coordinates as just (x, y), and then worry about if it’s a positive or negative number once you’ve figured out what the coordinates are.

So far we’ve talked about reflections across the x or the y-axis, but what about other lines of reflection? It really doesn’t specify in the standard whether or not this should be included, but just to be on the safe side, well, let’s see what happens when we reflect across a vertical or horizontal line, something other than the x or the y axis. So, let’s say we have a line of reflection that’s a vertical line with the equation x = 9. Let’s start off with this triangle with these coordinates, and let’s see what happens when reflect it across this line. Here are our new coordinates. So, let’s see what happened. Let’s see. On the left, we have 6 and 6, we have 6 and 6, and we have 12 and 12. It looks like our y coordinates are still the same value, which makes sense because it was actually horizontal shifts that happened here. Nothing happened vertically. So, if we generalize this, if we have original coordinates of y, our new coordinates will also be y’s, the same thing, no change.

Now, let’s see, what happens with the x coordinates? Now, this is a little bit different because we’re not at an axis. Now, notice that the x coordinates for those top two vertices, one shifted 2 to the right and the other shifted 2 to the left. So, we have a 7 versus an 11. Then for these two vertices, we shifted 5 to the right on one, and we have a shift of 5 to the left on the other, comparatively speaking with respect to this new line of reflection. Then for the outermost vertices here, let’s see, we went 8. We were 8 away from the line of reflection on one in the positive direction, and we’re 8 away in the opposite direction for the other.

So, this is what we have. X is the x value for our line of reflection. In this case it was 9. When we generalize, the little subscript, r, that’s what that indicates, that it’s the x coordinate for our line of reflection. So, for the original triangle, I was a certain horizontal distance from the line of reflection, and that’s reflected with the variable h. So, for the original, I was so much of a distance to the right of the line of reflection, and then notice that once the reflection happened, I was the same distance from my line of reflection, but in the opposite direction. And that’s how come you have the opposite of h as far as the distance. So, in one case, you’re going to reflect to the right, and in the other case, you’re going to reflect to the left.

Let’s try a horizontal line of reflection. Let’s say y = 5, and let’s start off with this triangle. Let’s reflect it across the line y = 5. Here are our new coordinates, and we color code our segments. What happened with the x coordinates? Hmm, no change—2 and 2, 5 and 5, 8 and 8, and again, that makes sense because there was no shift to the left or to the right. All the shifting took place vertically. So, if our original coordinates are x, then the new coordinates for the x values will stay the same. Nothing changes there.

So, now let’s look at the y coordinates. We have our line of reflection being y = 5. Let’s see, for that one pair of coordinates, one is 9 away in one direction. One is 9 away in the other direction. Let’s see. We’re 3 away on this set, and we’re 3 away from the line of reflection in that set, just in opposite directions. So, here’s how we state that algebraically. If this y subscript r is our y coordinate for the line of reflection, then in this case, the original y coordinates were a certain vertical distance away, and it was above it. So, that’s how come we have the plus, and then, when we reflect across that horizontal line, then the new coordinates would be whatever the y coordinate is for the line of reflection, and we would subtract the vertical distance from the line of reflection.

One of the things that we have to be careful about here is that students can see the numbers and everything, but when we generalize and we use variables, it’s very, very important that students understand what those variables represent. It needs to be clear as to what goes with what and what these different letters and subscripts and so forth represent.

Now let’s look at what happens with rotations. We will only be dealing with rotations about the origin. When it comes to other types of rotations, that’s more at the high school level. Let’s say our original point is at the coordinates (2, 5). If we try to do this with rotations using a circle on the graph and leaving it in the same position, that’s going to get pretty complicated. It’s going to be very difficult for students to figure exactly where they end up after a rotation. So, here’s an idea. This is a lot easier. It’s a lot less confusing. It’s best to just take your whole graph paper and just rotate it. In essence, you’re going to be changing the positions of the x and the y axes, but, it will be very easy to tell exactly what happens after a certain rotation takes place.

So, let’s do that. Let’s say we’re going to rotate 90 degrees counterclockwise. That’s what the little CC stands for. Okay, so let’s just take our paper and rotate it 90 degrees in a counterclockwise direction, and here is where we end up. So, let’s see what our values are for our coordinates now. Okay, it looks like we would have to go 5 to the left and up 2 to be in our new location. So, that’s what happens with a 90-degree rotation counterclockwise.

Now, let’s keep going. What if we go another 90 degrees for a total of 180-degree rotation in a counterclockwise direction? So, let’s do that. Okay, so in essence we have actually switched the location of the x and the y-axes again. So, let’s see. Where are we? Okay, we go 2 to the left and 5 down. Hmm, so we’re at the point (-2, -5). Now let’s do a little bit more in the counterclockwise direction. Let’s go another 90 degrees so that it will be a total of a 270-degree counterclockwise rotation. So, let’s do that. Let’s just rotate our whole graph paper around. We stop here. We figure out that our new coordinates are (5, -2)

But, let’s not stop there. That’s just going in a counterclockwise direction. Now let’s try clockwise. So, let’s start over again with our original location for our point being (2, 5), and now let’s go 90 degrees clockwise. So let’s do that. So, here we are. Let’s figure out our new location. We went five to the right and down two. So, our new coordinates here are (5, -2). Let’s keep going. Let’s go another 90 degrees for a total of a 180-degree rotation clockwise. Figure out our new coordinates. They will be (-2, -5), and one more time, let’s rotate clockwise 90 degrees more. Here’s our new location, five to the left and up two. So, our 270-degree rotation clockwise had us end up at (-5, 2) for the coordinates.

Now, let’s double-check ourselves. Let’s see what happens, and let’s try to see what things we might have in common and whatever patterns we might have. Well, let’s see. If we look at our original coordinates of (2, 5), there’s a similarity between a 90-degree counterclockwise rotation and a 270-degree clockwise rotation. Well, let’s see. Let’s say that was our starting location, and let’s look at what these actual rotations would involve. Let’s see. If I start there, and let’s take our 90 degrees counterclockwise. So, we start here and go 90 degrees counterclockwise, and we end up here. So, for the second option, 270 degrees clockwise, we start here and we clockwise around to here and we end up at the same place. So, that checks out.

There’s a commonality here. It looks like we have the exact same thing if we go 180 degrees counterclockwise, or if we go 180 degrees clockwise. Well, that makes sense, but let’s double check it to make sure. Here’s our starting point. So, if we go 180 degrees counterclockwise, we start here. Go 180 degrees so we ‘ll end up here. Then if we start here and go 180 degrees clockwise, yes, we’ll be at the same location, so that one makes sense. It looks like we have this pairing here. We started off with the coordinates (2, 5) and we ended up with coordinates (5, -2), so let’s check that.

Okay, here’s our starting location. Let’s see, let’s try going 270 degrees counterclockwise, so it would be like so, and we end up here, and then we start here and we go 90 degrees clockwise. Yes, we end up at the same location with either one of those rotations. If we let our original coordinates be (x, y), what is the pattern here? Well, let’s see. For our 90 degree counterclockwise and 270 degrees clockwise, these are our new coordinates. How do they compare to the original? Well, it looks like our x coordinate here is now the y coordinate after that rotation, and our new x coordinate is what used to be the y originally, and it’s also the opposite sign, and if we examine the numbers as to what happened here, that is, in fact, the case.

Let’s look at this pairing, 180 degrees counterclockwise or 180 degrees clockwise. Well, it looks a little bit simpler here. It looks like the x and the y coordinates stayed the same as far as the actual numbers, but notice that in both cases it’s the opposite signs. Okay, so that makes sense. So, if the original location has coordinates (x, y), then with a 180-degree rotation either clockwise or counterclockwise, you’ll end up at the opposite of x and the opposite of y for your new coordinates. Let’s take this third pairing. What happened here? Let’s see. We have fives for the x’s. We have 5 for the y over here, so, okay, so now what happens here, my x coordinate is now what used to be the y coordinate, and then for our new y coordinate, it’s what used to be the x, and it’s also the opposite sign.

Let’s see if this works. Let’s try this with some type of geometric figure. So, let’s take our graph and let’s say this is the figure. It’s a triangle, and here are the coordinates. So, these are our originals. Let’s say we decide to do a 90-degree clockwise rotation. So, we do that. Now what we need to do is figure out our new coordinates, and so here they are. They would be (4, 5), (4, 8), and (10, 2). Now let’s compare that to the original. Well, let me see. Here, the original x is now our y coordinate, and it is the opposite sign, so that makes sense. Then the y coordinate originally was 10 and our x coordinate over here is a 10, so that makes sense. Our x coordinate now is what used to be the y originally. Let’s compare, let’s say, the (-5, 4) original starting point. What happens here? The new y coordinate, it is the opposite of the original x, and then our x coordinate for the new location used to be the y coordinate over here. If we examine this other set of points, the (-8, 4), the exact same thing happens. So, this is valid as far as the generalization.

If we’re talking about translations, rotations, and reflections, note that in all of these, the only changes involve the position and/or the orientation of a geometric figure. The measures of the angles and the sides remain constant. There were no changes in size or shape. Now let’s look at dilations, and that’s a little bit of a different animal. Dilations differ from the other listed transformations. Although shape is preserved, the size of the new figure is different due to the changes resulting from multiplication by the scale factor. So, again, the shape is going to be the same. It’s not going to change at all, but the size is what changes. It’s either going to be a larger or a smaller image of the original. Mathematically, the way that we would say that in much simpler fashion would simply be that the resulting figure from a dilation will be similar to the original. Of course, similarity simply says that you would have the exact same shape but not exactly the same size.

Let’s look at what happens with a dilation, and of course, we need to connect that to the idea of coordinates. Let’s start off with this triangle ABC. Let’s figure out and label our coordinates for each of the vertices. Let’s say we want to do a dilation, and our scale factor is two. So, what we need to do is simply multiply the x and the y coordinates of each of those vertices by 2. So, we take our original vertex A and we multiply both x and y coordinates by 2. We get the new location A' being (6, 10). For our B, if we multiply again, everything by 2, we get the new location for B as (12, 4) for the coordinates, and then for C, double the x and the y. We get (22, 14) for that new location. Connect the vertices and here’s our new triangle, triangle A'B'C'.

Earlier we talked about this idea of the center of dilation. It’s not mentioned anywhere in this standard, but it won’t hurt to maybe go ahead and lay the foundation for this so that we don’t wait until high school to do it. What makes this a lot simpler and really not that complicated is that our center of dilation is the origin. So, that really simplifies things a lot. So, here’s what happens. If we start at the origin and we draw a line that goes through our first vertex A and we keep it going, it’s going to also go through the new vertex A'.

Same thing happens here. When we draw a ray that goes through B, it’s going to go through the new location B', and the same thing with C. Now, here’s where the scale factor comes in. The scale factor was 2, so what’s going to happen is that the distance from the origin to A' is, in fact, twice the distance as it is from the origin to A. The same thing here–this segment to C' from the origin is going to be twice the distance as the segment from the origin to point C, and the same thing happens with the segment that is drawn from the origin to B'. That would be twice the length as from the origin to point B.

Let’s try a second example. Let’s do another dilation. This time let’s go with a quadrilateral ABCD. Here are our coordinates. Let’s go in the opposite direction. Let’s have a dilation that makes the original figure smaller. Let’s say our scale factor is 1/3. So, just like before, all we’re doing to get the new coordinates is multiplying all of our x and y coordinates by 1/3. So, for our A coordinate, our new one will be A'. We divide both of those by 3, so we get the point (1, 4). For point B' we divide by three again, which is, of course, taking 1/3. So, we get (2, -1). Our new C coordinate is (4 2/3, 2), and then for the new location of point D, that’s going to be 3 for the x and 3 2/3 for the y.

Let’s connect those to get our new figure, and sure enough, it’s the exact same shape and everything, but it’s definitely a different size. Just like before, our center of dilation is going to be the origin, which makes things a lot simpler. Let’s start at the origin and draw a ray that goes through both the new location and the original location of our A vertex. Let’s do the same thing for our point B. Let’s do it for vertex C and C', and let’s also do it for D. If we were to compare the respective distances to the new locations for the four vertices, being that the scale factor is 1/3, each one of these distances from the origin will be 1/3 as much as the distance was to the original locations. Again, the reason is because the scale factor is 1/3.

The idea of dilations is relatively new probably to most students and even to some teachers, so it’s real important that we get these basic facts down. But again, it all connects back to the idea of similarity. We’re going to have the exact same shapes of these figures but the sizes will change, and of course, that is determined by our scale factor, and as you can tell, it would be a very easy connection to the idea of proportionality.

If we look at our standards for mathematical practice, by doing the activities that are involved with this standard, students would reason abstractly and quantitatively. They will construct viable arguments and critique the reasoning of others. If we look at the remaining standards for mathematical practice, by doing what’s involved with this standard, they will use appropriate tools strategically. Of course, they do need to attend to precision as you could tell, and last but not least, they will look for and express regularity in repeated reasoning.