Transformations change the size or position of shapes. Congruent shapes are identical but may be reflected, rotated or translated. Scale factors can increase or decrease the size of a shape.
Equations are incredibly important in transformations as they help determine lines of symmetry, centres of rotation and many more...
In this lesson, we are working through transformations. The learning objective is to be able to reflect, rotate, enlarge and translator shape. And through this lesson, we are going to go through the difference between lines of symmetry, tree and rotational symmetry, understand what is meant by a plane of symmetry and also be able to rotate, reflect, translate and enlarge an image. Let's make a start on line of symmetry and just line symmetry in general. So if you can draw any shape and you can draw a line through that shape so that both halves are identical or they're mirror images of the other, the shape has a reflection. And that line there that we've just split the shape into half is called a line of symmetry. Now we'll come back to lines of symmetry when we actually do reflection. So we'll move on to rotational symmetry now, which is slightly different. So an object has rotational symmetry if it fits exactly into itself when it's turned about a point at its centre. So, for example, here we've got this image here. If I take the top right corner and I rotate it once and it becomes positional here, the shape looks identical. If I go again, the shape looks identical, and if I go third time it looks identical, and then a fourth time it's back to the original place. So you would say that this shape has rotational order symmetry. For that top corner moved 1234, and in all four times they looked the same, the shape did not change in any way at all. So that's rotational symmetry. The next one is the planes of symmetry. So it's essentially when you have a 3D shape and you divide it into two symmetrical parts. Now, a cube is quite a special type of shape because it's more than one plane. So if I slice this right down the centre like that, you can see that the two halves made are identical. So that's a plane of symmetry. But with a square, sorry, with a cube, you can also slice it down a centre like that, or place your plane down here. There's loads of different versions where the two halves look identical. Think about slicing it in the two. Are the two halves made going to look the same? Okay, so that's what a plane of symmetry is. And you can do that with all sorts of shapes, like a cone, cuboid, a sphere. So planes tend to be dealing with 3D shapes. Let's now move on to the four different types of transformations. So reflection is the first one. As we said before, an object can be reflected in a mirror line or axis of reflection to produce an image. If you have a look at this image here, we've got a direct reflection. The wording, the letters here actually backwards as well. Now, when you are explaining in an exam about reflections, you need to say that it's a reflection. You always need to give the type of transformation, but also you need to give the equation of the line. So this reflection on this mirror line, if it's on an XY grid, will have an equation. So for example, if it's a vertical line like this, your line equation will be x equals something because it's crossing the x axis. If your reflection line was actually horizontal because it's crossing the y axis, this time your equation would be y equals something. Now, if your line was going diagonal, then you would say that the equation is y equals x. And if it's the other way around, it would be y equals minus x. So here, if you look at the original object and this is the mirror image, the distance between each corner or each part of the shape is always equal from the point to the line back from the line to the new point. For example, if this was six squares in, I need to go six squares out. So that has to be equal. And that's the case for all of these corners. Okay? Now one key word is congruency or the shapes are congruent. And that basically means they are exactly the same. So you can flip a shape and rotate it. As long as the size of it does not change, then you can say they are congruent. Okay, so the next one is rotations. So a rotation occurs when an object is turned around a fixed point. So just to describe a rotation, we need the angle of rotation. So if it's a half turn, a quarter turn or a three quarter turn, you need to give the angle of that rotation, which is 180 90 degrees for a quarter and 270 degrees for three quarters. We need to give the direction of the rotation, which would be anticlockwise or clockwise. The only time that doesn't matter is if we're dealing with a half turn or 180 degree turn, because the image would be the same going clockwise or anticlockwise. So the centre of rotation is basically the pivot point of your shape. So where you're placing your pencil when you're rotating your chasing paper around, here is a diagram. And you've got your triangle here, and that is my centre of rotation. So if I place my pencil in that spot, draw my shape onto my chasing paper, and if I wanted to do a 90 degree clockwise rotation, the image would end up here. So when you're talking about rotations, you need to give the centre of rotation as a coordinate. That would be one comma. Two, you need to give the degree of turn. So 90 degrees, 180 degrees or 270 degrees. And you would also need to say, don't forget to say it's actually a rotation. And whether it's clockwise or anticlockwise. Okay, the next type of transformation is the translation. This is essentially where an object moves in a straight line from one part of your grid to another. So the distance between each point on your original to your translated image would be the same. So here I'm going two up and 12345 across to get a to go to the new position. So all of these points would be going two up and five across. C is going two up, five across. So that's your translated shape. When you're describing a translation, you give that movement as a vector. So the top number indicates a left or right movement and the bottom number indicates an up or down. So if you think about it like a coordinate x comma y, it's the same thing x here, because x left or right on your x y axis and the bottom number is the y, that would be up or down. Whether it depends on up, down, left or right, it depends on whether it's positive or negative. So if your numbers are positive, we're thinking about movements to the right or movements up. And when we're dealing with negative numbers, it's down or left. Here are some examples. These two numbers are positive. So we're going five to the right because your x axis goes positive the more you go to the right and three up the second one. This time this number is negative. So we're actually going to go four to the left. This time it's the opposite direction, but this number is still positive. So we're still going six up minus nine and minus four. So we're going nine to the left and four down. So in the opposite direction. So that's how you would write a vector. So in this example, you've got two scenarios really. You're either going to be told what the vector of movement is and you will need to move the shape, or they'll all ready have done the translation and you'll need to say it's translation by vector and then give the vector of movement. What you don't want to be writing is these bits. These are just explanations. You don't want to say five right to be up. They want it as a vector. Okay? And then enlargement. So enlargement is basically where the shape gets bigger or smaller, but the actual shape itself is the same and the angles stay the same. So congruency meant the shapes are exactly the same, the original and the new image. But with an enlargement, the shapes are said to be similar and similar shapes are basically the lengths of the sides change to get bigger or smaller. So here it was one and it became two, but the angles in the shape say the same. So if this is 90 degrees on the top left, it's still 90 degrees top left here. Okay? So when you are describing an enlargement, you need to give something called the scale factor, which is what are you multiplying the original side by to get the enlarged side. So it's not adding, it's not dividing, it's always multiplying. So here one of the sides was one and here it became two. So I'm multiplying by two. So you would say that the scale factor is two and you can cheque that with another one. There's three at the bottom here and it became six. So three times two six. So here, if we look at some of these lines, I always go for the straight lines rather than diagonals because sometimes diagonals can be a bit awkward. So it's two across at the top or even two down, and then we've got six down. So two times what makes six, three. So that's by scale factor of three, sometimes the shape can actually get smaller. So that's the original and that's the new shape. So if we look at this base here, it's two and it became one. Don't forget we're always dealing with multiplying with scale factors, even if the shape is being divided by two. Another way to say divided by two is actually two times by a half. So the scale factor here would be a half. So anytime your scale factor is between zero and one, so it's bigger than zero but less than one, your shape is getting smaller. If your scale factor was exactly one. Well, if you multiply all the sides by one, then you're going to get the exact same image. So that doesn't change the image at all, but anything greater than one. So one and a half, two, you're doubling your shape times by three, you're tripling your shape, then your shape gets bigger. Now, I'm going to actually touch up on two things here, negative scale factors, but also how to find your centre of rotation, sorry, your centre of enlargement as well. So negative scale factor basically shows that the shape translates, sorry, the shape enlarges in a different direction. So where you would essentially say this. So let's say your enlargement was minus three. Okay, so A, I'm going to find the original distance between here and the centre and then I'm going to triple that because times it by three. But you're also going in a different direction. So where you would be going right and down, you'd actually be going the other way down and then right that way. And that's how your image would be enlarged with a negative. So what you'll find is the image actually ends up flipping over and being in a different direction as well as getting larger. Now, to find the centre, you'll see all these dotted lines and you can do this whether it's got a negative scale factor or a positive, if you join each adjacent corner. So A here became the a down here for apostrophe. If I join that with a straight line and if I do that for all of the lines, I've also do that for the D. So again, join the D's together, join the CS together, join the BS together. You'll notice they all meet at one point here. That's called your centre of enlargement and that's what you would start off with when you are enlarging your shape. And that's pretty much it. So when you are explaining an Enlargement, you need to say it is an Enlargement, you need to give the scale factor of Enlargement as well, and the centre of Enlargement. So when you're enlarging a shape, if I go back just to one of our original diagrams, and let's say our Enlargement centre of Enlargement was here, you would need to travel from the corner to your centre and then have those distances. Essentially, that's how this works. So where from your centre, let's say it was actually your centre would probably be down here somewhere. Let's say your centre was here and the distance was ten, you would then need to go, okay, well, from this corner, I'm actually now only going to travel five, and that's your new position. Likewise, if the centre is the same again here, and let's say this distance was six centimetres, you would say, well, this time I'm going to travel three and position your new point. So that's where the centre of Enlargement comes from. Okay, so again, the centre here, if I join this, is just on a negative scale effect this time. So if you join this corner, top left to this top left, and you just keep going, some students go up there as well, that's fine. But if you keep going all of the corners, even if I just third line here and join these corners, they would all meet here. So you would give your centre of Enlargement as a coordinate. So these are the four main transformations and this is the key information for each part. The reflections, you would need to say that it's a reflection. Get the mirror line. Rotations, you would need to give the angle of rotation. So 91, 80 or 270 degrees. You would give the direction, which is clockwise or anticlockwise. And you would also give that centre of rotation as well. A translation. You would say it's a translation. And you would also need to give the vector of translations. In this case, it would be four across and three up. So you would say four, three as your vector. And Enlargement, you would need to say it's an Enlargement. You would need to see what the Enlargement scale factor is, and also the centre of Enlargement, which I think in this case would be zero. If we joined all three corners, they all meet at the origin. So in another lesson, I would be actually applying all of these transformations to exam style questions that look more like these grids down here. So this lesson was really just to focus on what each transformation does to a shape and what key information you give. And in another lesson we would be going through exams are questions of actually rotating shapes, translating shapes, enlarging shapes and actually describing them as well.
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Sanaa is currently Deputy Head of Maths at a secondary school and has been tutoring GCSE Maths students for over 10 years. She tailors her approach so that no student is left behind.