Lesson video

Hello, I'm Mrs. Lashley, and I'm gonna be working with you as we go through the lesson today.

I really hope you're ready to try your best and ready to learn.

So our learning outcome today is to be able to perform a reflection and also to be able to describe a reflection that has taken place on an object.

The screen shows some keywords that we will be using during this lesson and you have learned previously, but you may wish to pause the video to ensure that you are happy and familiar with them before we make a start.

So the lesson is checking and securing understanding of reflection, and we're gonna break the lesson into two learning cycles.

The first learning cycle is where we're gonna describe a reflection that has taken place, and the second learning cycle is where we will perform a reflection.

So let's make a start at looking how to describe a reflection.

A reflection is a type of transformation.

So you have met this before.

There are four transformations that we tend to use, and that is rotation, translation, reflection, and enlargement.

When a reflection has taken place, the object is flipped over to create the image, and this is something we're gonna look at as we go through.

If the object and image are not congruent, then it cannot be a reflection.

So what that means is if they are not exactly the same size and shape, if you couldn't cut the two shapes out the object and the image and place them on top of each other with no overlap, you may need to flip them or rotate them, if you cannot do that, then they are not congruent and therefore it cannot be a reflection.

So which of these words have been reflected and which have been rotated? So if you look at the four pairs, they all have FAST, but some of them have rotated and some of them have reflected.

So just take a moment to think if you can sort that out.

So how can you tell which ones are rotated and how can you tell which ones are reflected? What strategies could you use to make some of the words more readable? Well, it might be, and you may have done this, to make some of them more readable just by tilting your head slightly.

So that would be the ones that are rotated.

The ones that have just been rotated, you can tilt your head, so sort of rotate your head, to make the word look exactly the same, to make it horizontal again.

Whereas the ones that are reflected, you cannot tilt your head.

It doesn't matter how many times or how many ways you tilt your head, it will not read FAST, left to right.

So why does reflection not maintain sense? Well, so when a word has been reflected, the letters of that word will have been transformed in a way that makes it impossible to identify as the same letter, no matter how much you tilt your head.

So if you look at FAST on the bottom left, it's been reflected in two different directions: vertically and horizontally.

And the letter S is quite clearly not the same as we see an S, whereas on the T in one of the reflections, the T looks to be the same.

You know that is the letter T, whereas in the other reflection, the T is clearly now not a T.

The sense of those letters has changed, because it will never look the original way again without having to do a further reflection.

That's not true for all of those letters, but for most letters, just to get the idea of what sense means.

So with F, can you tilt your head to make the letter F look like this? So here's a check.

In which of these pairs of words has the sense changed? So you've got PLAN, AXIS, SIGN, COSH, and ONES.

Which of those has the sense changed? Which of those have been reflected? Pause the video, and when you're ready to check, press play.

So PLAN has been reflected, the sense has changed.

One of the letters that you can see that on is the L.

If you try to tilt your head, you can't make the L on the right hand word look like the L on the left hand word.

COSH.

Another one.

The letter that you can use there is the S.

And SIGN.

S, the G, and the N, no matter how much you tilt your head, it will not go back to looking the way the top word of SIGN does.

Whereas AXIS and ONES are a rotation.

If you imagine doing a handstand, the word axis on the bottom, you can read it quite easily.

So when an object is reflected, the lengths of the sides are in variant.

So now we're thinking more about a shape as opposed to a word.

So the lengths of the sides are invariant, which means they do not change, but the sense of the sides will always change.

So it doesn't matter what the object is, whether it's a word or a shape, when a reflection has taken place, the sense will change.

So which of the examples in a moment is an example of a reflection and which is the non-example? So have a look at those.

Which one is a reflection and which one is not? So the left one is an example of a reflection, and that's because the object and image are congruent, the lengths of the edges are in variants, but the sense has changed.

Whereas on the right-hand side, the height has changed.

If you look at the maximum, the object has a maximum height of five, whereas the image has a maximum height of four, so it's no longer congruent.

The object and the image are not congruent, therefore it cannot be a reflection.

And then working still with the idea of congruency, when an object is reflected, the object and image are congruent.

So the angles within the shape will also be invariant, which means unchanged.

So again, which of these is an example of a reflection and which is the non-example? So take a moment to look at those two pairs of object and image.

The left one was the example of the reflection.

You can see that all of the angles within the quadrilateral are the same on both the object and the image.

Also, importantly, that the order is the same.

If you chose to start at 135 degrees, you could work clockwise or anti-clockwise, and they'd go to 63 degrees, then 107 degrees, and then 55 degrees.

So not only are the angles within the shape the same on both the object and image, they're also in the same order.

On the non-example, it's quite clear that the angles are not the same.

There is a 90-degree on the image, but there isn't a 90-degree angle on the object.

So when the object is reflected, its orientation will always change even if the image looks the same as its object.

And this is to do with certain shapes.

Even if you reflect them, will not look to be reflected.

For example, a square.

So if I have told you that these are reflected, then that's fine, but if you didn't know they had been reflected, you may thought that's just a translation.

It's just moved to the right.

Whereas if we were to label the vertices, so here, if we make this A, B, C, D, if I now reflect it, then the vertices will also reflect.

The image of each vertex is in a different position and we can now see that a reflection has taken place as opposed to a rotation.

So a check are to do with looking at the labelling of vertices after a reflection for you.

So I'd like you to complete the labels on the image.

Press pause whilst you do that, and then when you're ready to check your answer, press play.

So hopefully you've got them in the positions on the screen.

Thinking about that this is a reflection.

So there are three types of reflection.

There's a horizontal reflection.

And so this is where the reflection has happened from left to right or right to left.

There's a vertical reflection, and this is where it's reflected up to down or down to up.

And lastly, a diagonal reflection.

And so it's reflected diagonally.

Both of the object and the image in all three types of reflection are equidistant from the line of reflection.

The equi means equal and distant means distance.

So a corresponding point on the object and the image are an equal distance from the line of reflection.

So if we go back to labelling our vertices, vertex A and the image of A will both be the same distance, same perpendicular distance, from the line of reflection.

So here, this point is six squares from the line of reflection, so the image of that point is also six squares from the line of reflection.

Similarly, with this pair of corresponding points, four squares from the line of reflection, and so its image is also four squares from the line of reflection.

So equidistant from the line of reflection is really important as we move through reflection.

So this is an example where we can see that the dashed line is the line of reflection because it is equidistant from both the object and the image.

Whereas this is a non-example.

The line of reflection, or that dashed line, is not equidistant from both the object and the image.

So a check.

Which of these are examples of lines of reflections? Press pause whilst you have a go, and then when you want to check, press play.

A and D are our lines of reflection.

If you look, they are the same distance, equal distance, from both the object and the image.

So we are looking at how to describe a reflection.

And within a description of a reflection, we will need to locate the line of reflection.

So how can we do this? Well, we can make use of the idea that the line of reflection is equidistant from both object and image.

So I'm gonna talk you through the stages to locate the line of reflection.

So first of all, you need to pick a pair of corresponding vertices and join them up with a line segment.

So I've joined these two corresponding vertices.

We want the distance to be exactly the same on both sides of the line of reflection, then we need to find the midpoint.

So if this is four squares, the line segment is four squares long, then the midpoint would be at two squares.

We can mark that.

We are now gonna repeat this with another pair of corresponding vertices on the object and the image.

We'll count that.

There are 12 squares, so the midpoint would be six squares along from either end.

Now, we've got two midpoint marked, that means that they are exactly halfway, equidistant, we can join the mark to make our line of reflection.

This method works even if we don't have a grid line.

And this time, we just have to use our ruler.

Instead of counting squares to know where half is, we'll just use our ruler to measure where the half is.

So it's necessary to use the ruler accurately to make sure we are finding the midpoint.

So once again, you're gonna connect two corresponding points to make a lot with a line segment.

The difference is we're not counting squares; instead, we're gonna measure it.

Then we will half that distance.

So here, it's five, so we're gonna mark it at 2.

5.

Repeat the process with another pair of corresponding points.

Measure it, 10, half it, and mark the midpoint, and then we can join that up with a ruler or straight edge to be the line of reflection.

That line of reflection is now equidistant from both object and image.

So a check.

There are five words in only three markers, so two of those words, you do not use.

You please can you label the markers with the correct words.

Press pause whilst you're making your decisions on A, B, and C, when you are ready to check, press play.

So A is a line segment, B is the midpoint, and C is our line of reflection.

So we've spoken that there are three types of reflection: horizontal, vertical, and lastly, diagonal.

So which of these is the correct reflection? So the line of reflection is marked in grey dashed line.

You've got your object and you've got your image.

So which of these is correct? Don't worry if you're struggling to know which one is correct.

And this is where it might be easier if you were to rotate the diagram, which I'll do in a moment.

Or if you have this on a piece of paper or a worksheet or in your exercise book, you could rotate the piece of paper.

So by rotating the piece of paper, we can then have the line vertical.

We could have rotated it to make it horizontal as well.

And this becomes a lot easier then to see where a correct reflection has taken place.

Has the sense changed? On the left, the sense has changed, and on the right, it hasn't.

So being careful when we are looking to describe a reflection that we don't think it's a reflection when it isn't, and making sure that we have our line of reflection, and potentially, to make it easier, rotate it.

We can also check if a diagonal reflection has been done correctly by drawing line segments and marking the midpoints in a similar way that we did before.

But instead of counting along the horizontal or along the vertical, this time we're gonna count along the diagonals of each square on our grid background.

So if we join up a pair of corresponding points like we did previously, and we are now gonna count the diagonals.

So that's two diagonals as the midpoint would be one diagonal.

Choose another pair of corresponding points.

This is four diagonals, so midpoint would be at the two.

We can then connect this up to be our line of reflection.

Once again, note that these are perpendicular to the line of reflection.

However, if we had chosen a different pair of corresponding points such as these two, then when we count the diagonals, we'd have nine.

If we half that, we get 4.

5.

So sometimes your count will mean that you need to include half a diagonal and your position of the midpoint would therefore be within a square.

And you can still connect them up.

So a quick check.

Which of the three lines marked is the line of reflection? Press pause, and then when you're ready to check, press, play.

I'm hoping you went for B because all corresponding vertices are equidistant from that line.

So when we want to describe a reflection fully, we need to state that the reflection has taken place.

So we need to say which transformation.

In this case, it is a reflection.

And then we need to give the equation of the line of reflection.

So first of all, you need to identify whether it is a reflection, and really, you need to check that they are congruent firstly, and then has the sense changed.

So in this diagram, yes, the sense has changed.

Then we need to identify where the line of reflection is.

So going through the process of finding the line that is equidistant from both object and image, and you can do that with the methods we've seen.

Once you've located that, draw it on, then we need to come up with the equation of that line.

And this may be where you need to pause and go and look up and remind yourself a little bit about how to find the equation of a line.

Here is a vertical line, and this line could be extended both upwards and downwards.

It's drawn as a line segment currently, but it is a line.

All of the coordinates that align on this line have something in common.

And what they have in common is that the x-coordinate is five.

And so this, the equation of this line is x = 5.

So our full description would be: the object has been reflected in the line x = 5.

We have stated a reflection has taken place and we have given the equation of the line of reflection.

Once again, if we wanted to describe this reflection, is it a reflection? Well, yes, the object and image are congruent and the sense has changed.

Where is the line of reflection? Well, it's there.

Now, we need to know what the equation of that line is.

Once again, you can do that by thinking about the coordinates that lie on the line.

So I could say that 1,1 is a coordinate on the line.

5,5 is on the line.

7,7 is on the line.

So what is common to all of those pairs of coordinates? Well, that the x- and the y-coordinate are the same.

So the object has been reflected in the line y = x.

So that's the equation of that diagonal line.

So for you, can you write down, can you complete the equation of the line that is located on this diagram to fully complete the description? Press pause, and then when you're ready to check, press play.

So this one is y = -x.

Once again, if you looked at some coordinates, you would note that the value, the digit is the same, 6,-6, 2,-2, but one is positive and one is negative.

So up to the first task of the lesson, where on question one, I want you to identify the following pairs that might be a reflection.

So press pause whilst you're doing that and think about why the word is might as opposed to the following pairs are a reflection.

When you're ready to move on to question two, press play.

Okay, so on question two, you need to identify where the line of reflection is, so draw it on using your pencil and ruler, and also write down the equation of that line.

So press pause whilst you're working on question two, when you're ready for question three, press play.

Question three, the descriptions were started to take place, but there are some missing elements to each description, so you need to fill the blanks for all five parts.

Press pause whilst you're working on that, and then when you're ready for question four, press play.

And finally, question four, you need to fully describe each transformation that has taken place on this diagram.

So use the labels, A and A' are the pair of corresponding object and image, and make sure that you are fully describing the transformation.

Press pause to do that, and then when you press play, we're gonna go through our answers to Task A.

So question one was which of the following pairs might be a reflection? So you were looking, and checking, are they congruent, firstly? And then if they are congruent, has the sense changed? So on A, they were congruent but the sense had not changed, so that couldn't be a reflection.

On B, they were both congruent and the sense had changed, so that might be a reflection.

C, they were congruent, but the sense hadn't changed, so that one couldn't be a reflection.

Then on D, they were congruent and the sense might have changed.

So this is the one where we have to use the word might.

Because this could have just been a rotation as opposed to a reflection.

But it might be a reflection.

So that one is one you should have written down.

And then finally, E, are they congruent? Yes.

Has the sense changed? Yes.

So it might be a reflection.

Question two, you needed to draw on and identify where the line of reflection is and then write down the equation.

So you're gonna do that by finding corresponding pairs and marking the midpoint.

It may be that you could see by counting where the sort of line of reflection would be.

So the first one, A, is the line y = 3.

5, B, the line is x = 4.

5, and C, it is y = x.

Question three, you need to complete the descriptions.

So the first part, A, you was missing the word reflected.

We need to state the transformation that has taken place.

On B, the line was y = -2, on C, it was y = x, and on D, you had to write the full equation of the line, which was x = -3.

5, and E was y = -x.

Question four, you needed to fully describe each transformation that has taken place.

I've just written reflected in, and then the equation.

If you've written shape A has been reflected in x = 4.

5, that's great, that's a fuller description, but you must have said reflected and you must have given the correct equation for the line.

So x = 4.

5 was the equation of the line for A, y = -1 was the equation of the line for B, y = x was the equation for the line for C, x = -3 was the equation of the line of reflection for D, and finally, y = -x was the equation for E.

So we're now gonna move on to the second learning cycle.

And in this second learning cycle, we will actually be performing the reflection.

To perform a reflection, a piece of tracing paper can be used.

And so a piece of tracing paper can be used in quite a few transformations and it can also be used for reflection.

So if we were trying to reflect the object in the line y = 1, these are the steps that we would take in order to do it.

So first of all, you would locate and draw the line y = 1.

So if you are.

Hopefully, you're feeling more confident of what that equation of the line means, you've reminded yourself about that.

So we'd draw that on.

Then we're gonna mark a point along that line.

It doesn't matter where that point is, that is a choice of yours, it's going to be an anchor point, and I'll come back to that in a moment.

So we are going to have located the line that we know we're gonna reflect our object in and mark a point on that line.

Then we're gonna get our tracing paper.

So this is where our tracing paper is gonna be helpful, and we are going to trace the object, the line of reflection, and that anchor point that you chose onto your tracing paper.

So then what we do next is we are going to literally flip over the tracing paper.

And then we are going to align it back on top of the line of reflection and that point that you chose, and that's why I'm calling it an anchor point.

It tells you where to place your tracing paper to make sure it's all lined up perfectly.

Once you've lined it up, you can see where your image will be located.

And from here, you now need to get the drawing of your image onto your sheet of paper.

And that would be the complete of your reflection.

So our tracing paper has been helpful because it allows us to literally flip over the object to become our image, and we use a point to anchor it to the right position.

So here is a check.

What is the first step when reflecting using tracing paper? So pause the video and read the options, and then when you've decided on what the correct option is, press play to check it.

So it is B.

It doesn't have to be a cross that you mark on that line of reflection, it could be a dot, but an anchor point of some sort in order to make sure that after the flip of the tracing paper that you align it into the correct position.

Another check.

What three things do you need to draw onto the tracing paper? So if I place my tracing paper over this, I want to perform a reflection, what do I need to draw onto it? Pause the video to think about that, and then when you're ready to check, press play.

So you need to draw the object, you need to draw the line of reflection, and importantly, you need that anchor point, the point that you chose on the line of reflection.

So reflecting using tracing paper is always helpful because it will work regardless of the complexity of the object.

However, it sometimes is time consuming and unnecessary if you can see visually where the reflection will end up.

So the method of reflection by counting squares uses the property that the point on the object and its corresponding point on the image are equidistant from the line of reflection.

So this is a second method for performing a reflection.

So if we look at this one here, if we want to reflect this object in the line of reflection by counting of squares, we're gonna first choose a vertex.

We know that the vertex, the image of that vertex, is equidistant from the line of reflection.

So firstly, we need to see how far it is.

So count how far it is to the line of reflection perpendicular to it.

So it's two squares away, so our image of that vertex will also be two squares away.

Now, we can repeat this for every vertex of our object.

So for this one, it's four squares, so the image of that vertex will be four squares on the opposite side of the line of reflection to ensure that it is equidistant.

And then lastly, the last vertex of the triangle is also four squares.

So four squares on the opposite side, we can mark our third and final vertex on the image.

And we can connect them up using line segments to form our image.

So this method of counting is a method that will work on diagonals as well.

The only change is that we're no longer gonna count horizontally or vertically, we're gonna count using the diagonals.

So if we had this object and the line of reflection, we would choose a point and count perpendicular using the diagonals to the line of reflection.

So here, it would be three diagonals.

However, it can be a not a whole number of diagonals.

So this one here would be two diagonals and a half.

It's important then that when we reflect on the other side, when we count on the other side, we sort of do it in reverse.

We would count the half and then the two.

I would advise you to count for all vertices, not try to skip any steps, especially on the diagonal.

So if we go through this example, this first vertex is one full diagonal from the line of reflection.

So the image of that vertex is one full square on the other side, whereas this one is one and a half way from the line of reflection.

So when we replicate it, remember we're gonna sort of do it in reverse, we're gonna do the half, and then the one before we mark our image of the vertex.

We then repeat this for all of the vertices.

So this one here is two diagonals, so the image would be two diagonals.

This one is two and a half, so it'd be half, and then two to mark the image.

We can then join them up with line segments to create and form the image in our reflection.

So for you, a check.

Which of these is the correct location of the reflected image of vertex A? Is it A, B, C, or D? So press pause whilst you make a decision, and then when you're ready to check, press play.

So hopefully you went for B because it is two diagonals away from the line of reflection, the same as two diagonals for A.

So now, to the last task of the lesson where you are gonna complete the reflections.

So on A and B, you are reflecting in the coordinate axis.

So A is reflect shape A in the x-axis, and B is reflect shape B in the y-axis.

It's up to you whether you do it by counting or whether you use a piece of tracing paper.

Press pause whilst you complete those, and then when you press play, we'll move on to question two.

So question two is reflecting still, but this time you are gonna have to draw and locate the line of reflections that are not the coordinate axes.

So press pause whilst you work on A and B, and then when you press play, we're gonna move to question three.

Question three, we are now moving on to diagonal reflections.

Up 'til now, you've done vertical and horizontal in the practise, so these are now diagonal reflections.

So you're still completing a reflection.

You've got to reflect in y = x and y = -x.

So press pause, when you press play, we're gonna come back to question four.

Question four is also diagonal reflections.

What's different now is that the equation of the line of reflection is not y = x or y = -x.

It might be, if you're struggling to remember where they are, you could come up with a table of values or you could use Desmos as a website to unplot them to locate where they are.

Once you've drawn it on naturally, then the reflection part is exactly the same.

So press pause and work through question four.

When you finish that, we're gonna go through the answers to Task B.

So question one, were reflecting in the x-axis and the y-axis.

So on A, it was a vertical reflection.

We've reflected from the below and we've reflected up.

On B, it was a horizontal reflection.

We reflected from the left side to the right side.

Make sure that your object and your image are congruent, make sure that the sense has changed, and make sure that they are equidistant from the line of reflection.

Question two, you're still doing vertical and horizontal reflection.

x = 2 was the equation of the line of reflection on part A.

So that should have been a vertical line passing through the x-axis at 2.

Again, check that your object and image are congruent, check that they are equidistant from the line of reflection, and check that the sense has changed.

And on B, it was the line y = -1.

So that was a horizontal line passing through the y-axis at -1.

Check that they're congruent, check that the sensors changed, and check that they are equidistant from the line of reflection.

Question three was now working with diagonal lines.

It may have been that you took that hint and that tip to rotate your sheet of paper, but you'd still be counting along the diagonals if you didn't use tracing paper.

So check, pause the video if necessary, and look at the vertices on the image to check that you have your image in the correct location.

Question four, as I say, the process of reflection is exactly the same, but you needed to locate and draw the line of reflection in the relevant place.

So the first one was a parallel line to y = x because it has a y-intercept of +3.

So that's where the line of reflection should have been.

And then if you use tracing paper, remember you should have marked a point on that line to get an anchor, but if you've counted diagonals, counting from the object to the line of reflection and doing the same on the opposite side.

On part B, the line was y = 2 - x, so this line was parallel to y = -x and pass through 2 on the y-axis.

Once again, you may wanna pause the video to check your coordinates of your image against the image on the screen.

So just to summarise today's lesson on checking and securing understanding of reflection: Reflection is a type of transformation.

If a reflection has taken place, the object and image are congruent, the sense and orientation changes.

And the sense is the one that's most obvious when a reflection has taken place if the shape sort of has an element that you can check.

The object and image will be equidistant from the line of reflection, and so that's really important.

And the line of reflection might be vertical, horizontal, or diagonal.

It could touch or pass through the object.

Really well done today and I look forward to working with you again in the future.