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 looking forward to the lesson and willing to try your best.

Our learning outcome today is to be able to describe a translation and perform a given translation to an object.

There are some key words on the slide here that you may wish to pause the video so that you can re-remind yourself and familiarise yourself with those.

You've learned them before, but we are going to make use of them during the lesson.

So today's lesson of checking and securing understanding of translation is gonna be split into two parts.

The first part, the first learning cycle is to be able to perform a translation and then we'll move on to the second where we will describe one.

Let's make a start with the performing of a translation.

So a translation is a type of transformation.

When a translation takes place, the object is moved to create its image.

If the object and the image are not congruent, then a translation cannot have taken place.

So remember, congruent means that they are two identical shapes.

Not just the shapes are the same, so in terms of them, not just both rectangles but the size of them as well.

So if you were to be able to cut them out and place them on top of each other, they would be exactly the same.

There is a link there to a geode profile where you can look at translation a little bit closer and remind yourself about that.

You may wish to pause video and do the link first before continuing.

So on the screen, there are some examples of translated objects.

So A and A prime, one is the object and one is the image.

And we can see that they are congruent to each other.

So both of them are rectangles, but also the dimensions are the same.

One has just moved to become the other.

That's the same for the two triangles, N and M prime.

It doesn't matter that there is overlap between the two.

The object has moved down and that's become the image.

They are congruent to each other.

And then P and P prime, that's moved both left and down.

So the shape has slid or moved, it's translated.

The object and the image are congruent to each other.

Whereas these are showing non-examples of translated objects.

So once again, we've got rectangles, but they are different sizes.

They're not congruent, so they cannot be a translation.

The two triangles at the top right, they are congruent to each other, but it's still not a translation.

And the reason it's not a translation is because one could not just slide on top of the other and be an exact copy.

You'd have to flip one over.

And then the last pair of shapes, which are two right angled triangles, you could have marked there we've got different angles.

If you know that the angles within the shape are different, then they cannot be congruent.

So a quick check for you.

To be a translation, the image must be.

So A, B, or C.

Pause the video, read through the options, make a decision on what the correct end of the sentence is.

And when you're ready to check, press play.

So the answer is congruent to the object.

So to be a translation, the object and image must be congruent.

So when we want to perform a translation, then we need to translate an object or a single point.

We'd need to move it horizontally and or vertically, maintaining its orientation.

So we can think about moving the shape as translation.

So here is a translation.

You can see if you look at each vertex and a corresponding vertex, it's moved by the same amount.

You could slide one on top of the other and they would be exactly the same.

So they are congruent to each other and the orientation hasn't changed.

Whereas this is an example of not a translation.

So despite the fact that they are congruent shapes, the object, that purple one and the blue one are congruent, you can see that the orientation one is twisted slightly.

So this cannot be a translation.

And if we use tracing paper, so tracing paper is one way of performing a translation, we can replicate the idea of moving the whole object or translating the whole object.

So with this one, if we place a piece of tracing paper on top and mark the three vertices because it's a triangle, if we wanted to translate this triangle and that means to move it or to slide it horizontally and vertically or just horizontally or just vertically, but with no rotation or flipping, then we can see all the different locations that it may end up being.

And here is where it stopped.

And so this is where the location of the image after a translation would be.

Another way that we can perform a translation is by considering the vertices and translate them one by one.

So if we translate the vertex X, then we could translate it to this point here.

We have translated it by going three to the right, so that's a horizontal displacement, and two up, which is a vertical displacement.

For the object to be moved without any rotation or reflection, and also for the image to be congruent, then every vertex needs to translate by that same amount.

So three to the right and two up.

So if I do that for the Y vertex, then Y prime, the image of Y would be there, three to the right and two up from the corresponding point Y.

Do the same for Z and we get out image.

So that is a congruent triangle and it is a translation because the orientation hasn't changed nor has the sense.

Another way we could do this is we could translate one vertex.

So here I've translated that X vertex again.

So by the same amount, so three to the right and two up.

And then we could just draw in the remaining parts of the image, taking care to think about the size of it so that it maintains being a congruent shape.

And this would be the location.

So we could do it vertex by vertex, translating each point the same amount so that we get in the right location or just translate one vertex and draw the rest of the image in.

Sometimes the complexity of the object may support one method more than another.

So if you look at these two, if these were the object that you needed to translate, maybe you would prefer using tracing paper or maybe you would prefer translating one point and drawing the remaining part of the of the image, or you're gonna translate every 0.

1 by one.

So which method might you use for each of these? Have just a moment to reflect on that.

There's not really a right or wrong answer it.

It's a more of a personal preference.

So when we are asked to perform a translation, we need to know how much by, so the description of how far to translate the object could be given to you as words.

So here translates shape A three units to the right and five units down.

So we now know how far we are moving or sliding the object.

But it could be given as a column vector.

And so the same description is given here.

So a column vector tells us how much we are going to move it horizontally.

That's the top value in the column vector.

And also how much we're gonna move it vertically, which is the bottom value.

The number will be positive or negative.

And if the top value is positive, then we are moving to the right.

And if the top value is negative, then we'd be moving to the left.

And similarly on the bottom value, which remember is our vertical, when it is a positive value, that's up, and when it's negative, that's down.

You might want to remind yourself or think about the coordinate axis to support with that.

From the origin, 0:0 if you moved up the Y axis, they are positive numbers.

If you move down the Y axis, they are negative numbers.

So negative means down, positive means up for the vertical.

And again with the X axis, if you do the same thing when you moved along and the numbers increase, it goes to the right, they become positive on the right hand side relative to the origin, and on the left they are negative.

So left is a negative value.

The vector is a more succinct way to explain the movement and that's where how you're gonna see it more often than not using a column vector.

So a check.

The bottom value in a column vector tells you what displacement.

So that's horizontal or vertical.

You need to make a choice there.

And if the horizontal displacement is negative, then you translate which direction.

So pause the video to figure out those words.

And then when you're ready to check, press play.

Bottom value is vertical.

And the horizontal displacement when negative means move to the left.

So just like a pair of coordinates, X before Y and X is the horizontal, Y is the vertical, it's exactly the same on a column vector.

The top number is your horizontal and the bottom number is your vertical.

So I'm gonna go through how to translate this once as an example, and then there'll be one for you to do.

So if I've been asked to translate the object, which is the trapezium, A, B, C, D, by column vector -2:2, first of all, I need to sort of translate in my mind what -2:2 means.

Well the top value is our horizontal, horizontal first.

<v ->2, where would that be on the X axis</v> compared to the origin? It would be on the left.

The negative values are on the left.

So I'm going left by two.

And then that bottom value, which means vertical is positive two.

So if I'm at the origin, where would positive two be on the Y axis, would be up from it.

So I'm going left by two and up by two.

So every coordinate here is translating left by two and up by two.

So A would become a prime in the position 1:3.

Let's just think about the coordinate.

So the coordinates of A is 3:1.

The coordinates of a prime is 1:3.

It's a coincidence that they have just transposed.

Why have they become? Why has 3:1 become 1:3? Because if you do three takeaway two, that top value in the column vector is about the X coordinate.

So three takeaway two becomes one.

And the Y coordinate of A is one.

Ane add two 'cause it's a positive two is three.

So the Y coordinate on the image is now three.

So that column vector, the top value is changing the X coordinate and the bottom value is changing the Y coordinate.

We can do the same thing for all of the vertices.

They all going to move left by two and up by two.

If they don't all move by the same amount, all translate by the same amount, then our image would not be congruent and therefore a translation hasn't taken place.

So I have then translated point by point, vertex by vertex, and I can draw my image, I can join those vertices up to create my quadrilateral.

So here's the one for you to try.

You need to translate the object.

Your object is a Pentagon, by the column vector by the vector 2:-3.

So pause the video whilst you try that, and then when you're ready to check, press play.

The image of A will be here.

If we just think about the coordinate again, the A coordinate was 0:4.

So if I need to add two to the zero, the X coordinate, then it becomes two.

And four, I need to take away three.

So four, takeaway three becomes one.

So the coordinate of the image of A is 2:1.

I can do the same for all of my vertices and then draw it in.

And this is the image after that translation.

Well done if you manage to get it in the right location.

Remember you don't have to do it by doing vertex by vertex.

You could have found the location of one vertex and drawn the image in.

So we're up to the first task for this lesson.

There are three questions in task A.

So here is question one.

On question one, there is a diagram with four different objects, A, B, C, and D.

And you need to complete the translation using the given vector.

So pause the video whilst you complete those four translations, and then when you press play, we'll move on to question two.

Question two, their diagram has been removed.

So now you've got the vertex and its image and you need to check to see whether a translation has taken place.

Remember, a translation will have taken place if the same movement, horizontal and vertical has happened to all of the vertices within the object.

So press pause whilst you go through those and check.

And then when you're ready to move on to question three, press play.

Question three is now sort of in the opposite direction.

So I want you to translate the three vertices by the given vector.

So again, we've removed a diagram, you're just doing this by coordinates.

Remember that a column vector is telling you what change there is to the X coordinate.

That's the top value.

And what change there is to the Y value, that's the bottom value.

There are some algebraic terms here as well.

So your answers will have expressions because you don't have the value of A or B as it might be.

So press pause whilst you're working through this.

And then when you press play, we're gonna go through our answers to task A.

So question one, you needed to translate the four objects by the given vector in each description, and they are shown there on the diagrams. So the locations of the image of A, the image of B and the image of C and the image of D.

It may be that they're in the incorrect space when you've tried it yourself, and that's probably just a mistake on the vector.

So a vector 2:3 means you needed to move right by two and up by three.

A vector of -2:1 meant she should have moved left by two and up by one.

Part C, 0:-2, zero means there's no horizontal displacement.

It means that the object hasn't translated left or right, but the -2 is meaning it has translated down by two.

So it's directly below because there is no horizontal change.

On D, the vector is 3:0.

So there is a horizontal change that's three to the right, but there is no vertical change.

So the image of D is directly to the right of D.

On question two, you needed to determine whether a translation had taken place, and if so, what the vector.

And that was done by comparing the coordinates.

If the same thing had happened to each pair of corresponding coordinates, then a translation had taken place because the image would be a congruent shape to the object and its orientation, et cetera wouldn't have changed.

If the same thing has not happened for each pair within the object, then a translation hasn't taken place.

So on A, if we go through A, the vector was 2:2 because each coordinate, the X coordinate had increased by two and the Y coordinate had increased by two.

So as a translation it has translated right by two, up by two.

So if you look A to A prime, 3:2 has become 5:4, three add two gives you the five and two add two gives you the four.

And that works for B to B prime and C to C prime.

So the same thing has happened to each corresponding pair.

Whereas B is not a translation.

So if you look at coordinate A, it's -1:4 and the image of A is -2:5.

So the X coordinate has decreased by one and the Y coordinate has increased by one.

When we get to vertex B, the X coordinate of the image of D hasn't decreased by one.

The Y coordinate has increased by one, but the X coordinate hasn't.

So the translation has not been the same through all pairs of corresponding points.

C was a translation and that was by -2:3.

So it translated left by two and up by three.

And part D was also a translation.

And that was by the vector 2:-5.

So that means it is translated right by two and down by five.

Question three, you needed to write the corresponding coordinate for the vertex given the vector.

So for part A, you needed to translate by 1:2.

So that means move to the right by one and up by two.

Moving to the right by one means that the X coordinate will have increased by one.

And moving up by two means that the Y coordinate would increase by two.

So A, the coordinate of A was 2:5, so the image of A is 3:7.

The X has increased by one, the Y has increased by two.

On B, the coordinate was a three.

So to increase by one we need to write an expression, A plus one, three add two is five.

And then for the vertex C, it was A B.

So we're gonna have two algebraic expressions on the image, A plus one and B plus two.

Pause the video and check through the answers to the other three parts to question three.

Really well done if you've been highly successful on this question.

So the second learning cycle is describing a translation.

So when a shape has been transformed, you need to identify which transformation has taken place.

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

In this lesson we are focusing on translation.

So a translation may have taken place if the object and image are congruent.

Remember, it cannot be a translation if they are different shapes and sizes.

The object and image have the same orientation.

So once again, if one of the shapes, if the image was rotated slightly, if the orientation had changed, then it wouldn't be a translation because a translation is all about moving horizontally and vertically.

And finally, if the object and image are not reflections of each other, then a translation may have taken place.

So the sense of the image needs to be the same.

So in this check, which of the following could be a translation? So pause the video, look at the three transformations and which one might be a translation.

Press play when you're ready to check.

So B.

A, the sense has changed, one is reflected of the other, and so it cannot be a translation.

And on C, the orientation has changed, whereas B, all that's changed is its position.

It has translated as moved as slid vertically and horizontally.

A good description of a translation should state that it has been translated, so we should use the word translated rather than moved or slid.

Whether it has been translated left or right and how far, this is the horizontal displacement, it either will have moved, left, translated left, or it will have translated right or it may not have moved in that direction horizontally.

And also we need to state whether it is translated up or down.

And how far, and this is the vertical displacement.

So which figure shows the description? The object has been translated right by three squares and down by two squares.

So here we're using a worded description and which of the figures matches it? Press pause whilst you decide.

And then press play when you're ready to check.

So C is the match.

We can see A and the image of A or A prime so we know which one's the object and we know which one's the image.

It's important that we are counting from the object to the image, otherwise we may get our description the wrong way round.

So similarly, if a shape is translated, all the vertices need to move by the same amount for it to be congruent.

And we've seen this in the earlier part of the lesson.

So here is a translation.

And we know it's a translation because we can see from the vertices, if we just look at the coordinates of the vertices, that the same thing has happened for each corresponding pair.

And the thing that has happened is that they have all moved to the right by eight.

The vertical, there is no vertical change to these corresponding pairs.

Whereas on this one, this is not a translation.

We can see quite clearly this is not a translation because the object and image are not congruent.

But if we choose to look at the coordinates, imagine you didn't have the diagram, then we can see that the 6:4 and the 14:4 they're the corresponding coordinates.

The same thing hasn't happened for that pair as the other two.

The other two have moved right by six, or the X coordinate has increased by six.

Whereas on the 6:4, 14:4 pair it's increased by eight.

So because there has been a change in the translation, then the image is not congruent.

So by looking only at coordinates, we can check to see if a translation has taken place.

So on the left hand column is gonna be the coordinates of the object.

And on the right hand side it's gonna be the coordinate of the image.

So two minus four is on the object and two minus two is on the image.

These are a corresponding pair.

So what's happened? Well the X coordinate hasn't changed, so horizontally there is no change.

Vertically, the Y coordinate has, and that Y coordinate has changed by increasing by two.

And we check this pair the same thing.

The X coordinates have not changed.

The Y coordinates have.

And this pair, the X coordinates have not changed, the Y coordinates have.

And more importantly, the Y coordinate of each has increased by two.

If they had increased by different amounts, then this wouldn't be a translation.

But because they have increased by two on each pair, then it has been translated and it's translated up by two units.

How do we know that an increase by two means it's moved up by two? All of you think about the Y axis.

And if you go from a -4 up to -2, you don't go from -4 down to -2 on the axis.

So we have moved from -4 up to -2.

Here we've got another pair.

Let's check to see if a translation has taken place.

So what's happened here, we've got -3:2 is the coordinate on the object and it's corresponding vertex is 2:-1.

So the X coordinate has changed and the Y coordinate has changed.

The X coordinate has changed by increasing by five and the Y coordinate has changed by decreasing by two.

So if this has happened on this pair.

So has the X coordinate increased by five? Yes.

Has the Y coordinate decreased by two? Yes.

So this pair has also followed the same translation.

And then on this pair, has the X coordinate increased by five? Yes.

Has the Y value decreased by two? Yes.

So in this case, the X coordinate of each has increased by five, and the Y coordinate has decreased by two.

So a translation has taken place on this triangle.

It's translated five units to the right because if you were to be on the X axis starting at negative three and you've moved to positive two, then you've gone right, the direction of that is to the right.

So it's an increase of five on the X coordinate means that you have moved right by five and two units down.

So a check for you.

Has a translation taken place on these corresponding coordinates.

Press pause while you work through it.

And then when you're ready to check, press play.

The answer is yes, a translation has taken place.

And it's taken place by two units to the left and one unit up.

The X coordinates have decreased.

So you would be moving to the left and the Y coordinates have increased, so you'd be moving up.

So these vectors, we had those words there left and up.

Again, we can use the vector.

So just to remind ourselves that a very succinct way of describing a translation is by writing translated.

So we've stated the transformation by, and then we are giving our horizontal displacement and our vertical displacement.

But by writing it as a column vector, that's as much as we need to do.

So translated by -2:3 means the description is saying that it's left by two units and up by three units.

Translated by 4:-5 means four to the right, four units to the right and five units down.

And translated by 0:2 means there is no horizontal displacement, but it has translated by two up.

So up by two.

So here's a quick check on reading a column vector.

So translated by -5:3 means up by three and left by five, left by three and down by five or right by three and down by five.

Pause the video, and when you're ready to check your answer, press play.

So it's the top one.

However, the top one I've written sort of back to front if you like.

I've written the vertical before the horizontal.

It's probably a good habit to get into writing it horizontal before vertical to remind ourselves that the order of the column vector is horizontal before vertical and that actually matches with our pair of coordinates that we do horizontal movement and then vertical.

What about this one.

Translated by 0:1, what does that mean? Pause the video, and then when you're ready to check, press play.

So this means up by one part, C.

Zero tells us it doesn't move left or right.

There's no translation left or right horizontally, but there is a vertical displacement.

So if we wanted to describe the transformation here, so I've got an object, PQR, it's a triangle and the image P prime, Q prime, R prime.

How do I describe the transformation? Well, I basically need to work out how it has translated, how it has moved.

So I'm gonna choose a pair of vertices.

I'm gonna choose P and P prime, and I need to figure out how P has translated horizontally and or vertically to become in the same position as P prime.

So I can think draw arrows if I necessary, and it's three to the left and down by one.

I'm then gonna write my description as translated by.

And then the column vector.

Three to the left means all the X coordinates have decreased by three.

So that's negative three.

And if it's one down, that means all the Y coordinates have decreased by one, so minus one.

So here's one for you to do.

So pause the video and describe the transformation.

When you are Ready to check it, press play.

So hopefully you've written translated.

It's important that you state the transformation and then the column vector 3:1.

You could have chose any pair of corresponding.

Make sure they are corresponding points and count the horizontal displacement and the vertical, right by three and up by one.

So both numbers are positive.

So onto the last task of the lesson.

So question one, you need to match up the diagram to the correct description.

So A, B, C, and D are the diagrams and E, F, G and H are the descriptions.

Match them up.

Press pause whilst you're doing question one.

When you finish question one, and you're ready to move to question two, press play.

Question two.

There are copies of the same shape.

They are all congruent shapes on that diagram.

And I need you to complete the descriptions, fill in the blanks basically.

So press pause whilst you're doing that.

And when you're finished, we move to question three.

Question three, you've got five parts, A, B, C, D, E.

So work through those five parts, writing four descriptions for each of the following.

And there is a diagram to make use of.

Press pause whilst you're working on question three.

When you press play, we'll go through the answers to task B.

So task B, question one, you needed to match diagrams to the descriptions.

So E matched with A, is moved right by one and up by four.

F matched with D, it's moved left by two and up by three.

G matched with C, is moved right by three and up by one.

And H matched with B, which is left by one and up by two.

Question two, you needed to fill the blanks for the descriptions using the diagram to support you.

So on part A you needed to fill in shape C.

On part B, it was also shape C.

Part C was the vector and the vector needed to be 2:-2.

So two to the right and down by two.

And part D, the two shapes were D and F.

It's important that shape D onto shape F.

Shape D was your object and F was your image because it moved to the right and up.

Question three, you had five parts to answer.

So A and B are on the screen here.

You needed to write full descriptions.

It may have been that that you just wrote translated by and then given the vector, and that's okay because the question stated C onto shape F.

So shape C has translated by 0:-3.

That means it's just three below three down.

Part B shape E has translated by -4:0 onto shape D.

And so that's left by four and then no vertical change.

C, D and E are now on the screen.

So C, shape H has translated by -3:7 onto shape B.

So that was left by three and up by seven.

Remember, it's really important that you are counting between corresponding points.

Part D, shape G has translated by 3:-1 onto shape H.

So that's three to the right and down by one.

And finally, E was not asking for a full description.

It was a question, which shape cannot translate onto another shape.

And that is shape A.

So Shape A, although it is congruent to all the other shapes, it has a different orientation.

So if you were to translate shape A, it wouldn't land on top of any of the other shapes on the diagram.

So to summarise today's lesson on checking and securing understanding of translation, it's one of the transformations.

So it's a type of transformation.

When you're describing a translation, you need to state that the object has been translated, write down the type of transformation that it is, which would be translation, and give the vector that's translated by.

So that's the horizontal displacement and then the vertical displacement.

When you are asked to perform a translation, you can do it in a variety of ways.

So one way is to either translate just one of the vertices and get it into the correct position and draw the rest of the image, or you can translate all the vertices to make the image.

So do them one by one.

Really well, well done today.

I hope you've enjoyed the lesson and I look forward to working with you again in the future.