Lesson video

Hello, I'm Mrs. Ashley.

I'm looking forward to guiding you through your learning today.

So today's lesson we're gonna look at the minimum information that's required to describe a translation.

So displacement is a word that we'll be using during the lesson, and it's the distance from the starting point when measured in a straight line.

A vector is another word that we're gonna be using during the lesson and it can be used to describe a translation, and as I said we're gonna look at that in more depth during the lesson, but that's one of our key words today.

So we've got two parts to our lesson, and the first part is gonna be looking at describing a translation and then we're gonna really focus on the vector notation that we can use to describe translations.

So lets make a start.

So here we've got an image with lots of L shapes and they are translations from object A, so because they are translations, that means that they are congruent to A and Alex has chosen one of those shapes.

He's going to describe, and we are gonna try and locate which of those other L shapes, which of those is the image to A.

So he has chosen one of them and he's gonna describe it for us and we're gonna try and locate it.

So his first description says that A, shape A has moved across and up.

Before I reveal it, can you have a think about which ones that means it cannot be? Okay so you're bit of a detective right now.

You're trying to work out which ones it cannot be, and therefore by process of elimination we should find the one that he was originally thinking of.

Okay so A has moved across and up.

That means anything below it cannot be, because it's moved up, and anything that hasn't moved across, so anything that's not to the left or the right wouldn't be, so that leaves these five, so because A has moved across and up, anything that hadn't moved up couldn't have stayed and anything that hadn't moved across couldn't have been his identified image.

We need a little bit more information.

We've got five left, we've done well to get to five.

We need a little bit more information now.

So he's now said that A has moved right and up so again, before I reveal which ones are gonna disappear, which ones cannot be his selected shape? So hopefully you removed the ones on the left, because he had told us that A moved to the right, so anything that had moved left couldn't have been his selection.

So we're now down to three.

He now gives us a little bit further information.

So A has moved right by three and up.

So again, think about which one cannot be the answer.

So hopefully you got rid of that one there because that had moved more right than just three, it had gone across more than that to the right.

So we've only got two left.

Let's hope Alex gives us a very specific description that allows us to work out which of the two he chose at the start.

So he's now said A has moved right by three and up by seven, so which of the two did he select? And hopefully you chose that one.

So from that original diagram, we managed to get down to one.

His descriptions got better over time which allowed us to really hone in on the one that he chose, so if he'd just told us that last description, would we have known the correct image? So if he had just at the very start said, A has moved right by three and up by seven, would you have located that L shape? And yes, it would lead you to the correct one.

He gave us all of the information that we needed.

We were told it had moved both horizontally, originally he used the word across, and across could mean either left or right, then he changed it to right, so we had a definitive direction.

And then he gave us a number, so we had a scale as well.

So we were told how much it had moved horizontally, and how much it had moved vertically.

Vertically in this case seemed to be up, but it could have been down as well.

So a good translation, a good description, sorry, of a translation should state that it has been translated, so the idea of translating is it moves or it has slid, but we need to use that mathematical term of translated, so translated infers that it has been moved.

So it has been translated.

You need to be told whether it has translated left or right and how far.

This is known as the horizontal displacement.

So displacement is the distance from the start point, okay, so horizontally how much has it moved, how much has it translated from the start? And horizontal is left or right.

And then the same idea but now vertical displacement.

So how has it translated up or down and how far? So just to check, choose the figure that matches the description, the object has been translated left by three squares and down by two squares.

Pause the video and come back when you're ready to move on.

So hopefully you went for B.

So if you look B and A are very similar.

The only difference is its labels and the colours, so remember that that label, A prime means the image of A, so the purple or the one with the A vertex is the one we need to count from because that's the original, that's the object.

So the object has been translated left by three and down by two.

So you're now given a little bit of practise to think about this describing of translations.

So first question, you need to complete the description for each image.

Again, make sure you've identified which one is the object and which one is the image before you start.

Pause the video and come back when you're ready to move on.

Okay question two, Izzy is given an incomplete description of a translation, so she has said that shape has moved across and down.

I would like you to write a full description for the translation shown.

So think about what makes a good description of a translation.

Izzy's is not quite good yet.

Pause the video and come back when you're ready.

And then for question three, write full descriptions for both of these translations.

So remember you need to discuss the horizontal displacement and the vertical displacement and the word translated.

Pause the video and come back when you're ready.

So for question one you needed to complete the descriptions.

So identify the object.

The object is the one with the vertex A and identify the image, the one which is the image of A vertex.

So the objects are being translated right by one square and up by two squares.

So right by one and up by two, so the horizontal displacement is right by one and the vertical displacement is up by two.

On part B, again, on object and image, make sure you've identified which way 'round.

The object has been translated left by two squares and up by one square.

If you had got them the wrong way 'round, if you didn't think about object and image at the beginning, then for B you probably have written the object has been translated right by two and down by one.

Okay so if you've got those directions sort of the opposite, that's because you've not identified which one is the object and which one is the image.

So always start with your description of identifying the object and the image.

So question two, we had Izzy's quite vague description of a translation, so the shape has moved across and down.

So we don't want to use the word moved and we don't like across because it doesn't actually tell you a direction.

The down is a direction but we haven't got any kind of size, any magnitude, so hopefully you've written something like the object has been translated, so you've used the word translated rather than moved, right by one square and down by one square, so then we've got our horizontal displacement and our vertical displacement.

Question three you now needed to write the descriptions for both of these.

So for A the object is being translated right by two squares and down by one square.

So counting between those corresponding vertices.

So R and the image of R, it's two to the right and down by one.

And then B, the object has been translated left by two squares.

There hasn't got a vertical displacement so we don't need to mention it.

You don't need to say it hasn't moved up or down.

By just saying it's moved left, translated left by two squares, that is what's happened.

That is the description of the translation.

Again being very careful about which one is the object and the image.

Okay so we're onto this part of the lesson and that's all to do with vectors and the vector notation.

So we've looked at how to describe a translation using quite worded descriptions.

So we're now gonna look at more mathematical notation, which is vectors.

So we're gonna step back a little bit and just make sure we really got this idea of translation.

The translation is about moving all vertices the same distance and the same direction.

So when a length is translated it remains unchanged, but in a different position.

So the H edge length of your shape when it's translated doesn't change size, the lengths do not change size, but it does end up in a different position.

And we can consider the distance between two numbers such as two and five.

So if it was on a number line and you've got two and you've got five, the distance between them is three.

Okay, you can think of that as the difference, but the distance would be three.

If we now add five to both of those numbers then the distance between them would remain unchanged because we've added five to the two and we've added five to the five, so because they've both been translated up by five, they've both had five added to them, the distance between them stays unchanged.

However, if we were only to add five to the two, and you'd add three to the five, then that distance would change.

The distance between them has now got smaller in this case.

It may have got larger depending on what we'd added, but because we didn't add the same amount to both, then its distance has changed.

And so this is similar to a shape.

It's really important that when you do a translation that the distance the horizontal displacement is the same on all of the vertices.

The vertical displacement is the same on all of the vertices.

If not, the image will not be congruent, because the lengths will have changed.

So here we have got an example of a translation on a coordinate axes grid.

So if you look at the coordinates of the vertices, we've got four, four, four, eight and six, four, the image is 12, four, 12, eight and 14, four, and so all of the Y coordinates have not changed.

They are the same as their corresponding and that's because if you look it hasn't moved up or down.

It has no vertical displacement, but it has moved, it's translated to the right.

And that is because the X coordinates have changed, but more importantly, the X coordinates have changed by the same amount, so the four has become a 12 and that's plus eight.

The six has become 14, that is also plus eight.

The horizontal displacement has been the same on every vertex, and hence its image is congruent to its object.

Whereas on this one, this isn't a translation, because the, again, the Y coordinates on each of the vertices hasn't changed, there's no vertical displacement here, but the horizontal displacement is different between the vertices, so the fours have become 10, that is add six, so they are translated to the right by six, whereas the six four coordinate is now 14, four.

And that's plus eight.

So there is a difference, and hence it's not a translation.

A translation is a transformation where every point or vertex is moved by the same distance in the same direction.

So here they haven't all moved by the same distance.

So by looking at the coordinates of the objects and its image, we can check to see if a translation has taken place.

A translation will have taken place if the same value has been added or subtracted to the X and the Y throughout the shape.

So if the objects coordinate was two, five, and the corresponding coordinate, its image, was two, seven, what can we say has happened here? Well the X coordinate is unchanged, so remember the X is horizontal, so then horizontally it hasn't moved, it hasn't changed, but it's Y coordinate, the vertical coordinate has changed.

It's gone from five to seven, so that's add two.

So if we've now got a second vertex of this shape and that's at three, six, if a translation has happened, what would you expect this image coordinate to be? Hopefully you expected it to be three, eight, because the first pair had a Y coordinate change of two.

So we need this one to have a Y coordinate change of two so that a translation may have taken place.

And then if the objects coordinate was four, five, what would we expect its image to be for this translation to have taken place? We'd expect it to be four, seven, because the Y coordinate must change by two, it must increase by two.

So this is a triangle, it's got three vertices, and we now know that a translation has occurred because the same distance in the same direction has happened to all three of the vertices.

So the Y coordinate has increased by two, so we now know that the translation has moved the triangle up by two, because Y coordinates are the vertical axis, so this now the vertical displacement.

It's gone up by two, so it's gonna go up.

It's increased by two so they are higher than it was before.

Let's go for another one like this then.

So the objects coordinate is negative three, zero, and its image is two, negative two, so in this case, neither the X nor the Y have remained in the same position.

So we've got both horizontal displacement and vertical displacement.

So the negative three has turned into two.

That has gone up by five, negative two, negative one, zero, one, two, so that's an increase by five on the X coordinate.

The Y coordinate has gone from zero to negative two, so this one has decreased by two.

So we've had an increase of five on the X coordinate and a decrease by two on the Y coordinate.

So we've now got second vertex, negative one, zero, so if a translation has taken place you are expecting the same horizontal displacement, which was to increase by five, and the same vertical displacement which was to decrease by two.

So what would that image coordinate be? It would be four, negative two.

So negative one plus five, zero, one, two, three, four, so four, and zero decreased by two gets you to negative two.

Okay so we've got another coordinate negative one, three.

Again, if a translation has taken place then the same horizontal displacement and the same vertical displacement needs to have happened on all of its vertices.

Otherwise, the image will not be congruent.

So this one, the image coordinate, hopefully is five above the X and two below the Y, increased by five on the X, the horizontal, and decreased by two on the Y.

And that is four, one, negative one, add five is four, three, take away two, is one.

So again this is a triangle, it's got three vertices and the X coordinate of each has increased by five.

The Y coordinate has decreased by two so this triangle has translated.

The same thing has happened to each of the vertices, so it has translated and it's translated five units to the right because if you have increased by five on the X coordinate, if you're going up the X axis, then you are moving to the right.

And then the Y value has decreased by two, so if you are moving down by two or decreasing by two on the Y axis, that would be a translation down.

So just to check, same idea, so what translation has taken place if a translation has taken place? Pause the video and come back when you're ready to move on.

So hopefully you said yes that there was both horizontal and vertical displacement in this example.

The shape will have translated left by two and that's because all of the X coordinates have decreased by two.

If you take away two, you get from the object to the image.

So three take away two gives you the one.

Five take away two gives you three.

Six take away two gives you four.

And it's also moved up by one.

It's translated up by one unit and that's because the Y coordinates have all increased by one.

So zero add one gives you the one.

One add one gives you the two, and five add one gives you the six.

So it is a translation because the same horizontal displacement and vertical displacement has occurred on each of the vertices.

So here we've now got the image or figure on the coordinate axes to show you.

So we've got our purple object and our sort of green image, and Andeep says that the description for this transofration is that it moves left by two squares and moves up by one square, and we've got arrows there to indicate that movement.

So that horizontal movement and that vertical movement.

Andeep's correct.

It has translated by two to the left and up by one, if we chose coordinates, you'd be able to see just like we've done previously that the X coordinates have decreased by two and the Y coordinates have increased by one.

Jacob said that Andeep didn't need to say all the he said.

Jacob thinks that you could have just written translated left by two squares, up by one square.

And Jacob's right, because we know that translated means move, so rather than say moved by, we just say translated.

Aisha said that actually Jacob still had too much information translated left by two, up by one.

So she has dropped the squares part and that's because we're on a coordinate axes so there is a grid and we therefore assume we are talking about units on the grid.

If there was no grid, we'd still need a scale.

We'd have to give some sort of unit so centimetres or millimetres if you have to physically measure the distances.

Okay the displacement.

But here we don't need it because we're on this coordinate grid.

Jun said well you don't need any word other than translated.

So he is thinking about the fact that the X coordinates have gone down by two.

We've taken away two and the Y coordinates have gone up by one, so if we think about the X coordinates, so when you have a coordinate pair, X before Y, so we are always gonna talk about the horizontal before the vertical.

So it's minus two on the X coordinates and up by one, so that negative two is telling you that the X coordinate has gone down by two which infers that it's moved to the left.

Sofia said actually she knows about some notation.

So translated by negative two, one in these brackets.

And this is a vector.

So the vector notation is what Sofia has correctly used here.

So this is a column vector, and it's a really succinct helpful way of describing a translation.

We don't need many words at all.

The only word we need or two words are translated by.

Translated tells us that it is a movement and that vector tells us how.

So column vectors have a general form of horizontal is the top figure and vertical is the bottom.

And the way to try and remind yourself is that's exactly the same as coordinates.

The coordinates go horizontal, the X value, and then the Y value, which is the vertical.

So the column vector works in exactly the same way as that horizontal on the top, vertical on the bottom.

There's no line, it's not fractured.

You don't got a line between the two numbers.

It's actually a matrix, which is something that in sort of the future you may look at in a little bit more detail.

And then we need to understand what the values actually tell us, so on that top number, because that's about the horizontal displacement, if it is a negative number, all of the X coordinates are being subtracted by that amount, and therefore it will move to the left.

So the negative tells you it's a horizontal displacement to the left.

If it's a positive number, we don't need to write plus five, but if it's a positive five then that will tell you it's moving to the right, and that's the same for the vertical.

The down, because if you are taking away from the Y coordinate, then you would be moving down the Y axis, so if it's a negative number on the bottom then that is a translation down and if it's a positive number, again, no need for a plus sign, but if it's six then that's positive six, that means it's moved up by six.

So some examples, translated by negative two, three.

So that negative two is the top number, so that is the horizontal X values first, so horizontal, the X axis is horizontal.

The negative tells you it's going to the left.

The three is the vertical and the vertical displacement is a positive three so we're gonna be moving up.

If you go up by three or if you add three, you'd go up on the Y axis.

So this means, left by two units and up by three units.

Okay so four, negative five.

You have a think what it is before I reveal it.

So this would be four to the right and down by five, horizontal first, so that four, it either means left or it means right.

Because it's positive, it's to the right.

Another way to think about this is called an axes.

You've got the origin in the middle where they cross, the Y and the X axis cross, and the numbers that go to the right are positive, and the numbers that go to the left are negative.

So if it's a positive it would be on the right hand side of your axis.

So that might be another way to think about it, and it works for the vertical as well.

So where would minus five be? It would be down from the origin.

So it means right by four and down by five.

And then what would zero, two mean? So this one means it's just moved up by two.

So when we use words we wouldn't say zero to the left or zero to the right, but in a column vector you need to have two numbers.

There needs to be a value in the horizontals place and a value in the vertical, and so if it hasn't moved horizontally, then you need to put a zero, and that works the same if it hadn't moved vertically, you would put zero in the second in the bottom number.

So a check, translated by three, minus five means which of those in words? Pause the video and come back when you're ready to move on.

Hopefully you've gone for right by three and down by five.

So three is positive, would be on the right hand side of the X axis, minus five would be on the bottom of the Y axis so that's down from the origin.

Okay what about one, zero? What does the vector one, zero mean? Pause the video and come back for the answer.

And so that just means right by one.

There's no vertical displacement.

The zero in that second space means there's no vertical displacement, it's only moved across, and then the positive one tells you it's moved to the right.

Here we've got a translation, we've got two squares, purple and green, I and A prime, the image of A.

Laura says that the object has been translated by one, minus two.

So that's her vector that she thinks the translation's been taken by.

And Izzy thinks the object is being translated by five, negative two.

Who's correct or is neither of them correct? So Izzy is correct, and that's because you need to count between the corresponding vertices.

Doesn't matter which pair, so if you had done the bottom two corresponding vertices, you'd still get the vector five, minus two.

Because your vector needs horizontal and then vertical, it's always good to count in that way as well, so count horizontally, count vertically.

So, I'm gonna go through one and then you're gonna do one as a check.

So describe the transformation.

So again, make sure you've identified to yourself which one's the object and which one's the image.

So the object is triangle PQR, the purple one, and the image is triangle P primed, Q primed, R primed, the image of that triangle.

So we need to choose corresponding vertices to count.

And I've used the R's, so it's three to the right so that's a positive three and one up, so that's a positive one, so it's translated by three, one.

That is the description.

It feels very short and that's what we like about it, it's succinct, it tells you exactly what it needs to tell you without having lots and lots of words to write.

Okay, you have a go at this one.

Pause the video and when you're ready, come back for the answers.

So hopefully you went for translated by negative three, negative one.

So this one has gone to the left by three, which is why it's a negative three, and it's also gone down by one, so that's a negative one.

You now can do your practise.

So question one, you need to match up the descriptions to the correct column vector.

Pause the video, do the matching and then come back.

Question two, you need to write a description using a column vector for each translation, so translated by and then which column vector.

Remember, count between the corresponding vertices.

Pause the video.

When you're ready, come back.

Question three, we did examples like this during the lesson, you need to describe the translation by looking at what has changed between each pair and it should be the same throughout.

Pause the video and when you're ready, come back for the answers.

Okay so question one, hopefully you've matched it in the same way as this so left by five and up by three means negative five, positive three, right by three and up by five is three, five, both of those are positive to the right and up.

Right by five and up by three, five, three, and right by five and down by three is five, negative three.

Your descriptions hopefully for the first one negative four, negative two, it's gone to the left by four and down by two.

Did you choose the correct object? And then for B translated right by four and down by one so the vector is four minus one.

And question three looking at the coordinate pairs, what translation has happened? Well the coordinates in A went up by five horizontally so a positive five, and went down by one vertically, so minus one, and for B, the X coordinates went down by three, so a negative three, that means it's moved to the left, and the Y coordinates have increased by two, so it's a two on the bottom.

So during today's lesson we have looked at describing a translation and know that we need to give both direction and distance from the start.

That is displacement, whereas a column vector is another way of doing this, much more succinct and it gives you the horizontal displacement and the vertical displacement in one notation.

Well done today and I look forward to working with you again.