Lesson video

Hi there, my name's Ms. Lambell.

You've made a superb choice deciding to join me today to do some maths.

Let's get going.

Welcome to today's lesson.

The title of today's lesson is "Checking and Securing Understanding of Translations," and that's within the unit, "Vectors." By the end of this lesson, you'll be able to describe a translation using column notation, and carry out the translation of a point.

And we'll also actually look at carrying out translations with different shapes too.

Some key words that we'll be using in today's lesson are: translation, congruent, vector, vertex, and displacement.

A translation is a transformation in which every point of a shape moves the same distance in the same direction.

Congruent, if one shape can fit exactly on top of another using rotation, reflection, or translation, then the shapes are congruent.

A vector can be used to describe a translation.

A vertex is a point where two or more line segments meet, the plural is vertices.

And displacement, displacement is the distance from the starting point when measured in a straight line, effectively, the shortest distance between two points.

Today's lesson is split into two separate parts.

In the first one, we will look at describing translations, and in the second part, we will look at carrying out translations.

Come on, let's get going with that first one, so we'll be describing translations.

Aisha and Jacob are playing a game.

They have to describe how to pin the tail onto the elephant.

So, we can see on the grid, that there's a tail and an elephant, and they need to get the tail to the green cross.

Aisha says, "Well, that's easy.

Move it four units to the right and two units up." Jacob says, "I agree, Aisha, but I'm sure that there is a more efficient way of writing that." Aisha's response is, "You are right, Jacob.

I think you mean writing it in vector form." "That is it." Can you remember how to write four units right and two units up in vector form? I know you remember how to do that, and I'm sure you've written down your answer to this.

Well, let's firstly check that Aisha's description was right.

We need to get the tail onto the green cross.

To do that, we are going to move four places to the right, and then we will need to move two places up.

So Aisha's description was right.

Jacob was also right, we can write this in vector form.

And like I said, I'm pretty certain that this is what you'll have written down.

Column vectors have the general form, horizontal displacement on the top and vertical displacement on the bottom, and we write this in a pair of brackets, remembering, if we're moving to the left, it's going to be a negative value, and to the right, a positive value, and if we're moving down, it's going to be a negative value, and if we're moving up, that's going to be a positive value.

Therefore, we will be moving four to the right, so this is gonna be positive four.

Remember, you don't have to put the plus there, because we know that any number without a symbol is positive, but you may choose to write it.

And then two up.

Again, here, I've decided not to put the plus symbol, but you could if you wanted to.

Now the tail and the elephant have moved, let's look at how we are going to get the tail onto the elephant.

How are we going to do that? I'll give you a moment to decide what movement you think needs to happen.

And I know you said that we need to move five places to the right, or five units to the right, and two units down.

In vector form, we're going to move five right, so five right is going to be the top number, and that's five, and then we're going to move two down to the bottom number, it's gonna be two down.

And what do we include if it's down? That's right, we include that negative, negative two.

Let's take a look at this one.

Again, I'm gonna pause a moment to give you a chance to work out what movement you think you need to make to get the tail into the correct place.

And let's take a look.

We would need to move it horizontally seven to the left, and vertically, we would need to move it three up.

The tail is now in the correct place.

Seven to the left is negative seven, and three up is positive three.

Remember, you don't have to write that positive symbol.

And one more together.

This time, we need to move six places to the left, and we need to move five places down, or units, I should really say.

As a vector, the horizontal movement is left, so is it gonna be positive or negative? Yeah, you're right, it's gonna be negative, so it's gonna be negative six.

And then the vertical displacement is five down, so is that gonna be positive or negative? Yeah, it's gonna be negative, isn't it? So we end up with the vector negative six, negative five.

Now it's your turn, pause the video, work out which is the correct vector to get the tail onto the correct place on the elephant.

Good luck with that, I know you'll smash it, and I'll see you back here in a moment, we'll check your answer.

What did you decide, A, B, C, or D? The correct answer, we'll have moved one place to the right, or one unit to the right, and three units down.

And that is represented by the vector, D, one, negative three.

How did you get on? Yeah, of course you've got that right.

We're still playing the same game, and we can see here that we need to move the tail four down.

Jacob says, "So, this one must be negative four." Do you agree with Jacob? Unfortunately, Jacob is not right, he has not described the horizontal displacement.

Aisha says, "Jacob, you need to describe the horizontal displacement." And Jacob's response to that is, "But it did not move horizontally." The correct vector is zero, negative four.

It's really important we place the zero in our column vector to show that there is no movement horizontally, represented by the zero on the top of the column vector.

Now you are ready to have a go at this check for understanding.

I'd like you, please, to pause the video, you need to match each description to the correct vector.

Good luck with this, but I know you'll smash it.

I'll be here waiting when you get back, and we'll check those answers for you.

Good luck.

How did you get on? Yeah, of course you did.

Six units right is the column vector, six, zero.

Two units right, six units down, is the column vector, two, negative six.

Six units right, two units down is the column vector, six, negative two.

Six units down is the vector, zero, negative six.

Two units left, six units up is negative two, six.

And then, obviously, the final one, six units left, two units down is negative six, negative two.

How did you get on? Six out of six? Brilliant.

I knew you would.

Wow, that was a quick learning cycle wasn't it? We are now ready, or you are now ready, to have a go at this first task, Task A.

For question number one, I'd like you please to write the vertical and horizontal displacements represented by those vectors.

There are five to do.

Pause the video, write down your answers, and then come back when you are ready.

Good luck.

Well done.

Question number two, we're back to pinning the tail on the elephant.

I'd like you please to write a vector to describe how to translate the tail into the correct position.

This time, you'll notice that the cross is black, makes no difference, it's still in the same place on the elephant.

You need to have a go at these five questions, and I know you are gonna get five out of five.

Pause the video, count really, really carefully, and I would always double check these, it's really easy to miscount, especially if you are doing this from a little distance back.

Pause the video now, and then when you come back we'll check those answers for you.

Well done on those.

Let's check those answers.

Question one, part a, negative five, zero, is a displacement of five units left.

b, vector, eight, negative three, is a displacement of eight units right, three units down.

Vector c, zero, four, is a displacement of four units up.

d, the column vector, four, negative two, is showing a displacement of four units to the right and two units down.

And then, finally, e, negative three, negative one, is representing a displacement of three units left and one unit down.

Five out of five? Of course you did.

Question number two, a is, four, negative one.

b, zero, five.

c, two, three.

d, negative two, negative three.

And e is six, zero.

And again, how did you get on? Brilliant.

Well done.

I knew you'd get 10 out of 10 on Task A.

We're ready now then to move on to our next learning cycle, and that is carrying out translations.

So, if you've smashed that first learning cycle, you're definitely going to do the same with this one.

Now that we've recapped how translations are represented with a vector, let's consider some more translations.

So, here, we've got some different images and objects.

Aisha says both of these show a translation as the triangles are congruent.

And Jacob says, "Remember, Aisha, a translation only moves a shape horizontally and vertically." Aisha responds by saying, "Of course, the second one shows a translation." So, the first one, we can see that they are congruent, the triangles are identical, but the purple one has been rotated.

Whereas if we look at the right-hand one, we can see that the shape has not turned at all, so it's not been rotated, it's not flipped, it's not been reflected, it's just slid, is the word I like to think of, into a different position on the grid.

Now we're going to translate point A by the vector, negative three, four.

What does that mean? It means we're going to move three places to the left, and then, from that point, we're going to move four places up.

That is where point A is going to be, and we mark it with A', a little dash.

Now let's try the next one.

Translate point A by, one, negative three.

This time, the one, the horizontal displacement, the top number, remember, is one.

I'm going to move one place to the right.

And then negative three, so, from this point, I'd need to move three down.

And again, I'm gonna label this as A'.

Let's just do one more together, and then you can have a go at the one on the right-hand side independently.

Translate point A by, one, negative two.

I'm going to move one to the right, because it's a positive number, and I'm going to move two down.

Remember to label your new point A'.

Your turn.

Pause the video, have a go at this one.

Remember to label your point when you type where it's going to be, and I'll be waiting when you get back.

Okay, let's check that answer.

This vector, negative two, top number is our horizontal displacement, so I move two places to the left, and then the bottom number is our vertical displacement, so I am moving one place down.

And that is the new point which we're going to label A'.

Often, we need to translate an entire shape, rather than just a point.

We can consider the vertices of a shape and translate them one by one.

Here, I'm moving A two places to the right, one place up.

I would then do the same with B, and I can do the same with C.

And then I can join my points together to give me the triangle.

Remember, the triangle is going to look exactly the same, it won't have been reflected or rotated, it would've just slid to a different position on the grid.

We can also use tracing paper, and it can help us with translating the whole object.

Here's an example of how we would use the tracing paper.

We're going to translate this triangle.

We place the tracing paper, and we draw the vertices, marked with crosses.

We're then going to move the piece of tracing paper three places to the right, and one place up, because we were translating using the vector, three, one.

I can now see where my points are going to be, and then I can transfer those onto my grid.

Sometimes, the complexity of the object may support one of the methods more than the other.

Which method might you use for each of these? I would probably use the counting method for the first one, but I would probably use tracing paper for that second one, because it's got multiple vertices within the shape.

We're going to translate this shape by the vector, three, negative two.

I'm going to start with point A.

I'm going to move it three places to the right, two places down, and label it A'.

B, again, I'm gonna do the same thing, three places to the right and two down, label it B' prime.

Do the same with C, and then with D.

And then I can join my points together, and my trapezium, we can see, is congruent.

It's exactly the same, so I know that I've not made an error as I've done this.

Now, your turn.

I'd like you please to translate this shape by the vector, negative five, one.

Off you go.

Let's check and make sure you've got it in the correct position.

Firstly, what was the vector asking us to do? Negative five is five to the left, and one, on the bottom, is our vertical displacement, so that's one up.

Five left, one up.

Just check that all of your vertices have all moved five places to the right and one place up.

If they have, your triangle will look identical.

You may have decided to use some tracing paper here.

Ready now, then, for Task B.

You're going to be carrying out some translations.

I've given you some different shapes, next to each of those shapes is a vector.

What you need to do, using your favourite choice, so, counting or using tracing paper, you're going to take each shape and translate it by the vector next to it.

If you do this, and you do it correctly, there will be a word in the middle of your page.

Take your time over this, count really, really carefully, particularly those ones with large numbers.

So, for example, the negative 11, negative eight, and the negative 16, negative four, and the 11, eight, it's really easy just to miscount by one square in any direction.

Pause the video.

Like I said, take your time over it, double-check carefully that you've moved it, use tracing paper if you need to.

And then, when you come back, you'll be able to tell me what the word was.

Pause the video now, and I'll be waiting when you get back.

Good luck with this one.

Great work.

I wonder if you've got the right word? What word have you got? That's right, it was vector.

And why was it the word vector? Well, because we use a column vector when we're translating shapes.

Your image should look exactly the same as mine, and I'm sure that it does.

Summarising our learning from today's lesson then.

Vectors are used to describe translations of points and shapes.

Column vectors are used to do this, and they have the general form, where the horizontal displacement is represented by the top number in the vector, and the vertical displacement is represented by the bottom number in the vector, remembering, if we move to the left, it's a negative number, and to the right, is a positive number, and if we move down, it's a negative number, and up is positive.

The way I think of that is how the numbers look on a set of axes, the negative ones are at the bottom, so I'm going to be moving down, and the negative ones are on the left-hand side, so I'm going to be moving left.

Couple of examples there.

If I'm moving four to the right, and two up, that's represented by the vector four, two.

And the second one, if I'm moving seven to the left and three up, that's represented by the vector, negative seven, three.

What a fantastic job you've done with today's learning.

I hope you enjoyed translating those shapes and spelling out that word, vector.

Like I said, superb job.

Well done.

I really do hope that you'll join me again soon to do some more maths.

Take care of yourself.

Goodbye.