Lesson video

Hello, everyone, I'm Mr. Gratton and thank you so much for joining me for today's math lesson.

In this lesson, we will be learning practical techniques to rotate objects by very specific rotation instructions.

We will be using a range of mathematical language when following descriptions of rotations.

Pause here if you need to refamiliarize yourself with some of these words.

First up in this lesson, we'll be learning the basics of rotating with tracing paper and focusing on rotations where the centre of rotation is on the vertex of the object.

It is crucial that you will have tracing paper for today's lesson, as well as a ruler and a pencil.

Right, let's go straight into an example of rotation where we have to use tracing paper.

Step one, place tracing paper over your object.

Make sure the tracing paper covers all of your object and the centre of rotation as well.

Do not leave any part of the shape uncovered by the tracing paper.

Step two, using a pencil, sketch over both the object and put across at the location where the centre of rotation is.

Both of these must be drawn onto the tracing paper.

Step three, push your pencil firmly into the centre of rotation to keep the tracing paper fixed in place.

Step four, rotate the tracing paper, in this case, 90 degrees anti-clockwise, making sure the pencil is still acting as a pivot or an anchor to keep the tracing paper still and in place like so.

Step five, keep the tracing paper as still as possible and firmly push the pencil into each of the vertices of the image on the tracing paper like so.

This will imprint the vertex of the image onto the tracing paper underneath.

Whilst not always true, it is likely that the vertices will line up with the lines on the grid on the piece of paper behind and not just in the random middle of one of the squares.

So make sure when you are pushing your pencil down through the tracing paper onto the paper, the pencil is being pushed down into one of those crossed points and not the empty space in a square around it.

Step six, when all vertices are visible on the tracing paper, it is safe to remove the tracing paper itself.

You should be able to see all of the marks left by you pushing that pencil firmly into the paper through each of the vertices.

Step seven, join up all of the vertices that you've imprinted onto the paper to complete your shape.

This is the image after the rotation.

We can see that this image is a 90 degree anti-clockwise rotation of the object.

Let's have a look at this example now using real tracing paper and mathematical equipment.

So here's a practical demonstration of how I would rotate using tracing paper.

Step one, place this tracing paper over the object.

Grab a ruler and then trace over each and every one of the sides of your object onto the tracing paper like so.

After you have fully drawn the object, making sure to keep the tracing paper in place, also explicitly draw on the centre of rotation.

After that, push your pencil firmly onto the tracing paper so that it is fixed and then rotate your tracing paper 90 degrees.

What indicates 90 degrees to me is rotating the tracing paper approximately 90 degrees and then ensuring that all of the points on the object line up with the point on the grid lines on the paper underneath.

Push firmly onto each of the vertices of your object to leave an imprint on the paper underneath.

Now after I'm happy that I've made an imprint on the paper underneath, I can remove my tracing paper and with a pencil, go over those imprints so that I can see the vertices on the piece of paper itself.

Right, after the vertices have been drawn onto the paper, you can grab a ruler and connect all of the vertices together to create your final rotated image.

Here you can see that this image is a 90-degree clockwise rotation of my object.

Check time, after placing the tracing paper over the paper, what two things must I draw onto the tracing paper itself? Pause the video and have a think about what those two things are.

The two things you need to put on the tracing paper are the object and the centre rotation like this.

And onto the next check.

The object needs to be rotated 90 degrees around the centre of rotation that is marked on each diagram.

Which of these diagrams shows the correct placement of the tracing paper after a 90-degree rotation has happened? Pause the video to have a look at each of the three diagrams and see which one is correct? B is correct.

A is a 180-degree rotation, so it is the wrong size of rotation whilst C did not push the pencil down to keep the centre rotation in the same place and so the image is in a different location to where it should be.

And the final check.

After which step do you push your pencil hard onto the tracing paper to leave an imprint on the paper? Pause the video and have a look at which of those three options is correct for this particular scenario.

And the answer is after rotating the tracing paper.

This is to leave an imprint on the rotated shape, not the original image, so that you can draw over the imprint on the piece of paper itself.

We can create interesting shapes by repeatedly rotating an object several times by the same-sized angle.

In this particular task, we want to rotate this shape by 90 degrees, and then another 90 degrees, and then another 90 degrees, a total of three times.

After each rotation, we draw the image, once each time.

Here's how we'll do it.

Step one, place the tracing paper over the object and sketch the object and the centre of rotation onto the tracing paper.

Step two, rotate the tracing paper 90 degrees clockwise.

Then push down on the paper at each vertex to leave an imprint for you to use to draw the image.

Draw the rotated image onto the paper afterwards.

As you can see here, we have drawn our first image onto our paper.

Right, next step, step three.

Let's do it all again, but this time treat the image that you just drew as your new starting point, as your new object.

Place the tracing paper back on and line it up.

Then rotate it again following the same steps as before.

There we go, we've done our second rotation.

We have the object that we started with and then the first and second images after the first and second rotations.

Our final step, as you might be able to figure out, is to do it one more time, the third time.

Let's have a look, and this is where that third image, the third rotation will end up.

Let's have a look at what the shape actually is now.

What does this shape actually look like? Well, usually it will look like a windmill or a pinwheel.

The details will change depending on what the starting object looks like.

Are each of the images congruent or different from each other? Well, every image is congruent, identical to each other and also to the object you started with.

Rotations always produce congruent images, no matter how many times a rotation has occurred or what the rotation is.

Tracing a shape is duplicating a shape exactly.

When you rotate it, it gives a different orientation, but the shape itself does not change.

What would happen if we did a fourth rotation? Well, four lots of 90 degrees is 360 degrees which is a full turn of rotation.

A full turn is the only condition where the orientation is always preserved, along with its position, and so the fourth image would fully overlap the original object, see? Keep in mind that all images will have a different orientation to its object until that 360 degree rotation has occurred.

Okay, time for our next set of checks.

In this question, who is correct, Alex or Izzy? Has the tracing paper been rotated or not? Pause the video to select your answer, A, B, C, or D, and an explanation to go with it.

And the answer is either could be correct.

The diagram may have been rotated by a full turn of rotation or maybe it has not even been rotated at all.

Okay, onto independent practise.

For question one, use tracing paper to rotate this object by 90 degrees clockwise.

After your rotation, one of the vertices of the image will be the dot already on the drawing space.

This is to check that your image is in the right location.

Pause the video, grab some tracing paper, and go with practising some rotation.

Right, question number two.

Make sure you have more tracing paper.

Rotate each of these three objects by the descriptions underneath them.

Take your time and rotate each shape one by one.

Pause the video to give each of them a go.

And the final question of task A.

Here is where the repeated rotations technique will be used.

Rotate each of these objects three extra times to get three extra images plus the original object.

Each result should look like a pinwheel or a windmill.

Grab some more tracing paper if you need to, pause the video, and enjoy rotating.

Now for the answers.

For question one, the top left of your image should lie on top of that pre-drawn dot.

The top right of your image should lie on top of the centre of rotation.

For question number two, here are your three images.

Take your time to check each of those images and whether they match the ones on screen.

Question three, here is what each of the final shapes should look like.

Each of them again should look like a pinwheel or a windmill.

Amazing work on all the rotations that you've drawn so far.

But for now, onto our next set of rotations, one where the centre of rotation is outside of the object and not on the vertex like the ones we have just dealt with.

Let's have a look.

Notice how the centre of rotation is nowhere to be found, but I'm telling you it is somewhere and it will be outside the object.

You will have to draw on the centre first and it will usually be a centre described to you by a coordinate.

For example, rotate this object by 90 degrees anti-clockwise around the centre of rotation at the coordinate 5, 4.

For every question that doesn't have the centre of rotation already plot for you, your first step always must be to plot the coordinate for the centre of rotation first.

In this example, plot the coordinates 5, 4.

After this, the process is exactly the same as before.

Place the tracing paper over the object and the centre of rotation, sketch the shape onto the tracing paper, and rotate.

Make sure your tracing paper is big enough that it can fit all of this information on it.

It may need to be a bigger piece than before as the centre of rotation is further away from the object.

Push your pencil onto the vertices, remove the tracing paper, and draw your rotation onto the grids.

It is a misconception that your image will never overlap the object, especially if the centre of rotation is outside of the object itself.

That is wrong, it will.

Not all the time, but it does happen.

In this example, if I place my tracing paper over the shape and rotate it 90 degrees anti-clockwise, you can see that there is one square of overlap between the object and the image.

Here are a few checks to see how much you know about these rotations.

What is the first step of rotating an object when the centre of rotation is given to you as a coordinate? Sketch the object onto the tracing paper, plot the centre of rotation, or rotate the tracing paper.

Pause here to consider which step comes first.

The answer is B.

You must always check if the centre of rotation is drawn on.

If not, you must draw it for yourself.

If the centre of rotation is outside the object, then the object and image will not overlap.

Is that statement of Andeep correct? Pause the video here to decide if you agree with him and think of an explanation to why, justify your answer.

Andeep is incorrect, the object and image may overlap no matter where the centre of rotation is.

Next, check for understanding.

Imagine the object on screen.

Imagine it has been rotated with the centre of rotation at the coordinate 8, 6.

I have imprinted three of the four vertices of the image onto the paper.

Those are represented by the three dots, but I have missed one vertex.

What is the coordinate of the missing vertex of my image? Pause the video to think of which coordinate it could be.

The answer is the coordinate 11, 1.

Right, grab some more tracing paper and let's get going with some more independent practise.

For both of these questions, you must plot the coordinate for the location of the centre of rotation first before you use your tracing paper.

Perform two rotations, both starting at object A, but using different centres of rotation.

Pause the video now to perform both of these rotations.

Follow the instructions nearby each object and rotate each object one by one.

If all the rotations are correct, the images of all three will combine to create a picture.

Pause the video now to perform all three rotations and good luck making that picture.

Here are your images for questions one and two.

Notice how both of the images are in the same orientation, but they are in a different location.

This is because we have rotated both times by 90 degrees clockwise, but the centre of rotation has changed to redefine the location of the image.

And for question two, the three images should have created a boat shape.

Have a look to see if all three of your images match those on this boat.

And that concludes the second cycle where we have rotated with the centre of rotation outside the object.

But what about the final option where the centre of rotation is on or inside the object? This will be an opportunity for more rotation practise but providing a slightly different context to make sure you are familiar with all possible types of rotation.

The centre of rotation may be on the perimeter of the object or inside it.

Thankfully the method we have used all along does not change.

However you must be careful.

The image is more likely to overlap the object in some way and so it can be more confusing to try and plot the image correctly whilst it's overlapping the picture that is already there.

Many of the vertices of the image will overlap the object either on the perimeter or inside the object somewhere.

So the big advice here is you must mark the centre of rotation and the vertices onto the tracing paper very carefully and firmly push your pencil onto the paper so it leaves a mark visible enough that you can see it over whatever object is already on the paper.

So let's approach this exactly the same way as before.

Plot the coordinate onto the centre of rotation first.

Make sure this is clear as it will be overlapping your object.

Then trace the shape and the centre of rotation onto the tracing paper and rotate like before.

Make sure to press your pencil onto the paper firmly enough so that you can see the vertices over the object.

When you're done with that, remove the tracing paper, check your vertices are being marked clearly enough, and then draw your image.

Let's check your knowledge once more.

The object here is rotated by 180 degrees around the point 4, 3.

What is the first step to drawing this rotation? Is it rotating the tracing paper, sketching the shape onto the tracing paper, or marking the centre of rotation? Pause the video to make your decision.

The answer is C.

You must always plop the centre of rotation onto the grid before anything else.

Right, same shape, same rotation of 180 degrees.

Where does the vertex labelled T end up after the 180 degrees rotation? Pause to figure out which of the coordinates T ends up at.

Let's solve this in parts.

Let's connect T to the centre of rotation with a line segment and rotate it 90 degrees, 180 degrees.

T will end up at the bottom of that line segment and the location of the bottom of the line segment is C, the location for one.

Remember the vertices can end up inside the shape after a rotation.

That is fine, that is allowed.

And onto the next check.

The object is rotated 90 degrees clockwise around the coordinate 6, 5.

That is the centre of rotation.

By connecting A to the centre of rotation by a line segment, pause the video to figure out where A will end up after that 90 degree clockwise rotation.

Right, this direction's clockwise and that is the rotation location after the 90 degree rotation.

So A is the correct answer in this question.

Exactly the same shape and rotation again, but this time we are focusing on vertex B.

Where will vertex B end up? Pause the video to see which coordinate it will end up at.

As with before, connect B to the centre of rotation with a line segment.

Here it is after that 90-degree rotation and therefore the answer is the coordinate 8, 2.

Brilliant work so far.

Here are the final practise questions.

Using tracing paper to rotate this shape as instructed, count the number of unit squares on the grid where the object and reflected image overlap.

Pause this video to start your rotation and count the number of unit squares of overlap after you've drawn your image.

Rotate each of these rectangles by the description given where the image will either be overlapping or next to its object.

One last time, pause the video to practise the rotations for these rectangles.

Here are the answers for those three questions.

This is the overlap image for question one.

I count 12 unit squares of overlap.

Can you spot where those 12 squares are? And the three rotated images for question three create the letters L and I and then a plus shape as they overlap or sit adjacent to its object.

Great work for persevering with all of those rotations.

You have completed a lesson where we have learned to use tracing paper to perform actual rotations, where the centre of those rotations can either be on the vertex of a shape, inside or outside the shape, or on the perimeter of the object itself.

We've also looked practically at rotations where the orientation of an object is only preserved after a full 360 degrees has occurred.

Thank you so much for joining me for today's lesson.

Have a great rest of your day, and I hope that you join me again for some more maths in the future.