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Rotate a Shape

In this worksheet, students will rotate shapes around a given point and solve problems involving rotations which have taken place.

'Rotate a Shape' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Geometry and Measures, Congruence and Similarity

Curriculum subtopic:   Properties and Constructions, Plane Vector Geometry

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

What is rotation?

 

Rotation in mathematics means turning around a centre.

The distance from the centre to any point on the shape stays the same.

 

 

What information do we need to carry out a rotation?

 

If we tried to describe something that was turned in real life, we would need to give three pieces of information: which way it was turned, how far it turned, and what it was turned around.

 

In maths terms, these would be called:

 

1) Distance: How many degrees the object was turned. This is usually limited to 90°, 180° and 270°;

2) Direction: This is referred to as either clockwise (CW) or anti-clockwise (ACW);

3) Centre of rotation: In any rotation, there is a point that doesn't move, around which the turn is applied. We define this point as a coordinate.

 

 

How do we rotate an object or shape?

 

There are a couple of pieces of equipment you will need here: a pencil (sharpened), some squared paper and, ideally, some tracing paper.

If you don't have access to tracing paper, then regular white paper should work too. 

 

Let's put these tools to work using an example now. 

 

 

 

e.g. Rotate the triangle shown below 90° clockwise around the point (1,1):

 

Step 1: Draw out the information we have been given:

 

Four quadrant grid with a triangle and point plotted

 

Sometimes, the shape will be drawn for you (as in this case), but if not, draw the shape and the centre of rotation first.

 

 

Step 2: Trace the shape and centre:

 

Put your tracing paper over your shape, making sure you also cover the centre of rotation:

 

Four quadrant grid with a triangle, point and tracing paper overlay

 

We would recommend that when you trace the centre, draw a cross over it.

This makes it easy to see how far you have turned.

 

 

Step 3: Complete the rotation:

 

Place the point of your pencil on the centre of rotation (this makes sure it doesn't move while you turn the tracing paper) and turn the tracing paper 90° clockwise:

 

Four quadrant grid with two triangles, point and tracing paper overlay

 

 

Step 4: Transfer the image and label:

 

The easiest way to transfer the shape back onto the paper is to lightly press on the corners of the shape with your pencil.

This will create indentations in the paper below that you can then join up to create the new shape.

 

Remember that if the image is labeled a, the rotation will be labeled a':

 

Four quadrant grid with two triangles and a point

 

And it's as simple as that! 

 

 

 

In this activity, we will rotate shapes around a given point using the method described above and solve problems involving rotations which have taken place. 

 

You will find it helpful to have a sharp pencil, some squared paper and some tracing paper handy to support you in this activity. If you do not, you will need to be able to visualise these rotations in your mind's eye. 

Read the process below, then type one word in each gap to summarise it accurately. 

Why does a rotation of 180° not require a direction?

 

Because the shape will end up in ____ position. 

 

Which word from the options below accurately completes this statement? 

The same

A different

The opposite

Rotating 90° clockwise is the same as rotating ____ anti-clockwise.

 

Which word from the options below accurately completes this statement? 

180 °

240 °

270 °

260 °

Shape a has been rotated anti-clockwise below to create shape a':

 

 

How far has it been rotated?

 

Type only numbers in the gap in the sentence below to express your answer. 

180 °

240 °

270 °

260 °

Review the rotation shown below:

 

 

 

Which of the transformations below could describe this transformation?

90° clockwise

90° anti-clockwise

180° either way

270° clockwise

270° anti-clockwise

Review this rotation again:

 

 

Is shape a' a clockwise or anti-clockwise rotation of shape a?

Clockwise

Anti-clockwise

We can't tell

Rotate this shape 90° clockwise around the point of origin (0,0):

 

Four quadrant grid showing a triangle with a point labelled b

 

What are the new coordinates of point b'?

Clockwise

Anti-clockwise

We can't tell

Rotate this shape 180° clockwise around the point of origin:

 

Four quadrant grid showing a triangle with a point labelled b

 

What are the new coordinates of point b'?

Clockwise

Anti-clockwise

We can't tell

Shape a has been rotated 180° around the point (2,1):

 

Four quadrant grid showing three triangles

 

Which of the rotations shown is correct?

Shape 1

Shape 2

Neither

Shape a has been rotated 180° around the point (1,-1):

 

Four quadrant grid showing three triangles

 

Which of the rotations shown is correct?

Shape 1

Shape 2

Neither

  • Question 1

Read the process below, then type one word in each gap to summarise it accurately. 

CORRECT ANSWER
EDDIE SAYS
Could you recall this information from the Introduction? We need to remember these three key definitions to get really good at using rotation in our maths work! In order to make or interpret a rotation accurately, we need to know how much to turn, in which direction to turn, and around which point the turn is occurring. When we are asked to describe a rotation, we need to provide these same three pieces of information to get full marks. Let's put these features into action in the rest of this activity now...
  • Question 2

Why does a rotation of 180° not require a direction?

 

Because the shape will end up in ____ position. 

 

Which word from the options below accurately completes this statement? 

CORRECT ANSWER
The same
EDDIE SAYS
Let's give it a try... Take an object (like a book, a pen, etc.) and turn it 180° clockwise, then 180° anti-clockwise. What do you observe? It will end up in the same position! As 180° is the same as a half turn, it doesn't matter which way we turn, we will end up at the same point. Did you spot that?
  • Question 3

Rotating 90° clockwise is the same as rotating ____ anti-clockwise.

 

Which word from the options below accurately completes this statement? 

CORRECT ANSWER
270 °
EDDIE SAYS
When thinking about direction, it is important to note that we can always turn either clockwise or anti-clockwise, so long as the distance or angle we use matches accordingly. Turning a quarter turn clockwise is exactly the same as turning three-quarters of a turn in the opposite direction. If 90° represents a quarter, then we need to subtract this from 360° (the total amount around a point) to find the alternate amount: 360 - 90 = 270° As before, if you want to test it out by turning an object, then give it a go now!
  • Question 4

Shape a has been rotated anti-clockwise below to create shape a':

 

 

How far has it been rotated?

 

Type only numbers in the gap in the sentence below to express your answer. 

CORRECT ANSWER
EDDIE SAYS
Did you read the question carefully? We are asked to state the distance travelled (in degrees) when turning anti-clockwise, so we need to imagine the original shape turning to the left. If we are turning anti-clockwise, the shape has been turned three-quarters of a full turn, which is 270°.
  • Question 5

Review the rotation shown below:

 

 

 

Which of the transformations below could describe this transformation?

CORRECT ANSWER
90° clockwise
270° anti-clockwise
EDDIE SAYS
This is the same rotation from our previous question, so we know that 270° anti-clockwise is a viable answer, but are there any others? Did you recall the key fact linking rotation and direction? Whenever we enact a rotation, there are always two ways we can choose to turn (so long as the distance or angle matches accurately). So a three-quarters turn anti-clockwise is the same as a quarter turn clockwise, which is 90°.
  • Question 6

Review this rotation again:

 

 

Is shape a' a clockwise or anti-clockwise rotation of shape a?

CORRECT ANSWER
We can't tell
EDDIE SAYS
This was a tricky one... The question asks us if we can tell which direction the rotation has occurred in. In the previous question, we decided that either a three-quarters turn anti-clockwise or a quarter turn clockwise could have created this rotation. So, without more information, we can't say for sure which way it has been turned, as either of these options are viable.
  • Question 7

Rotate this shape 90° clockwise around the point of origin (0,0):

 

Four quadrant grid showing a triangle with a point labelled b

 

What are the new coordinates of point b'?

CORRECT ANSWER
EDDIE SAYS
You will find your pencil, squared paper and tracing paper handy here. If you have these, copy the grid, triangle and centre of rotation onto your squared paper before you start. Next, put your tracing paper over the shape, making sure you also cover the centre of rotation, and trace the shape and centre. When you trace the centre, draw a cross over it. Place the point of your pencil on the centre of rotation (this makes sure it doesn't move while you turn the tracing paper) and turn the tracing paper 90° clockwise. The easiest way to transfer the shape back onto the squared paper is to lightly press on the corners of the shape with your pencil. This will create indentations in the paper below that you can then join up to create the new shape. If you don't have the resources to draw this out, try to imagine this rotation taking place in your mind's eye. Now you have completed the rotation successfully, where is point b' located on the new shape?
  • Question 8

Rotate this shape 180° clockwise around the point of origin:

 

Four quadrant grid showing a triangle with a point labelled b

 

What are the new coordinates of point b'?

CORRECT ANSWER
EDDIE SAYS
Did you visualise it this time or draw it out? It is the same grid as the previous question, so you can re-use your grid and tracing if you drew these out last time. This time, place the point of your pencil on the centre of rotation and turn the tracing paper 180° clockwise. Lightly press on the corners of the shape with your pencil, then join up these indentations on your page below to create the new shape. Now you have completed the rotation successfully, where is point b' located on this new rotation?
  • Question 9

Shape a has been rotated 180° around the point (2,1):

 

Four quadrant grid showing three triangles

 

Which of the rotations shown is correct?

CORRECT ANSWER
Neither
EDDIE SAYS
If you have resources, copy the grid, starting triangle and centre of rotation onto your squared paper before you start. Next, trace the shape and centre. When you trace the centre, draw a cross over it. Take note that this time the centre is not at (0,0), but at (2,1) instead. Place the point of your pencil on the centre of rotation and turn the tracing paper 180° clockwise or anti-clockwise. Lightly press on the corners of the shape with your pencil, then join up the indentations on the paper below to create the new shape. If you don't have the resources to draw this out, try to imagine this rotation taking place in your mind's eye. Now you have completed the rotation successfully, does the position of your rotation match either Shape 1 or Shape 2 shown on the diagram above? It should not match either, so the correct answer is Neither. Shape 1 is a rotation of the shape around the point (0,0), whilst Shape 2 is a rotation around the point (2,0.5).
  • Question 10

Shape a has been rotated 180° around the point (1,-1):

 

Four quadrant grid showing three triangles

 

Which of the rotations shown is correct?

CORRECT ANSWER
Shape 1
EDDIE SAYS
If you have resources, copy the grid, starting triangle and centre of rotation onto your squared paper before you start. Next, trace the shape and centre. When you trace the centre, draw a cross over it. Take note that this time the centre is not at (0,0), but at (1,-1) instead. Place the point of your pencil on the centre of rotation and turn the tracing paper 180° clockwise or anti-clockwise. Lightly press on the corners of the shape with your pencil, then join up the indentations on the paper below to create the new shape. If you don't have the resources to draw this out, try to imagine this rotation taking place in your mind's eye. Now you have completed the rotation successfully, does the position of your rotation match either Shape 1 or Shape 2 shown on the diagram above? It should match with Shape 1, so this is correct answer. Shape 2 has been rotated around the point (-1, 1), which clearly shows how important it is to get our coordinates the right way around! Excellent progress - that's another activity completed! You can now rotate shapes around a given point and solve problems involving rotations which have taken place.
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