Math resources Geometry

Transformations

Rotations

# Rotations

Here you will learn about rotations, including how to rotate a shape around a fixed point, and how to describe clockwise rotations and counterclockwise rotations.

Students will first learn about rotations as part of geometry in 8 th grade.

## What are rotations?

Rotations are transformations that turn a shape around a fixed point by a certain angle measure. This movement changes the shape’s orientation but not its shape or size.

To rotate a shape, you need:

• a center of rotation
• an angle of rotation (given in degrees)
• a direction of rotation – either clockwise or counterclockwise
(Counterclockwise direction is sometimes known as anticlockwise direction.)

For example,

Rotate shape A \; 90^{\circ} clockwise around a fixed point.

Shape A has been rotated a quarter turn clockwise to give shape B.

For example,

Rotate shape A \; 180^{\circ} around a fixed point.

Shape A has been rotated a half turn to give shape B.

Whether the direction is clockwise or counterclockwise is irrelevant.

Using tracing paper can be very useful when using rotations.

The original figure is called the object or the preimage, and the rotated figure is called the image or rotated image.

For rotations, the object shape and the image shape are congruent because they are the same shape and the same size. As the lengths of the shapes have been kept the same, the shapes are said to have isometry.

## Common Core State Standards

How does this relate to 8 th grade math and high school math?

• Grade 8 – Geometry (8.G.A.3)
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

• High School – Geometry – Congruence (HS.G.CO.A.5)
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

• High School – Geometry – Congruence (HS.G.CO.B.6)
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

## How to rotate a shape around a fixed point

In order to rotate a shape around a fixed point:

1. Trace the shape.
2. Rotate the tracing paper around the center of rotation.
3. Draw the rotated shape onto the grid.

## Rotations examples

### Example 1: rotate a shape around a fixed point

Rotate the shaded shape 90^{\circ} clockwise around the fixed point:

1. Trace the shape.

Use a pencil and trace the shape onto a piece of tracing paper.

2Rotate the tracing paper around the center of rotation.

Use the pencil and put the tip onto the fixed point. Pivot the tracing paper a quarter turn clockwise.

3Draw the rotated shape onto the grid.

Carefully lift the tracing paper and draw the rotated shape in the correct position.

Note – one of the vertices of the triangle has not moved. So the original triangle and the rotated triangle share a point. This is also known as an invariant point of the shape.

### Example 2: rotate a shape around a fixed point

Rotate the shaded shape 180^{\circ} around the fixed point:

Trace the shape.

Rotate the tracing paper around the center of rotation.

Draw the rotated shape onto the grid.

### Example 3: rotate a shape around a center of rotation

Rotate the shaded shape 90^{\circ} counterclockwise around (3,3)\text{:}

Before we can start, we need to mark the center of rotation on the diagram.

Trace the shape.

Rotate the tracing paper around the center of rotation.

Draw the rotated shape onto the grid.

### Example 4: rotate a shape around a center of rotation

Rotate the shaded shape 180^{\circ} around O\text{:}

Before we can start, we need to mark the center of rotation on the diagram.

O stands for the Origin of the coordinate grid and has the coordinates (0,0).

Trace the shape.

Rotate the tracing paper around the center of rotation.

Draw the rotated shape onto the grid.

## How to describe a rotation

In order to describe a rotation:

1. Trace the shape.
2. Rotate the tracing paper.
3. Write down the description.

## Describing rotations examples

### Example 5: describe a rotation

Describe the rotation of shape A to shape B .

Trace the shape.

Rotate the tracing paper.

Write down the description.

### Example 6: describe a rotation

Describe the rotation of shape A to shape B

Trace the shape.

Rotate the tracing paper.

Write down the description.

### Teaching tips for rotations

• Provide students with transparent paper or tracing paper to manually rotate shapes. This helps them visualize how each rotated point moves.

• Use different colors to show how vertices of polygons move between quadrants after rotation.

### Easy mistakes to make

• Rotating by an incorrect angle
Adding a small arrow onto your tracing paper can help to rotate the shape by the correct angle.

• Confusing clockwise and counterclockwise
You may need to look at a clock to remind yourself of the difference between clockwise and counterclockwise.

• The origin
The origin of a coordinate grid has the coordinates (0,0). It is commonly denoted as O. It is used often as the center of rotation.

• Misunderstanding the position of the center of rotation
The center of rotation can be within the object shape.

For example,

• Not understanding alternative angles and directions
A rotation of 270 degrees in a clockwise direction is a correct alternative to a 90 degree rotation in a counterclockwise direction.

A rotation of 270 degrees in a counterclockwise direction is a correct alternative to a 90 degree rotation in a clockwise direction.

### Practice rotation questions

1. Rotate the shaded shape 180^{\circ} around the center of rotation:

The object shape has to be rotated a half-turn. It needs to have been rotated around the center of rotation. It can not have been reflected.

2. Rotate the shaded shape 90^{\circ} counterclockwise around the center of rotation:

The object shape has to be rotated 90^{\circ} counterclockwise. The center of rotation should be used. The additional extra dotted lines may help to make this rotation clearer.

3. Rotate the shaded shape 90^{\circ} clockwise around (0,0)\text{:}

The object shape has to be rotated 90^{\circ} clockwise. The center of rotation should be the origin. As you can see, the shape starts in the 1 st quadrant but the rotated image is in the 4 th quadrant (having rotated across the x -axis.) The additional dotted lines help to make this rotation clearer.

4. Rotate the shaded shape 180^{\circ} around (- \, 1,0)\text{:}

The center of rotation should be the (- \, 1,0). The additional dotted lines help to make this rotation clearer.

5. Describe the rotation of shape A to shape B.

Rotation of 90^{\circ} counterclockwise around the origin

Rotation 90^{\circ} clockwise around (1,1)

Rotation of 90^{\circ} clockwise around (1,1)

Rotation of 90^{\circ} clockwise around the origin

Make sure you know which is the original object shape and which is the image shape. The additional dotted lines help to make this rotation clearer. The center of rotation is (0,0), the origin.

6. Describe the rotation of shape A to shape B.

Rotation of 180^{\circ} around (- \, 1,- \, 1)

Rotation of 180^{\circ} around (1, 1)

Rotation of 90^{\circ} counterclockwise around the origin

Reflection of 180^{\circ} around (0,0)

Make sure you know which is the original object shape and which is the image shape. Since this is a half-turn the direction of the 180^{\circ} is not needed. The shape rotates across the y -axis. The additional dotted lines help to make this rotation clearer. The center of rotation is (- \, 1,- \, 1).

## Rotations FAQs

What is a rotation in math?

A rotation is a transformation that turns a figure around a fixed point, known as the center of rotation or point of rotation, by a specified angle.

Can rotations be combined with other types of transformations?

Yes, rotations can be combined with translations, reflections, and other rotations to form more complex transformations.

Can a rotation be clockwise or counterclockwise?

Yes, rotations can be in either direction. Positive angles typically represent counterclockwise rotations, while negative angles represent clockwise rotations.

How is a rotation in math similar to the rotation of earth?

A rotation in math is similar to Earth’s rotation because both involve turning around a fixed point or axis while maintaining the shape and size of the object. In math, a figure rotates around a point on the coordinate plane, while Earth rotates around its axis.

Is a rotation a rigid transformation?

Yes, a rotation is a rigid transformation because it turns a figure around a fixed point without changing its shape or size.

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