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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.

**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.

Prepare for math tests in your state with these Grade 3 to Grade 6 practice assessments for Common Core and state equivalents. 40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

DOWNLOAD FREEPrepare for math tests in your state with these Grade 3 to Grade 6 practice assessments for Common Core and state equivalents. 40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

DOWNLOAD FREEHow 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.

In order to rotate a shape around a fixed point:

**Trace the shape.****Rotate the tracing paper around the center of rotation.****Draw the rotated shape onto the grid.**

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

**Trace the shape.**

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

2**Rotate 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.

3**Draw 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.

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

**Trace the shape.**

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

**Rotate the tracing paper around the center of rotation.**

Use the pencil and put the tip onto the fixed point. Pivot the tracing paper.

**Draw 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. This is also known as an **invariant** point of the shape.

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.**

It can be useful to add a line on the diagram extending from the shape to the center of rotation. Use a pencil and trace the shape onto a piece of tracing paper.

**Rotate the tracing paper around the center of rotation.**

Use the pencil and put the tip onto the center of rotation. Pivot the tracing paper. It may be useful to add a line connecting the shape and the center of rotation.

**Draw the rotated shape onto the grid.**

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

Those dotted lines are extra, but they help to show more clearly that the shape has been rotated correctly.

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.**

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

Those dotted lines are extra, but they help to show more clearly that the shape has been rotated correctly.

In order to describe a rotation:

**Trace the shape.****Rotate the tracing paper.****Write down the description.**

Describe the rotation of shape A to shape B .

**Trace the shape.**

Use a pencil and trace the object shape (shape A ) onto a piece of tracing paper.

**Rotate the tracing paper.**

Think about where the center of rotation might be. Use the pencil and put the tip onto that point. Pivot the tracing paper to check. It may take a few tries until you find the correct center of rotation.

**Write down the description.**

Make sure you state that it is a rotation. Then give the angle of rotation and if necessary the direction of rotation. Also, give the coordinates of the center of rotation.

Rotation of 180^{\circ} around the point (0,1) .

Since the rotation is a half-turn, no direction is needed.

Since the rotation was 180^{\circ}, if you connect the vertices in corresponding pairs, these lines all cross at the center of rotation.

Describe the rotation of shape A to shape B

**Trace the shape.**

Use a pencil and trace the object shape (shape A ) onto a piece of tracing paper.

**Rotate the tracing paper.**

**Write down the description.**

Make sure you state that it is a rotation. Then give the angle of rotation and if necessary the direction of rotation. Also, give the coordinates of the center of rotation.

Rotation of 90^{\circ} clockwise around the point (- \, 1,- \, 1) .

- 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.

**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.

- Transformations
- Scale factor
- Translation math
- Reflection in math
- Dilations
- Center of dilation

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).

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.

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

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

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.

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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