# Rotating shape: What is rotation of shapes?

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## How to Rotate a Shape

### What Does it Mean to Rotate a Shape?

Rotating a shape means to change its direction by turning it. The size and shape do not change during rotation. A shape is often rotated about a specific point called the centre of rotation. We also need to know what angle and direction the shape is rotated in.

For example, this triangle has been rotated 90° counter-clockwise about the point.

You can tell that a shape has been rotated because it is not facing the same direction as it was originally. The original shape is called the object and the rotated shape is called the image.

When rotated, a shape remains the same distance away from the centre of rotation. It is just in a different direction.

### Rules for Rotating a Shape About the Origin

The rules for rotating shapes using coordinates are:

Clockwise rotation angle Counter-clockwise rotation angle Rule
90° 270° (x, y) → (y, -x)
180° 180° (x, y) → (-x, -y)
270° 90° (x, y) → (-y, x)
##### How to Rotate a Shape by 90 Degrees

To rotate shape 90° clockwise about the origin, all original coordinates (x, y) becomes (y, -x). To rotate a shape 90° counter-clockwise about the origin, the coordinates (x, y) become (-y, x). Simply switch the x and y coordinates and multiply the coordinate with the negative sign by -1.

For example, use the rule (x, y) to (y, -x) to rotate the shape 90° clockwise.

To use this rule, simply switch the (3, 1) to (1, 3) and then make the 3 negative to get (1, -3).

##### How to Rotate a Shape by 180 Degrees

To rotate a shape by 180° clockwise or counter-clockwise, the rule is to replace the (x, y) coordinates with (-x, -y). For example, a coordinate at (3, 1) will move to (-3, -1) after a 180° rotation.

Simply multiply each coordinate by -1 to rotate a shape 180°.

If a coordinate is negative, it will become positive after a 180° rotation. For example, the coordinate (-1, -4), will move to (1, 4) after a 180° rotation.

##### How to Rotate a Shape by 270 Degrees

To rotate shape 270° clockwise about the origin, all original coordinates (x, y) becomes (-y, x). To rotate a shape 270° counter-clockwise about the origin, the coordinates (x, y) become (y, -x). Simply switch the x and y coordinates and multiply the coordinate with the negative sign by -1.

For example, to rotate the point (3, 1) 270° clockwise, it becomes (-1, 3). Simply switch the x and y coordinates to get (1, 3) and then multiply the 1 by -1 to get (-1, 3).

### How to Rotate a Shape Using Tracing Paper

To rotate a shape using tracing paper:

1. Place the tracing paper over the shape and draw around the shape.
2. Draw an arrow from the centre of rotation pointing upwards.
3. Keep the pen over the centre of rotation and rotate the tracing paper.
4. Stop when the arrow is facing either right (for 90° CW / 270° CCW turn), down (for 180° turn) or left (for 270° CW / 90° CCW turn).
5. Draw the shape in this new position below the tracing paper.

The easiest way to rotate a shape is to use tracing paper.

For example, use tracing paper to rotate the shape 90° clockwise about the point.

After a 90° clockwise rotation, the upwards arrow is facing right. Use the grid lines of the paper to help line up the arrow correctly, ensuring that it is completely horizontal.

Then draw in the shape below.

Here is another example. Rotate the shape 270° clockwise about the point using tracing paper.

Every lot of 90° is equivalent to a quarter turn. 270° is 3 lots of 90° and so, 270° is equivalent to three-quarters of a turn.

The upward-facing arrow will be facing to the left after a 270° clockwise rotation.

We draw the shape in below.

### How to Rotate a Shape Without Using Tracing Paper

To rotate a shape without tracing paper, draw horizontal and vertical arrows from the centre of rotation to each corner of the shape. The new corner can be found by rotating each of these arrows according to the following rules:

Original Direction 90° CW / 270° CCW 180° CW / 180 ° CCW 270° CW / 90 ° CCW

For example, rotate the shape 90° clockwise without using tracing paper.

The first step is to draw horizontal and vertical arrows connecting the centre of rotation to a corner on the shape.

The corner selected below is one square right and one square up from the centre of rotation.

Using the rules above, a right-facing arrow will be facing down following a 90° clockwise rotation. An upwards-facing arrow will be facing right after a 90° clockwise rotation.

Instead of the corner being one right and one up, it will now be one down and one right.

Using the new position of this corner, the rest of the shape can be drawn in. The same process can be repeated for all corners to check the result.

Here is another example. Rotate the shape 270° clockwise without using tracing paper.

Draw horizontal and vertical arrows from the centre of rotation to each corner.

After a 270° clockwise rotation all upwards-facing arrows will be facing left and all left-facing arrows will be facing down.

Once all corners are drawn in their new positions, the rotated shape can be drawn by connecting these together.

## 3 Ways to Rotate a Shape

These 3 methods to rotate a shape were super helpful for my students!

Rotating a shape can be a difficult concept for both students and teachers. After seeing my students struggle with this topic, I came up with a few strategies to make rotations easier.

These 3 strategies work with different levels of learners. I encourage you to try all 3!

### 1. Use Patty Paper

Patty paper or wax paper is useful for so many math concepts! I always had plenty of patty paper in my classroom. This is the patty paper I use from Amazon.  I love using patty paper to rotate a shape because it helps visual learners “see” the rotation before actual graphing it.

For this method, you will place the patty paper over the graph and trace the shape. You will also plot the origin (0,0) on your patty paper.

Next, rotate the patty paper. For this example, I wanted to rotate 90 degrees clockwise. So, I turned the patty paper one quarter turn to the right. Make sure you keep the origin on the patty paper lined up with the origin on your graph.

Next, write down the coordinates of your new shape.

Finally, remove the patty paper and graph your image.

### 2. Use Coordinate Rules

Coordinate rules are a great tool for transformations. There are three coordinate rules for rotating about the origin.

Using the coordinate rules to rotate a shape are great if your students aren’t allowed to use patty paper on the test. Check out the rules below.

The first step for using coordinate rules to rotate a shape is to write the coordinate rule on your paper. For this example, I wrote the coordinate rule for 180 degrees.

Next, write the coordinates of your pre-image.

Then, use the coordinate rule to get the coordinates for your image.

For 180 degrees, the rule is (-x, -y). This means to change the signs of both the x value and the y value.

BE CAREFUL: The negative DOES NOT mean that the number must be negative. It means change the sign. So, if my pre-image coordinates are negative, they will change to positive.

Finally, graph the coordinates for your image.

This method is a variation of method #2. It’s my favorite method! It uses the four quadrants of the graph to rotate the shape.

First, write down the coordinate rule and the coordinates of your pre-image.

In this example, the pre-image is in the second quadrant. If I rotate 90 degrees clockwise, the shape will be in the first quadrant. Rotating 180 degrees, will put the shape in the fourth quadrant. If I rotate 270 degrees, the shape will be in the third quadrant.

So, all points should be in the third quadrant.

The quadrant tells you what the signs should be for all of your coordinates. In the third quadrant, the signs are (- , -).

All of my coordinates for my image will have those signs.

Finally, look back at the coordinate rule. For 270 degrees (and for 90 degrees), the rule tells me to switch the x and y values.

Since I already have my signs, I just need to switch the x and y values. So, since point W is (-4, 5), then W’ will be (-5, -4).

Do that for all the coordinates. Then graph your image.

I hope you liked these 3 strategies for rotating a shape!

Would any of these methods help you with rotations? Let me know below!

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