Triangles on a Cartesian plane

Explore the following grids with triangles on them. What is happening on each grid? How does the triangle’s position change?

Record your ideas in a notebook or a method of your choice.

What is a transformation?

Each image in the Minds On section represents a transformation of a triangle.

Task 1: Translations of a triangle

Translations move a shape without rotating, reflecting, or changing the size.

Imagine you place a sign on the floor and move it somewhere else in the room. As long as the sign is not rotated (or flipped), this would be a translation. We use directional words like right or left and up or down to describe the direction a shape is moved.

Let’s explore this video to learn how to translate a shape.

Translations, Copyright © 2021 The Ontario Educational Communications Authority. All rights reserved.

Practice

Translate the following triangle 3 units to the left and 2 units down. Record your answer using a method of your choice.

When you are done translating the triangle on your own, press ‘Answer’ to access the solution.

Answer

Task 2: Reflections of a shape

Reflections flip a shape. An example of a reflection is to flip a sign to show the back of the sign.

Let’s explore this video to learn how to reflect a shape.

Reflections, Copyright © 2021 The Ontario Educational Communications Authority. All rights reserved.

Practice

Reflect the rectangle over the dotted line which is located at y = 3. Record your solution using a method of your choice.

When you are done reflecting the rectangle on your own, press ‘Answer’ to access the solution.

Answer

How else could you reflect the rectangle?

Task 3: Rotations of a Shape

A rotation is when you turn a shape around. Let’s explore a video to learn how to rotate a shape.

Rotations, Copyright © 2021 The Ontario Educational Communications Authority. All rights reserved.

Steps to rotating a rectangle

Consider the following steps when rotating a rectangle.

In order to rotate a rectangle, you need:

• a centre of rotation either on the shape itself or off the shape
• the angle
• the direction (clockwise or counterclockwise) at which the shape rotates

If using a protractor to rotate the rectangle:

1. Place the centre of the protractor on the centre of rotation.
2. Determine the angle of rotation and where the new points will be (for example, if we were rotating 90° counter-clockwise, we would identify where the 90° mark is).
3. Examine the distance of the points from the centre of rotation when the 0° line intersects the points and count the number of spaces.
4. Place the points at their new location, using the angle of rotation and the distance from the centre of rotation, and then connect the points of your shape.

Practice

Triangles are great shapes to play with if we want to rotate or reflect them. Let’s practice.

Use a pattern block triangle, one made of paper, or an interactive triangle. Decide on a point of rotation. Which point will stay still while the rest of the shape rotates?

Now rotate the triangle. You should notice that the point that is held in place does not move, but the other points will move places.

Now let’s try it on a Cartesian plane. When rotating any shape, you can use a protractor to get the angle correct. Also, the distance to the centre of rotation is the same for the original point as well as the rotated point.

The following triangle is being rotated 90° clockwise with point A as the centre of rotation.

Point C has already been rotated. Using the same rotation, try plotting the image of point B.

Record your answer using a method of your choice.

You may use the following grid. Press the ‘Activity’ button to access the Triangle Rotation activity.

Press the ‘Activity’ button to access Triangle Rotation. Activity (Open PDF in a new window)

When you are done rotating the triangle, press ‘Answer’ to check if you rotated the triangle correctly.

Answer

Now, let’s try rotating a whole triangle.

Use the following graph or record your answers using a method of your choice.

• Identify each coordinate of triangle ABC.
• Rotate ABC 90° clockwise using point C as the centre of rotation.
• Identify each coordinate of the reflected triangle A′B′C′.

When you are done rotating the triangle on your own. Press ‘Answer’ to access a solution and check your answer.

Answer

Task 1: Reflecting a shape

Consider an empty Cartesian plane. (0,0) is at the origin where the x and y-axes intersect, and both the x and y-axes are numbered 0 to 8.

Place a shape on the plane. You can use pattern blocks, draw on paper, or use online manipulatives. Using what you have learned about coordinates, describe the location of the shape.

Reflect the shape on a horizontal reflection line AND a vertical reflection line. You can place the reflection lines anywhere you like.

Record your ideas in a notebook or a method of your choice.

Press the ‘Activity’ button to access Cartesian Plane. Activity (Open PDF in a new window)

Task 2: Translating a shape

Consider an empty Cartesian plane. (0,0) is at the origin where the x and y-axes intersect, and both the x and y-axes are numbered 1 to 8. Place a shape on the plane.

You can use pattern blocks, draw on paper, use online manipulatives, or create an audio recording.

Using what you have learned about coordinates, describe the location of the shape. Translate the shape 3 up and 4 across (left or right, your choice).

Now describe the new coordinates of the shape.

Record your thoughts on paper, on the computer, using speech-to-text, in an audio recording, or using a method of your choice.

Press the ‘Activity’ button to access the Cartesian Plane. Activity (Open PDF in a new window)

Task 3: Transforming shapes

Now let’s practice transforming three different shapes and describing them in our own words.