E. Spatial Sense · Spatial Sense: Shapes, Grids, Angles, Area and Surface Area

Reflections and Rotations in Four Quadrants

E1.4About 45 minutes

Learning goals

I can reflect a shape in the x-axis or the y-axis and give the coordinates of its image.
I can rotate a shape 90°, 180°, 270° or 360° around the origin, clockwise or counterclockwise.
I can predict where an image will land before I draw it.

Words to know

transformation
A change in a shape's position: a translation (slide), a reflection (flip) or a rotation (turn).

image
The shape after a transformation. We often label its vertices with a prime mark: A′ (say "A prime").

reflection
A flip over a mirror line. Each image point is the same distance from the line as the original point, on the other side.

rotation
A turn around a fixed point, called the centre of rotation.

clockwise / counterclockwise
Clockwise is the way a clock's hands turn. Counterclockwise is the opposite way.

Remember from Grade 5

In Grade 5 you translated (slid), reflected (flipped) and rotated (turned) shapes on a grid. You rotated shapes a quarter turn (90°) or a half turn (180°), using the counting method: right becomes down, down becomes left, and so on. This year the grid has four quadrants, and turns can go all the way to 360°.

Let's start with a mirror lake

On a calm summer evening, a lake reflects the trees on the shore. Each treetop's reflection is just as far below the waterline as the real treetop is above it. A reflection on a grid works the same way: the axis is the waterline.

Reflect in the x-axis: each point jumps to the other side of the x-axis, the same distance away. The x-coordinate stays the same. The y-coordinate changes sign: (1, 3) → (1, −3).
Reflect in the y-axis: each point jumps to the other side of the y-axis. The y-coordinate stays the same. The x-coordinate changes sign: (1, 3) → (−1, 3).

Triangle ABC with A (1, 1), B (4, 1) and C (1, 3) in quadrant I. Its reflection in the y-axis is in quadrant II, at (−1, 1), (−4, 1) and (−1, 3). Its reflection in the x-axis is in quadrant IV, at (1, −1), (4, −1) and (1, −3).

Worked example

Example 1: Reflect in the y-axis

Reflect triangle ABC, with A (1, 1), B (4, 1), C (1, 3), in the y-axis.

A is 1 unit right of the y-axis. Its image is 1 unit left: A′ (−1, 1).
B is 4 units right. Its image is 4 units left: B′ (−4, 1).
C is 1 unit right. Its image is 1 unit left: C′ (−1, 3).
Check: every y-coordinate stayed the same, and the image faces the other way, like a mirror.

A′ (−1, 1), B′ (−4, 1), C′ (−1, 3), in quadrant II.

Rotating around the origin

To rotate around the origin, use the Grade 5 counting method. The coordinates already tell you how far each point is from the origin!

For a 90° clockwise turn, each direction moves one step clockwise: right → down → left → up → right.

For a 90° counterclockwise turn, go the other way: right → up → left → down → right.

For 180°, every direction flips to its opposite, so both signs change: (x, y) → (−x, −y).

For 270°, that's three quarter turns. A 270° clockwise turn ends in the same place as a 90° counterclockwise turn. And a 360° turn brings every point back where it started.

Triangle ABC rotates 90° clockwise around the origin. A (1, 1) goes to A′ (1, −1), B (4, 1) goes to B′ (1, −4), and C (1, 3) goes to C′ (3, −1).

Worked example

Example 2: Rotate 90° clockwise

Rotate triangle ABC 90° clockwise around the origin.

A (1, 1): 1 right, 1 up. Clockwise, right → down and up → right. So 1 down, 1 right: A′ (1, −1).
B (4, 1): 4 right, 1 up. Becomes 4 down, 1 right: B′ (1, −4).
C (1, 3): 1 right, 3 up. Becomes 1 down, 3 right: C′ (3, −1).
Check with tracing paper: trace ABC and the axes, hold the origin with a pencil tip and turn the paper a quarter turn clockwise.

A′ (1, −1), B′ (1, −4), C′ (3, −1), in quadrant IV.

Worked example

Example 3: A 270° turn

Rotate the point (2, 5) 270° clockwise around the origin.

270° clockwise ends in the same place as 90° counterclockwise (because 270° + 90° = 360°, a full turn).
(2, 5) is 2 right, 5 up. Counterclockwise, right → up and up → left. So 2 up, 5 left.
Check: going clockwise three times, (2, 5) → (5, −2) → (−2, −5) → (−5, 2). Same answer!

(−5, 2)

Just for fun

Why did the triangle stand in front of the mirror for so long?

It was taking time to reflect.

Predict first

Before you plot, predict which quadrant the image will land in. A 90° turn moves a shape to the next quadrant (clockwise: I → IV → III → II; counterclockwise: I → II → III → IV). A 180° turn moves it to the opposite quadrant (I ↔ III, II ↔ IV). If your answer lands somewhere else, check your work.

Hands-on activity

Tracing-paper turns

You need: grid paper, a ruler, a pencil, coloured pencils and some tracing paper (or thin paper such as baking parchment).

Draw axes from −6 to 6. Draw a small flag shape in quadrant I with a corner at (1, 1), for example (1, 1), (1, 4), (3, 4), (3, 3), (1, 3).
Predict where it will land after a reflection in the x-axis, a reflection in the y-axis, and rotations of 90° clockwise, 180° and 270° clockwise around the origin. Write your predicted coordinates.
Check each one with tracing paper. For reflections, flip the tracing paper over the axis. For rotations, hold the origin with your pencil tip and turn.
Draw each image in a different colour.

Math talk: "Which transformations made the flag face the other way? Which ones kept it facing the same way?"

Your turn

Reflect (5, 2) in the y-axis.

Reflect (−3, 4) in the x-axis.

Rotate (2, 3) 180° around the origin.

Rotate (4, 1) 90° clockwise around the origin.

Rotate (4, 1) 90° counterclockwise around the origin.

Translate the triangle with vertices (1, 1), (4, 1), (1, 3) 5 units left and 2 units down.

More practice: warm-up

When you reflect in the y-axis, which coordinate changes sign?

Reflect (0, 3) in the x-axis.

How many degrees is a three-quarter turn?

Where does a point end up after a 360° rotation?

Reflect (−2, −5) in the y-axis.

More practice: on your own

Rotate (−3, 2) 90° clockwise around the origin.

Rotate (−3, 2) 270° clockwise around the origin.

The image of (6, −1) is (−6, −1). What transformation was used?

The image of (6, −1) is (1, 6) after a rotation around the origin. Which rotation?

Reflect (0, 4) in the y-axis. What do you notice?

Challenge

A square has vertices A (1, 1), B (3, 1), C (3, 3), D (1, 3). Rotate it 90° clockwise around the origin. Then compare with its reflection in the x-axis. What do you notice?

Without plotting, predict the quadrant and coordinates of (−4, 7) after a 180° rotation around the origin.

Is rotating 90° clockwise three times the same as rotating 90° counterclockwise once? Test it with (2, 5).

Check your understanding

Quiz

Reflect (3, −2) in the x-axis.

Reflect (3, −2) in the y-axis.

Rotate (5, 2) 180° around the origin.

Rotate (3, 0) 90° clockwise around the origin.

Which turn ends in the same place as 270° clockwise?

A shape in quadrant I is rotated 180° around the origin. Where is the image?

For parents and teachers

Big idea: The Ontario curriculum (E1.4) asks Grade 6 students to describe and perform combinations of translations, reflections and rotations up to 360° on a grid, and to predict the results. This lesson covers single reflections and rotations in all four quadrants; the next lesson combines them.

Common difficulties:

Mixing up the axes. Reflecting in the x-axis flips up and down (y changes sign). Reflecting in the y-axis flips left and right (x changes sign).
Turning the wrong way. Have your child trace the shape and physically turn the paper.
270° turns. Remind them that 270° one way lands in the same place as 90° the other way.

How to help: Tracing paper (or baking parchment) is a great checking tool. Ask your child to predict the quadrant before plotting.