E. Spatial Sense · Spatial Sense: Angles, Triangles, Transformations and Area

Rotations: Quarter Turns and Half Turns

E1.5 A1About 40 minutes

Learning goals

I can rotate a shape 90° (a quarter turn) clockwise or counterclockwise, or 180° (a half turn), around a point.
I can find the coordinates of a rotated shape's vertices on a grid.
I can predict what a rotated shape will look like.
I can tell whether an image was made by a translation, a reflection or a rotation.

Words to know

rotation
A turn of a shape around a fixed point.

centre of rotation
The point that stays still while the shape turns around it.

clockwise
Turning the same way as the hands of a clock.

counterclockwise
Turning the opposite way to the hands of a clock.

quarter turn
A 90° rotation.

half turn
A 180° rotation.

Remember

You know two transformations already: the translation (slide) and the reflection (flip). You also know that 90° is a quarter of a full turn and 180° is half a turn. Today you will learn the third transformation: the rotation, or turn.

Let's start with a door

Watch a door open. The hinges stay still, and the rest of the door swings around them. The hinge is the centre of rotation. A rotation turns a shape around a fixed point.

To describe a rotation, you need three things:

The centre of rotation (the point that stays still).
The amount of turn: in Grade 5, a 90° quarter turn or a 180° half turn.
The direction: clockwise (the way a clock's hands move) or counterclockwise (the other way).

A right triangle with vertices (4, 4), (7, 4) and (4, 6) turns a quarter turn clockwise around the point (4, 4). The image has vertices (4, 4), (4, 1) and (6, 4).

Rotating on a grid: the counting method

For each vertex, count how far it is from the centre: how far right or left, and how far up or down.

For a quarter turn clockwise, each direction turns one step clockwise:

right becomes down
down becomes left
left becomes up
up becomes right

For a quarter turn counterclockwise, go the other way: right becomes up, up becomes left, left becomes down, down becomes right.

For a half turn, everything flips to the opposite: right ↔ left and up ↔ down. (A half turn clockwise and a half turn counterclockwise land in the same place!)

Worked example

Example 1: A quarter turn clockwise

Rotate the triangle (4, 4), (7, 4), (4, 6) 90° clockwise around (4, 4).

(4, 4) is the centre. It stays at (4, 4).
(7, 4) is 3 right of the centre. Right becomes down: 3 down from (4, 4) is (4, 1).
(4, 6) is 2 up from the centre. Up becomes right: 2 right from (4, 4) is (6, 4).
Check: the long side used to point right. After a quarter turn clockwise, it points down. ✓

The image is (4, 4), (4, 1), (6, 4).

Worked example

Example 2: A half turn

Rotate the same triangle 180° around (4, 4).

(4, 4) stays.
(7, 4) is 3 right. Opposite: 3 left → (1, 4).
(4, 6) is 2 up. Opposite: 2 down → (4, 2).

The image is (4, 4), (1, 4), (4, 2). It looks upside down.

All the rotations of the triangle around (4, 4). The start points right. The quarter turn clockwise points down. The half turn points left. The quarter turn counterclockwise (dashed outline, not shaded) points up. Together they make a pinwheel.

The tracing paper method

Trace the shape and the centre point on thin paper. Hold your pencil point on the centre. Turn the paper a quarter turn or a half turn, watching one side of the shape so you know when you've turned far enough. The tracing now shows the image.

Worked example

Example 3: Name the transformation

A flag shape points right. Its image points down, and both share the same bottom corner. Which transformation was it?

A translation keeps the flag pointing right. ✗
A reflection in a vertical or horizontal mirror line would make it point left, or keep it pointing right but upside down. It wouldn't point down. ✗
A quarter turn clockwise around the shared corner turns "right" into "down". ✓

A rotation of 90° clockwise around the shared corner.

Stuck? Act it out

Rotations can twist your brain! If you get stuck, turn the page, use tracing paper, or act it out with your own arm: point right, then make a quarter turn clockwise, and you're pointing down. Using your body or a tool is a smart strategy, not cheating.

Just for fun

A clock's minute hand makes a quarter turn clockwise every 15 minutes. That's 4 quarter turns an hour and 96 quarter turns a day. No wonder clocks never sit still!

Hands-on activity

Pinwheel art

You need: grid paper, a pencil, coloured pencils, tracing paper (or thin paper) and a ruler.

Draw coordinate axes from 0 to 10. Mark the centre point (5, 5).
Draw a small shape that touches the centre, like a triangle with vertices (5, 5), (8, 5), (5, 6).
Rotate it 90° clockwise around (5, 5) using the counting method. Write the coordinates.
Rotate the original 180° and 90° counterclockwise too.
Check one of them with tracing paper.
Colour your pinwheel!

(Answers: 90° clockwise: (5, 5), (5, 2), (6, 5). 180°: (5, 5), (2, 5), (5, 4). 90° counterclockwise: (5, 5), (5, 8), (4, 5).)

Math talk: "Which rotation was easiest to predict? Why?"

Your turn

Which way is clockwise?

How many degrees is a quarter turn? A half turn?

Rotate the point (6, 4) 90° clockwise around (4, 4).

Rotate the point (4, 7) 180° around (4, 4).

Rotate the point (4, 6) 90° clockwise around (4, 4).

Is a half turn clockwise the same as a half turn counterclockwise?

An image looks upside down and points the opposite way, with the same centre point. Which rotation could it be?

More practice: warm-up

What is another name for a rotation?

What do we call the point a shape turns around?

A quarter turn is how many degrees?

Is a rotated shape congruent to the original?

Which way is counterclockwise: the way a clock's hands go, or the opposite way?

More practice: on your own

Rotate the triangle (3, 4), (6, 4), (3, 6) 90° clockwise around (3, 4).

Rotate the same triangle 180° around (3, 4).

Rotate the same triangle 90° counterclockwise around (3, 4).

Rotate the point (7, 5) 90° clockwise around (5, 5).

Rotate the point (5, 8) 90° counterclockwise around (5, 5).

Challenge

Rotate the point (6, 3) 90° clockwise around (4, 4).

What single rotation is the same as two quarter turns clockwise around the same centre?

A square is rotated 90° around its own centre. Why does the image look exactly like the original?

Check your understanding

Quiz

How many degrees is a half turn?

Rotate (7, 2) 90° clockwise around (5, 2).

Rotate (2, 6) 180° around (4, 4).

Which point does NOT move in a rotation?

A flag points up. After a quarter turn clockwise, which way does it point?

Which transformation turns a shape around a point?

For parents and teachers

Big idea: The Ontario curriculum (E1.5) asks Grade 5 students to describe and perform rotations up to 180° on a grid and predict the results, along with translations and reflections. In practice this means 90° (quarter) and 180° (half) turns, clockwise or counterclockwise, around a point.

Common difficulties:

Mixing up clockwise and counterclockwise. Keep an analog clock or a drawing of one nearby.
Forgetting the centre of rotation. The result depends on where the centre is. Mark it clearly.
Thinking a rotation is a reflection. Use tracing paper: you turn the paper for a rotation but flip it over for a reflection.

How to help: Tracing paper (or parchment paper) is very helpful. Let your child act out turns with their arm or a toy. Social-emotional learning (A1) is woven in: using tools and strategies when stuck, and persisting with a challenging idea.