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Rotations

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visibility 198update 2 months agobookmarkshare

🎯 In this topic you will

Rotate 2D shapes 90° around a vertex

🧠 Key Words

anticlockwise
centre of rotation
clockwise
rotate

Show Definitions

anticlockwise: A direction of rotation opposite to the movement of a clock’s hands.
centre of rotation: The fixed point around which a shape turns during a rotation.
clockwise: A direction of rotation that follows the same direction as the hands of a clock.
rotate: To turn a shape around a fixed point without changing its size or shape.

Shapes All Around Us

When you go to a play park, have you ever looked at the floor? Many play parks now have rubber flooring, which is good in all weathers and it doesn’t hurt very much if you fall over! Have you ever thought about who designs the flooring, and how they do it? This is where knowing how to turn, reflect and translate shapes comes in really useful. Have a look at the flooring in this picture. Can you see any shapes that have been turned, reflected or translated?

📘 Worked example

Rotate triangle A $90^\circ$ clockwise about the centre of rotation marked C. Label your answer triangle B.

Step 1.

Trace the shape, then put your point of your pencil on the centre of rotation.

Step 2.

Start turning the tracing paper $90^\circ$ (a quarter turn) clockwise.

Step 3.

Once the turn is completed make a note of where the new triangle is.

Step 4.

Draw the new triangle onto the grid and label it B.

Answer:

The rotated triangle is labeled B and is the image of triangle A after a $90^\circ$ clockwise rotation about point C.

To rotate a shape, fix the centre point (C) and turn the shape around it.

A $90^\circ$ clockwise turn means a quarter turn to the right.

Each point of the triangle moves the same distance from the centre and keeps its shape and size.

❓ EXERCISES

1. Copy each diagram and rotate the shapes $90^\circ$ clockwise about the centre of rotation C.

👀 Show answer

The shapes should be rotated $90^\circ$ clockwise about point C. Each square moves to a new position the same distance from C, forming the rotated images.

🧠 Think like a Mathematician

This is part of Sita’s homework.

Question: Rotate shape A $90^\circ$ anticlockwise about centre C. Label your shape B.

Sita has correctly rotated the shape but has forgotten to label the shapes A and B.

Follow-up Questions:

a. Which is shape A and which is shape B? Explain how you know.

b. Describe the difference between a $90^\circ$ rotation clockwise and a $90^\circ$ rotation anticlockwise.

c. Think about ways to remember which way is clockwise and which way is anticlockwise.

Show Answers

a: Shape A is the original shape, and shape B is the rotated image after turning $90^\circ$ anticlockwise about point C.
b: A clockwise rotation turns to the right, while an anticlockwise rotation turns to the left. Both are quarter turns when the angle is $90^\circ$.
c: You can remember clockwise by thinking of the direction a clock’s hands move. Anticlockwise is the opposite direction.

❓ EXERCISES

2. Copy each diagram and rotate the shapes $90^\circ$ anticlockwise about the centre of rotation C.

👀 Show answer

Each shape should be rotated $90^\circ$ anticlockwise about point C. All points move the same distance from C in a quarter turn to the left, keeping the size and shape unchanged.

3. The diagrams i to vi all show shape A rotated to shape B. The centre of rotation is shown by a dot $(\bullet)$.

a. Sort the diagrams into two groups. Describe the properties of each group.

b. Compare your answers and discuss your choices. Did you choose the same groups or different ones?

👀 Show answer

The diagrams can be grouped by the direction of rotation: clockwise and anticlockwise. In each group, the shapes have been turned around the centre point by the same angle and distance. The main difference between the groups is the direction of rotation.

🧠 Think like a Mathematician

Hamila draws a triangle on a grid. She translates it $3$ squares right and $2$ squares down. She joins the corresponding vertices with straight lines. This is what her diagram looks like.

Tip: Corresponding vertices are vertices that are in the same position on the shape before and after a transformation.

Follow-up Questions:

a. Copy this diagram. Reflect the triangle in the mirror line. Join the corresponding vertices with straight lines.

b. Copy this diagram. Rotate the triangle $90^\circ$ clockwise about centre C. Join the corresponding vertices with straight lines.

c. What do you notice about the lines that join the corresponding vertices of the triangles after a $i$ translation $ii$ reflection $iii$ rotation?

d. Decide which word is missing from this rule: When you translate or reflect a triangle, the lines joining the corresponding vertices will always be ________, but when you rotate a triangle the lines joining the corresponding vertices will never be ________.

e. Does the rule in part d only work for triangles, or does it work for any 2D shape? Explain your answer.

Show Answers

a: After reflection, corresponding vertices are mirrored across the line, and joining lines are parallel and equal in length.
b: After rotation, the triangle turns around point C and corresponding vertices move in arcs, so joining lines are not parallel.
c: Translation and reflection produce parallel joining lines, but rotation does not.
d: The missing word is parallel. The lines will always be parallel for translation and reflection, but never parallel for rotation.
e: This rule works for any 2D shape because all shapes follow the same transformation rules for translation, reflection, and rotation.

❓ EXERCISES

4. Copy each diagram and rotate the shapes $90^\circ$ about the centre of rotation C, using the direction shown.

👀 Show answer

Each shape is rotated $90^\circ$ about point C. In part a, the shape turns clockwise (to the right), and in part b, it turns anticlockwise (to the left). The rotated shapes keep the same size and shape but change orientation.

📘 What we've learned

We learned how to rotate 2D shapes by $90^\circ$ about a centre of rotation.
A $90^\circ$ rotation represents a quarter turn around a fixed point.
Clockwise rotation turns shapes to the right, while anticlockwise rotation turns shapes to the left.
During rotation, the shape keeps the same size and shape but changes its orientation.
Each vertex of a shape moves the same distance from the centre of rotation.
We also explored how translation, reflection, and rotation affect corresponding vertices differently.

Rotations

🎯 In this topic you will

🧠 Key Words

Shapes All Around Us

❓ EXERCISES

🧠 Think like a Mathematician

❓ EXERCISES

🧠 Think like a Mathematician

❓ EXERCISES

📘 What we've learned

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