A shape can move without becoming a different shape. It can travel across the page, spin around, or turn over like a card.
In this lesson, you will experiment with three moves and work out what stays the same.
Which one does not belong?
The same little shape appears four times below. Three pictures only move it. One actually changes its shape.
Move a shape yourself
Cut out a simple shape from scrap paper — an arrow, an L-shape or anything else that is clearly different on its two sides. Put it on the table and try these moves before learning their names.
Push the shape across the table without turning it.
Keep one point fixed and spin the shape around.
Turn the shape over as if a mirror line were beside it.
Transformation lab
Use the interactive lab to move the same asymmetric shape around a grid. Try making a move, then ask: what changed, and what stayed the same?
Now give the moves their mathematical names
You have already been sliding, turning and flipping shapes. Mathematicians have special names for these moves.
Translation
A translation moves every point the same distance in the same direction.
The shape does not turn, flip or change size.
Rotation
A rotation turns a shape through an angle around a fixed point.
That fixed point is called the centre of rotation.
Reflection
A reflection makes a mirror image on the other side of a line.
Matching points are the same distance from the mirror line, but on opposite sides.
They can move or turn a shape, but they do not stretch it, squash it or change its size.
Which move was it?
Imagine a little arrow pointing up. Decide which transformation happened before opening each answer.
Which move was it?
Look at the shape on the left and the shape on the right. What happened in between?
Was it a translation, a rotation, a reflection — or a combination of two moves?
Some pictures are made by just one move. Others need two moves together. In mathematics, doing one transformation after another is called a composition.
Be the robot
Follow transformation instructions
Stand facing the front of the room. Someone else is the programmer.
Try instructions such as:
Slide two steps right → turn a quarter turn clockwise → slide three steps forward → flip the paper arrow you are holding.
Can your programmer describe your final position before you move? Then swap roles.
Does the order matter?
Suppose you want to do two things. Does it matter which one you do first?
Socks and shoes
Imagine getting dressed.
This works.
Not quite the same result!
The same two actions can give a different result when we change their order.
Transformations can behave like this too. Try a turn followed by a flip, then compare it with a flip followed by a turn.
Before starting the animation, make a prediction: will the shape finish in the same place and facing the same way?
In general, doing a rotation and then a reflection does not give the same result as doing the reflection first and then the rotation.
Mathematicians say that these transformations do not always commute: changing the order can change the final result.
Try inventing your own pair of transformations.
For example: turn then slide, slide then turn, or flip then slide. Does reversing the order always change the result?
Go further: make a transformation pattern
Take one small asymmetric motif — perhaps a lightning bolt, fish, leaf or invented symbol. Repeat it across a page using just one rule:
What pattern appears?
Can you make a rosette?
Can you make alternating pairs?
In the next lesson, we will look for these ideas outside our diagrams — in art, plants, animals, buildings and other patterns around us.
Lesson 4: Slide, turn, flip
Download the combined handout and activities to practise translations, rotations and reflections away from the screen.
Get the Lesson 4 resources →What did you think?
Whether you explored this together or worked independently, we would love to hear from both learners and parents. Tell us what you enjoyed, what was confusing, or what you would like to see next.
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