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Slides, flips and turns

Geometry · level 2, lesson 6 of 6

A question to start

Write a small letter b on a piece of see-through paper (or tracing paper). Now try three things.

  • Flip the paper over, left to right, and look through it. What letter do you see?
  • Flip it over top to bottom instead. What letter now?
  • Do not flip it. Turn it upside down, half a turn, on the table. What letter now?

You get d, then p, then q. Four different letters, b, d, p and q, are really one shape, moved in different ways! No wonder they are so easy to mix up when you learn to read. In this lesson you will learn the three moves of geometry, and how to tell them apart.

The idea

Three moves that keep a shape the same

In geometry there are three simple ways to move a shape without changing its size or its shape:

  1. Slide: move it along, without turning it.
  2. Flip: flip it over a line, like a mirror image.
  3. Turn: turn it round a point.

After any of these moves, the shape is exactly the same size and the same shape as before. If you cut out the old shape, you could lay it exactly on the new one. Only its position (where it is) or its direction (which way it faces) has changed.

Grown-ups call these moves translation (slide), reflection (flip) and rotation (turn). The shape you get after a move is called the image.

Slides

In a slide, every point of the shape moves the same distance in the same direction. The shape does not turn and does not flip. It is like pushing a book across a table without twisting it.

slide: 7 squares right and 1 square up

On squared paper, describe a slide by counting squares: how many right or left, and how many up or down. Here the orange L slides 7 squares right and 1 square up to land on the blue L. Pick any square of the orange L and its matching square of the blue L: it is always 7 right and 1 up. That is what "every point moves the same way" means.

A slide keeps the shape facing the same way. The foot of the L still points to the right.

Flips

In a flip, the shape is flipped over a mirror line, like the mirror lines in Lines of symmetry. The image is the mirror image of the shape.

flip: a mirror image across the dashed line

The rule for a flip: every point of the image is the same distance from the mirror line as the matching point of the shape, but on the other side. The orange L's long stem is 3 squares from the line, so the blue L's stem is 3 squares from the line on the other side. The orange L's foot is closer to the line, and so is the blue L's foot.

Something important happens in a flip: the shape is turned inside out, the way your right hand looks like a left hand in a mirror. The orange L's foot points right. The blue L's foot points left. No slide and no turn could make that happen: an L can never be slid or turned into its mirror image while it stays flat on the table. It has to be lifted and flipped over.

That is why b and d are different letters: d is a flip of b. And it is why a right glove will never fit your left hand, however you turn it.

Turns

In a turn, the shape spins round a fixed point, called the centre of turning. The centre stays still and everything else goes round it, like the hands of a clock going round the middle of the clock.

turn: a quarter turn around the dot

To describe a turn, say three things:

  1. Where the centre is (here, the dot).
  2. How much: a quarter turn (like a clock hand going from 12 to 3), a half turn (12 to 6), or a three-quarter turn (12 to 9). A full turn brings everything back where it started.
  3. Which way: clockwise (the way clock hands go) or anticlockwise (the other way; also called counterclockwise).

Here the orange L makes a quarter turn clockwise around the dot and lands on the blue L. Its tall stem, which was pointing up, now points to the right, just as a clock hand pointing at 12 points at 3 after a quarter turn clockwise.

A turn does not turn a shape inside out. The blue L is the orange L lying on its side, not its mirror image. If you cut out the orange L, you could turn it on the table to fit the blue one exactly, without lifting it.

Letters again

Now we can explain the puzzle from the start.

From b Move You get
b flip left to right (up-and-down mirror line) d
b flip top to bottom (side-to-side mirror line) p
b half turn q

And here is something lovely: a half turn is the same as doing both flips, one after the other. Flip b left to right to get d, then flip d top to bottom: you get q, the same as the half turn. Try it with your tracing paper.

Some letters look the same after a half turn: H, I, N, O, S, X, Z. These letters have turning symmetry. Look at N and S: they have no mirror line at all, but they still look the same upside down.

Moves and tilings

The moves explain the tilings from the last lesson. In a square tiling, you can slide the whole floor one square to the right and it lands on itself. In the tiling made from a lopsided triangle, every second triangle is a half turn of its neighbour. Every tiling is a shape repeated by slides, flips and turns.

Worked examples

Example 1. Describe a slide. A shape's top corner is at a point. After a slide, the same corner is 4 squares left and 2 squares down. Another corner of the shape was at a point 3 squares right of the first. Where is it after the slide?

  1. In a slide, every point moves the same way: 4 left and 2 down.
  2. So the second corner also moves 4 left and 2 down.
  3. It started 3 squares right of the first corner. After the move it is still 3 squares right of the first corner's new position. Slides keep shapes exactly as they were, just somewhere else.

Example 2. Which move? An arrow points up. After a move, the arrow points down. Was it a slide, a flip or a turn?

  1. A slide never changes direction. So not a slide.
  2. A half turn changes up to down. So it could be a turn.
  3. A flip over a side-to-side mirror line also changes up to down. So it could be a flip too!
  4. An arrow is symmetric (it has an up-and-down mirror line), so both moves give the same picture. We cannot tell which move it was. With a lopsided shape, like an L or the letter F, you could tell: a half turn and a flip give different results.

Example 3. Quarter turns add up. A clock hand points at 12. It makes a quarter turn clockwise, then another quarter turn clockwise, then a half turn clockwise. Where does it point?

  1. A quarter turn clockwise from 12 goes to 3.
  2. Another quarter turn goes from 3 to 6.
  3. A half turn from 6 goes to 12.
  4. It points at 12, back where it started. Quarter + quarter + half = one full turn, and a full turn brings everything back.

Try it

  1. Is a slide, a flip or a turn: a lift going up in a building?
  2. Is a slide, a flip or a turn: a door opening on its hinges?
  3. Is a slide, a flip or a turn: your face in a mirror?
  4. What letter do you get if you flip p left to right?
  5. What letter do you get if you give d a half turn?
  6. On squared paper, a square is moved 5 squares left and 3 squares up. How far did each of its 4 corners move?
  7. A clock hand points at 3. Where does it point after a three-quarter turn anticlockwise?
  8. Which capital letters look the same after a half turn: A, H, M, N, S, T, Z?
Answers
  1. A slide. The lift moves straight up without turning. Every point of it moves the same distance in the same direction.
  2. A turn. The door turns round its hinges, which are the centre of turning (seen from above, the hinge is a point, and the door swings round it).
  3. A flip. A mirror image is a reflection. That is why the writing on a T-shirt looks backwards in the mirror.
  4. q. The bump of p is on the right; after flipping left to right it is on the left, and the stick still hangs down: q.
  5. p. Turn d upside down: the stick that went up now goes down, and the bump that was on the left is now on the right: p.
  6. Every corner moved 5 squares left and 3 squares up. In a slide, every point moves the same way.
  7. Anticlockwise from 3: a quarter turn reaches 12, a half turn reaches 9, a three-quarter turn reaches 6. (Check another way: a three-quarter turn anticlockwise is the same as a quarter turn clockwise, and a quarter turn clockwise from 3 also reaches 6.)
  8. H, N, S and Z. A, M and T turn upside down: they have a mirror line but no turning symmetry.

Common mistake

Mixing up a flip and a half turn. Both can make a shape "upside down", but a flip makes a mirror image and a half turn does not. Test: does the shape now face the other way, like a left hand becoming a right hand? Then it was a flip. A turn never turns a shape inside out.

Counting a slide from the wrong points. To count a slide on squared paper, pick one corner of the shape and the matching corner of the image. Counting from the left edge of one shape to the right edge of the other gives the wrong answer.

Where it leads