Translating shapes on a grid

Translating shapes on a grid

Translating shapes on a grid

In maths, a translation is a fancy word for a slide. You move every point of a shape the same distance in the same direction. The shape doesn't turn, doesn't flip, doesn't grow — it just slides to a new place.

The four directions

On a coordinate grid, a translation always has two parts:

  • a horizontal move — left or right along the x-axis,
  • a vertical move — up or down along the y-axis.

A typical instruction looks like:

Translate the shape 3 squares right and 2 squares up.

That means every point of the shape moves +3 along x and +2 along y.

How to translate a shape

  1. Pick one corner of the shape and read its coordinates.
  2. Add the horizontal move to the x-coordinate, and the vertical move to the y-coordinate.
  3. Plot the new corner.
  4. Repeat for every corner of the shape.
  5. Join the new corners in the same order as the original.

The new shape is exactly the same size and shape as the old one — just in a different spot.

A worked example

A triangle has corners at (1, 1), (4, 1) and (1, 5). Translate it 3 squares right and 2 squares up.

For each corner, add (+3, +2):

  • (1, 1) → (1 + 3, 1 + 2) = (4, 3).
  • (4, 1) → (4 + 3, 1 + 2) = (7, 3).
  • (1, 5) → (1 + 3, 5 + 2) = (4, 7).

The new triangle has corners at (4, 3), (7, 3) and (4, 7). Same triangle, slid right and up.

Going left or down

A move "to the left" subtracts from x. A move "down" subtracts from y. In Year 4 we make sure the shape stays in the first quadrant, so we never subtract enough to go below zero.

Translate the point (5, 4) two squares left and one square down.

  • New x: 5 − 2 = 3.
  • New y: 4 − 1 = 3.
  • New point: (3, 3).

What translation does NOT do

A translation is just a slide. It does not:

  • rotate the shape (turn it),
  • reflect it (flip it like a mirror),
  • stretch or shrink it.

After a translation the shape is identical in size, shape and orientation — only the position changes.

A puzzle

A square has corners at (1, 1), (3, 1), (3, 3) and (1, 3). After a translation it sits at (4, 5), (6, 5), (6, 7) and (4, 7). What was the translation?

Compare any one corner: (1, 1) became (4, 5).

  • x changed from 1 to 4: that's +3 (3 right).
  • y changed from 1 to 5: that's +4 (4 up).

Translation: 3 right and 4 up. The same change applies to all four corners.

Common mistakes

  • Translating only one corner and forgetting the others.
  • Adding to x but subtracting from y by accident. Read the question carefully — "3 right and 2 up" is +3 to x and +2 to y, both positive.
  • Confusing translation with rotation. A translation never turns the shape. The orientation of the original and the translated copy is the same.

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