Transformations on a coordinate grid

What the tool does

A transformation moves a shape. Translations slide it, reflections flip it over a line, and rotations turn it about a point. All three keep the shape's size and angles the same. On this page, choose a triangle, an L-shape or a quadrilateral. Arrow buttons translate it one unit at a time, or you can drag it. Two buttons reflect it in the x-axis or the y-axis, and two rotate it 90° clockwise or counterclockwise about the origin.

A dashed copy shows where the shape started, with corners labeled A, B, C. The moved shape is labeled A′, B′, C′, read as A prime. The table lists each vertex's coordinates at the start and now, and the readout gives the rule for the last move, such as (x, y) → (x, −y) for a reflection in the x-axis.

Why the coordinates matter

Moving a shape is easy to picture; describing it exactly is the skill. Watching the table change while the shape moves shows the patterns: a translation adds the same amount to every coordinate, a reflection in the x-axis changes the sign of y, a reflection in the y-axis changes the sign of x, and a 90° counterclockwise rotation sends (x, y) to (−y, x). The Match the target game asks students to plan a sequence of moves, which is how transformations are tested in grade 8 and at GCSE.

Common mix-ups

• Reflecting in the wrong axis. A reflection in the x-axis flips the shape up and down, across the horizontal axis, so the y-coordinates change sign.
• Mixing up clockwise and counterclockwise rotations. Track one vertex and check which quadrant it lands in.
• Thinking the shape changes size. Translations, reflections and rotations never change lengths or angles.
• Assuming the order of moves never matters. A reflection followed by a translation can end somewhere different from the same translation followed by the reflection.

A 10-minute transformations lesson

• Reflect the triangle in the y-axis. Ask what happened to the x-coordinates and the y-coordinates.
• Press Start again, then rotate 90° counterclockwise. Ask the child to predict A′ before looking at the table.
• Translate 2 right and 3 up and write the rule as (x, y) → (x + 2, y + 3).
• Press Match the target twice and talk through each plan before moving.
• End by asking which moves change the direction the shape faces and which do not.

Where it fits by grade

• Grades 5 and 6: plot points in the first quadrant and then all four quadrants, using negative numbers.
• Grade 8: verify that rotations, reflections and translations keep lengths and angles (8.G.A.1) and describe their effect using coordinates (8.G.A.3).
• England: Year 5 describes the position of a shape after a reflection or translation, Year 6 translates shapes on the coordinate plane and reflects them in the axes, and Key Stage 3 adds rotations.

Learning more

Negative numbers are on Eurekami's skill map, and Daily Practice brings them back when they start to fade. For older students, Exam Prep includes GCSE Maths practice papers, and the AI tutor asks guiding questions when a transformation problem is tricky.