Transformations: translations, reflections and rotations
Describing and performing translations, reflections and rotations of points and shapes on a coordinate grid
About three lessons of 45 to 60 minutes
Moving a shape without changing it
A video game character slides sideways, a kaleidoscope mirrors a pattern, a wheel turns a quarter-circle: every one of these is a transformation, a precise way of moving a shape to a new position. What makes them mathematically interesting is what does NOT change: the shape's size and its angles stay exactly the same. Only its position or orientation moves.
There are 3 core transformations to master: a translation slides a shape along a vector (so many units right/left, so many up/down), a reflection flips it across a mirror line, and a rotation turns it about a fixed centre point. Each one can be described with a precise coordinate rule, so 'where does this vertex end up?' always has an exact answer, never a guess.
- A sprite sliding across the screena translation: every pixel moves by the same vector
- A kaleidoscope's mirrored patterna reflection: each point flips to the same distance on the other side of the mirror line
- A ferris wheel car turning about the centrea rotation: every point turns through the same angle about 1 fixed centre
- A congruent copy of a triangle, moved but not resizedall 3 transformations preserve size and shape exactly (they are 'rigid')
What students will be able to do
Students will translate a point or shape by a given vector, reflect a point or shape in the x-axis, y-axis or the line y = x, rotate a point or shape about the origin by 90 or 180 degrees, and explain why every 1 of these transformations preserves size and shape.
- I can translate a point or shape by a given vector (dx, dy).
- I can reflect a point or shape in the x-axis, the y-axis, or the line y = x.
- I can rotate a point or shape by 90 degrees clockwise, 90 degrees counterclockwise, or 180 degrees about the origin.
- I can explain why a translation, reflection or rotation never changes a shape's size.
- I can find the image of every vertex of a shape under a given transformation.
Standards this unit teaches
- KS3 Maths: Geometry and measuresUK National Curriculum (England)Geometry and measures
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Geometry and measures" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "identify properties of, and describe the results of, translations, rotations and reflections applied to given figures".
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Translation
- a transformation that slides every point of a shape the same distance in the same direction, described by a vector (dx, dy)
- Reflection
- a transformation that flips a shape across a mirror line, so every point lands the same distance from the line on the opposite side
- Rotation
- a transformation that turns a shape about a fixed centre point by a given angle
- Vector
- an instruction with a direction and a size, written (dx, dy) for a translation: dx right/left, dy up/down
- Image
- the shape or point produced after a transformation, usually labelled with a prime mark, e.g. A'
- Congruent
- having exactly the same size and shape; every translation, reflection and rotation produces an image congruent to the original
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Translations
ConcreteA translation slides every point of a shape the same distance in the same direction, described by a vector (dx, dy): dx tells you how far to move right (positive) or left (negative), and dy tells you how far to move up (positive) or down (negative).
Because every vertex moves by the SAME vector, the shape never rotates, flips or resizes, it simply slides to a new position. To translate a shape, add the vector to every vertex's coordinates, 1 vertex at a time.
Triangle ABC has vertices A(1, 1), B(3, 1), C(2, 3). Translate it by the vector (3, 2). Find A', B' and C'.
- Add (3, 2) to every vertex.
- A(1, 1) -> A'(1 + 3, 1 + 2) = A'(4, 3).
- B(3, 1) -> B'(3 + 3, 1 + 2) = B'(6, 3). C(2, 3) -> C'(2 + 3, 3 + 2) = C'(5, 5).
Answer: A' = (4, 3), B' = (6, 3), C' = (5, 5).
- Translate the point (0, 0) by the vector (-2, 5). What is the image?
- If every vertex of a shape moves by the same vector, does the shape's size change? Explain.
2. Reflections
PictorialA reflection flips a shape across a mirror line: every point lands the same distance from the line, on the opposite side. 3 common mirror lines give simple coordinate rules: the x-axis, the y-axis, and the line y = x.
Reflecting in the x-axis keeps x the same and flips the sign of y: (x, y) -> (x, -y). Reflecting in the y-axis keeps y the same and flips the sign of x: (x, y) -> (-x, y). Reflecting in the line y = x swaps the 2 coordinates entirely: (x, y) -> (y, x).
Triangle ABC has vertices A(2, 1), B(5, 1), C(3, 4). Reflect it in the y-axis. Find A', B' and C'.
- Reflecting in the y-axis uses (x, y) -> (-x, y).
- A(2, 1) -> A'(-2, 1). B(5, 1) -> B'(-5, 1). C(3, 4) -> C'(-3, 4).
Answer: A' = (-2, 1), B' = (-5, 1), C' = (-3, 4).
- Reflect the point (6, -3) in the x-axis.
- Reflect the point (4, 7) in the line y = x.
3. Rotations
AbstractA rotation turns a shape about a fixed centre point by a given angle. In this unit, every rotation is centred on the origin (0, 0). Turning counterclockwise (the positive mathematical direction) by 90 degrees uses the rule (x, y) -> (-y, x).
The other 2 common rotations about the origin are 90 degrees clockwise, (x, y) -> (y, -x), and 180 degrees (a half-turn, which is the same either direction), (x, y) -> (-x, -y). Just like translations and reflections, a rotation never changes a shape's size, only its orientation and position.
Triangle ABC has vertices A(1, 1), B(4, 1), C(1, 3). Rotate it 90 degrees counterclockwise about the origin. Find A', B' and C'.
- Rotating 90 degrees counterclockwise about the origin uses (x, y) -> (-y, x).
- A(1, 1) -> A'(-1, 1). B(4, 1) -> B'(-1, 4). C(1, 3) -> C'(-3, 1).
Answer: A' = (-1, 1), B' = (-1, 4), C' = (-3, 1).
- Rotate the point (5, 0) by 180 degrees about the origin.
- Rotate the point (2, 6) by 90 degrees clockwise about the origin.
Common misconceptions and how to address them
MisconceptionReflecting in the y-axis flips the y-coordinate, since 'y-axis' sounds like it should change y.
Why it happens: Students match the axis NAME to the coordinate that changes, rather than to the coordinate that stays fixed.
How to address it: Reflecting OVER a line keeps points on that line fixed. Every point on the y-axis has x = 0, so reflecting in the y-axis keeps x = 0 points fixed, meaning y stays the same and x flips sign: (x, y) -> (-x, y). The axis name tells you the MIRROR, not which coordinate changes.
MisconceptionA rotation or reflection makes a shape smaller or bigger, since the shape 'looks different' in its new position.
Why it happens: Students associate any visual change in a diagram with a change in size, rather than distinguishing position/orientation from size.
How to address it: Translations, reflections and rotations are all RIGID transformations: every one of them preserves every side length and every angle exactly. The image is always congruent to the original, just moved. Compare corresponding side lengths directly to prove this.
MisconceptionRotating 90 degrees clockwise and 90 degrees counterclockwise give the same image.
Why it happens: Students treat 'a quarter turn' as a single idea without tracking which direction the turn goes.
How to address it: The 2 directions give DIFFERENT images (except for a shape with special symmetry). Compare (2, 0) rotated 90 clockwise, giving (0, -2), against 90 counterclockwise, giving (0, 2): these are different points. Always state and check the direction.
MisconceptionA translation vector with a negative number, like (-3, 2), means the shape shrinks.
Why it happens: Students read a minus sign as 'less' in a general size sense, rather than as a direction (left instead of right).
How to address it: In a translation vector (dx, dy), the sign only shows DIRECTION: negative dx means move left, negative dy means move down. The shape's size is completely unaffected by a translation, whatever the sign of dx or dy.
Guided practice (with answers)
1. Translate the point (3, -2) by the vector (-1, 4).
Answer: (2, 2), because (3 - 1, -2 + 4) = (2, 2).
2. Reflect the point (0, 5) in the x-axis.
Answer: (0, -5), because reflecting in the x-axis flips the sign of y.
3. Reflect the point (-3, 2) in the line y = x.
Answer: (2, -3), because reflecting in y = x swaps the coordinates.
4. Rotate the point (4, 0) by 90 degrees clockwise about the origin.
Answer: (0, -4), using (x, y) -> (y, -x).
5. Rotate the point (-2, -2) by 180 degrees about the origin.
Answer: (2, 2), using (x, y) -> (-x, -y).
6. A translation moves every vertex of a triangle by the vector (5, -3). Does the triangle's area change?
Answer: No. A translation is a rigid transformation: it preserves every side length and angle, so the area stays exactly the same.
Independent practice worksheets
Practise translations, reflections and rotations with computed, never-wrong answer keys.
Differentiation
- Use squared/grid paper (or a printed coordinate grid) and physically count squares for every translation before writing the coordinate rule.
- Keep all 3 reflection rules and all 3 rotation rules on a reference card, colour-coded, until they are memorised.
- Practise 1 transformation type at a time before mixing them in the same practice set.
- For every image point, say the rule out loud before applying it, e.g. 'x stays the same, y flips sign'.
- Investigate what happens when you apply 2 transformations in a row (e.g. a reflection followed by a rotation), and whether the order matters.
- Rotate about a centre OTHER than the origin, and derive how the coordinate rule changes.
- Explore reflecting in the line y = -x, and derive its coordinate rule from first principles.
- Investigate which single transformations, if any, can be built by combining 2 reflections.
Assessment: exit ticket
A three-question exit ticket sampling a translation, a reflection and a rotation.
1. Translate the point (-4, 6) by the vector (7, -2).
Answer: (3, 4), because (-4 + 7, 6 - 2) = (3, 4).
2. Reflect the point (5, -1) in the y-axis.
Answer: (-5, -1), because reflecting in the y-axis flips the sign of x.
3. Rotate the point (0, 3) by 90 degrees counterclockwise about the origin.
Answer: (-3, 0), using (x, y) -> (-y, x).
Teacher notes and timings
- Rough timing across 3 lessons: Lesson 1 translations (section 1), Lesson 2 reflections (section 2), Lesson 3 rotations plus the exit ticket (section 3).
- The transformation figures in every section reuse the site's shared Transformation diagram engine (also used in the Grade 5-6 coordinate-plane unit), which draws a real reflection-over-the-y-axis and a real 90-degree-counterclockwise rotation, so the worked examples in sections 2 and 3 are deliberately written to match that exact mirror line and rotation direction.
- Language to keep repeating: a translation, reflection or rotation NEVER changes size (they are rigid transformations); reflecting OVER a line keeps that line's own points fixed; always state the DIRECTION of a rotation.
- The worksheets extend beyond the figure's fixed cases to also cover reflecting in the x-axis and the line y = x, and rotating 180 degrees and 90 degrees clockwise, all exact coordinate-rule applications independent of the diagram engine.
- Present and print both work: build each transformation live on a grid with the class, then print the 3 linked worksheets for independent practice.