Geometry & Measures
Grade 2-5Transformations
Transformations questions test two separate skills: carrying out a reflection, rotation, translation or enlargement accurately on a grid, and describing a transformation fully using the correct mathematical language. This lesson covers all four transformations, describing a single transformation fully, and working out the overall effect of combining two transformations.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Reflections
A reflection flips a shape over a mirror line, so that each point and its image are the same distance from the line, on opposite sides. Reflecting in the x-axis negates the y-coordinate of each point, and reflecting in the y-axis negates the x-coordinate. Reflecting in a line such as x = 3 or y = 1 reverses the distance of each point from that line.
Reflecting the point (2, 3) in the x-axis gives (2, -3). Reflecting the point (1, 1) in the line x = 3 gives (5, 1), since 1 is a distance of 2 to the left of the line, so the image is a distance of 2 to the right.
Rotations
A rotation turns a shape a given angle around a fixed centre point. A 90° clockwise rotation about the origin maps (x, y) to (y, -x), a 90° anticlockwise rotation maps (x, y) to (-y, x), and a 180° rotation maps (x, y) to (-x, -y). To rotate about a point other than the origin, translate that point to the origin first, apply the rotation rule, then translate back.
Rotating the point (3, 1) by 90° clockwise about the origin gives (1, -3), using the rule (x, y) → (y, -x).
Translations and enlargements
A translation slides a shape without turning or resizing it, described by a vector showing the movement right/left and up/down. An enlargement changes the size of a shape by a scale factor from a centre of enlargement: a scale factor greater than 1 makes the shape bigger, and a scale factor between 0 and 1 makes it smaller.
Translating the point (2, 3) by the vector (3, -2) gives (5, 1). Enlarging the point (2, 4) by scale factor 2, centre the origin, gives (4, 8), since each coordinate is multiplied by the scale factor.
Describing transformations fully
A full description must name the correct transformation and give all of the required details: a reflection needs the mirror line, a rotation needs the angle, direction and centre, a translation needs the vector, and an enlargement needs the scale factor and centre. Combining two transformations can sometimes be described as a single equivalent transformation.
Reflecting a shape in the x-axis and then in the y-axis has the same overall effect as a single rotation of 180° about the origin, which can be checked by testing what happens to one point under both processes.
Worked Examples
Three exam-style questions, fully solved.
Triangle A has vertices (1, 1), (3, 1), (1, 4). Reflect triangle A in the y-axis. Find the coordinates of the image.
Easy- 1.Reflecting in the y-axis negates the x-coordinate of each point
- 2.Apply this to each vertex of the triangle
Answer: (-1, 1), (-3, 1), (-1, 4)
Triangle A has vertices (2, 1), (4, 1), (2, 3). Rotate triangle A 90° clockwise about the point (2, 1). Find the coordinates of the image.
Medium- 1.Translate so that (2, 1) becomes the origin: (0, 0), (2, 0), (0, 2)
- 2.Rotate 90° clockwise using (x, y) → (y, -x): (0, 0), (0, -2), (2, 0)
- 3.Translate back by adding (2, 1) to each point
Answer: (2, 1), (2, -1), (4, 1)
A shape is first reflected in the x-axis, then reflected in the y-axis. Describe the single transformation that has the same overall effect.
Hard- 1.Test with the point (1, 1): reflecting in the x-axis gives (1, -1)
- 2.Reflecting (1, -1) in the y-axis gives (-1, -1)
- 3.Compare this to the original point (1, 1) to identify the equivalent single transformation
Answer: A rotation of 180° about the origin
Avoid These
The most common mistakes students make.
Mixing up the rules for 90° clockwise and 90° anticlockwise rotation about the origin, since one negates the x-value and the other negates the y-value.
Forgetting to translate a rotation centre to the origin first when the centre is not the origin, and applying the rotation rule directly to the original coordinates instead.
Giving an incomplete description of a transformation, such as saying "rotation" without stating the angle, direction and centre, or "enlargement" without stating the scale factor and centre.
Confusing translation with reflection, for example flipping a shape instead of sliding it when a vector is given.
For an enlargement with scale factor less than 1, assuming the shape gets bigger instead of smaller.
FAQ
Questions parents and students ask.
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