Transformations
Stage 10 of 23 Strand 3 of 9 6 lessons
6 illustrated lessons, each teaching the why before the how.
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Reflection #
Every point crosses the mirror to the same distance.
Reflection carries each point straight across the mirror line to the same distance beyond
The gold line is the mirror, and the shape beyond it is the image.
Each vertex lands as far past the line as it started before it — count the squares.
Reflected in the y-axis, the vertex (4, 3) goes to (−4, 3): only the x-coordinate changes sign.
The mirror y = x swaps the coordinates: (x, y) lands on (y, x).
Now you
After a reflection, where does the image sit?
Reflect the point (3, 2) in the y-axis. Where does it land?
Lesson complete. Continue your journey in the app — your progress saves there.
Rotation #
The whole shape turns about one fixed point.
A rotation turns every point of the shape about one fixed center
This is a rotation of 90° counterclockwise about the origin, and the gold dot is the center.
A rotation of 90° counterclockwise sends (x, y) to (−y, x), so (2, 1) goes to (−1, 2).
A half turn sends (x, y) to (−x, −y): both signs flip, either way you turn.
Every point circles the center — except the center itself, which never moves.
Now you
Rotate (3, 0) by 90° counterclockwise about the origin. Where does it land?
Rotate (2, 1) by 180° about the origin. Where does it land?
Lesson complete. Continue your journey in the app — your progress saves there.
Translation of Shapes #
Every point slides by the same vector.
A translation moves every point of the shape by the same vector
Every vertex moves the same distance in the same direction, shown by the gold arrow.
The column vector [a, b] adds to every vertex: a across, then b up.
From (1, 2), the vector [3, −1] means 3 right and 1 down: it lands on (4, 1).
Nothing rotates and nothing stretches: the image is the same shape in a new position.
Now you
Translate the point (2, 3) by the vector [4, −2]. Where does it land?
Translate the point (−1, 2) by the vector [3, 3]. Where does it land?
Lesson complete. Continue your journey in the app — your progress saves there.
Enlargement with a Center #
Rays from one point stretch the shape.
An enlargement stretches every vertex along its ray from the center by the scale factor
Rays run from the center through each vertex, and the image sits farther out.
Scale factor k: each vertex lands k times as far from the center, along its ray.
With every vertex moves halfway to the center. The image is smaller, but it is still called an enlargement.
With a negative k, each vertex is taken through the center to the other side, and the image is upside down.
Now you
What does a scale factor of do?
Where does a negative scale factor put the image?
Lesson complete. Continue your journey in the app — your progress saves there.
Combined Transformations #
Two moves, and sometimes one name for the pair.
Two transformations run in turn, and sometimes one single move does the same job
First move: reflect the shape in the y-axis.
Second move: reflect that image in the x-axis. That is two reflections, one after the other.
One single move has the same effect: a 180° rotation about the origin.
Swapping this pair changes nothing — but for many pairs the order does matter.
Now you
Is a translation then a reflection generally the same as reflecting first?
Reflecting in the y-axis, then in the x-axis, equals which single move?
Lesson complete. Continue your journey in the app — your progress saves there.
Rotational Symmetry #
A turn that lands the shape on itself.
The order of rotational symmetry counts how many turns land a shape on itself in one full turn
A rotation of a third of a turn maps this shape onto itself: the three parts look the same.
The order of rotational symmetry counts how many times the shape maps onto itself in one full turn: here, 3.
A square maps onto itself at every quarter turn, four times in a full turn, so its order is 4.
Every shape returns after one full turn, so order 1 means no extra symmetry.
Now you
What is the order of rotational symmetry of an equilateral triangle?
A shape has rotational symmetry of order 1. What does that mean?
Lesson complete. Continue your journey in the app — your progress saves there.