Transformations

Here you will learn about transformations, reflections, translations, rotations and dilations.

Students will first learn about transformations as part of geometry in 7th and 8th grade and continue to learn about them in high school.

What are transformations?

Transformations change the size and/or the position of a shape.

The first type of transformation students are introduced to in school is usually scale factor. The scale factor describes how much the size of a shape has been scaled up or down.

The scale factor is multiplied by every side length of a shape to increase or decrease the size. Changing a shape by a scale factor greater than 1 will make the shape a larger figure.

For example,

Shape A that has been enlarged by scale factor 2 to give shape B.

Step-by-step guide: Scale factor

There are four geometric types of transformations:

  1. Rotations involve a center of rotation, an angle of rotation, and a direction of rotation (clockwise or anticlockwise).

For example, Rotate shape A 90º clockwise around a fixed point.

  1. Translations involve a horizontal shift or vertical shift.

For example, Shape A has been translated to shape B by the column vector (3, 2).

  1. Reflections involve a mirror line, also known as a line of reflection.

For example, Triangle P has been reflected across the line x=4 to give Triangle Q.

  1. Dilations make a shape bigger or smaller. They must have a scale factor and they may involve a center of enlargement.

For example, Shape A has been reduced by scale factor 1/2 to make shape B.

The center of dilation is the fixed point from which all points of a figure are scaled during a dilation.

Common Core State Standards

How does this relate to 7th grade math, 8th grade math, and high school math?

  • Grade 7 – Geometry (7.G.A.1) Solve problems involving scale drawings of geometric figures.
  • Grade 8 – Geometry (8.G.A.3) Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures.
  • High School – Geometry – Congruence (HS.G.CO.A.5) Draw the transformed figure using graph paper, tracing paper, or geometry software.

How to perform transformations

To perform transformations:

  1. Identify the transformation required.
  2. Complete the transformation.

Transformations examples

Example 1: using a scale factor to enlarge a shape

Enlarge this shape by scale factor 3:

  1. Identify the transformation required.
  2. Complete the transformation.

Example 2: rotate a shape around a center of rotation

Rotate the shape 180º about the origin.

Example 3: translate a 2D shape

Translate shape A by the column vector (0, -3).

Example 4: reflect a shape on a coordinate grid

Reflect the triangle across the line y=-1.

Example 5: dilation on a coordinate plane

Enlarge the shaded shape by scale factor 2 about the point (1, 1).

Example 6: dilation on a coordinate plane

Shrink the shaded shape by scale factor 1/2 about the point (1, 0).

Teaching tips for transformations

  • A good intro into this topic is to give students definitions of transformations and simple 2D shapes to practice with.

Easy mistakes to make

  • Not using the grid for accuracy. Use the lines on the grid to help position the new shape.

  • Not adding details to your diagram. Make use of your pencil and ruler to help you.

Practice transformations questions

  1. Enlarge this shape by a scale factor of 2.

  2. Rotate the shaded shape 90º clockwise around (0, 0).

  3. Translate the shaded shape by the column vector (2, 1).

  4. Reflect the shape across x=-2.

  5. Dilate the shaded shape with scale factor 1/2 about the point (1, 1).

  6. Dilate the shaded shape with scale factor 3 about the origin.

Transformation FAQs

What is a rigid transformation? A transformation that does not change the dimensions of the figure.

What is a quadratic function? A function with a variable that is squared.

What is the absolute value of a number? The distance that number is from 0.