Transformations: Translations, Reflections, Rotations: Fifth Grade Mathematics

Geometric transformations move a figure in a plane without altering its size or shape, producing a congruent image.

The Three Rigid Transformations

A rigid transformation (or isometry) changes the position or orientation of a figure, but preserves all side lengths and angle measures. The original figure is called the pre-image, and the moved figure is called the image.

+--------------------------------------------------------------------------+
|                  THE THREE RIGID TRANSFORMATIONS                         |
+---------------------+----------------------------+-----------------------+
| Transformation Name | Everyday Name              | Geometric Action      |
+---------------------+----------------------------+-----------------------+
| Translation         | Slide                      | Glides a shape a fixed|
|                     |                            | distance in a straight|
|                     |                            | horizontal/vert path. |
| Reflection          | Flip                       | Flips a shape across a|
|                     | (Mirror Image)             | line of reflection.   |
| Rotation            | Turn                       | Rotates a shape around|
|                     |                            | a fixed center point  |
|                     |                            | by a degree angle.    |
+---------------------+----------------------------+-----------------------+

Translation (Slide) on a Coordinate Grid

In a translation, every vertex moves the exact same horizontal distance (left or right) and the exact same vertical distance (up or down).
- Rule: (x, y) -> (x + h, y + v).

- Translating point (2, 3) right 4 units and up 5 units: (2 + 4, 3 + 5) = (6, 8).

Visualizing Transformations:
  Translation (Slide):   [ Shape ]  -------->  [ Shape ]
  Reflection (Flip):     [ F ]  |  [ | ] (Reverses orientation)
  Rotation (Turn):       [ A ]  ---> rotated 90° clockwise

Reflection (Flip)

A reflection flips a shape across a line of reflection. Every point on the image is the exact same distance from the reflection line as the corresponding point on the pre-image, but on the opposite side. Reflection reverses the clockwise/counterclockwise orientation of the vertices.

Rotation (Turn)

A rotation turns a shape around a fixed pivot point called the center of rotation. Rotations are specified by two parameters:
1. Direction: Clockwise (CW) or Counterclockwise (CCW).

2. Angle: 90° (quarter turn), 180° (half turn), or 270° (three-quarter turn).

Exercises and Solutions

Exercise 1: Identifying Transformations. State which transformation (translation, reflection, or rotation) moved Figure A to Figure B in each case: Case 1: A triangle at (1, 1) moves to (5, 4) without turning or flipping. Case 2: A footprint on the left foot matches a footprint on the right foot. Case 3: A minute hand moves from pointing at 12 to pointing at 3.

Solution 1: Case 1 is a translation (slide) because the triangle slid 4 units right and 3 units up without changing its orientation. Case 2 is a reflection (flip) because the left foot and right foot are mirror images of each other. Case 3 is a rotation (turn) because the hand turned 90° clockwise around a central pivot point.

Exercise 2: Coordinate Translation Calculation. A triangle has vertices at A(1, 2), B(4, 2), and C(2, 6). The triangle is translated 5 units to the right and 3 units down. Calculate the new coordinates for the image vertices A', B', and C'.

Solution 2: Apply the translation rule: add 5 to each x-coordinate (right 5) and subtract 3 from each y-coordinate (down 3): For vertex A(1, 2): A' = (1 + 5, 2 - 3) = (6, -1). (Note: in Quadrant I only, up 3 would keep it positive: 2 + 3 = 5; if translated down 3, it reaches y = -1). For vertex B(4, 2): B' = (4 + 5, 2 - 3) = (9, -1). For vertex C(2, 6): C' = (2 + 5, 6 - 3) = (7, 3). The new image vertices are A'(6, -1), B'(9, -1), and C'(7, 3).