1. Is there a TRANSFORMATION in your future?
TRANSLATIONS
REFLECTIONS
ROTATIONS
Fix effects

2. What is a Transformation?
In lay terms, a transformation is a change.
Instinctively, in geometry a transformation means movement.
In the formal sense, a transformation of a plane is a one-to one mapping (function) of a plane into itself.
Sun RightMove Right_Sun Wh
Sun F(Sun) = Right_Sun

3. Translations
A translation “slides” the original figure (or preimage) to a new position creating a new one called image.
A B
A’ B’
D C
D’ C’
A” B”
D” C”

4. Translations Translations
In other words, a translation of a plane shifts all points on the plane in the same direction and in the same distance.
A B
A’ B’
D C
D’ C’
A” B”
A’” B’”
D” C”
D”’ C”’

5. Translations in the Coordinate Plane
An arrow (or vector) also shows a translation. The vector AA’ (1,4) translates point A to point A’ as for every point of the figure, in particular the vertices. A’ B’
What does this mean?
Given a vector, v, the image P’ of a point P is the point for what PP’ is parallel to v and PP’= v C’ D’ A B
Discuss: Translations preserve distance, angles measure and collinearity.
Also, the lines which were parallel remain parallel (parallelism) and the labelling sequence is also preserved (betweenness.)
Teacher stops here. Students will complete worksheets. C D

6. Translations in the Coordinate Plane
Let’s practice Translations!

7. Reflections
Do you see yourself in the mirror every morning? That image is a reflection of your face.
When a shape is reflected, it does so over a reflecting line, “flipping” over it.

8. Reflections
Then, a reflection is a transformation where each point in a shape appears at an equal distance on the opposite side of a given line - the line of reflection.

9. Reflections In Geometric terms, a reflection is described like this:
Given a point P, let m’ be the straight line through P that is perpendicular to m. Then P’ is the point on m’ on the opposite side of m to P that is equidistant from m. m’

10. Reflections Now, an experiment: Reflecting your hand!
And some practice too!

11. Rotations
A rotation can be seen as a ‘spin’ around a center.

12. Rotations
So, a rotation is a transformation that turns a figure about a center of rotation.  Said ‘turn’ is made in a specific angle, called angle of rotation.

13. Rotations
A formal definition defines rotation about a point P through angle α is a transformation such that:
(1) If a point A is different from P, then PA=PA’ and the measure of <APA’= α and
(2) If point A is the same as point P, then A’=A
(2) Point rotating over itself is transformed in the same place. α α

14. Rotations
Now take turns practicing rotations!

15. Where transformations can be seen, When transformations can be used.
Native American Art and Video Games
Note for the teacher: Translations, rotations and translations can be seen in Native American art, The basket shows a translation and the jacket has basically reflections. Video games uses translation to reproduce scenes in most of the cases.