1. Translations

2. Translation is a movement in a straight line.
Translation can be on the left or right, up or down.
In order to translate a figure, the translation vector is needed.
The x-component means to move horizontally and the y-component means to move vertically.

3. Right means Positive X Left means Negative X Up means Positive Y
Down means Negative Y
These X and Y values make a pair of coordinates and they are called vectors.
For example, the vector describes a translation 3 right and 4 down. –4 3

4. Translation - Rule If the translation vector is (a, b) then
P(x, y) P’(x + a, y + b)

5. Properties of a Translation
• The orientations of the original and the image are the same.
The inverse of any translation is an equal move in the opposite direction.

6. Translate Triangle A throughx 1 9 8 7 6 5 4 3 2 y
4 right
2 up B A
Label the image B
4 right
2 up
2 up
4 right

7. Translate Hexagon A through1 9 8 7 6 5 4 3 2 y
3 left
4 down A
3 left
4 down
Label the image B B

8. x 1 9 8 7 6 5 4 3 2 y

9. y Translate triangle ABC by the vector ( ). x 6 5 A 2 4 2 -3 3 -3 2 1
( ). 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 A 2 2 -3 -3 B C x

10. y Translate triangle ABC by the vector ( ). x 6 5 4 3 2 1 -1 -2 -3 -4
( ). 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 -3 -3 4 A 4 B A C x B C

11. Vector of Translation
To find the translation vector, we find the difference of coordinates of the image and the original object point. C’ A’ B’
image C A B
object C A B
object C A B
object C A B
object C A B
object C A B
object C A B
object C A B
object
object
Introduce the key terms and ask pupils to give examples of translations from everyday situations. For example, walking in a straight line from one side of the room to the other.
Discuss how to define a translation. There are two ways to mathematically describe movement in a straight line in a given direction. One is to to give the direction as an angle and a distance (see S7 Measures – bearings), the other is to give the direction using a square grid. A movement in a straight line can be given as a number or units to the left or right and a number of units up or down.
Point out that when a shape is translated each vertex (or corner) is moved through the given translation.
The movement across is always given before the movement up or down.
Ask pupils how we could return the translated shape back to its original position.

12. Find the translation vector
A TO B 1 2 3 4 5 -1 -2 -3 -4 -5 6 7 8 -6 B C A D

13. A TO C 1 2 3 4 5 -1 -2 -3 -4 -5 6 7 8 -6 B C A D

14. D TO A 1 2 3 4 5 -1 -2 -3 -4 -5 6 7 8 -6 B C A D

15. D TO C 1 2 3 4 5 -1 -2 -3 -4 -5 6 7 8 -6 B C A D

16. B TO D 1 2 3 4 5 -1 -2 -3 -4 -5 6 7 8 -6 B C A D

17. R (5, 2) 1 2 3 4 5 -1 -2 -3 -4 -5 6 7 8 -6

18. R (-3, -5) 1 2 3 4 5 -1 -2 -3 -4 -5 6 7 8 -6

19. R (-6, 1) 1 2 3 4 5 -1 -2 -3 -4 -5 6 7 8 -6

20. R (4, -5) 1 2 3 4 5 -1 -2 -3 -4 -5 6 7 8 -6

21. Translations on a coordinate grid
Stress that when we translate shapes on a coordinate grid we can find the vertices of the image shape by adding (or subtracting) the value of the translation to the right or left to the x-coordinate (add for right and subtract for left) and adding (or subtracting) the value of the translation up or down to the y-coordinate (add for up and subtract for down) .

22. Translation golf
In this game pupils are required to estimate the required translation to move the ball to the hole. The length of one unit is shown in the top right-hand corner.
To translate the ball to the left or down, a negative number must be used.
Challenge pupils to get the ball into the hole in as few moves as possible.

23. Translations
When a shape is translated the image is congruent to the original.
The orientations of the original shape and its image are the same.
An inverse translation maps the image that has been translated back onto the original object.
What is the inverse of a translation 7 units to the left and 3 units down?
The inverse is an equal move in the opposite direction.
That is, 7 units right and 3 units up.

24. Describing translations
When we describe a translation we always give the movement left or right first followed by the movement up or down.
We can describe translations using vectors.
For example, the vector describes a translation 3 right and 4 down. –4 3
As with coordinates, positive numbers indicate movements up or to the right and negative numbers are used for movements down or to the left.
Compare a translation given as an angle and a distance to using bearings. For example, an object could be translated through 5cm on a bearing of 125º.
A different way of describing a translation is to give the direction as an angle and the distance as a length.

25. Translations on a coordinate gridy
A(5, 7)
The vertices of a triangle lie on the points A(5, 7), B(3, 2) and C(–2, 6).
C(–2, 6) 7 6 5 4 3 2
B(3, 2)
Translate the shape 3 squares left and 8 squares down. Label each point in the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1
A’(2, –1) –2
C’(–5, –2) –3
Point out that when we translate shapes on a coordinate grid we can find the coordinates of the vertices of the image shape by adding to the original’s x-coordinates the size of the translation to the right (or subtracting the size of a translation to the left); and adding to the original’s y-coordinates the size of the translation up (or subtracting the size of a translation down). –4
What do you notice about each point and its image? –5 –6
B’(0, –6) –7

26. Translations on a coordinate gridy
The coordinates of vertex A of this shape are (–4, –2). 7 6 5 4 3
When the shape is translated the coordinates of vertex A’ are (3, 2). 2
A’(3, 2) 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1 –2
A(–4, –2)
What translation will map the shape onto its image? –3
Tell pupils that we can work this out by subtracting the coordinate of the original shape from the coordinate of the image.
The difference between the x-coordinates is 7 (3 – –4) and the difference between the y-coordinates is 4 (2 – –2). The required translation is therefore 7 right and 4 up. –4 –5 –6 –7
7 right
4 up

27. Translations on a coordinate grid1 2 3 4 5 6 –2 –3 –4 –5 –6 –7 –1 y x 7
The coordinates of vertex A of this shape are (3, –4).
When the shape is translated the coordinates of vertex A’ are(–3, 3).
A’(–3, 3)
What translation will map the shape onto its image?
The difference between the x-coordinates is –6 (–3 – 3) and the difference between the y-coordinates is 7 (3 – –4). The required translation is therefore 6 left and 7 up.
If required this slide can be copied and modified to produce more examples. To change the shapes, go to View/toolbars/drawing to open the drawing toolbar. Select the shape and using the drawing toolbar, click on draw/edit points to drag the points to different positions. Edit the new coordinates as necessary.
A(3, –4)
6 left
7 up

28. Translation
When an object is moved in a straight line in a given direction we say that it has been translated.
For example, we can translate triangle ABC 5 squares to the right and 2 squares up. C’ A’ B’
image C A B
object C A B
object C A B
object C A B
object C A B
object C A B
object C A B
object C A B
object
Every point in the shape moves the same distance in the same direction.
Introduce the key terms and ask pupils to give examples of translations from everyday situations. For example, walking in a straight line from one side of the room to the other.
Discuss how to define a translation. There are two ways to mathematically describe movement in a straight line in a given direction. One is to to give the direction as an angle and a distance, for example, as a bearing. The other is to give the direction using a square grid. A movement in a straight line can be given as a number of units to the left or right and a number of units up or down.
Stress that when a shape is translated each vertex (or corner) is moved through the given translation.
The movement across is always given before the movement up or down (compare with coordinates).
Ask pupils how we could return the translated shape back to its original position.
We can describe this translation using the vector 5 2

29. Describing translations
When we describe a translation we always give the movement left or right first followed by the movement up or down.
We can also describe translations using vectors.
For example, the vector describes a translation 3 right and 4 down. 3 –4
As with coordinates, positive numbers indicate movements up or to the right and negative numbers are used for movements down or to the left.
Compare a translation given as an angle and a distance to using bearings. For example, an object could be translated through 5 cm on a bearing of 125º.
One more way of describing a translation is to give the direction as an angle and the distance as a length.

30. Describing translations
When a shape is translated the image is congruent to the original.
The orientations of the original shape and its image are the same.
An inverse translation maps the image that has been translated back onto the original object.
What is the inverse of a translation 7 units to the left and 3 units down?
The inverse is an equal move in the opposite direction.
That is, 7 units right and 3 units up.

31. Inverse translations What is the inverse of the translation ? –3 4
This vector translates the object 3 units to the left and 4 units up.
The inverse of this translation is a movement 3 units to the right and 4 units down.
The inverse of is –3 4 3 –4
In general,
The inverse of the translation is a b –a –b

32. Translation - Rule If the translation vector is (a, b) then
P(x, y) P’(x + a, y + b)